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Essays in asset pricing
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Content
Essays in Asset Pricing
Samuel M. Hartzmark
University of Southern California
A Dissertation Presented to the
Faculty of the USC Graduate School
University of Southern Cailfornia
In Candidacy for the Degree
Doctor of Philosophy
Advisor: Wayne Ferson
ii
Abstract:
In the first chapter of my dissertation “The Worst, the Best, Ignoring All the Rest:
The Rank Effect and Trading Behavior” I document a new stylized fact about how investors
trade assets: individuals are more likely to sell the extreme winning and extreme losing positions
in their portfolio (“the rank effect”). This effect is not driven by firm-specific information or the
level of returns itself, but is associated with the salience of extreme portfolio positions. The rank
effect is exhibited by both retail traders and mutual fund managers, and is large enough to induce
significant price reversals in stocks of up to 160 basis points per month. The effect indicates that
trades in a given stock depend on what else is in an investor’s portfolio.
The second chapter of my dissertation is “Economic Uncertainty and Interest Rates.” A
number of asset-pricing models predict a positive relation between the risk-free interest rate and
expected economic growth, and a negative relation between the interest rate and the uncertainty
(i.e. the conditional variance) of growth. I document that uncertainty and the interest rate have a
strong negative relation. This holds when examining up to 140 years of data, using various
measures of economic growth and uncertainty, and after controlling for inflation. The result has a
number of implications for models such as habit and long-run risks. A negative relation between
habit and the interest rate disappears after controlling for uncertainty. Previous research presents
a puzzle as to the lack of relation between the macroeconomy and the real interest rate which this
paper partially resolves.
iii
Table of Contents
Chapter 1. The Worst, the Best, Ignoring All the Rest ............................................................................ 1
1.1. Introduction ........................................................................................................................................ 1
1.2. Literature Review ............................................................................................................................... 6
1.2.1. Relative Evaluation and Rank ................................................................................................... 6
1.2.2. Salience ..................................................................................................................................... 6
1.2.3. Previous Studies of Portfolios ................................................................................................... 6
1.3. Data and Summary Statistics .............................................................................................................. 9
1.4. The Rank Effect ................................................................................................................................. 11
1.4.1. Univariate Results ..................................................................................................................... 11
1.4.2. Controlling For Firm-Specific Factors ..................................................................................... 13
1.4.3. Controlling For Performance Since Purchase .......................................................................... 15
1.4.4. Rank Effect Mechanism ........................................................................................................... 21
1.5. Price Effects.................................................................................................................................... 25
1.6. Rank Effect Robustness .................................................................................................................. 30
1.6.1. Tax-Based Explanations .......................................................................................................... 30
1.6.2. Covariate Balance .................................................................................................................... 30
1.7. Conclusion ...................................................................................................................................... 33
Chapter 2. Economic Uncertainty and Interest Rates ........................................................................... 56
2.1. Introduction ...................................................................................................................................... 56
2.2. A Basic Model ................................................................................................................................... 60
2.3. Data and Summary Statistics ............................................................................................................ 61
2.4. Empirical Results............................................................................................................................... 64
2.4.1. Baseline Results ..................................................................................................................... 64
2.4.2. The Effects of Extreme Volatility Periods ............................................................................ 65
2.4.3. The Long Historical Record .................................................................................................. 66
2.4.4. Quarterly Data ....................................................................................................................... 67
2.4.5. Different Interest Rates ......................................................................................................... 67
2.4.6. International Data ................................................................................................................... 68
2.5. Using Alternative Forecasts .............................................................................................................. 68
2.5.1. Using Ex-Post Growth ........................................................................................................... 68
iv
2.5.2. Using Professional Forecasts ................................................................................................ 69
2.5.3. Using the VIX ........................................................................................................................ 70
2.6. Inflation Risk ..................................................................................................................................... 71
2.7. Econometric Issues ........................................................................................................................... 72
2.8. Implications of Specific Models ........................................................................................................ 73
2.8.1. External Habit ........................................................................................................................ 73
2.8.2. Long-Run Risks ..................................................................................................................... 76
2.9. Using Alternative Forecasts .............................................................................................................. 78
Chapter 1
The Worst, the Best, Ignoring All the Rest: The Rank Effect and Trading Behavior
1.1. Introduction
Portfolio theory has long advocated that investors combine multiple stocks to create
optimally diversified portfolios (Markowitz 1952). While such models are good normative
descriptions of how investors ought to behave, it is not clear that investors actually do behave
this way. A number of papers in behavioral finance have examined real world investor trading
and documented several stark departures from these theoretical predictions. Individuals often
hold too few stocks (Barber and Odean 2000), use naïve diversification strategies (Benartzi and
Thaler 2001; Goetzmann and Kumar 2008) and engage in behavior like the disposition effect
(the tendency to sell gains more than losses as in Odean 1998; Shefrin and Statman 1985) or
overconfidence (trading too much as in Barber and Odean 2000). However, these studies have
considered investor trading primarily on a stock-by-stock basis. One consequence of this is that
the empirical literature largely examines investors as if they are ignoring the portfolio problem in
its entirety. Between one extreme (complex optimization) and the other (stock-by-stock trading)
there is the large and largely unexplored middle ground of how investors actually deal with the
portfolio problem.
In this paper, I explore one aspect of the investor’s portfolio choice problem, namely how
the relative performance of stocks within the portfolio impacts trading decisions. I document a
new stylized fact about trading in a portfolio setting: investors are more likely to sell both their
best and their worst positions, based on return from purchase price. I term this the rank effect.
Using data from a large retail brokerage, I show that on a day an investor sells a position
in their portfolio, the investor has a 31% chance of selling the stock with the highest return in the
2
portfolio and a 26% chance of selling the stock with the lowest return, after controlling for a
number of factors discussed below. In contrast, a position in the middle of the portfolio (not the
top two or bottom two returns) has only an 11% probability of being sold. I define the rank effect
as the difference in probability of sale compared to the middle. Thus, in this specification the
rank effect is 20% for best-ranked stocks and 15% for worst. Mutual funds also exhibit the rank
effect: best-ranked stocks are 12% more likely to be sold than the middle, and worst-ranked
stocks are 17% more likely to be sold.
An obvious concern is that rank may be proxying for many possible firm-specific factors.
For example, stocks with best and worst ranks may differ in their information flows, returns,
volatility and any number of other attributes. I control for this possibility in several ways.
Perhaps the cleanest is to examine the behavior of two investors who both hold the same stock
on the same day, but differ in whether the stock is extreme-ranked in their portfolio. This sample
of stocks, which on the same day are extreme-ranked in one investor’s portfolio but not in the
other’s portfolio, shares the same firm-specific information, such as recent returns, earnings
announcement, analyst forecasts, etc. Even in this sample, both individual investors and mutual
funds are more likely to sell the stock that is at the extreme rank. This suggests that the rank
effect is not driven solely by any firm-specific attributes of extreme-ranked stocks, as the same
variation exists purely comparing across investors.
In addition to this broad test of firm-specific information, I examine whether rank proxies
for stock performance over the period an investor holds a stock. To rule out a simple explanation
based on the disposition effect, i.e., the tendency to sell gains rather than losses, I analyze
portfolios where all stocks are at a gain or all stocks are at a loss. I find a greater propensity to
sell the best and worst-ranked stocks in both subsamples. Stocks with best and worst ranks have
3
generally experienced high or low returns from purchase. Without controlling for rank, investors
display a propensity to sell such positions (Ben-David and Hirshleifer 2012). After controlling
for the return of the stock, stock volatility and the length of time the stock is held, the rank effect
becomes stronger. For investors, the propensity to sell stocks with high or low returns disappears
or reverses after accounting for rank. This suggests the propensity to sell stocks with large
returns in absolute value may be better understood as a propensity to sell extreme-ranked stocks.
Tax law may also create incentives to realize extreme positions in systematic ways. For
example, by realizing losses to offset previously realized gains. To test whether this explains the
effect, I examine trading in both tax deferred accounts and taxable accounts separately and find
similar results. Further, a significant rank effect is present every calendar month, whereas other
tax-based phenomena tend to manifest themselves towards the end of the year. The rank effect is
also not explained by a lack of covariate balance as matching utilizing entropy balancing yields a
rank effect. A number of further robustness checks are presented in the internet appendix.
Finally, I show that, in addition to being a robust trading pattern, the rank effect has
economically important pricing consequences. In particular, the selling of extreme positions by
mutual funds is sufficiently widespread that it induces predictable price pressure. When the
information from each mutual fund’s report is public, I estimate which stocks are likely to be
extreme-ranked for the fund and thus susceptible to price pressure from rank induced selling. In
contrast to both a short-term reversal and a momentum strategy, the strategy purchases both best
and worst-ranked stocks, as both are predicted to have high subsequent returns.
The portfolio comprised of purchasing worst-ranked stocks has a monthly four-factor
alpha of 136 basis points, which increases to 161 once a short term reversal factor is included.
The portfolio comprised of purchasing best-ranked stocks has a four-factor alpha of 19 basis
4
points, which increases to 36 when a short term reversal factor is added. This is consistent with
my finding that funds sell worst-ranked positions much more heavily than best-ranked positions.
Therefore, worst-ranked positions experience more price pressure and subsequently larger
returns. Weighting the portfolios towards stocks predicted to have higher selling pressure
increases the alphas to as high as 65 basis points for best-ranked stocks and 222 basis points for
worst-ranked. Using Fama-Macbeth regressions, I also find that best and worst-ranked positions
have positive and significant returns after controlling for a number of other return predictors.
Given that the standard finance explanations above do not appear to explain the rank
effect, a natural question arises as to what aspect of extreme-ranked positions is causing
investors to sell. One broad class of explanations for the individual investors is that extreme
positions are more salient in the investor’s portfolio. This may occur because relatively larger
returns are more attention-grabbing, or because the investor chooses to research certain stocks
based in some part on relative performance in the portfolio. In either case, the extreme positions
will be a greater focus of investor attention, potentially resulting in trade.
One prediction of a salience explanation is that the effect ought to occur on the purchase
side as well - there is no reason to presume that investors paying more attention to an extreme-
ranked stock will only choose to sell it rather than buy more. Consistent with this, I show that
investors are more likely to re-purchase more of positions with extreme returns. In addition, a
salience explanation suggests that investors will react not only to the rank ordering of stocks but
also to how much that stock stands out relative to the rest of the portfolio. Empirically, I find that
both effects are present: a direct rank order effect (of being the most extreme position) and a
relative size effect (the difference in returns between the extreme position and the next position).
5
Because stock returns are necessarily correlated with many economic factors, it is
difficult to show precisely that rank in returns has a psychological effect, rather than an
economic one. To establish that rank and salience in general can have purely psychological
effects (even if the cause of a return-based rank effect itself is difficult to isolate precisely), I
consider the effects of an alternative rank order sorting that is likely to be salient to investors, but
less likely to have an economic explanation. Specifically, I examine the tendency of investors to
trade based on the alphabetical order of company name - an ordering unique to the portfolio and
orthogonal to economic variables. Stock performance is often presented in this order, such as
online or in a brokerage statement. I show the first and last positions by alphabetical order are
more likely to be sold after controlling for stock by day probability of sale. This lends indirect
support to the view that part of the effect of extreme-ranked returns is their increased
psychological salience to investors.
In the context of models that assume narrow framing, the results suggest that investors
use a reference point based on performance relative to other stocks in their portfolio. This view
provides an extension to these theories that assume investors trade based on a specific portion of
wealth, such as a stock’s return, in isolation.
1
For example, realization utility assumes an investor
receives a burst of utility when selling a stock based on the size of the gain or loss (Barberis and
Xiong 2012). A large and ongoing debate is what reference point investors actually use to define
a gain or loss.
2
Theories that rely on narrow framing alone will struggle to explain why investors
trade differently according to the relative performance of positions within the portfolio.
1
Such as Barberis, Huang and Thaler 2006; Barberis and Huang 2001; Barberis and Xiong 2012; Ingersoll and Jin
2013; Frydman, Barberis, Camerer, Bossaerts and Rangel 2013; Ben-David and Hirshleifer 2012.
2
A number of reference points have been suggested, such as a zero return (Odean 1998), the risk free rate (Barberis
and Huang 2001), or expectations of performance (Kőszegi and Rabin 2006, Meng 2012).
6
The rank effect provides evidence for a fundamental aspect of investor behavior –
namely, that the evaluation given to any particular stock depends on what else the investor is
holding. This fact may provide a micro-foundation for models that rely on disagreement across
investors, such as those attempting to explain the level of trading volume in the stock market
(Milgrom and Stokey 1982). In particular, investors may be acting differently towards the same
piece of information about the same stock due to effect that other stocks in their portfolio have
on the way they perceive the given information.
1.2. Literature Review
1.2.1. Relative Evaluation and Rank
Investors typically hold multiple stocks and see their performance presented together, for
example online or in a brokerage statement. Behavioral economics provides evidence that when
information is presented together, individuals evaluate the information and make decisions
jointly, by comparing all the information, rather than separately, where each piece of information
is examined on its own. Further, the choices made are often different when decisions are
considered jointly, as these tend to highlight differences between the proposed alternatives
(Bazerman, Loewenstein and White 1992; Hsee 1996; Hsee, Loewenstein, Blount and Bazerman
1999; Kahneman 2003; List 2002). Thus, in the context of a portfolio, if investors use joint
evaluation when making trading decisions, their choices may be quite different than if they
evaluated each stock in isolation.
1.2.2. Salience
The relative performance of a stock in the portfolio may make investors pay more or less
attention to a position. When faced with a large number of possibilities, individuals typically do
7
not pay equal attention to each one, but spend more time examining the most salient. According
to Taylor and Thompson (1982), “salience refers to the phenomenon that when one’s attention is
differentially directed to one portion of the environment rather than to others, the information
contained in that portion will receive disproportionate weighting in subsequent judgments.” A
number of stock-specific events have been shown to be attention-grabbing.
3
This paper
complements these studies by considering salience not just as a firm-specific attribute, but rather
as an individual-specific measure relative attention.
Jointly evaluating positions in a portfolio allows rank to be a meaningful attribute in
decision making. Psychology has a long history examining rank, both describing its influence in
a variety of situations and explicitly incorporating it into decision theory.
4
As an example, rank-
dependent utility models, such as cumulative prospect theory (Tversky and Kahneman 1992),
predict that rank ordering impacts salience and the best and the worst-ranked positions are the
most attention grabbing. The intuition of rank-dependent utility is that extreme ranks are salient,
attention grabbing positions (Diecidue and Wakker 2001).
Individuals often simplify complex decisions by using a basic rule of thumb to decide
which options to pay attention to. This is called forming a consideration set and the method used
to form the consideration set is called a heuristic. Varying attention within the portfolio is
consistent with investors forming consideration sets as a part of their trading decision – for
instance, deciding which stocks to sell by first narrowing down the full portfolio to a smaller list
of potential stocks to focus on more closely.
5
In the current context, my results suggest that
3
Such events include extreme returns (Barber and Odean 2008), extreme abnormal trading volume (Gervais, Kaniel
and Mingelgrin 2001; Barber and Odean 2008), mentions in the news (Barber and Odean 2008) and Google search
volume (Da, Engelberg and Gao 2011).
4
For an overview of the literature see Diecidue and Wakker (2001) and Wakker (2010).
5
There is a large literature in marketing on consideration sets and heuristics (see Hauser 2013 for an overview), as
well as literatures examining costly information acquisition and consideration sets (Gabaix, Laibson, Moloche and
8
extreme-ranked positions are more likely to enter the consideration set when investors are
choosing which stocks to trade.
The discussion of the cause of the rank effect focuses on individual investors rather than
mutual funds. The behavior of mutual funds contains an additional complication, how managers
believe their public disclosure will be viewed. Mutual funds could exhibit the rank effect through
the same mechanism as individual investors, or because managers strategically trade based on
how they think investors will respond to their report (Musto 1999; Solomon, Soltes and Sosyura
2012). Most likely it is a combination of both. Describing how these two effects interact is
beyond the scope of the paper, but there is some evidence that managers engage in rank-based
window dressing. Section 5 shows funds liquidate worst-ranked stocks at a much higher rate than
other positions. While this could represent fund manager preferences, it is consistent with
window dressing. Stocks that are liquidated before a report date are not present in the next report.
Managers may hope that liquidating the position obfuscates the poor performance.
1.2.3. Previous Studies of Portfolios
The empirical literature examining investors’ portfolios generally focuses on portfolio
composition rather than trading behavior. It finds that investors do not hold properly diversified
portfolios (Odean 1998, Bernartzi and Thaler 2001; Goetzmann and Kumar 2008), that the
degree of diversification varies with investor characteristics (Goetzmann and Kumar 2008) and
that investors exhibit a preference for local or familiar stocks (Huberman 2001; Frieder and
Subrahmanyam 2005). There has been very little research examining how investors trade in a
portfolio setting (Subrahmanyam 2008).
Weinberg 2006), how firms respond to consumers’ use of consideration sets (Ellison and Ellison 2009) and models
of how such heuristics impact asset prices (Gabaix 2013).
9
1.3. Data and Summary Statistics
The analysis in this paper is based on two main datasets. The first contains data on
individual investors trading on their personal accounts. These data have the benefit of a large
number of accounts with information on the exact day that trading occurs. The analysis is also
conducted on mutual funds. Funds control significantly more money traded by professional
investors and the data also offer a longer time series. Unfortunately, holdings are only available
infrequently, not on the day a position is traded, and may be subject to window dressing.
The individual investor data are the same as that used in Barber and Odean (2000) and
Strahilevitz, Odean and Barber (2011). The data describe the date, price and quantity of trades
made by retail investors from a large discount brokerage from January 1991 through November
1996. The data is augmented using CRSP data on price as well as adjustments for splits.
The analysis examines the portfolio of stocks that an investor could sell on each day that
they do sell at least one position (a sell day). Section 4.4 also examines the decision to buy more
of a stock each day an investor purchases a stock (a buy day). I use the term sell to indicate a
decrease in the number of shares held while I use the term liquidate to refer to a sale where all
shares of a given stock are sold. The time of trade is not observed, so if an investor trades the
same stock multiple times within a day, the price and quantity used in the analysis is the value
weighted average price across transactions and the net quantity. On days that a position is opened
(purchased when previous holdings were 0) it is not included in the portfolio as available to be
sold because it is unclear if it is in the portfolio at the time a different position is sold. Short
positions are excluded from the analysis (about 0.3% of holdings).
6
Some trades have negative
6
For accounts not present in the first month of holdings data, a position is considered short when it is sold with prior
holdings of the position less than or equal to zero, or when a position is purchased and the resulting holdings are
zero or negative.
10
commissions which could indicate canceled trades (about 0.5% of observations). All
observations of a stock that ever have a negative commission for a given investor are dropped.
7
Returns at time of sale are calculated between the purchase price and the closing price on
the day prior to the sell day.
8
If a position is already in the portfolio and additional shares are
purchased, the purchase price used to calculate returns after this date is the value weighted
average of the multiple purchase prices.
In order to study complete portfolios I drop portfolios containing positions for which the
purchase price is unknown. This is accomplished by excluding accounts with holdings in the first
month of the position files. These positions were purchased before the start of the dataset, thus
the purchase price is not known. This excludes roughly 25% of accounts. As the paper examines
portfolio rank, investors must hold at least five stocks to be included in the analysis. This
excludes about 19% of observations.
Table 1.1 Panel A shows summary statistics describing the data on individual investors.
After the filters are applied the data includes 10,619 unique accounts, 94,671 sell day by account
observations with a sale, and 1,051,160 positions held on those days. The average portfolio is
comprised of 11.1 stocks. On a sell day, on average 12% of positions in a portfolio are sold, of
which 9.6% are liquidated.
The mutual fund analysis combines holdings data from Thompson-Reuters and fund price
and volume information from CRSP, and stock return information from CRSP. The Thompson-
Reuters and CRSP files are merged by the Wharton Financial Institution Center number
7
Results are robust to including these positions, or to using a cleaning algorithm that matches the two sides of a
cancelled trade and keeps the remaining observations.
8
This is chosen to keep consistency between the calculation of returns for actual sales and paper gains and losses.
The analysis is not materially different if the actual sale price is used on positions that are sold and the CRSP closing
price used for those that are not, or if the closing price of the sell day instead of the day before the sell day is used.
The analysis ignores trading fees, but is nearly identical if returns net of fees are used instead (see Appendix Table
IA.22). A stock is included only if there is CRSP information for every observation.
11
(WFICN). Returns are calculated analogously to the investor data, but using report dates because
sell dates are not known. A sale is defined as a decrease in the number of shares from the
previous report, while a liquidation is defined as not holding any shares in a position which was
held in the previous report. To be included in the analysis a fund must hold at least 20 CRSP
merged stocks on a report date. I apply the Frazzini (2006) filters to the Thompson-Reuters data
to exclude observations that appear to be errors.
9
The sample period studied is 1990-2010,
though to construct the purchase price history I use data starting from 1980.
Table 1.1 Panel B shows summary statistics for the mutual fund data. There are 4,730
funds holding 120 stocks on average. There are 129,415 report day by account observations with
15.6 million positions held on these days. 38.9% of positions are at least partially sold between
report dates and 15.1% of positions are liquidated.
The paper mainly examines the decision to sell a stock, rather than trading generally. An
investor is limited to selling stocks in their portfolio, while they can buy any stock in the market.
An investor purchasing a stock will in practice consider a small subset of all the stocks in the
market, but this subset is not known. Thus the set of possible positions to be sold is considerably
smaller and better defined than the positions considered when buying a stock. Section 4.4
examines trading in general including the decision to buy more of a position already held.
1.4. The Rank Effect
1.4.1 Univariate Results
Table 1.2 shows the rank effect, the tendency to sell the best and worst-ranked stocks, in
the simplest specification, with no controls. Stocks are ranked by return from purchase as best,
9
Holdings are set to missing when: 1) the number of shares a fund holds is greater than the number of shares
outstanding of that stock, 2) the value of a holding is greater than the fund’s total asset value, 3) the value of the
fund’s reported holding is different from the CRSP value by more than 100%.
12
second best, worst, second worst and middle. Middle includes all stocks not ranked in the top or
bottom two positions. A stock is ranked best if it has the highest return from purchase price in
the portfolio, and worst if it has the lowest. For investors, only days where at least one stock is
sold are examined, thus each observation is a stock (j) for an investor (i) on a sell day (t). For
mutual funds report days are examined. Thus each observation is a stock (j) for a fund (i) on a
report day (t). Each row is the proportion of positions of the indicated rank that are sold. The best
and worst rows are:
# Best Sold is the number of best positions on a sell day (report day for funds) that had their
positions decreased and # Best Not Sold is the number of best positions where the number of
shares increased or stayed the same. Other ranks are defined similarly. The measure is analogous
to the proportion of gains (losses) realized used in Odean (1998).
The rank effect can be seen by examining the bottom portion of the table. Best – Middle
is the difference between the best and middle rows with the t-statistic for the test that they are
equal, clustered by date and account, underneath. For individual investors a best-ranked position
is 16.3% more likely to be sold than a middle-ranked position and a worst-ranked position is
8.5% more likely to be sold, both with highly significant t-statistics. The mutual fund results are
of a similar magnitude, but the worst-ranked position is more likely to be realized than the best.
For these funds a best-ranked position is 11.9% more likely to be sold than a middle-ranked
position, and a worst-ranked position is 19.1% more likely to be sold.
This analysis aggregates across investors, thus it may mask heterogeneity in their
tendency to sell extreme positions. For example, the same averages would be observed if two
traders are more likely to sell both best and worst-ranked positions, or if one trader sells only
13
best positions, with the other sells only worst. This could occur if certain investors over-
extrapolated past performance and always sold their worst performing stocks, while other traders
believed in mean reversion and always sold their best.
Figure 1.1 graphs a heat map of the joint density of the probability of selling best and
worst positions for each investor and fund with at least five sell days or report days in the data.
The x-axis is the investor specific proportion of best positions realized, while the y-axis is the
corresponding proportion of worst positions. The lighter the cell, the more density it has. If half
of the investors sold only best-ranked positions, and half sold only worst, the bottom right and
top left corners would be white. For both types of traders these are the areas with the lowest
density and are black. The correlation between the proportion best sold and proportion worst sold
is positive, 0.37 for individual investors and 0.41 for mutual funds. Thus traders tend to sell both
best and worst-ranked stocks.
Many theories posit that investors trade to rebalance their portfolio, or after they update
their beliefs changing the composition of the optimal portfolio. If beliefs are held constant, a
best-ranked stock has increased from its optimal portfolio weight and a worst-ranked stock has
decreased. To rebalance, the best-ranked stock should be sold and the worst-ranked stock should
be purchased. The selling of worst-ranked stocks rules out simple rebalancing with fixed weights
and constant beliefs as the cause of the rank effect.
1.4.2 Controlling For Firm-Specific Factors
One possibility is that rank is simply proxying for firm-specific information. For
example, extreme-ranked stocks may be more likely to have had high or low recent returns,
increases in volatility, or mentions in the news. Rather than attempting to control for all possible
stock-specific characteristics, I limit the analysis to stocks that on the same day are extreme-
14
ranked in one investor’s portfolio, but not extreme-ranked in another investor’s portfolio. This
group of extreme-ranked stocks and non-extreme-ranked stocks have the identical stock-specific
information set and differs only in rank across investors. If investors respond to firm-specific
factors the same way regardless of rank, this sample will not exhibit the rank effect.
Table 1.3 Panel A examines whether there is a rank effect for this subsample. Taking the
best row as an example, the sample is limited to stocks that on the same sell day are best-ranked
in one investor’s portfolio, and not best-ranked in another investor’s portfolio. Each observation
has a Sell variable equal to one if the position is sold and zero if it is not. The difference between
this variable for stock ranked best and stocks not ranked best for the same stock on the same day
is taken, and the average is reported:
10
( )
∑ ∑ (
( )
( )
)
( )
( )
Where t indexes each day and j( ) indexes each stock that on day t is best-ranked in one
portfolio and not best-ranked in another. # Pairs is the number of unique day by stock
observations. The worst row contains the same measure for worst-ranked positions.
Extreme-ranked stocks are more likely to be sold than the same stocks in portfolios where
they are not extreme-ranked. Individual investors are 10.2% more likely to sell a best-ranked
stock than the same stock that is not best-ranked, and are 6.3% more likely to sell a worst-ranked
stock than the same stock that is not worst-ranked. Mutual funds are 7.4% more likely to sell a
best-ranked stock and 12.6% more likely to sell a worst-ranked stock.
Panel B conducts a similar analysis, utilizing a linear probability model and stock by day
fixed effects. A sell dummy variable is regressed on a best dummy variable, a worst dummy
10
If the same stock on the same day is best-ranked (or not best-ranked) in multiple portfolios, the average of sell is
taken over these investors with the same stock on the same day with the same rank.
15
variable and interaction fixed effects for each stock and date pair. With this interaction fixed
effect, the regression is identified based on variation from stocks that have different ranks on the
same day across investors’ portfolios. Thus the results are quite similar to Panel A as both
individual investors and mutual funds are more likely to sell extreme-ranked stocks.
I also consider the question of whether the rank effect can be driven by changes in
investor expectations of means, variance, or covariances. Table 1.3 casts doubt on this
explanation, nonetheless, I directly test the effect in Appendix Table IA.16 and Appendix Table
IA.17 explicitly controlling for changes to expectations of a stock’s mean, variance and
covariance. These proxies do not materially impact the magnitude or significance of the rank
effect.
1.4.3 Controlling For Performance Since Purchase
One of the largest literatures on trading behavior focuses on the disposition effect. Coined
by Shefrin and Statman (1985), the disposition effect refers to investor’s predilection for closing
out positions at a gain relative to a loss. Odean (1998) examines data from a large US discount
brokerage and finds evidence consistent with the disposition effect. The effect is a robust
empirical result that has been found in a variety of settings.
11
The disposition effect is based on a
stock’s performance from the time it was purchased, so without further controls rank could be
proxying for such performance and driving the rank effect.
The sharpest distinction between the rank effect and simple descriptions of the
disposition effect is the tendency for investors to sell their most extreme-ranked losing positions,
which the disposition effect suggests they should hold on to. Finding that the worst-ranked
position (typically a loss) is the second most likely stock (for investors) or the most likely stock
11
Settings include individual investors (Odean 1998; Feng and Seasholes 2005; Kaustia 2010), mutual fund
managers (Wermers 2003; Frazzini 2006), futures traders (Locke and Mann 2005) and prediction markets
(Hartzmark and Solomon 2012). Kaustia (2010) provides a recent review of the disposition effect literature.
16
(for mutual funds) to be sold makes it unlikely that the simple disposition effect can account for
the rank effect.
While the disposition effect is often described purely in terms of gains and losses, the
magnitude of the gain and loss can impact trade as well (Ben-David and Hirshleifer 2012).
Empirically, investors are more likely to sell a position as it becomes a larger gain, or a larger
loss. Investors trading on the magnitude of past returns could drive the rank effect as portfolio
rank is best with the largest gain and worst with the largest loss.
Return levels and other factors are controlled for to show that relative performance, not
return levels, is responsible for the rank effect. Using logit regressions, a dummy variable Sell,
equal to one if a stock is sold and zero otherwise, is regressed on variables for rank and a number
of controls taken from Ben-David and Hirshleifer (2012) and listed in Equation 1:
12
( )
( )
( )
( )
( √ )
( √ )
(√ )
( )
( )
(1)
Rank Variables
represents the various measures of the rank effect. To control for the likelihood
of closing out a gain versus a loss I include a dummy variable equal to one if the position has a
positive return relative to the purchase price (Gain). To control for an increasing probability of
sale in returns I include variables for the return from purchase price interacted with a dummy for
a positive return relative to the purchase price (Gain*Return), and a separate variable of the
return interacted with a dummy for non-positive return from the purchase price (Loss*Return).
12
This paper’s focus is slightly different than Ben-David and Hirshleifer (2012), so the variables have been
modified for conciseness. I have omitted log(buy price), included zero returns in the Loss dummy instead of a
separate dummy and included only one variable for holding days. The results are robust to using the original
specification (see Appendix Table IA.25).
17
Including these variables allows for the probability of sale to increase with returns (in absolute
value) and to have a different slope in the positive and negative domain.
13
Mechanical effects due to holding period and volatility are also controlled for. All else
equal, a stock held for a longer time is more likely to achieve an extreme rank. To control for the
number of days a position is held from purchase and sell date, the square root of the days
(√ ) and interactions with the gain and return (Gain*Return*√ )
and loss dummy by return (Loss*Return*√ ) are included. A stock with a higher
variance is more likely to achieve a best rank or a worst rank. To control for this effect, the
variance over the previous year, is interacted with the gain dummy (Gain*Variance) and loss
dummy (Loss*Variance).
14
All results are presented as marginal effects.
Table 1.4 column [1] presents the regression for individual investors, with no rank
variables. The propensity to sell bigger gains and larger losses is apparent as the Gain*Return
coefficient is positive and significant, while the Loss*Return dummy is negative and significant.
Further, the Gain dummy is positive and significant indicating a higher likelihood of selling
gains rather than losses, even with the controls. Without controlling for rank, consistent with
Ben-David and Hirshleifer (2012), investors are more likely to sell a gain than a loss, and also
more likely to sell both gains and losses as their returns become larger in absolute value.
Columns [2] and [3] add variables to examine the rank effect for individual investors.
Column [2] adds dummy variables for the highest return in the portfolio (Best) and the lowest
(Worst). The best-ranked stock is 15.7% more likely to be sold and the worst-ranked stock
(Worst) is 10.7% more likely to be sold, both with large t-statistics. After including the two
13
The Internet Appendix contains robustness checks of these specifications and shows that the results are robust to
examining subsets of portfolio size (Appendix Table IA.3), days between trade (Appendix Table IA.5) and allowing
for non-linear patterns in the level of returns (Appendix Table IA.1).
14
The variance is calculated over the preceding 250 trading days, if there are at least 50 non-missing observations.
18
dummy variables for rank, the Loss*Return and Gain*Return coefficients are insignificant and
the Gain dummy coefficient decreases.
Column [3] includes dummies for the Best and Worst-ranked stocks as well as 2
nd
Best
and 2
nd
Worst-ranked stocks. The coefficient on Best increases from 16.3% in Table 1.2 without
controls to 20.5% with controls. The coefficient on Worst increases from 8.5% without controls
to 14.7% with controls. Rather than explaining the rank effect, the inclusion of disposition effect
controls makes the rank effect larger. After including rank controls, the coefficients on
Loss*Return and Gain*Return switch signs. Together, this suggests that the propensity to sell
positions at a larger gain and larger loss observed in Ben-David and Hirshleifer (2012) is better
understood as a propensity to realize extreme-ranked positions.
Further, adding rank variables to the basic specification increases the explanatory power
of the regressions. Examining the R
2
presented below columns [1] and [3] there is an increase
from 0.010 to 0.047. While this R
2
indicates much more is needed to fully explain trading
behavior, including controls for rank increases the explanatory power more than threefold.
Examining columns [4] through [6] yields similar patterns for mutual funds. In column
[6] the coefficient on Best is 11.9% and the coefficient on Worst is 16.9%, both are highly
significant. The coefficients on Loss*Return and Gain*Return decrease in size after the rank
variables are added, but, unlike the individual investor specification, retain their sign and
significance. The Best and Worst coefficients for the funds are large and highly significant, with
a similar magnitude to those of the individual investors.
In Table 1.2, the two most likely positions to be sold for individual investors are best and
2
nd
best, while for the mutual funds it is worst and second worst. After including controls, it is
best and worst for both types of investors. The asymmetry remains in the ordering of best and
19
worst, where best is more likely to be sold for investors, and worst is more likely to be sold for
funds, but with the addition of the controls the importance of being the highest, or lowest return
becomes clear.
To provide a more precise control for the disposition effect, Table 1.5 repeats the
analysis, restricting the sample to individual investor portfolios where all positions are at a gain
and all positions are at a loss (omitting the gain and loss dummies).
15
This restricted sample still
exhibits the rank effect. Traders with all positions at a gain are 6.2% more likely to sell their
worst stock and 11.7% more likely to sell their best compared to a middle stock. Traders with all
positions at a loss are 5.8% more likely to sell their worst and 4.5% more likely to sell their best
stock compared to a middle-ranked position.
In the context of models of trade based on narrow framing, these results suggest that the
reference point needs to incorporate performance relative to the portfolio. The Ingersoll and Jin
(2013) realization utility model with a reference point of the purchase price offers an illustrative
example. In this model investors narrowly frame on each stock and sell a stock when its price is
above an upper cutoff or below a lower cutoff. If such an investor holds a portfolio with all
positions at a gain, each position is above the lower cutoff and so the worst-ranked position will
never be sold. Similarly, when all positions are at a loss the best-ranked position is not sold.
Table 1.4 and Table 1.5 shows this is not the case.
16
Alternatively, a standard Ingersoll and Jin investor can use a reference point based on
relative evaluation within the portfolio and display the rank effect in such situations. When all
positions are at a loss, such a reference point would allow the investor to sell a best-ranked
15
The test is limited to individual investors as there are very few mutual fund reports where all positions are at a
gain or a loss. Funds have 129,415 report days of which 301 are all at a gain and 159 are all at a loss.
16
Table 1.5 shows this is true for a zero return reference point. As an alternative example, an Ingersoll and Jin
investor could use the market return as the benchmark to measure gains and losses. Appendix Table IA.2 presents a
similar analysis based on when all holdings beat the market, or do not beat the market and finds a rank effect.
20
position, and similarly sell a worst-ranked position when all positions are at a gain. Thus the
empirical results suggest that the reference point cannot be based solely upon those mooted in
the literature and must vary with relative performance in the portfolio.
Section 4.2 examines whether rank proxies for common information, while this section
examines whether trading based on past returns can explain the rank effect. By utilizing both
controls for past performance and also including fixed effects for the interaction of stock and
day, the results in Table 1.6 rule out both simultaneously. Further it controls for investor specific
effects by adding fixed effects for the interaction of investor and day.
Even with these fixed effects and controls, the rank effect remains a major determinant of
trading behavior. Column [2] and [5] include stock by day fixed effects so the regression is
identified based on variation of rank across stocks on the same day. Individual investors are
14.1% more likely to sell best-ranked positions and 10.4% more likely to sell worst-ranked.
Mutual funds’ best-ranked positions are 9.4% more likely to be sold, and worst-ranked positions
are 12.3% more likely to be sold.
Columns [3] and [6] add dummy variables for investor by day fixed effects. This controls
for investor-specific characteristics on a given day. These two sets of dummy variables remove a
significant amount of the variation in the data, but the rank effect remains. Investors are 7.9%
more likely to realize a best-ranked position and 5.1% more likely to realize the worst-ranked.
Funds are 3.7% more likely to realize their best-ranked position and 7.7% more likely to realize
their worst. The results indicate that neither stock specific effects nor investor specific effects,
allowing for separate effects on each day, are responsible for the rank effect.
17
17
Appendix Table IA.12 and Appendix Table IA.13 present analysis using fixed effects without using the date
interaction. These specifications yield a similar magnitude, and if anything are slightly stronger.
21
1.4.4 Rank Effect Mechanism
The salience of extreme outcomes offers one possible explanation for the rank effect for
individual investors. If an investor holds multiple stocks they can pay more or less attention to
certain stocks in their portfolio. If investors pay more attention to extreme-ranked stocks they
will be more likely to trade these stocks which could account for the rank effect.
If extreme positions are salient, investors will not only be more likely to sell, but also
more likely to buy additional shares of a stock with the best or worst return. Investors decide to
buy more of a given stock far less frequently than they decide to sell a position, as only about
3.3% of purchases are of positions already held. Table 1.7 column [1] presents summary statistic
results similar to the difference rows in Table 1.2. A best-ranked position is 0.6% more likely to
be realized than a middle position and a worst-ranked is 3.8% more likely to be. Table 1.7
columns [2] and [3] present the same specification as Equation 1 controlling for returns since
purchase, but uses a dummy Buy, equal to one if additional shares are purchased, as the
dependent variable. Examining the top two rows, investors are more likely to purchase more
shares of the best and worst-ranked positions, with a Best coefficient of 1.7% and a Worst
coefficient of 2.1%. While small in magnitude, these coefficients suggest that best and worst
positions are roughly 50% to 80% more likely to be purchased than the baseline middle
probability of 2.6%.
The rank of a stock is correlated with other economically meaningful variables, making
clean identification of salience difficult. To test if investors are framing on their portfolio and
that the extremes of ordering are attention grabbing, ideally investors would see stocks presented
in an order uncorrelated with economic outcomes. While investors do not see stocks presented in
a random order, they often see their holdings presented alphabetically by company name in a
22
brokerage statement or online.
18
Alphabetical order is unlikely to exhibit significant correlation
with economic variables of interest, after controlling for stock-specific information.
Table 1.8 shows that the stocks with the names that come first or last in the portfolio by
alphabetical order are more likely to be both purchased and sold. The table presents regressions
of Sell or Buy on a dummy variable for the first and last name in the portfolio, along with stock
by day fixed effects. The fixed effect limits identification to variation from stocks that are
alphabetically first or last in one portfolio and not first or last in another portfolio on the same
day, thereby controlling for stock specific information on a given day. Columns [1] and [4] limit
the data to the first and second name by alphabetical order. The first name is 2.6% more likely to
be sold than the second name and 0.8% more likely to be purchased. Columns [2] and [5] limit
the sample to the last and second to last name. The last name is 2.9% more likely to be sold than
the second to last name and 0.8% more likely to be purchased. When the entire sample is
considered in columns [3] and [6] both the first and last name are 6.1% more likely to be sold
than a middle name and both are 1.7% more likely to be purchased. The unconditional
probability of sale is 11.3% and 3.3%, respectively for sales and purchases. In these regressions,
company names contain no economically meaningful information, but still have a meaningful
impact on the probability a position is sold because of the salience induced by being at the
extreme of an ordering.
To empirically understand what aspect of rank is important for the rank effect, I examine
three potential mechanisms suggested by various theories of salience:
1) Rank Extremeness: As discussed in Section 2, rank-dependent utility models (such as
Tversky and Kahneman 1992) predict that extreme returns receive the most attention.
18
Unfortunately I do not have brokerage statements associated with this dataset. Alphabetical by name is consistent
with the internal stock identifier used by the brokerage. Similar results are obtained using ticker instead of company
name (see Appendix Table IA.19).
23
2) Average Extremeness: Models such as Bordalo, Gennaioli, and Shleifer (2012)
predict a position becomes more salient as it becomes more extreme relative to a
portfolio benchmark, such as the average holding return in the portfolio.
3) Outlier Extremeness: Certain models of consideration sets predict that a position is
more salient when it is best or worst-ranked and also as it becomes more extreme
versus the next closest return in the portfolio (Hauser 2013).
Empirically, both rank extremeness and outlier extremeness are significant aspects of the rank
effect. This is consistent with the theory of consideration sets suggesting that the choice of what
to pay attention to is an important aspect of the trading decision.
Average extremeness is suggested by the model of Bordalo, Gennaioli, and Shleifer
(2013) where certain stocks are salient because they are most different from a benchmark.
Salient assets, such as those considered safe or with positive skewness, receive attention because
they are “most different or salient relative to the average” in the market. A stock-specific
benchmark is ruled out in section 4.2, but if this model is extended to include a portfolio specific
benchmark, such as the average return in the portfolio, it predicts the salience of extreme
positions because they are the most different from this average.
Two possible benchmarks for average extremeness are utilized, the difference from the
average and the difference from the median. The average variable is scaled by the standard
deviation of portfolio returns ((Return-Avg Return)/SD Portfolio) while the difference from the
median return (Return-Med Return) is included as a level variable without scaling.
19
Scaling by
standard deviation normalizes the measure across portfolios to examine extremeness in the
distribution of portfolio returns. The unscaled measure controls for a level effect.
The third measure captures outlier extremeness, how much of an outlier a position is
compared to the next closest return, motivated by models of consideration sets. An individual
19
Similar results are obtained if the mean return measure is not scaled by the standard deviation or the median is
scaled by the standard deviation or interacting either variable with Best or Worst. See Appendix Table IA.20 for
these specifications.
24
utilizes a consideration set if they simplify a complex problem by first using a rule of thumb, a
heuristic, to consider only a subset of the initial options. A common heuristic is to consider any
option that has at least one attribute that an investor cares about, and ignore options that lack all
of these attributes (Hauser 2013). This is known as a disjunctive decision rule which in a
portfolio setting can lead to outlier extremeness.
For example, assume an investor utilizes such a rule and that one attribute that warrants
consideration is the investor viewing a stock as having performed well or poorly.
20
When
deciding which stocks have done well or poorly the investor is deciding what stocks are in the
middle so they can be ignored. If the performance of the 2
nd
ranked stock appears different
relative to the extreme stock it, along with other stocks in the middle, is excluded from the
consideration set. Thus, outlier extremeness between the extreme-ranked stock and the 2
nd
most
extreme stock can impact the attention an extreme-ranked position receives. Outlier extremeness
is measured as the difference between the best and second best return Best*(Best Return - 2nd
Best Return) and the worst and second worst return Worst*(Worst Return - 2nd Worst Return).
Table 1.9 presents the results for different specifications to determine which measures of
extremeness best capture the rank effect.
21
Examining columns [1], [2] and [3], each of the
variables on its own is associated with more selling. However, after adding dummy variables for
the top two and bottom two positions in columns [4] and [5] the average return variables switch
signs while the median return variables become smaller and insignificant for the negative
coefficient. The variables for outlier extremeness decrease with the addition of rank dummies,
but remain economically meaningful and significant. Columns [7] and [8] include both measures
20
In addition they can also use other “excitement” criteria such as stocks that were in the news, had high recent
returns or stocks that caught their eye at the top of the brokerage statement. This paper focuses on rank, but that does
not preclude other avenues that may induce salience.
21
Appendix Table IA.15 presents the analogous results for buying, finding a similar pattern.
25
of extremeness and the variables proxying for average extremeness are unable to account for the
rank effect. Thus both rank and outlier extremeness are significant aspects of the rank effect.
This is consistent with investors utilizing consideration sets as a part of their trading decision.
1.5. Price Effects
The rank effect has a large impact on trading behavior, but is there enough rank based
selling to induce predictable returns? Given heavy selling and downward sloping demand, selling
pressure from rank-based trade will impact the returns of extreme-ranked stocks leading to a
decrease and subsequent reversal. Using the mutual fund data, a trading strategy is constructed to
exploit this predictable impact on prices.
First, I examine the selling intensity of mutual funds and show that mutual funds sell a
larger proportion of their best and worst-ranked positions, and are more likely to liquidate entire
positions for both best and worst-ranked positions. Further, the selling is more intense for worst-
ranked stocks compared to best-ranked stocks.
Table 1.10 shows the results. The Fraction Sold columns present linear regressions, with
the number of shares sold divided by the number of shares held on the left hand side. The All
Holdings column includes all holdings regardless of whether they are sold, and the Sell Only
column includes only positions where some fraction of a holding is sold. Best-ranked positions
have 7.8% more of their position sold compared to all stocks and 2.3% of the best-ranked are
sold compared to all stocks that are sold. Worst-ranked stocks have a significantly larger fraction
sold of 17.5% more compared to all other stocks (14.1% compared to stocks that are sold).
Best and worst-ranked stocks are also more likely to be completely liquidated than other
positions. The Liquidate column presents marginal effects from a logit regression similar to
26
Equation 1, where the sell dummy is replaced with a dummy equal to one if a position is
liquidated. Best-ranked stocks are 3.8% more likely to be liquidated than all other positions, and
0.1% more likely to be liquidated versus other stocks that are sold. The worst-ranked stock has a
larger effect where it is 12.7% more likely to be liquidated than all other positions and 23.7%
more likely to be liquidated than other positions that are sold. Worst-ranked stocks are sold the
heaviest, so they should receive more price pressure and exhibit larger price effects.
The analysis shows that mutual funds sell extreme-ranked positions before the next report
date and they sell these positions heavily, especially for worst-ranked positions. The trading
strategy focuses on the reversal subsequent to the sale, rather than the initial price pressure for
two reasons. First, the mutual fund data does not report precisely when a position is sold.
Second, as discussed in Schwarz and Potter (2012), there is a lag from the date that a fund holds
a portfolio of positions and the date the information becomes public. By law a fund has 60 days
to report the information to their shareholders, and must make it publicly available 10 days after
this report. Schwarz and Potter show that the information is public within 71 days for almost all
observations, but that it is typically not available much earlier. Thus an implementable strategy
must be based on information at least 71 days after the report date.
To capture returns from the reversals, portfolios are formed in the following manner. All
holdings reported in a given month, denoted month M, are examined ten trading days into month
M+3. This is more than 70 trading days past the report day and almost a full quarter from the
report date in most cases. In the simplest specification, all stocks that are ranked best in at least
one fund from purchase price to ten trading days into month M+3 are put into an equal-weighted
portfolio. The same is done with worst-ranked stocks. I subsequently examine portfolios
27
weighted more heavily towards stocks predicted to have received more price pressure. Portfolios
are held from the 11
th
trading day of month M+3 until the 10
th
trading day of month M+4.
For example, if a fund reports its holdings on March 25, 2006, the stocks in that fund are
ranked on their returns between the purchase price and June 14, 2006, 81 days after the report.
Based on these ranks, portfolios are formed the next trading day and held until July 17, 2006. All
holdings from all of the funds reporting in March, 2006 are ranked from purchase price to June
14, 2006. Thus the portfolio held from June to July is based on funds reporting in March.
The stocks in the best portfolio will generally have done well recently and the stocks in
the worst portfolio will have done poorly.
22
Thus both portfolios are impacted by momentum
(Jegadeesh and Titman 1993) and one month reversals (De Bondt and Thaler 1985) although in
opposite directions. If momentum is responsible for the returns in these portfolios, the best-
ranked stocks will have positive returns while the worst will be negative. Similarly, if one month
reversals (De Bondt and Thaler 1985) are responsible for the returns, the best-ranked stocks will
have negative returns while the worst-ranked will have positive returns. To control for the two
effects I include a momentum factor (UMD) and a short term reversal factor (ST_REV).
Table 1.11 presents CAPM, Fama-French three factor, four-factor and five-factor
regressions (including a one month reversal factor) for the equal-weighted best and worst-ranked
portfolios. Examining the worst portfolio, the CAPM and three factor models have insignificant
alpha coefficients. The surprise is not the insignificance, but rather the fact that it is not
significantly negative, as in general the worst-ranked portfolio will be a negative momentum
portfolio. After including a momentum factor the alpha becomes 137 basis points and significant
with a t-statistic of nearly five. It loads heavily and inversely on the momentum factor. Adding
the short term reversal factor yields an alpha of 161 basis points.
22
See Appendix Table IA.14 for summary statistics of the stocks that comprise these portfolios.
28
The best-ranked portfolio has a positive and significant alpha for both the CAPM and
three-factor alpha, as it is in general a positive momentum portfolio. After including the
momentum factor, the alpha decreases to 20 basis points and becomes insignificant. This
portfolio will in general also be a high one month reversal portfolio, indicating it should have a
low return the month after ranking. After including controls for a short term reversal factor the
alpha is 36 basis points and is significant with a t-statistic of 1.98. The worst-ranked portfolio has
a higher alpha than the best-ranked, consistent with the Table 1.10 finding that worst-ranked
stocks are sold more intensely.
These equal-weighted portfolios can be refined to put more weight on stocks predicted to
experience greater selling pressure. Table 1.12 examines the price effects in the four and five-
factor regressions using two such weighting schemes. First, portfolios are weighted by the
number of funds holding the stock at an extreme rank. The worst-ranked portfolio has a four-
factor alpha of 182 basis points with a t-statistic of 4.57, which decreases to 163 with a t-statistic
of 3.57 after including the short term reversal factor. The number-of-funds-weighted best-ranked
portfolio has insignificant abnormal returns in the four-factor model, but increases to 65 basis
points with a t-statistic of 2.39 after the short term reversal factor is added.
The second strategy weights the portfolios based on the fraction of a stock’s market cap
that is extreme-ranked. Stocks with a higher proportion of their market cap sold are more likely
to experience greater price pressure. Worst-ranked positions have a four-factor alpha of 211 basis
points with a t-statistic of 4.10 and an alpha of 222 basis points with a t-statistic of 3.75 after the
short term reversal factor is added. Best-ranked stocks have an insignificant alpha of 21 basis
points before the short term reversal factor is added, which increases to 42 basis points with a t-
statistic of 2.15 with the short term reversal factor.
29
Table 1.13 uses Fama-Macbeth regressions (Fama and Macbeth 1973) to control for
additional factors in the cross section. Regressions are run on the same monthly time periods
(from 11 trading days into a month until ten trading days into the next month) with return the
month after ranking (multiplied by 100) as the dependent variable. Column [1] presents the
results from the regression including only dummy variables for being the best or worst-ranked
position on the right hand side. The coefficients are positive, but significant only for the worst
portfolio. Column [2] adds controls for momentum, one month reversals, market capitalization,
and book to market (calculated as in Fama and French 1992). The coefficients on Best and Worst
are now significant with Best having a coefficient of 0.253 and a t-statistic of 2.04 and Worst
having a coefficient of 0.743 and a t-statistic of 2.64.
Next controls are included for five anomalies for which rank may proxy - the earnings
announcement premium (Beaver 1968; Frazzini and Lamont 2007), the Gervais, Kaniel and
Mingelgrin (2001) high-volume return premium, the dividend month premium (Hartzmark and
Solomn 2013), idiosyncratic volatility (Ang, Hodrick, Xing and Zhang 2006) and share issuance
(Pontiff and Woodgate 2008).
23
After adding these controls, the coefficients on best and worst
increase in both point estimate and significance.
23
Earnings are controlled for by a dummy variable equal to one if the stock has an earnings announcement during
the period. The high-volume return premium is controlled for using a dummy equal to one if the ranking day was in
the top five of the previous fifty trading or in the bottom five by volume. Predicted dividend is a dummy equal to
one if the company paid a dividend 3, 6, 9 or 12 months ago, where “month” is defined from 11 trading days into the
current month until 10 trading days into the next month. Idiosyncratic volatility for stock j on day t is calculated as
the standard deviation of the residuals from a regression of excess return on MKT, SMB and HML from t-1 to t-21.
Share Issuance for stock j with holding period ending in month M is the log of shares outstanding at M-6 minus the
log of shares outstanding at M-17 using the same month definition as the predicted dividend variable.
30
1.6. Rank Effect Robustness
1.6.1 Tax-Based Explanations
Tax-based incentives may induce the trade of extreme-ranked positions. U.S. tax law
creates time varying incentives to sell positions throughout the year, especially for losses
(Constantinides 1984). Figure 1.2 repeats the specification from Table 1.4 column [3] separately
for each calendar month. It graphs the coefficients on Best and Worst along with their 95%
confidence intervals as dotted lines. As expected, there is a seasonal pattern where losses are
more likely to be realized in November and December due to tax considerations. The reverse
pattern is apparent for the best positions which are most likely to be sold in January. Even so, the
95% confidence intervals are always far above zero. Thus even though the coefficients exhibit
seasonality, the effect in each month for both Best and Worst is large and significant.
Roughly 20% of the observations in the data come from deferred tax accounts. Such tax
deferred accounts lack many of the tax induced incentives that might account for the rank effect.
Thus finding the rank effect in these accounts suggests trading based on short term tax incentives
does not account for the effect. Table 1.14 separates the analysis by the tax status of the
accounts. Tax deferred accounts exhibit a rank effect of a similar magnitude to the standard
taxable accounts. Thus tax incentives do not appear to be responsible for the rank effect.
1.6.2 Covariate Balance
If the data lacks the requisite covariate balance, logit regression may not identify the
effect of becoming extreme-ranked. Covariate balance refers to the similarity of the empirical
distributions of the covariates for extreme rank and middle rank. A lack of balance can increase
model dependence and bias the estimation of the effect of being extreme-ranked (for example
Abadie and Imbens 2007; Ho, Imai, King and Stuart 2007). There is a mass of high return values
31
that are ranked best with few positions at those levels not ranked best (and vice versa for worst-
ranked positions). This could lead to improper inference that rank is responsible for the pattern,
when it is a spurious relation due to model dependence and a sample where the treatment group
(best or worst rank) is not balanced with the control group (not ranked best or worst).
To identify the effect of becoming best or worst rank, I utilize entropy balancing
(Hainmueller 2012). To my knowledge this is the first paper to employ entropy balancing in a
finance setting. Entropy balancing offers a number of advantages over methods such as nearest
neighbor matching, propensity score matching or propensity score weighting.
24
These indirectly
attempt to achieve covariate balance using an estimated probability of treatment from the
covariates of interest. They match on this probability and not on the covariates themselves.
Entropy balancing directly achieves covariate balancing by matching moments of the covariates
between the treatment and control group. It weights to achieve this balance while keeping the
weights as close to the original values as possible.
Table 1.15 presents the results. Entropy balancing is for binary treatments, thus best and
worst variables are examined separately. The first column labeled “Unweighted” contains:
The first column contains similar, though slightly smaller numbers to Table 1.2.
25
The second column contains these values utilizing weights from entropy balancing. The
balancing is conducted on return, square root of holding days, variance and the interaction of the
return and holding days. The balancing is conducted separately for each day which indirectly
24
Similar results are obtained using these more traditional methods. Appendix Table IA.24 presents a similar
analysis using a nearest neighbor propensity score match.
25
The magnitude is smaller as the comparison groups now contains extreme-ranked positions (i.e. Not Best contains
worst-ranked stocks) and the 2
nd
best and 2
nd
worst positions making the comparison group mean slightly higher.
32
controls for changes over time of the effect of the covariates. To make sure there is enough data
for balancing, in the investor data I exclude days in the bottom quartile of observations per day
(626 observations). Taking the Best-Not Best row in the Entropy Balanced column as an
example, the Best proportion sold has a Not Best proportion sold subtracted from it where the
weighted average sample comprising Not Best has the same average return, number of holding
days, interaction term and variance each day as the data that comprise Best.
The 2
nd
column labeled Entropy shows that after entropy balancing, the rank effect
increases and remains significant for both individual investors and mutual funds. For individual
investors, the values of 12.8% for best and 8.4% for worst are both slightly smaller than the
coefficients with simple controls in the logit. For funds, the values of 11.6% for best and 16.5%
for worst are of a similar magnitude to the previous coefficients. These tests measure the same
difference as the logit regressions of Table 1.4 columns [2] and [5] yield similar results. Thus
the entropy balancing results underscore the significance of the effect and suggest the logit
results are not driven by a lack of covariate bias.
As a final example of the robustness of the rank effect I show that it is large and robust
when the entropy balanced sample is examined separately day by day. Figure 1.3 graphs the
mean difference of the entropy balanced sample for each day. The x-axis indicates the daily
entropy balanced mean for Best-Not Best or Worst-Not Worst, with the dotted red line indicating
a 0 difference. The y-axis graphs the number of days with that difference. Thus mass to the right
of the red line represents days with a positive rank effect, and mass to the left indicates days
without a rank effect. Almost every observation with significance has a positive mean, indicating
that looking at each day separately the extreme-ranked positions are more likely to be sold.
33
Examining the investor charts, for most days (719 of 756 for Best and 693 of 756 for
Worst) extreme-ranked stocks were more likely to be sold, indicated by a positive value on the x-
axis. For best-ranked stocks, 524 days have a positive and significant difference (the red bars),
while no day has a negative and significant difference. For worst-ranked positions, 327 days
have a positive and significant difference, while no day has a negative and significant difference.
The bottom two figures present the results for mutual funds, showing a similar, strong
effect. In 182 of the 200 report days, best-ranked positions are more likely to be sold, and in 190
of the days worst-ranked positions are more likely to be sold. For best-ranked, 103 days are
positive and significant, while no days are negative and significant. For worst-ranked, 155 days
are positive and significant while only two are negative and significant (the purple bars). The
results suggest that a lack of covariate balance is not a significant factor in the initial analysis.
1.7. Conclusion
This paper documents a new stylized fact which I term the rank effect. The effect refers
to the finding that best and worst-ranked positions are more likely to be sold compared to
positions in the middle of the portfolio. This effect is present for both individual investors and
mutual funds and is robust to a variety of controls for common information, past returns, tax
motivations and model dependence. The rank effect is associated with heavy selling of extreme
positions by mutual funds which induces predictable returns of 40 basis points per month for
best-ranked stocks and 160 basis points per month for worst.
The paper shows that portfolio specific salience is an important component to the trading
decisions of individual investors. The alphabetical order in the portfolio by company name is not
related to a stock’s fundamentals, and thus offers a clean test of the salience induced by ordering.
34
Demonstrating the importance of portfolio specific salience shows that an integral portion of the
trading process that has hithertofore not been emphasized, is deciding what stocks to pay
attention to through the formation of consideration sets.
The rank effect illustrates a fundamental component of investor behavior, namely that
how a stock is viewed is based in part on how it compares to other holdings in a portfolio. This is
in stark contrast to the commonly used assumption of narrow framing, where each stock is
assumed to be evaluated in isolation without regard to the other holdings in a portfolio. While
investors are not forming optimal portfolios, the portfolio is a relevant and important component
to trading behavior.
35
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Figure 1.1 - Heterogeneity of Rank Based Selling
These figures present the joint density between the investor specific proportion worst realized (#Worst Sold/(#Worst Sold+#Worst Not Sold)) and the
investor specific proportion best realized (#Best Sold/(#Best Sold+#Best Not Sold)) for individual investors and fund managers that have at least five
sell days, or report days in the data. Lighter shading corresponds to a higher the density with white indicating the densest portion. Darker shading
corresponds to less density with black indicating the lowest density. Investor data covers January 1991 to November 1996 and mutual fund data covers
January 1990 to June 2010.
Panel A: Individual Investor Panel B: Mutual Fund
39
Figure 1.2 - Seasonality of the Rank Effect for Individual Investors
Marginal effect from logit regression as specified in Table 1.4 column [3]. Regressions are run separately for each
calendar month. Best is the coefficient on the dummy variable for the highest return and worst is the coefficient
from the dummy variable for the lowest ranked return. The dotted line is the upper and lower bound of the 95%
confidence interval. This figure contains individual trading data from January 1991 through November 1996.
40
Figure 1.3 – Daily Rank Effect after Entropy Balancing
This table presents the entropy balanced regression coefficients from daily regressions of a dummy variable equal to
1 if a stock is sold regressed on a dummy variable for best (worst) versus not best (worst). Each day the sample is
entropy balanced and these weights are used in the regression. Pos indicates a coefficient greater than 0 and Neg
signifies a coefficient less than or equal to 0. Only days where a stock is sold are included and an investor must hold
at least 5 stocks to be included in the sample. Stocks are not included on the day that position is opened. A stock is
considered best (worst) if the stock has the highest (lowest) return in the portfolio. Investor data covers January 1991
to November 1996 and mutual fund data covers January 1990 to June 2010. Standard errors are clustered by account
or fund and indicated significance is at the 5% level.
41
Table 1.1 – Summary Statistics
This table presents summary statistics. Panel A presents information on investors from January 1991 to November 1996. Only days where a
stock is sold are included and an investor must hold at least 5 stocks to be included in the sample. Stocks are not included on the day their
position is opened. Panel B is based on mutual fund data from January 1990 to June 2010. Dates examined are report dates. A fund must hold
at least 20 CRSP merged securities to be included in the analysis.
Observations Mean Std. Dev. Minimum 25th Pctile Median 75th Pctile Maximum
#Accounts 10,619
#Sell Days 94,671
Proportion Sold 1,051,160 0.120
Proportion Liquidated 1,051,160 0.096
Portfolio Size 94,671 11.103 13.887 5 6 8 12 429
Holding Days 1,051,160 340.299 381.702 1 66 194 477 2,148
Observations Mean Std. Dev. Minimum 25th Pctile Median 75th Pctile Maximum
#Funds 4,730
#Report Days 129,415
Proportion Sold 15,604,501 0.389
Proportion Liquidated 15,604,501 0.151
Portfolio Size 129,415 120.577 224.437 20 40 61 103 3,282
Holding Days 15,604,501 946.622 1056.988 11 243 548 1,277 11,048
Panel A: Individual Investor
Panel B: Mutual Fund
42
Table 1.2 – Proportion of Positions Realized by Rank
This table presents summary statistics of the ratios of stocks that are sold in the indicated group
divided by all stocks in that group. For example, the Best row reports #Best Sold/(#Best
Sold+#Best Not Sold). The last four rows present the difference between the indicated groups with
a t-statistic (clustered by date and account) on the null hypothesis that the difference is 0 in
parenthesis. Only days where a stock is sold and an investor holds at least 5 stocks are included in
the sample. Stocks are not included on the day the position is opened. Individual investor data
covers January 1991 to November 1996. Mutual fund data covers January 1990 to June 2010.
Dates examined are report dates. A fund must hold at least 20 CRSP merged securities to be
included in the analysis.
Individual Investor Mutual Fund
All Ranks 0.121 0.389
Worst 0.169 0.576
2nd Worst 0.137 0.529
Middle 0.084 0.384
2nd Best 0.195 0.487
Best 0.247 0.503
Worst-Middle 0.085 0.191
(15.20) (20.97)
Best-Middle 0.163 0.119
(28.36) (15.36)
Observations 1,053,065 15,604,501
43
Table 1.3 – The Rank Effect for Stocks that on the Same Day are Extreme-Ranked in one
Portfolio and Not Extreme-Ranked in Another Portfolio
Panel A presents the difference in probability between a best (worst) ranked stock and a not best
(worst) ranked. The sample includes only stocks on the same day are extreme-ranked for at least
one investor and not extreme-ranked for at least one other investor. First the average sale
probability is taken for each stock date by extreme rank observation. Next the difference between
this value for extreme rank and not extreme rank is taken. The Best – Not Best row reports this
difference in probability for best-ranked, and the Worst – Not Worst row reports this difference for
worst. The number in parenthesis is the t-statistic clustered by date and cusip and the number
below that is the number of cusip by day observations. Panel B presents a linear regression of sell
on a best and worst dummy with a fixed effect for each stock by day pair. The top number is the
coefficient, and the lower number in parenthesis is the t-statistic. Standard errors are clustered by
date and account for the investors and date and WFICN for the mutual fund data. For the investor
data only days where a stock is sold are included and an investor must hold at least 5 stocks to be
included in the sample. Stocks are not included on the day that position is opened. Data covers
January 1991 to November 1996. For the mutual fund data is analyzed on report dates and funds
hold at least 20 stocks. Data covers January 1990 to June 2010.
Individual Investor Mutual Fund
Best - Not Best 0.102 0.074
(20.77) (25.45)
37,374 48,079
Worst - Not Worst 0.063 0.126
(16.94) (30.64)
30,219 46,260
Individual Investor Mutual Fund
Best 0.094 0.075
(15.81) (15.09)
Worst 0.064 0.125
(11.36) (12.69)
Stock by Date FE X X
Observations 1,048,549 15,603,394
R
2
0.111 0.053
Panel A: Same Stock Match
Panel B: Stock by Day Fixed Effects
44
Table 1.4 –Rank Effect with Controls for Past Performance
This table presents marginal effects from logit regressions. The dependent variable is a dummy variable equal to 1 if
a stock is sold. Best (Worst) is a dummy variable equal to 1 if the stock has the highest (lowest) return in the
portfolio and 2
nd
Best (2
nd
Worst) is a dummy for the second highest (lowest) return. Gain (Loss) is a dummy
variable indicating a positive (non-positive) return. Return is the return since purchase. Investor data covers January
1991 to November 1996. Only days where a stock is sold are included and an investor must hold at least 5 stocks to
be included in the sample. Mutual fund data are from report dates from January 1990 to June 2010. A fund must
hold at least 20 CRSP merged securities to be included in the analysis. The top number is the marginal effect, and
the lower number in parenthesis is the t-statistic. Standard errors are clustered by date and account for the investors
and date and WFICN for the mutual fund data.
[1] [2] [3] [4] [5] [6]
Best 0.157 0.205 0.109 0.119
(20.15) (21.10) (11.61) (12.01)
Worst 0.107 0.147 0.163 0.169
(19.93) (20.06) (12.17) (12.25)
2nd Best 0.125 0.105
(16.51) (12.61)
2nd Worst 0.085 0.122
(14.37) (10.40)
Return*Gain 0.045 -0.002 -0.019 0.034 0.024 0.017
(4.55) (-0.28) (-2.58) (6.93) (4.57) (3.15)
Return*Loss -0.155 -0.036 0.004 -0.272 -0.242 -0.222
(-7.47) (-1.84) (0.19) (-12.39) (-11.21) (-10.27)
Gain 0.037 0.029 0.026 -0.013 -0.014 -0.014
(9.60) (8.31) (8.00) (-3.92) (-4.24) (-4.29)
-0.002 -0.001 -0.001 0.000 0.000 0.000
(-5.00) (-3.36) (-2.41) (-4.59) (-3.57) (-2.87)
0.004 0.002 0.002 0.008 0.007 0.007
(3.83) (2.82) (2.03) (11.09) (11.08) (11.05)
Variance *Gain 5.914 4.475 3.790 4.971 5.088 5.111
(2.94) (2.90) (2.81) (1.85) (1.90) (1.92)
Variance *Loss -3.644 -3.306 -3.047 -2.557 -2.329 -2.039
(-3.45) (-3.70) (-3.87) (-1.97) (-1.94) (-1.83)
√Holding Days -0.002 -0.002 -0.002 -0.002 -0.002 -0.002
(-8.90) (-9.64) (-10.37) (-5.29) (-5.27) (-5.28)
Observations 1,048,549 1,048,549 1,048,549 15,603,394 15,603,394 15,603,394
R
2
0.010 0.032 0.047 0.005 0.006 0.007
Return*Gain
*√Holding Days
Return*Loss
*√Holding Days
Individual Investor Mutual Fund
45
Table 1.5 – Rank Effect for Individual Investors with Controls for Past Performance
When All Positions in a Portfolio are at a Gain or Loss
This table presents the marginal effects from logit regressions of a dummy variable equal to 1 if a
stock is sold on characteristics of the stock being held. Only days where a stock is sold are
included and an investor must hold at least 5 stocks to be included in the sample. Stocks are not
included on the day that position is opened. The All Gain column includes investor day
observations where all positions in a portfolio have positive returns and All Loss contains
observations where all holdings are non-positive returns. Best (Worst) is a dummy variable equal
to 1 if the stock has the highest (lowest) return in the portfolio and 2
nd
Best (2
nd
Worst) is a
dummy for the second highest (lowest) return. Return is the return since purchase price, Data
covers January 1991 to November 1996. The top number is the marginal effect, and the lower
number in parenthesis is the t-statistic. Standard errors are clustered by date and account.
All Gain All Loss
Best 0.117 0.045
(8.31) (2.09)
Worst 0.062 0.058
(5.29) (3.10)
2nd Best 0.073 0.007
(7.19) (0.41)
2nd Worst 0.040 0.025
(3.88) (1.64)
Return 0.001 0.119
(0.04) (1.35)
Return*√Holding Days -0.001 0.003
(-0.81) (0.74)
Variance 18.006 -2.502
(5.16) (-1.49)
√Holding Days -0.001 -0.003
(-1.66) (-1.60)
Observations 23,679 8,898
R
2
0.013 0.012
46
Table 1.6 - Rank Effect with Fixed Effects for Stock by Day and Account by Day
This table presents coefficients from linear regressions of a dummy variable equal to 1 if a stock is sold on characteristics of the stock being
held. Only days where a stock is sold are included and an investor must hold at least 5 stocks to be included in the sample. Stocks are not
included on the day that position is opened. Account x Date FE indicates a fixed effect for each interaction of account and date. Stock x Date
FE indicates a fixed effect for each interaction of cusip and date. Best (Worst) is a dummy variable equal to 1 if the stock has the highest
(lowest) return in the portfolio and 2
nd
Best (2
nd
Worst) is a dummy for the second highest (lowest) return. Gain (Loss) is a dummy variable
indicating a positive (non-positive) return. Return is the return since purchase price. Additional controls are Gain, Return* Gain, Return* Loss,
Return*√Holding Days*Gain, Return*√Holding Days*Loss, Variance *Gain, Variance *Loss, and √Holding Days. Investor data covers
January 1991 to November 1996. Mutual fund data are from January 1990 to June 2010 where dates examined are report dates. A fund must
hold at least 20 CRSP merged securities to be included in the analysis. The top number is the coefficient, and the lower number in parenthesis
is the t-statistic. Standard errors are clustered by date and account for the investors and date and WFICN for the mutual fund data.
[1] [2] [3] [4] [5] [6]
Best 0.118 0.141 0.079 0.041 0.094 0.037
(27.52) (23.24) (10.74) (10.46) (11.69) (10.97)
Worst 0.060 0.104 0.051 0.110 0.123 0.077
(14.45) (17.83) (6.80) (21.45) (12.25) (17.22)
2nd Best 0.057 0.092 0.044 0.039 0.080 0.034
(21.63) (19.16) (7.82) (14.26) (12.51) (13.38)
2nd Worst 0.019 0.058 0.014 0.073 0.088 0.051
(7.45) (12.70) (2.59) (18.86) (10.78) (15.20)
Return*Gain 0.013 -0.050 -0.017 0.034 0.005 0.029
(1.78) (-4.58) (-1.26) (10.34) (1.05) (9.51)
Return*Loss -0.032 0.183 0.089 -0.226 -0.160 -0.143
(-1.40) (5.30) (1.94) (-12.72) (-5.11) (-11.56)
Gain 0.031 0.037 0.045 -0.007 -0.016 -0.009
(7.99) (8.20) (7.58) (-3.78) (-6.15) (-6.29)
Additional Controls X X X X X X
Account x Date FE X X X X
Stock x Date FE X X X X
Observations 1,048,549 1,048,549 1,048,549 15,603,394 15,603,394 15,603,394
R
2
0.128 0.677 0.769 0.326 0.108 0.389
Individual Investor Mutual Fund
47
Table 1.7 – Rank Effect for Buying
Column [1] presents the difference in sale probability between the indicated rank stock and a stock not in the top or
bottom two returns. Columns [2] and [3] present the marginal effects from logit regressions of a dummy variable
equal to 1 if more of a stock that is already held is purchased. Only days where a stock is purchased and an investor
holds at least 5 stocks are included in the sample. Stocks are not included on the day that position is opened. Best
(Worst) is a dummy variable equal to 1 if the stock has the highest (lowest) return in the portfolio and 2
nd
Best (2
nd
Worst) is a dummy for the second highest (lowest) return. Gain (Loss) is a dummy variable indicating a positive
(non-positive) return. Return is the return since purchase price, Data covers January 1991 to November 1996. The
top number is the marginal effect, and the lower number in parenthesis is the t-statistic. Standard errors are clustered
by date and account.
Summary
Statistics
[1] [2] [3]
Best 0.006 0.017 0.022
(2.20) (6.24) (6.51)
Worst 0.038 0.021 0.030
(13.91) (9.80) (9.57)
2nd Best 0.009 0.017
(3.37) (6.27)
2nd Worst 0.032 0.022
(12.22) (8.42)
Return*Gain -0.015 -0.017
(-5.61) (-6.46)
Return*Loss -0.062 -0.051
(-7.35) (-6.20)
Gain -0.010 -0.009
(-8.87) (-7.98)
0.000 0.000
(5.91) (6.64)
0.003 0.002
(7.00) (6.77)
Variance *Gain 1.068 1.036
(2.86) (2.86)
Variance *Loss -1.704 -1.533
(-3.21) (-3.29)
√Holding Days -0.001 -0.001
(-20.02) (-19.33)
Observations 1,440,981 1,440,981
R
2
0.041 0.046
Return*Loss
*√Holding Days
Regression
Return*Gain
*√Holding Days
48
Table 1.8 –Alphabetical Ordering by Company Name
This table presents regressions of a sell dummy (columns [1]-[3]) equal to 1 if a stock is sold or a buy dummy (columns [4]-[6]) on dummy variables based on
the alphabetical ordering by company name ordering and stock by day fixed effects. Only days where a stock is sold are included in columns [1]-[3] and only
days a stock is purchased are in cluded in columns [4]-[6]. An investor must hold at least 5 stocks to be included in the sample. Stocks are not included on the
day that position is opened. First (Last) name is a dummy equal to one if the stock name is the first (last) name by alphabetical order in the portfolio. Data covers
January 1991 to November 1996. The top number is the coefficient, and the lower number in parenthesis is the t-statistic. Standard errors are clustered by date
and account.
First and Second
Name Only
Last and Second
to Last
NameOnly All Names
First and Second
Name Only
Last and Second
to Last
Name Only All Names
[1] [2] [3] [4] [5] [6]
First Name 0.026 0.061 0.008 0.017
(3.80) (10.69) (2.31) (6.43)
Last Name 0.029 0.061 0.008 0.017
(3.52) (11.02) (2.22) (6.53)
Stock x Date FE X X X X X X
Observations 185,253 185,145 1,016,954 237,293 237,200 1,396,848
Selling Buying
49
Table 1.9 – Components of the Rank Effect using Various Measures of Extremeness
This table presents the marginal effects from logit regressions of a dummy variable equal to 1 if a stock is sold on characteristics of the stock
being held. Only days where a stock is sold are included and an investor must hold at least 5 stocks to be included in the sample. Stocks are not
included on the day that position is opened. Return is the return since purchase price. Best (Worst) is a dummy variable equal to 1 if the stock
has the highest (lowest) return in the portfolio. Average return, median return and Std. Dev. (standard deviation) is the given measure for a
portfolio on a given day. Pos is a dummy variable equal to one if the number in parenthesis is greater than 0, and Neg is a dummy equal to one
when the number is less than or equal to zero. Additional controls are Gain, Return* Gain, Return* Loss, Return*√Holding Days*Gain,
Return*√Holding Days*Loss, Variance *Gain, Variance *Loss, and √Holding Days. Columns [4]-[8] also include controls for 2
nd
best and 2
nd
worst return. Investor data covers January 1991 to November 1996. The top number is the marginal effect, and the lower number in parenthesis
is the t-statistic. Standard errors are clustered by date and account.
[1] [2] [3] [4] [5] [6] [7] [8]
Pos*(Return-Avg Return)/Std. Dev. 0.056 -0.022 -0.022
(19.75) (-11.17) (-11.28)
Neg*(Return-Avg Return)/Std. Dev. -0.032 0.027 0.032
(-8.34) (8.47) (10.47)
Pos*(Return-Median Return) 0.174 0.034 0.004
(14.94) (4.02) (0.44)
Neg*(Return-Median Return) -0.074 -0.016 0.009
(-4.80) (-1.33) (0.77)
Best*(Best Return - 2nd Best Return) 0.091 0.057 0.057 0.057
(14.63) (10.64) (10.49) (9.35)
Worst*(Worst Return - 2nd Worst Return) -0.308 -0.151 -0.167 -0.155
(-24.53) (-14.32) (-16.39) (-15.04)
Best 0.253 0.197 0.186 0.234 0.185
(23.74) (20.39) (20.94) (25.52) (21.09)
Worst 0.180 0.142 0.116 0.153 0.118
(17.78) (16.78) (18.00) (16.93) (16.43)
Additional Controls X X X X X X X X
Observations 1,047,057 1,047,057 1,047,057 1,047,057 1,047,057 1,047,057 1,047,057 1,047,057
R
2
0.020 0.014 0.017 0.048 0.047 0.048 0.050 0.048
50
Table 1.10 - Mutual Fund Fraction Sold and Liquidation by Rank
This table presents a linear regression in the Fraction Sold column where the dependent variable is
the number of shares sold on a report date divided by the number of shares held the previous
report date. The Liquidate column presents the marginal effects from logit regressions of a dummy
variable equal to 1 if all shares of a stock are sold. The All Holdings column includes all holdings
while the Sell Only column includes only accounts on days where some positions are sold.
Additional controls are Gain, Return* Gain, Return* Loss, Return*√Holding Days*Gain,
Return*√Holding Days*Loss, Variance *Gain, Variance *Loss, and √Holding Days. Mutual funds
must hold at least 20 stocks to be included in the sample. Stocks are not included on the day their
position is opened. Standard errors are clustered by fund (wficn) and date. Best (Worst) is a
dummy variable equal to 1 if the stock has the highest (lowest) return in the portfolio. Gain (Loss)
is a dummy variable indicating a positive (non-positive) return. Data covers January 1990 to June
2010.
All Holdings Sell Only All Holdings Sell Only
Best 0.078 0.023 0.038 0.001
(13.55) (3.76) (10.39) (0.23)
Worst 0.175 0.141 0.127 0.237
(16.00) (12.33) (14.91) (16.30)
Return*Gain 0.028 0.000 0.003 -0.047
(4.78) (-0.07) (1.41) (-7.46)
Return*Loss -0.264 -0.299 -0.155 -0.300
(-11.26) (-12.34) (-11.85) (-9.59)
Gain -0.007 -0.023 -0.011 -0.027
(-2.88) (-4.31) (-4.95) (-4.44)
Additional Controls X X X X
Observations 15,603,394 6,068,983 15,603,394 6,068,983
R
2
0.044 0.123 0.063 0.090
Fraction Sold Liquidate
51
Table 1.11 – Price Effects Based on Mutual Fund Portfolio Rank
This table presents Fama-French regressions on monthly equal-weighted portfolios. Portfolios are formed based on mutual fund holdings rank.
Rank is based on the return from purchase price to 3 months and 10 trading days after the report month. Portfolios are held from the 11
th
trading day of a month until the 10
th
trading day of the next month. A stock is included in the worst (best) portfolio if it is ranked worst (best) in
at least one fund. Data covers January 1990 to June 2010.
α (%) 0.669 0.407 1.366 1.612 0.355 0.448 0.199 0.357
(1.65) (1.10) (4.96) (5.11) (1.90) (2.69) (1.26) (1.98)
MKT 1.741 1.629 1.234 1.252 1.074 0.958 1.061 1.073
(21.09) (20.06) (19.15) (19.18) (28.22) (26.20) (28.78) (28.75)
SMB 0.833 0.895 0.900 0.306 0.290 0.293
(6.66) (9.89) (9.97) (5.43) (5.59) (5.67)
HML 0.594 0.108 0.088 -0.234 -0.108 -0.121
(5.14) (1.21) (0.98) (-4.49) (-2.10) (-2.34)
UMD -0.846 -0.865 0.220 0.208
(-14.89) (-14.94) (6.76) (6.27)
ST_REV -0.102 -0.066
(-1.58) (-1.77)
Worst Best
52
Table 1.12 – Weighted Price Effects Based on Mutual Fund Portfolio Rank
This table presents Fama-French regressions on monthly portfolios. “Number of funds where Stock is Ranked Worst (Best)” weights portfolios
by the number of funds where the stock is best (worst) rank. “Fraction of marketcap that is Worst (Best)” weights by the fraction of marketcap
for each stock that is ranked best (worst). Portfolios are formed based on mutual fund holdings rank. Rank is based on the return from purchase
price to 3 months and 10 trading days after the report month. Portfolios are held from the 11
th
trading day of a month until the 10
th
trading day
of the next month. A stock is included in the worst (best) portfolio if it is ranked worst (best) in at least one fund. Data covers January 1990 to
June 2010.
α (%) 1.816 1.632 2.110 2.223 0.330 0.646 0.209 0.418
(4.57) (3.57) (4.10) (3.75) (1.39) (2.39) (1.22) (2.15)
MKT 1.170 1.156 1.280 1.288 1.182 1.205 1.121 1.136
(12.58) (12.22) (10.63) (10.51) (21.29) (21.57) (28.06) (28.21)
SMB 0.877 0.874 1.391 1.393 0.228 0.234 0.174 0.177
(6.71) (6.68) (8.22) (8.21) (2.93) (3.02) (3.09) (3.18)
HML 0.097 0.112 0.509 0.500 -0.164 -0.189 -0.204 -0.221
(0.75) (0.86) (3.03) (2.94) (-2.11) (-2.44) (-3.66) (-3.96)
UMD -1.096 -1.082 -1.015 -1.024 0.337 0.313 0.269 0.253
(-13.37) (-12.89) (-9.56) (-9.41) (6.89) (6.31) (7.65) (7.09)
ST_REV 0.077 -0.047 -0.131 -0.087
(0.82) (-0.39) (-2.37) (-2.18)
Worst Best
Number of Funds
where Stock is
Ranked Worst
Number of Funds
where Stock is
Ranked Best
Fraction of
Marketcap That is
Worst
Fraction of
Marketcap That is
Best
53
Table 1.13 – Fama-Macbeth Price Effects Based on Mutual Fund Portfolio Rank
This table presents Fama-Macbeth regressions with monthly stock returns multiplied by 100 as the
dependent variable. A month runs from the 11
th
trading day of a month until the 10
th
trading day of the
next month. Rank is based on the return from purchase price to 3 months and 10 trading days after the
report month. Best (Worst) is a dummy variable equal to one if a stock is ranked best (worst) in at least
one fund. Momentum is the compounded returns from months M-2 to M-12 and Lag Return is the
return from M-1. Log(Book/Market) is the log of the book to market ratio. High (Low) Volume
measures the Gervais, Kaniel and Mingelgrin Premium using a dummy equal to one if the previous
day’s volume is in the top (bottom) decile of the past 50 trading days. Earnings is a dummy equal to
one if there’s an earnings announcement in the month. Predicted dividend measures the dividend
month premium from Hartzmark and Solomon using a dummy equal to one if the company paid a
dividend 3, 6, 9 or 12 months ago where month is defined from 11 trading days into a month until 10
trading days into the next month. Idiosyncratic volatility (Ang, Hodrick, Xing and Zhang 2006) for
stock j on day t is calculated as the standard deviation of the residuals of a regression of the excess
return on MKT, SMB and HML from t-1 to t-21. Share Issuance (Pontiff and Woodgate 2008) for stock
j with holding period ending in month M is [Log(Shares outstanding, M-6)- Log(Shares outstanding,
M-17)]. Only stocks held by mutual funds are included. Data covers January 1990 to June 2010.
[1] [2] [3]
Best 0.065 0.253 0.257
(0.30) (2.04) (2.22)
Worst 0.843 0.743 0.941
(2.16) (2.64) (3.82)
Momentum 0.290 0.270
(1.32) (1.28)
Lag Return -1.846 -2.063
(-3.08) (-3.53)
Log(Market Cap) -0.107 -0.162
(-1.79) (-3.59)
Log(Book/Market) 0.141 0.067
(1.29) (0.74)
High Volume 0.353
(4.57)
Low Volume -0.503
(-6.72)
Earnings 0.574
(7.45)
Predicted Dividend 0.217
(2.44)
Idiosyncratic Volatility -4.740
(-0.89)
Share Issuance -1.267
(-5.65)
Constant 1.106 2.328 3.032
(2.85) (2.46) (4.46)
Observations 722,157 658,662 631,518
54
Table 1.14 – The Rank Effect by Taxable Status of Account
This table presents the marginal effects from logit regressions of a dummy variable equal to 1 if a
stock is sold on characteristics of the stock being held. Only days where a stock is sold and an
investor holds at least 5 stocks are included in the sample. Deferred tax accounts are those
categorized as IRA or Keogh in the data. Stocks are not included on the day that position is
opened. Best (Worst) is a dummy variable equal to 1 if the stock has the highest (lowest) return in
the portfolio and 2
nd
Best (2
nd
Worst) is a dummy for the second highest (lowest) return.
Additional controls are Gain, Return* Gain, Return* Loss, Return*√Holding Days*Gain,
Return*√Holding Days*Loss, Variance *Gain, Variance *Loss, and √Holding Days. Data covers
January 1991 to November 1996. The top number is the marginal effect, and the lower number in
parenthesis is the t-statistic. Standard errors are clustered by date and account.
Deferred
Tax Account
Taxable
Account
[1] [2]
Best 0.194 0.205
(17.16) (17.90)
Worst 0.141 0.147
(12.90) (17.42)
2nd Best 0.124 0.125
(20.29) (13.72)
2nd Worst 0.084 0.083
(13.03) (12.04)
Return*Gain -0.001 -0.027
(-0.04) (-3.43)
Return*Loss -0.038 0.015
(-0.65) (0.84)
Gain 0.026 0.025
(4.63) (6.66)
Additional Controls X X
Observations 225,770 808,442
R
2
0.039 0.049
55
Table 1.15 – Rank Effect after Entropy Balancing
This table presents the proportion of best (worst) positions sold minus the proportion of not best
(worst) positions sold. The “Unweighted” column is the simple difference. “Entropy Balanced” is
the weighted average difference based on entropy balancing of the sample each day. Only days
where a stock is sold are included and an investor must hold at least 5 stocks to be included in the
sample. Stocks are not included on the day that position is opened. A stock is considered best
(worst) if the stock has the highest (lowest) return in the portfolio. Investor data covers January
1991 to November 1996. Mutual fund data are from January 1990 to June 2010. Dates examined
are report dates. A fund must hold at least 20 CRSP merged securities to be included in the
analysis. The top number is the difference, and the lower number in parenthesis is the t-statistic.
Standard errors are clustered by account and date.
Unweighted
Entropy
Balanced Unweighted
Entropy
Balanced
Best - Not Best 0.120 0.128 0.116 0.116
(19.74) (21.36) (15.12) (12.40)
Worst - Not Worst 0.059 0.084 0.188 0.165
(9.15) (18.85) (20.74) (10.03)
Individual Investor Mutual Fund
Chapter 2
Economic Uncertainty and Interest Rates
In the concrete world, the most conspicuous characteristic of the future is its uncertainty.
-Irving Fisher (1907); The Rate of Interest: Its Nature, Determination and Relation to Economic Phenomena
2.1. Introduction
While uncertainty about future economic growth is thought to have a broad impact on the
economy, it is often empirically and theoretically difficult to cleanly identify its influence. Thus
exploring the impact of time-varying uncertainty typically requires significant structure and
strong assumptions (e.g. Bloom 2009). A central prediction of finance (with roots as early as
Fisher (1907)) is that intertemporal smoothing induces a positive linear relation between the
interest rate and economic growth, while precautionary saving induces a negative linear relation
between the interest rate and uncertainty, the conditional variance of growth. This relation
describes an important link between uncertainty and a key economic variable that can be
estimated with a minimum of assumptions. However, it remains largely untested.
This paper examines this fundamental relation and finds that there is an economically and
statistically strong negative relation between the real interest rate and uncertainty. For example,
regressing the real annualized three month Treasury Bill rate on annual estimates of growth and
uncertainty, a one standard deviation increase in uncertainty is associated with a 1.2% to 2.3%
decrease in the level of the risk free rate when growth is measured by consumption, GDP or
Industrial production. In each case these coefficients have a t-statistic greater than three (in
absolute value). The adjusted R
2
of the model increases from roughly 0 when using growth alone
to 14%, 55% and 35% when both uncertainty and growth are included in the regression.
The early empirical work analyzing this relation was not supportive of the theory as it
found no relation between the real interest rate and forecasts of growth (e.g. Hall (1978)). These
studies did not analyze uncertainty as they assumed it was constant. While some attempt was
made to reconcile this empirical finding with the theory (e.g. Campbell and Mankiw (1989)), it
largely remained a puzzle as to why the interest rate and the macroeconomy have such a weak
empirical relation. I show that after accounting for uncertainty, in the US expected growth and
the interest rate still do not exhibit a significant association.
57
Nevertheless, this paper does help to at least partially resolve the puzzle. First, the strong
link between uncertainty and the risk free rate shows that the theory has explanatory power, but
that the uncertainty channel is the strongest. This finding is consistent with parameterizations of
the long-run risk model that put significantly more weight on the uncertainty term. Second,
coefficients on growth for the US are not significantly different from zero, but they are also not
significantly different from the small positive coefficients predicted by the long-run risk model.
Finally, extending the analysis to international data, a number of countries display both a
significant inverse relation between the interest rate and uncertainty and a positive relation
between the interest rate growth. Thus the data underscore the importance of this central relation
between the risk free rate, growth and uncertainty.
As this pattern arises in a number of models and applications, I do not examine it within
the context of a specific model. By using a reduced form approach, along with a number of
different econometric specifications, data sources and time periods, the paper yields a deeper
understanding of a central economic variable, the interest rate, and has implications for a number
of models including habit and long-run risks.
Measuring expectations of growth and uncertainty are key to the interpretation of the
analysis. The baseline estimates in the paper are from an ARMA-GARCH model fit in sample.
Annual macroeconomic data are used as it is available for a long and consistent time series
beginning in 1934. To show robustness I explore other data sources and methodologies.
One possible worry is that the time series of data is too short and does not allow enough
information to properly estimate the coefficients. While extending the time series necessitates
piecing together data from various sources, using historical data covering 140 years, an
economically and statistically significant negative relation between growth and uncertainty is
found.
Using a quarterly time series of macroeconomic data covering a shorter time horizon, but
with more frequent observations, also yields a negative and significant relation between
uncertainty and the interest rate. The main analysis utilizes the 3-month treasury bill rate, but the
results also hold for the 30 day, 6 month and 1 year interest rates as well.
The US may be unique either due to the realization of random shocks in a single time
series or because of its special position in international asset markets. Indeed, an interpretation of
the negative coefficient on uncertainty is that in uncertain times there is a flight to quality, where
58
money floods into the US. This would depresses the interest rate in the US and increases the
interest rate abroad. International data from eight developed economies, Belgium, Canada,
France, Germany, Italy, Japan, the Netherlands, and the United Kingdom are examined.
Measuring growth using consumption or GDP, nearly all of the countries display an inverse
relation between the interest rate and uncertainty. This is not consistent with the negative
coefficient on the US relation being caused by a flight to quality. Further the international data
lends more support to a positive coefficient of growth. Using consumption, six of the eight
coefficients are positive with three significant at the 5% level and using GDP five of the eight
coefficients are positive with three significant at the 5% level.
Forecasts from the time series model fit in sample may not accurately reflect the ex-ante
variables in the model either due to weak forecasts or a look ahead bias. Weak growth forecasts
could lead to the spurious finding of a negative coefficient on the uncertainty term or the finding
of no relation between expected growth and the interest rate. Using ex-post realized growth
instead of estimates from a time series model yield similar results. Ex-ante forecasts from the
Survey of Professional Forecasters are unrelated to the time-series model assumptions and also
yield a negative coefficient on uncertainty. As a final test for the look ahead bias, rolling time-
series estimates using only data known at the time or true ex-ante forecasts from the Survey of
Professional Forecasters yield similar results.
Perhaps the most commonly referenced ex-ante forecaster of uncertainty is the VIX
index. While only available for a short time period, the VIX is used to proxy for uncertainty
while both the Leading Index of the US and the Consumer Sentiment Index are used for monthly
growth forecasts. These measures are used in practice to forecast economic activity and do not
utilize the time series methods employed elsewhere in the paper. Testing the relation with these
measures yields a negative and significant association between the risk free rate and uncertainty.
I test empirical specifications suggested by the model for nominal interest rates, which
include the variance of inflation along with covariance terms between inflation and uncertainty.
Inflation risk is measured using a time-series model and also examining the dispersion of expert
forecasts. Controlling for these additional factors, the relation between growth and uncertainty
remains, suggesting that inflation risk is not responsible for the relation.
Econometric issues related to model choice and measurement error are examined and the
results are shown to be robust to many possible concerns. The pattern is not driven by outliers or
59
model choice regarding the number of lags in the ARMA process. The results are robust to using
longer sampling intervals and to econometric corrections for persistence in the regressors.
These results are based on a minimum of assumptions and thus are relevant to a number
of models. For example, a broad class of models, based on Campbell and Cochrane (1999),
examine the impact of time-varying risk aversion linked to external habit. Habit adds two new
terms to the interest rate process which could account for the relation between uncertainty and
the interest rate. Without controlling for uncertainty, habit has a negative and significant relation
to the risk free rate, but after uncertainty is accounted for, the relation between habit and the
interest rate is weak and insignificant. After controlling for habit, the coefficient on uncertainty is
negative, significant and roughly unchanged. The evidence is consistent with the risk free rate
having a strong relation with economic uncertainty, but not with time-varying risk aversion.
The long-run risks model (Bansal and Yaron (2004)) is one of the few models to include
time varying uncertainty. A key, somewhat controversial, parameter choice of the long-run risks
model is an elasticity of intertemporal substitution greater than one. Calibrations with the
elasticity less than one yield a coefficient on uncertainty that is positive, while the evidence in
this paper suggests that this coefficient should be less than zero. Thus, the results support the
assumption of an elasticity of intertemporal substitution greater than one.
The central contribution of this paper is to provide empirical support and a description of
the fundamental relation between uncertainty and the interest rate. While there is a large
literature exploring and modelling the properties of the interest rate, there is little research
exploring the economic causes of real interest rate behavior.
26
Further the literature focusing on
the economics of the real interest rate has found little support for the theory, by focusing on the
relation between growth and the interest rate. This paper shows that while growth and the interest
rate have a noisy empirical relation, the relation between uncertainty and the risk free rate is
large lending strong support to the basic finance theory.
This paper also contributes to the growing literature on the impact of uncertainty on the
economy. For example Bloom (2009) who examines firm level data and finds that shocks to
uncertainty correspond to rapid drops and rebounds in employment and output. Bloom, Floetotto
and Jaimovich (2010) study the impact of uncertainty on the business cycle and find that
increases in uncertainty lead to large drops in economic activity. Justiniano and Primiceri (2008)
26
For a recent survey see Neely and Rapach (2008)
60
who specify a dynamic stochastic general equilibrium model with parameter uncertainty. Boguth
and Kuehn (2013) find that consumption risk affects asset prices. This paper adds further
evidence that uncertainty is an important economic fundamental with a wide ranging impact.
2.2. A Basic Model
A number of asset pricing models imply a simple linear relation between the real interest
rate, growth and uncertainty. As an example, consider a simple discrete time representative agent
model with conditionally lognormal disturbances (as in Hansen and Singleton (1983), Ferson
(1983) and Harvey (1988)). The lognormal disturbances allow a closed form solution to be
obtained. Conditional lognormality allows for heteroskedasticity as a function of the
conditioning information. The consumer receives an endowment and chooses to consume it or
invest $P
i,j
in one of i=1,…,n securities with j=1,…k maturities. C
t
is consumption at time t and
the subscript of the conditional expectation indicates the time of the information the agent
conditions on. The consumer’s problem is:
{
{
}}
∑
[ (
)]
(2)
This yields the first order conditions (FOC):
[
(
)
(
)
(
) ] (3)
For each i,j security, where R
i,j,t
is the real return on asset i over j periods from t to t+j.
27
Now let
utility take the form:
( ) {
( )
(4)
Substituting the utility function into the FOC and then taking logs yields:
(
[
{
}
(
)])
[
{
}
(
)]
[
{
}
(
)]
(5)
27
Section 2.3 examines issues surrounding the real vs. nominal interest rates and inflation in more detail.
61
Re-arranging this equation, solving for the interest rate and assuming it is risk free (i.e. known at
time t) yields:
(
) ( )
[ (
)]
[ (
)] (6)
So in this model the risk free interest rate at maturity j is simply a linear combination of expected
consumption growth and the variance of the log of expected consumption growth. The variance
term is what I term uncertainty. To estimate this relation the constant model parameters are
subsumed into β
0
, β
1
and β
2
yielding my reduced form regression model:
[
]
[
] (7)
Where r
t
is the log of the time t to t+1 risk free rate, g
t+1
is the log consumption growth rate in
the subsequent period, E
t
is the expectation conditioning on information at time t, Var
t
is the
variance conditioning on information at time t and β
0
, β
1
and β
2
are constant coefficients to be
estimated.
Of course Equation 6 could be tested by imposing and testing restrictions on the β
coefficients. But the goal is not to test the specific model. Versions of Equation 6 can also be
derived from a number of different models and assumptions, each implying different restrictions
on the beta coefficients. Appendix A shows that a version of habit formation (Campbell and
Cochrane (1999)) and long-run risks (Bansal and Yaron (2004)) yield linear combinations of
growth and uncertainty that can be expressed using Equation 6. While the precise form of the
coefficients differs for the various models, they all imply β
1
>0 and β
2
<0. These general
restrictions are the focus of my analysis.
2.3. Data and Summary Statistics
Unless otherwise noted, growth is measured at an annual rate to avoid issues with
seasonal adjustment and extend the analysis to as many years as possible. All measures of
growth end in 2010. Growth is measured as the log of the ratio of the level measure
(consumption expenditures, GDP and output) at time t and the growth measure at time t-1. All
growth measures are in per-capita real dollars.
This paper uses three different measures of economic growth. One is real consumption
growth from the National Income and Product Accounts (NIPA), where consumption is defined
as non-durable goods plus services. Real NIPA GDP is an alternate measure of growth as well as
62
the Federal Reserve’s G.17 real industrial production index. Consumption and GDP growth data
begin in 1930, while data on the Industrial Production index begins in 1920.
To extend the analysis, the above series is augmented with historical data. For
consumption the Kuznets-Kendricks series (Kuznets (1961), Kendrick (1961)) compiled by
Shiller (1982, 1989) is used which starts in 1890. Romer has argued that the methodology used
to compile this data accentuates fluctuations, making it more volatile than the economy actually
was. Unfortunately there is not an alternative consumption series, but Romer (1989) has
compiled an alternative GNP series that is used from 1871. An industrial production index,
compiled by Miron and Romer (1990), is used which extends back to 1885. Results for quarterly
growth are presented as well where growth is measured as one plus the log of current level
divided by the level four quarters previous. The quarterly data begins in 1947.
Nominal interest rates are measured as the log of one plus the decimal fraction interest
rate. An annual rate is not available for the entire period of analysis so the rate is calculated by
annualizing the three month December rate, the last observation before the period of the growth
forecast.
28
The main interest rate used is the 3 month Treasury bill rate from the Federal Reserve
Bank of St. Louis (TB3MS series) converted from a bank discount basis to an effective yield
basis. This series is available starting in 1934. An alternative measure of the interest rate is the
one month Treasury bill rate from Ibbotson Associates compiled by Fama and French. The first
full year of data is available for 1927. The 1 year interest rate is the GS1 series starting in 1954
and from 1947 to 1953 is from Homer and Sylla (2005). A historical series of the 6 month rate
(again, the December rate from the previous year annualized) is used which extend back to 1871.
This series is compiled using the 4-6 month prime commercial paper rate from Macauly (1938)
from 1869 to 1937, the Federal Reserve from 1938 to 1970, the 6 month commercial paper rate
from 1971 to 1997 and the 6 month CD rate from 1997 to 2010.
29
In the analysis, the interest rates are net of expected inflation. Annual inflation is
calculated as the log of CPI in December of year t divided by CPI in December of year t-1. This
is modeled using an ARMA(1,1) process and the predicted value is used as the estimate of
28
The December rate is chosen because it is the last value is known when the forecasts of growth are made.
Appendix Table 1 shows for the 3 month interest rate results are not materially different using the January rate,
compounding the rate from March, June, September and December, forecasting the annual rate compounded from
January, April, July and October using an ARMA(1,1) process, or simply using the ex-post realized rate.
29
This is the same data sources as Shiller (1989), though Shiller uses the ex-post realized rate of rolling over in
January and July, while I use the December rate annualized to avoid any forward looking bias.
63
expected inflation (as in Constantinides and Gosh (2012)). This is subtracted from the log of the
interest rate to give the expected real interest rate. Section 4.5 analyzes whether the assumptions
necessary for this specification are supported in the data.
Table 2.1 presents summary statistics. All the measures of economic growth are
positively correlated, with GDP and consumption having a correlation of 0.53, industrial
production and consumption having a correlation of 0.47 and industrial production and GDP
having a correlation of 0.86. GDP and industrial production growth are more volatile than
consumption growth, with standard deviations more than twice as large. Also, consistent with the
Romer critique of the Kuznets series, consumption prior to 1930 is much more volatile than
consumption after, with double the standard deviation.
The Survey of Professional Forecasters, conducted by the Federal Reserve Bank of
Philadelphia, provides an alternative forecast of growth, presumably made using a richer
information set than simply the lagged values of growth. The survey is timed to coincide with the
release of the advanced estimates from the previous quarter, so the forecast of growth is made in
the 1
st
quarter of the year when the value for the 4
th
quarter from the previous year is known.
Thus the estimate of growth is the log of the forecast for the 4
th
quarter of year t+1 divided by
the known value of the 4
th
quarter in year t. The time series of data is relatively short with real
consumption growth calculated from 1981, and GDP and Industrial production calculated from
1968. Forecasts of inflation are calculated in the same manner as forecasts of growth using
forecasts of the GDP price index from 1968.
Results using ex-ante forecast data are presented using the VIX to proxy for uncertainty,
and the Leading Index for the United States and the University of Michigan Consumer Sentiment
Index to proxy for growth. Each variable is analyzed in logs. The VIX index begins in 1990 and
is constructed using options data to measure the 30-day expected volatility of the S&P 500.
30
To
deal with the short time series, growth forecasts that are available monthly are utilized. The
leading index corresponds to a 6 month forecast, constructed using forward indicators such as
30
For a more complete overview of the VIX see the CBOE white paper on its construction available at:
http://www.cboe.com/micro/vix/vixwhite.pdf
64
housing permits, and unemployment insurance claims, among others. The Consumer Sentiment
Index is based on a monthly telephone survey meant to gauge consumer confidence.
31
Finally I use data on 8 international developed countries, namely Australia, Belgium,
Canada, France, Italy, Japan, the Netherlands and the United Kingdom. These are the countries
with a long time series of data available. Consumption and growth is from Barro and Ursúa
(2008) the 3 month treasury rate and CPI for each country is from Global Financial Data.
32
2.4. Empirical Results
2.4.1. Baseline Results
To analyze the linear relation from Equation 6, estimates of expected economic growth
and the variance of economic growth are needed. As a baseline case I model growth as a time
series process. The most basic formulation is to model growth as ARMA(1,1):
33
(8)
After estimating the model, it is used to predict ĝ
t+1
which is then used as the estimate for
E
t
[g
t+1
] in Equation 6. The simplest way to estimate Var
t
[g
t+1
] is as the square of the residuals in
period t, so the estimate of the variance is (ĝ
t
-g
t
)
2
. Table 2.2 Panel A takes the estimates from
this model to estimate Equation 6.
To improve the estimate of uncertainty, Table 2.2 Panel B models growth as a
conditionally heteroscedastic process using a GARCH(1,1) model (Engle (1982), Bollerslev
(1986)). The specification used is:
(
)
(9)
This yields a forecast for growth and uncertainty that the agent can undertake at time t if the
agent knows the time series model of the economy.
Table 2.2 estimates Equation 6 using annual data on consumption, GDP and industrial
production as measures of growth. For all three measures, and using both methodologies for
31
For literature surrounding the leading index see Stock and Watson (1989, 1991), Hamilton and Perez-Quiros
(1996), Crone (2003). For literature surrounding the predictive power of Consumer sentiment see Carroll, Fuhrer
and Wilcox (1994), Matsusaka and Sbordone (1995) and Howrey (2001).
32
Where monthly data is available the interest rate is based on the December average. When only annual data is
available the annual measure is used.
33
Appendix Table 2 shows results of the basic estimation for different ARMA lags and shows lag choice does not
materially change the results.
65
measuring the variance, the uncertainty term is negative and statistically significant. Further, in
none of the specifications is growth significant.
Table 2.2 suggests that there is a strong relation between the interest rate and future
economic uncertainty. Examining the final row of Table 2.2 Panel B, “Impact of 1 SD
Var
t
[g
t+1
]”, a one standard deviation increase in the risk free rate is associated with a 0.012,
0.023, and 0.018 decrease in the interest rate for consumption, GDP and industrial production
respectively.
Further, the variance adds significant explanatory power to the model over simply
including growth. The row labeled “Adjusted R
2
”
gives the adjusted R
2
of the estimated model,
while the row labeled “R
2
Excluding Var
t
[g
t+1
]” gives the adjusted R
2
for the model regressing
the interest rate on the growth forecast while excluding the variance term. Using the residual
measure of variance in Panel A, the adjusted R
2
value significantly increases in all specifications
from zero without the uncertainty term to 0.08, 0.42 and 0.15 for consumption, GDP and
industrial production respectively. Similarly, using the GARCH specifications this value moves
from roughly 0 without uncertainty to 0.14, 0.55 and 0.35 with uncertainty included.
Table 2.2 Panel B contains the main results using, in my opinion, the best blend of
econometric specification and data available. Consistent data from the same source exists for the
entire period for all measures of the interest rate, inflation and growth. While the ARMA-
GARCH framework has its shortcomings, other options, such as expert forecasts or the VIX,
exist only for relatively short time periods and have their own set of issues. Nevertheless, in most
specifications examined, the relation between the interest rate and uncertainty remains robust,
negative and economically meaningful.
2.4.2. The Effects of Extreme Volatility Periods
Figure 2.1 graphs data from the ARMA-GARCH model in Table 2.2 Panel B. Predicted
consumption growth is the solid green line and the risk free interest rate is the dashed orange
line. Sometimes these lines seem to move together, for example during the 2000s and
surrounding WWII, while during other periods they follow very different paths, for example in
the early 1980s. There does not appear to be a strong relation between these two series.
Figure 2.2 graphs the variance of consumption growth as the solid green line versus the
interest rate as the dashed orange line. There are some periods with very high volatility. The
periods with the highest volatility occur during major events in US economic history. The first
66
period is the forecasts for 1937 (based on data from 1936) which coincides with the recession
within the great depression. The next period occurs in the forecasts for 1948 as the US emerged
from WWII. Another year with high volatility is 1975 as the US dealt with the first OPEC oil
embargo. The final high volatility period occurred in 2010 during the recent financial crisis.
To show the results are not driven by the large relative magnitudes of these periods, the
data is winsorised at the 90th percentile with the results presented in Table 2.3. All of the results
remain robust to this change. The significance of the variance coefficient, adjusted R
2
, and the
impact of a one standard deviation change are slightly stronger for consumption and industrial
production and slightly weaker for GDP. Table 2.3 suggests that the results are not driven by
outliers in the data.
2.4.3. The Long Historical Record
The preceding analysis begins with data starting in 1934, the first year that the 3 month
risk free rate is available. In this section the analysis is extended to the maximum extent possible.
This means using the historic interest rate data, which begins in the late 1800s. GDP using the
Romer series begins in 1871, and industrial production, using the Miron and Romer estimates
begins in 1885. For consumption, the Kendrick-Kuznets series is used, which begins in 1890.
Figure 2.1 graphs the extended Kendrick-Kuznets consumption series with the red
vertical line indicating 1930, the first year NIPA data is used for the growth rate. Prior to the
NIPA data, there is significantly more volatility. From 1934 to 2010, the period the initial
analysis was conducted on, the standard deviation is 0.024. Before this period, from 1890 to
1933, the standard deviation is 0.049, more than double the subsequent period. This is not true
with the interest rate, which actually has a higher standard deviation in the later period, 0.025
prior to 1934 and 0.31 after. Romer (1989) argues that the higher volatility of the Kendrick-
Kuznets series is due to the methodology of its construction, and not actual shifts in the
economy. Note that the Romer GNP series is slightly less volatile from 1871 to 1929, than the
NIPA series from 1930 to 2010. If the consumption series is artificially noisy, the estimates will
be attenuated. To partially offset this, the standard analysis is presented, as well as an analysis
where the estimates of growth and variance are winsorised at the 90
th
percentile.
Table 2.4 presents the evidence that the relation is strong over this long historical record.
First note that all the variance terms are again negative, and everything other than the
uncorrected consumption is significant at the 10% level. The relation with consumption is
67
weaker in terms of adjusted R
2
and overall impact, though it is improved slightly after
winsorising. This is unsurprising given Romer’s critique of the Kendrick-Kuznets data
artificially increasing the size of its fluctuations.
Using the Romer series, the pattern with GDP and industrial production remains robust.
The adjusted R
2
increases when uncertainty is included in the model. The R
2
is lower than it was
during the shorter period. This is unsurprising, as the regressions cover 140 years including many
different economic episodes, periods and policies. However, the general pattern remains
significant. The growth forecast term is positive in all specifications, but it is only significant at
the 5% level when using industrial production. The impact of a one standard deviation increase
in uncertainty is associated with a greater than 0.01 decrease in the interest rate for both GDP
and industrial production.
2.4.4. Quarterly Data
The previous analysis uses annual data as it is available for the longest time period and
does not suffer from being seasonally adjusted. As a further robustness check Table 2.5Table
presents the same specification as Table 2.2 panel B using quarterly data. The quarterly data
begins in 1947, so major events effecting uncertainty such as World War II and the great
depression are not included. Further concerns about persistent regressors are greater for these
regressions as the sampling occurs quarterly as opposed to annually. Finally the seasonal
adjustment may be taking out relevant variation from the data and biasing results (Ferson and
Harvey (1992)). All growth rates are annualized for comparability with the previous estimates.
Table 2.5 shows that in general the results remain robust to using the shorter time period
in the quarterly sample, though they are weaker for GDP. All three measures of variance are
negative, though GDP is no longer significant. Further, in all three specifications the adjusted R
2
increases versus the specification excluding the variance.
2.4.5. Different Interest Rates
The analysis up to this point utilizes the 3 month interest rate. Next the model is
estimated using a variety of alternative interest rates. The interest rates with data available for
this period are the 30 day interest rate, the 3 month interest rate, the 6 month interest rate and the
1 year interest rate. Table 2.7 presents estimates. Notice that the main patterns discussed in the
previous section are all present for each maturity. The expected growth is insignificant, while the
variance is negative and significant. The portion of the data explained using the variance versus
68
not, is significantly higher as measured by the adjusted R
2
. Thus the choice of short term interest
rate does not appear to be an important factor for the results.
2.4.6. International Data
Finally, I analyze the relation by examining countries other than the US. Extending the
analysis to other countries allows for more observations and alternate time series to analyze
whether the US relation is simply an anomaly. Eight different developed countries are examined
and estimates for growth and uncertainty are obtained by estimating a separate ARMA-GARCH
model for each country as specified by Equation 3.
Table 2.6 examines each country individually using consumption as a measure of growth
in Panel A and GDP in Panel B. Again the results are robust and similar to the US results. All
eight of the countries have negative coefficients on the uncertainty term and six of them are
significant at the 5% level. When examining GDP in Panel B, seven of the eight coefficients on
uncertainty have a negative sign and five of the eight are significant at the 5% level.
The international results are also offer some support for the link between growth and the
interest rate albeit not as robustly as that between the interest rate and uncertainty. Examining the
results utilizing consumption in Panel A six of the eight coefficients on growth are positive and
three are significant at the 5% level. Examining the results using GDP in Panel B four of the
eight are positive with three significant at the 5% level.
2.5. Using Alternative Forecasts
2.5.1. Using Ex-Post Growth
The estimates of growth to this point are from time series models which may not
accurately reflect the expectations of the market. The base interest rate relation implies that
growth should be positively correlated with the interest rate, but none of the β
1
coefficients are
significant and in some cases the point estimates are negative. Both the growth and uncertainty
term are measured with error which should attenuate the estimates if they are uncorrelated and
random. It is possible that the forecast of growth is measured with more error than the variance
term and that the significance of the variance term is driven by unmeasured variation in growth.
To test for this, Table 2.8 Panel A presents estimates of Equation 1 using ex-post realized
growth instead of the growth forecast. This introduces a different bias to the forecast term as the
agent does not know this value at time t, but if the results are being driven by systematically
69
failing to measure the agent’s expectations, the ex-post realized value should present a worst
case scenario for the errors in variables bias. All of the previous results remain materially
unchanged when using ex-post growth. Thus it is unlikely that measurement error of the forecast
term is driving the uncertainty coefficient.
2.5.2. Using Professional Forecasts
Table 2.8 Panel B presents another robustness check to the growth forecast by using an
alternative measure of growth from the Survey of Professional Forecasters. This table estimates
the panel regression:
[
]
[
] (10)
where i indexes each individual expert and t indexes time.
is an expert’s forecast of the real
risk free rate based on their estimate of inflation and
[
] is their estimate of growth. By
using a panel format it is possible to match a given expert’s estimate of the risk free rate with
their estimate of growth.
34
The variance term does not vary by expert and is specified by
Equation 3. Standard errors are clustered by both time and expert. Using the same measure of
uncertainty with different measures of inflation and growth serves as another test as to whether
systematically biased estimates from the ARMA model for growth are responsible for the
negative relation between the interest rate and uncertainty.
The results using the expert forecast data are consistent with those presented previously.
While GDP is no longer statistically significant, the β
2
coefficients for consumption and
industrial production are strong with a t-statistic greater than 4. All three are still economically
significant with a one standard deviation increase in uncertainty being associated with decreases
in the risk free rate of 0.011, 0.010 and 0.041 for consumption, GDP and industrial production
respectively. Even though there are fewer years, the main results are still present.
Aside from weak forecasts, another concern is that the time series model used induces a
look ahead bias that accounts for the results. The baseline model is estimated using data through
the entire sample. If the agent knows this model at time t, the agent can make forecasts using it,
but perhaps the model is not known and the results are biased.
Figure 2.2 graphs the forecasts from the time series model versus the median expert
forecast for the survey of professional forecasters. The expert forecasts were made by
34
Similar results are obtained using the median value of panelists at time t, estimating the relation analogously to
Equation 1.
70
professionals using only ex-ante knowledge. While not identical, the two series move closely
together throughout the period where there is data for both. This suggests that the forecasts from
the model are not driven purely by a look ahead bias, at least for these years.
As a further test, Table 2.8 Panel C estimates the relation where the forecasting model is
fit using data only for the years up to time t to forecast period t+1. The model is estimated
separately for each year adding in the new year of data and starting using a minimum of 15
observations. Using this rolling estimation technique negative and significant coefficients are
found for consumption and industrial production and GDP is no longer significant. While
slightly weaker, this suggests that a look ahead bias is not responsible for the empirical relation.
2.5.3. Using the VIX
This section examines the robustness of the previous results by using ex-ante forecasts of
both growth and uncertainty. The VIX index is utilized to proxy for uncertainty, though it has
two major shortcomings for this analysis. The first is that the data on the VIX begins in 1990.
Thus, there is only a short time series to estimate the relation. The second is that the VIX
corresponds to stock market volatility which, while related to economic uncertainty, is not a
perfect proxy.
To deal with the short time series of data, monthly forecasts from the Leading Index for
the United States and the University of Michigan Consumer Sentiment Index are used for
growth. This section, and this section only, utilizes nominal values for all variables. The VIX
measure is nominal volatility and the horizon of consumer sentiment is unclear. Given the stable,
low level of inflation over this period, similar results in both significance and magnitude are
obtained assuming a number of different specifications for inflation.
The VIX, the leading index and the consumer sentiment index are each constructed using
data available at the time they are reported and are used in practice for forecasting purposes.
Thus finding an inverse relation between the VIX and the interest rate underscores that the
previous analysis is not driven by assuming a specific time series structure.
Table 2.9 presents estimates for the relation utilizing the VIX. Panel A utilizes a Newey-
West regression framework, while Panel B utilizes the mARM correction (discussed in section 7)
to deal with persistent regressors. In all specifications the relation between the interest rate and
the VIX is negative and significant. The relation between the Leading Index and the interest rate
is not significant, while the consumer sentiment index is positive and significant. Thus the
71
interest rate and uncertainty retain a negative and significant relation, even when examining a
shorter period of time, utilizing different data and different techniques.
2.6. Inflation Risk
The previous discussion assumes the existence of a risk-free rate for which there is no
perfect analog in the data. For the initial analysis inflation is predicted using an ARMA(1,1)
process and the real rate is constructed by subtracting predicted inflation from the nominal
interest rate. Using this method, the data suggest that economic growth risk is important for
understanding the general equilibrium relation between economic uncertainty, economic growth
and the risk free interest rate. This ignores any inflation risk premium. The simple model can
also be re-stated using variables for nominal interest rates,
and inflation, I
t+j
, which is
unknown at time t as:
{
(
)
(
)} (11)
Re-arranging this and taking logs:
(
)
[ (
)]
( )
[ (
)]
[ (
)]
[ (
)]
[ (
) (
)]
(12)
With inflation uncertainty there are new terms for the variance of inflation and the covariance
between consumption growth and inflation. In order to examine the impact of inflation
uncertainty on the results I estimate:
[
]
[
]
[
]
[
]
(13)
where i
t+1
is the log of one plus inflation. The variance of inflation measures the uncertainty of
inflation from t to t+1. The covariance captures the real impact of inflation, through its relation
to real economic growth.
For this reduced form empirical model both inflation and consumption growth are
modeled using an ARMA(1,1)-GARCH(1,1) specification as described by Equation 8. The
covariance term is modeled by assuming the residual from the forecast of inflation multiplied by
72
the residual from the forecast of growth follows an ARMA(1,1) process. Thus the covariance
term is forecast using the equation:
(
)
(14)
Where
is the covariance between growth and inflation at time t.
The first column of Table 2.10 Panel A presents the estimates of Equation 12 using the 3
month interest rate and the three measures of growth. The regression includes the same variables
as Equation 1, but also the variance of the inflation rate and the covariance term. In all estimates
of Equation 4, the coefficient on uncertainty is negative and significant at the 10% level. Adding
the variance of inflation decreases the adjusted R
2
for all three measures of growth as compared
to Table 2.10 Panel B. A one standard deviation increase in growth is associated with decreases
in the interest rate of -0.011, -0.027 and -0.15 for consumption, GDP and industrial production
respectively. These are generally consistent with the numbers obtained when the inflation terms
are not included. Thus the results do not appear driven by simplifying assumptions related to
changing the nominal rate to real.
The estimates of inflation risk in Panel A are from a time series model which may not
yield appropriate forecasts. Table 2.10 Panel B repeats the panel analysis described by Equation
9 where inflation risk is proxied using the variance of expert forecasts of inflation. All
coefficients remain negative, though GDP is insignificant and a one standard deviation change in
uncertainty is associated with economically meaningful changes to the risk free rate.
2.7. Econometric Issues
All of the variables in the above regression have a degree of time series persistence.
Indeed the GARCH model was created to deal with data series that exhibited persistence in
volatility (Engle (1982), McNees (1979)). Using one year intervals and Newey-West standard
errors, the previous analysis attempts to control for this. This section examines the issue further
to see if it was sufficient. After a number of tests to examine the impact of persistent regressors,
all of the results remain.
First, a conservative, back of the envelope result is presented taking an observation once
every 5 years instead of annually. The 5 year sampling interval is chosen to be large enough to
mitigate concerns about persistent regressors. If the results disappear it raises the concern that the
empirical relation is driven by econometric bias rather than a true economic relation. Further, this
73
test discards a large amount of data to obtain a long sampling interval. Even if the full sample
results are not being driven by persistent regressors, it is possible that no relation will be found
simply due to the small sample size of 16 observations. Thus this is a conservative test that is
biased towards rejecting the relation even if persistence is not responsible for the results.
Table 2.11 Panel A presents estimates taking one observation every 5 years and shows
that the results are largely unchanged. The data starts in 1935, examining 1940, 1945 and so on
until 2010. The coefficient on the variance is more negative on consumption, less negative on
GDP and slightly more negative on industrial production compared to Table 2.2. All coefficients
are statistically significant, and all estimates of the impact of a one standard deviation change in
variance are economically significant. All adjusted R
2
are much higher when uncertainty is
included in the regression. Thus the back of the envelope test suggests that the results are not
being spuriously driven by persistent regressors.
Stambaugh (1999) shows that there is a finite sample bias when regressors are
autoregressive with errors that are correlated with the dependent variable. Amihud and Hurvich
(2004) and Amihud, Hurvich and Wang (2009) develop the mARM method for estimating
unbiased coefficients and test statistics of multi-variate models with persistent regressors.
Appendix B contains a description of the implementation used here.
Table 2.11 Panel B presents the adjusted regression. After the correction, all estimates of
the coefficient on uncertainty are negative and significant. The coefficients on consumption,
GDP and industrial production have t-statistics of -4.00, -6.22 and -5.63, respectively. Also note
that the order of magnitude of the estimates is similar, suggesting that bias is not greatly
impacting the coefficients.
2.8. Implications of Specific Models
2.8.1. External Habit
Campbell and Cochrane (1999), examine the impact of changing risk aversion caused by
deviations from slow moving external habit. Habit adds an additional component to the
stochastic discount factor, creating two new terms in the interest rate process. This section
examines whether these habit based components can account for the negative relation found
between uncertainty and the interest rate. After controlling for habit, a robust relation between
74
the interest rate and uncertainty remains. After accounting for uncertainty, the habit based terms
add little explanatory power.
In the Campbell and Cochrane (1999) model, the risk free rate takes the form:
( )
[
]
[
] ( )( ̅
)
[
]( (
) (
)
)
(15)
Where s
t
is the log of surplus consumption, defined as
(
),
is habit and (
) is
the sensitivity function. The first three terms of Equation 14 are the affine expression of growth
and uncertainty studied in this paper. The fourth term, ( )( ̅
) , represents the agent’s
desire to intertemporally smooth in habit. When agents are far below their habit, they want to
consume more and save less, driving up the interest rate. The final term indicates a precautionary
savings motive in habit. In bad times, agents become more risk averse, want to save more and
drive down the interest rate.
In the original model, (
) is specified so that these two motives exactly offset each
other, but under alternative specifications these two motives add together to form one term that is
linear in surplus consumption (Campbell and Cochrane (1995)). If this parameter on surplus
consumption, is less than 0, the intermporal smoothing motive dominates and the interest rate has
an inverse relation to surplus consumption. If the parameter is positive, the precautionary savings
motive is dominant, and surplus consumption and the interest rate have a positive relation.
The dominant motive can have important implications. For example, Wachter (2006)
builds a model of the term structure based on external habit formation utilizing this risk free rate
that is linear in surplus consumption. In order for the model to generate an upward sloping yield
curve and a positive risk premia on real bonds, the intertemporal smoothing motive must
dominate. In the model both growth and uncertainty are constant, so to calibrate the model and
test which motive dominates, the risk free rate is regressed on surplus consumption. The
regression yields a negative and significant coefficient, consistent with intertemporal smoothing
being the dominant habit based motive.
I replicate the Wachter result using annual data and the longer time period studied in this
paper to show that without further controls, the negative relation between surplus consumption
75
and the interest rate is robust. I use the same proxy for surplus consumption, a slow moving
weighted average of past consumption, and estimate:
∑
(16)
Surplus consumption is measured over a 10 year period as ∑
, where =(0.97)
4
.
Table 2.12 Panel A presents the results. The relation is negative and is significant for GDP
and industrial production. Thus, both uncertainty and surplus consumption are negatively
correlated with the level of the interest rate.
Uncertainty could capture the impact of surplus consumption, or surplus
consumption could capture the impact of uncertainty, so it is necessary to control for both
to understand the relation these variables have with the interest rate. To control for
surplus consumption and uncertainty, I utilize two regression specifications. First I follow
the specification used by Wachter where surplus consumption linearly enters the risk free
rate. I estimate:
[
]
[
]
∑
(17)
Next I allow for both the intertemporal smoothing and precautionary savings motives to
have separate effects and estimate:
[
]
[
]
∑
[
]∑
(18)
β 3 represents the intertemporal smoothing motive, so theory predicts that β 3<0. β 4
represents the precautionary savings term, so theory predicts that β 4>0.
Table 2.12 Panel B shows a strong relation between uncertainty and the risk free
rate and a weak relation between the risk free rate and habit. The first column under each
growth measure presents estimates utilizing Equation 17. After controlling for surplus
consumption, all three measures of growth yield negative and significant coefficients on
uncertainty that are a similar magnitude to those found without controlling for surplus
76
consumption. Compared to the specification with surplus consumption alone, adding
uncertainty and growth increases the adjusted R
2
from 0.05 to 0.17 using consumption,
0.17 to 0.57 using GDP and 0.15 to 0.47 using industrial production. The coefficient on
surplus consumption moves from negative and significant in Panel A, to roughly zero in all
three specifications in Panel B. The signs of the point estimates on surplus consumption are
both positive and negative across growth measures, and the largest t-statistic is -1.12.
The second column under each growth measure allows for a separate habit-based
intertemporal smoothing (proxied by surplus consumption) and precautionary savings
motive (proxied by the interaction of surplus consumption with uncertainty). After adding
these controls, the coefficient on uncertainty increases using consumption and GDP and
decreases using industrial production. The consumption coefficient is no longer statistically
significant, though both GDP and industrial production remain significant. The coefficients
on surplus consumption and the interaction term are consistent with the predictions of the
theory, but in no instance are the coefficients significant. The highest t-statistic on either of
the two surplus consumption terms is -0.90.
Surplus consumption is unable to account for the negative relation between
uncertainty and the interest rate. After uncertainty is added to the regressions, the relation
between surplus consumption and the interest rate is weak. This suggests that changes to
risk aversion based on habit are not a major factor for understanding the interest rate,
while uncertainty and the interest rate have a strong connection.
2.8.2. Long-Run Risks
The long-run risks model is one of the few models to include time varying uncertainty.
The interest rate is modeled as linear in growth and uncertainty, so the results in this paper add to
the empirical literature on long-run risks (Bansal and Yaron (2004), Bansal, Khatchatrian and
Yaron (2005), Bansal Kiku and Yaron (2007), Bansal, Kiku and Yaron (2012), Ferson
Nallareddy and Xie (2013) Jaganathan and Marakani (2011)). The strong negative relation
between uncertainty and the interest rate supports the assumption that the elasticity of
intertemporal substitution is greater than one. The results also suggest that standard calibrations
of the interest rate process place too much emphasis on growth and not enough emphasis on
uncertainty.
77
The long-run risks interest rate is linear in growth and uncertainty and, as discussed in
Appendix A, can be written as:
[
]
[
] (19)
The price consumption ratio is similarly affine and can be expressed as:
[
]
[
] (20)
Each of these As are constant parameters of the model. While the coefficients include different
model parameters than the motivating example in Section 2.1, the interest rate is the linear
function of growth and uncertainty examined in this paper.
In the long-run risks model, asset prices are impacted by a persistent predictable
component of consumption growth and a persistent predictable component of uncertainty. Thus,
similar to Beeler and Campbell (2012), this paper finds that the interest rate does not forecast
future economic growth, which is a puzzle for the long-run risks model. Importantly, this paper
demonstrates that failing to include uncertainty is not responsible for this lack of predictability.
The estimates in this paper suggest that calibrations of the long-run risks model should
place further emphasis on uncertainty in the interest rate. From the calibration of Bansal and
Yaron (2004) to the calibration of Bansal, Kiku and Yaron (2007), the coefficient on uncertainty
in the price consumption ratio (A
2
in Table 2.13) increased by roughly a factor of four from -449
to -1,632. The coefficient on uncertainty in the interest rate process remained roughly
unchanged, moving from -13.6 to -11.1. The estimate in this paper is -60.4, with other estimates
ranging from -126.2 to -43.1. Thus a fourfold increase of A
2,f
, in the interest rate process, would
move the calibration closer the estimates in this paper.
To obtain a negative relation between uncertainty and the interest rate, the elasticity of
intertemporal substitution must be greater than one, as illustrated empirically by Table 2.13. The
first two columns contain the calibrations from Bansal and Yaron (2004) and Bansal, Kiku and
Yaron (2007) with an elasticity, ψ, greater than 1. In each of these columns, the coefficient on
uncertainty, A
2,f
, is negative. The third column takes the calibration from Bansal, Kiku and Yaron
(2007), but changes ψ from 1.5 to 0.9. Using ψ equal to 0.9 results in a large positive coefficient
on uncertainty in the interest rate. The final two columns present examples from Constantinides
and Gosh (2012) with ψ equal to 0.9. The relation between the interest rate and uncertainty is
positive.
78
While an elasticity of intertemporal substitution greater than one is key to the model,
some papers have provided evidence against this assumption (for example Beeler and Campbell
(2012), Hall (1988), Campbell (2003)). Constantinides and Gosh (2012) attempt to estimate the
parameter directly, but the standard errors of their estimates are large. Thus the estimates lack the
precision to support an elasticity of substitution greater than or less than one.
This paper finds a robust and negative relation between uncertainty and the interest rate.
For the long-run risks model to yield this negative relation, the elasticity of intertemporal
substitution must be greater than one. Thus, within the context of the long-run risks model, the
results in this paper support an elasticity of intertemporal substitution greater than one.
2.9. Conclusion
Theory predicts that there is a relation between the interest rate, economic uncertainty
and economic growth. This paper demonstrates this relation empirically and shows that
economic uncertainty has a strong relation with the risk free interest rate. This relation holds
using a variety of specifications, datasets and models. The results imply that analyses of the
relation between interest rates and growth that leave out uncertainty may be seriously
incomplete.
79
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Appendix A
Using external habit formation from Campbell and Cochrane (1999) the specification of the risk
free rate becomes:
(
) ( )
[ (
)]
[ (
)]( (
))
( )(
̅
)
(1)
where s
t
is the log of the surplus to consumption ratio and λ(s
t
) is the sensitivity function.
Campbell and Cochrane Campbell (1999) specify λ(s
t
) so that the last two terms offset, yielding
the interest rate:
(
) ( )
[ (
)]
(
̅
)
[ (
)] (2)
81
Thus again this is simply a linear combination of expected growth and the variance of expected
growth and can be expressed in the form of Equation 1. In the original model expected growth
and the variance are constant, so the risk free rate is constant.
Another class of models that has a received much attention in recent years is the long-run
risks models introduced in Bansal and Yaron (2004). Constantinides and Gosh (2012) show that
the risk free rate can be written as:
(
)
(3)
Where A
0,f
, A
1,f
and A
2,f
, are time invariant parameters (see Constantinides and Gosh (2012)
Appendix A2.2 for explicit definitions), x
t
is the latent state variable and σ
t
is the variance of the
state variable’s innovation. Using the fact that:
[ (
)]
(4)
Where μ
c
is a constant, and defining the constant A
f
=A
0,f
+A
1,f
μ
c
one can write:
(
)
[ (
)]
[ (
)] (5)
Thus the long-run risks model also is consistent with Equation 1.
Appendix B
The implementation of the mARM procedure (Amihud, Hurvich and Wang (2009)) in
this paper follows closely to that described in Amihud and Hurvich (2004), the appendix of
Avramov, Barras and Kosowski (2012) and the appendix of Ferson, Nallareddy and Xie (2012).
Denote {x
t
} as a p-dimensional vector of predictors from t=0,…,T, where for this paper
p=2 and includes forecasts of growth and uncertainty. Also, to make the notation consistent with
Amihud, Hurvich and Wang, x
t
denotes the forecast made using data at time t for period t+1.
Further r
t
denotes the interest rate from t-1 to t. The model is thus given by:
(6)
82
(7)
[
] [
] [
]
[
] [
] (8)
Amihud and Hurvich (2004) show that one can write:
(9)
So the model with bias correction to be estimated is:
(10)
In order to implement the estimate the following procedure is used. Estimate the expression for
x
t
utilizing a VAR(1) regression.
35
This expression yields the preliminary estimate,
̂
and
̂
along with the covariance matrix from the residuals
denoted
̂
. To estimate the small sample
bias, the Nicholls and Pope (1988) estimate is used:
̂
(
̂
) [
̂
]
̂
[(
̂
)
(
̂
)
∑
(
̂
)
]
̂
(11)
Where I is a 2x2 identity matrix,
is the jth eigenvalue of
̂
and
̂
is estimated using the
formula (
̂
) [
( )]
(
̂
).
Using these initial estimates (denoted
̂
), an iterative procedure is used to construct
estimates as follows. The new estimate of coefficients is
̂
̂
̂
(
̂
) , the new estimate
of the constant is
̂
(
̂
)
̅
(where
̅
is the sample mean), and a new estimate of
̂
is
obtained from the residuals. If the model is non-stationary the iterations stop and values from
that iteration are used. If not, the procedure repeats itself using the previous steps and estimates
35
If the initial VAR model is non-stationary the Yule-Walker estimator is used
̂
[∑ (
̅̅̅
)(
̅̅̅
)
][∑ (
̅̅̅
)(
̅̅̅
)
]
where
̅̅̅
∑
83
to construct the next iteration for a maximum of 10 iterations. Taking the final values from this
procedure Equation 9 is estimated.
To construct test statistics, the variance of
̂
is calculated where all terms with hats are
the estimates from the final round of the iteration:
̂ (
̂
) ∑(
̂
)
̂ (
̂
)
∑∑
̂
̂
(
̂
̂
) ̂ (
̂
)
(12)
Where
̂
(β
̂
) is the OLS standard error from the augmented regression of Equation 9. To
implement the t-test the test statistic is calculated as:
̂
√ ̂ (
̂
)
(13)
84
Figure 2.1 - Expected Consumption Growth and 3 Month Treasury Rate
This Figure presents the real 3 month treasury bill rate in year t on forecasts of the log of one plus expected
economic growth in year t+1 using data from 1934 until 2010. Growth is measured by NIPA consumption,
Forecasts are made from an ARMA(1,1)-GARCH(1,1) time series model.
Figure 2.2 - Ex-Ante Variance of Consumption Growth and 3 Month Treasury Rate
This Figure presents the real 3 month treasury bill rate in year t on forecasts of the log of the variance of
growth in year t+1 using data from 1934 until 2010. Growth is measured by NIPA consumption, Forecasts
are made from an ARMA(1,1)-GARCH(1,1) time series model.
85
Figure 2.1 - Historic Consumption Growth
This figure graphs the log of one plus consumption growth from 1890 to 2010.
86
Figure 2.2 – Expert and Time Series Forecasts
This figure plots forecasts of the log of one plus expected economic growth from time series models and
from expert forecasts. Growth is measured by NIPA consumption, GDP and industrial production
respectively. The time-series model line shows forecasts from an ARMA(1,1)-GARCH(1,1). The Expert
line plots the median forecast of growth from the Survey of Professional Forecasters.
Table 2.1
Summary Statistics
This table presents summary statistics for the data analyzed in this paper. Panel A presents summary statistics for the various measures of the log of one
plus economic growth, and Panel B presents summary statistics of the log of nominal interest rates.
Mean First Year Last Year Std. Dev. Min
25th
Pctile Median
75th
Pctile Max
Consumption 0.018 1930 2010 0.022 -0.080 0.010 0.021 0.031 0.073
GDP 0.021 1930 2010 0.049 -0.146 0.002 0.023 0.043 0.158
Industrial Production 0.020 1920 2010 0.096 -0.282 -0.008 0.027 0.069 0.228
Historic Consumption 0.020 1890 1929 0.044 -0.085 -0.010 0.018 0.050 0.099
Historic GDP 0.038 1871 1929 0.035 -0.043 0.014 0.038 0.058 0.152
Historic Industrial Production 0.038 1885 1940 0.127 -0.351 -0.015 0.046 0.123 0.374
Mean First Year Last Year Std. Dev. Min
25th
Pctile Median
75th
Pctile Max
3 Month 0.040 1935 2010 0.032 0.000 0.012 0.037 0.057 0.160
30 Day 0.036 1927 2010 0.031 0.000 0.010 0.033 0.054 0.156
6 Month 0.049 1871 2010 0.028 0.003 0.032 0.049 0.061 0.172
1 Year 0.052 1947 2010 0.033 0.004 0.028 0.049 0.073 0.161
Panel A- Measures of Growth Rates
Panel B - Nominal Annualized Interest Rates
Table 2.2
Regression of Real Interest Rate on Forecasts of Growth and Uncertainty
This table presents regressions of the 3 month treasury bill rate in year t on forecasts of the log of one plus
expected economic growth and the variance of the growth in year t+1 using data from 1934 until 2010.
Growth is measured by NIPA consumption, GDP and industrial production respectively. In panel A
forecasts of growth are made from an ARMA(1,1) model and the variance of growth is the square of the
lagged residual. In Panel B forecasts are made from an ARMA(1,1)-GARCH(1,1) time series model. The
row labeled "Impact of 1 SD Var
t
[g
t+1
]" is the standard deviation of the variance of growth multiplied by
the coefficient on the variance of growth. Standard errors are Newey West using 3 lags. The top value is the
coefficient, the lower value in parentheses is the t-statistic.
Consumption GDP
Industrial
Production
E
t
[g
t+1
] -0.573 -0.127 -0.429
(-0.75) (-0.91) (-1.43)
Var
t
[g
t+1
] -23.906 -8.161 -1.002
(-3.01) (-8.14) (-1.98)
Constant 0.024 0.020 0.023
(1.47) (3.36) (2.44)
Observations 75 75 75
Adjusted R
2
0.08 0.42 0.15
R
2
Excluding Var
t
[g
t+1
] -0.01 -0.01 0.00
Impact of 1 SD Var
t
[g
t+1
] -0.010 -0.020 -0.012
Consumption GDP
Industrial
Production
E
t
[g
t+1
] -0.834 0.239 0.049
(-0.88) (1.14) (0.20)
Var
t
[g
t+1
] -60.389 -12.902 -2.441
(-3.30) (-11.22) (-3.04)
Constant 0.038 0.020 0.020
(1.72) (3.03) (3.72)
Observations 76 76 76
Adjusted R
2
0.14 0.55 0.35
R
2
Excluding Var
t
[g
t+1
] 0.00 -0.01 0.00
Impact of 1 SD Var
t
[g
t+1
] -0.012 -0.023 -0.018
Panel A - Residual Methodology
Panel B - GARCH Methodology
89
Table 2.3
Regression of Real Interest Rate on Forecasts of Growth and Uncertainty
Winsorised at 90
th
Percentile and using Ex-Post Realized Growth
This table presents regressions of the 3 month treasury bill rate in year t on forecasts of the log of one plus
expected economic growth and the variance of the growth in year t+1 using data from 1934 until 2010.
Growth is measured by NIPA consumption, GDP and industrial production respectively. Growth and
uncertainty forecasts are winsorised at the 90th percentile. Forecasts are made from an ARMA(1,1)-
GARCH(1,1) time series model. The row labeled "Impact of 1 SD Var
t
[g
t+1
]" is the standard deviation of
the variance of growth multiplied by the coefficient on the variance of growth. Standard errors are Newey
West using 3 lags. The top value is the coefficient, the lower value in parentheses is the t-statistic.
Consumption GDP
Industrial
Production
E
t
[g
t+1
] -0.681 0.422 -0.146
(-0.78) (1.04) (-0.56)
Var
t
[g
t+1
] -126.189 -17.249 -3.638
(-3.34) (-7.05) (-5.39)
Constant 0.046 0.019 0.028
(2.09) (2.00) (4.40)
Observations 76 76 76
Adjusted R
2
0.17 0.44 0.44
R
2
Excluding Var
t
[g
t+1
] -0.01 0.00 -0.01
Impact of 1 SD Var
t
[g
t+1
] -0.013 -0.020 -0.020
Table 2.4
Regression of the Real Interest Rate on Forecasts of Growth and Uncertainty Using Historic Data
This table presents regressions of the 3 month treasury bill rate in year t on forecasts of the log of one plus expected economic growth and the variance
of the growth in year t+1. Growth is measured by consumption (1890-2010), GDP (1871-2010) and industrial production (1895-2010) respectively.
Columns labeled "No Correction" use the raw forecasts while columns labeled "Winsorised 90%" winsorise the forecasts at the 90th percentile.
Forecasts are made from an ARMA(1,1)-GARCH(1,1) time series model. The row labeled "Impact of 1 SD Var
t
[g
t+1
]" is the standard deviation on the
variance of growth multiplied by the coefficient on the variance of growth. Standard errors are Newey West using 3 lags. The top value is the
coefficient, the lower value in parentheses is the t-statistic.
No
Correction
Winsorised
90%
No
Correction
Winsorised
90%
No
Correction
Winsorised
90%
E
t
[g
t+1
] 0.098 0.367 0.100 0.560 0.648 0.938
(0.18) (0.49) (0.28) (1.26) (2.19) (2.23)
Var
t
[g
t+1
] -6.150 -7.813 -6.262 -8.734 -1.487 -1.697
(-1.53) (-1.68) (-2.20) (-2.41) (-2.47) (-2.62)
Constant 0.026 0.022 0.034 0.026 0.024 0.021
(1.86) (1.16) (3.58) (2.24) (3.20) (2.11)
Observations 121 121 140 140 126 126
Adjusted R
2
0.02 0.03 0.11 0.13 0.15 0.16
R
2
Excluding Var
t
[g
t+1
] -0.01 0.00 -0.01 0.00 0.01 0.01
Impact of 1 SD Var
t
[g
t+1
] -0.005 -0.006 -0.011 -0.012 -0.011 -0.012
Industrial Production
1895 - 2010
GDP
1871 - 2010
Consumption
1890-2010
Table 2.5
Quarterly Regression of the Real Interest Rate
on Forecasts of Growth and Uncertainty
This table presents regressions of the 3 month treasury bill rate in year t on forecasts of the log of one plus
expected economic growth and the variance of the growth in year t+1 using data from 1947 until 2010.
Growth is measured by NIPA consumption, GDP and industrial production respectively. Forecasts are
made from an ARMA(1,1)-GARCH(1,1) time series model. The growth rate is annual and the interest rate
is annualized. The row labeled "Impact of 1 SD Var
t
[g
t+1
]" is the standard deviation on the variance of
growth multiplied by the coefficient on the variance of growth. Standard errors are Newey West using 4
lags. The top value is the coefficient, the lower value in parentheses is the t-statistic.
Consumption GDP
Industrial
Production
E
t
[g
t+1
] 0.086 0.062 0.019
(0.93) (0.70) (0.46)
Var
t
[g
t+1
] -64.377 -29.851 -14.882
(-3.30) (-1.56) (-5.20)
Constant 0.011 0.013 0.021
(2.15) (1.96) (6.36)
Observations 256 256 256
Adjusted R
2
0.08 0.02 0.26
R
2
Excluding Var
t
[g
t+1
] 0.01 0.00 0.00
Impact of 1 SD Var
t
[g
t+1
] -0.008 -0.005 -0.015
Table 2.6
Regression of Real Interest Rate on Forecasts of Growth and Uncertainty for Countries Outside the US
This table presents This table presents regressions for each country of the 3 month treasury bill rate in year t on forecasts of the log of one plus expected economic growth and the
variance of the growth in year t+1. Growth is measured by consumption and GDP. Forecasts are made from an ARMA(1,1)-GARCH(1,1) time series model for each country. The
row labeled "Impact of 1 SD Var
t
[g
t+1
]" is the standard deviation on the variance of growth multiplied by the coefficient on the variance of growth. Standard errors are Newey
West using 3 lags. The top value is the coefficient, the lower value in parentheses is the t-statistic.
Belgium Canada France Germany Italy Japan Netherlands UK
E
t
[g
t+1
] 0.7804 0.0453 -0.3590 0.3940 1.4501 0.8698 -0.0630 1.0797
(1.40) (0.06) (-0.91) (1.64) (2.40) (3.91) (-2.94) (3.98)
Var
t
[g
t+1
] -0.8067 -19.6832 -6.5062 -125.5366 -31.3694 -35.5100 -0.3447 -0.0002
(-0.03) (-3.91) (-4.21) (-2.07) (-3.77) (-3.72) (-11.64) (-1.85)
Constant 0.0138 0.0189 0.0256 0.0327 -0.0332 -0.0025 0.0059 -0.0056
(0.99) (0.97) (2.07) (3.30) (-1.47) (-0.45) (0.96) (-0.88)
Observations 61 75 49 56 69 49 68 109
Adjusted R
2
0.00 0.14 0.08 0.12 0.31 0.40 0.27 0.16
R
2
Excluding Var
t
[g
t+1
] 0.01 -0.01 0.03 -0.02 0.02 0.03 0.05 0.17
Impact of 1 SD Var
t
[g
t+1
] 0.000 -0.013 -0.008 -0.007 -0.083 -0.018 -0.015 -0.001
Belgium Canada France Germany Italy Japan Netherlands UK
E
t
[g
t+1
] 1.0901 -0.1262 -0.3438 0.4309 2.7429 0.5538 -0.0025 -0.1502
(4.66) (-0.50) (-1.13) (1.56) (3.02) (4.69) (-0.11) (-0.34)
Var
t
[g
t+1
] 10.5226 -15.9396 -3.5456 -2.7225 -5.8442 -9.3346 -0.3228 -9.0242
(0.61) (-3.63) (-2.55) (-1.07) (-4.52) (-3.47) (-5.54) (-0.42)
Constant -0.0036 0.0248 0.0243 0.0071 -0.1014 -0.0032 0.0049 0.0198
(-0.55) (2.49) (2.33) (1.11) (-3.01) (-0.65) (0.81) (1.01)
Observations 61 75 49 56 69 49 68 109
Adjusted R
2
0.17 0.09 0.07 0.00 0.41 0.40 0.19 -0.01
R
2
Excluding Var
t
[g
t+1
] 0.18 0.02 0.04 0.00 0.18 0.09 -0.01 -0.01
Impact of 1 SD Var
t
[g
t+1
] 0.001 -0.011 -0.007 -0.002 -0.077 -0.015 -0.015 -0.003
Panel A - Consumption
Panel B - GDP
Table 2.7
Regression of Various Interest Rates on Forecasts of Growth and Uncertainty
This table presents regressions various interest rates in year t on forecasts of the log of one plus expected economic growth and the
variance of the growth in year t+1 using data from 1934 until 2010. Growth is measured by NIPA consumption, GDP and industrial
production respectively using data from 1934 until 2010. The interest rate used is indicated by the column header. Forecasts are made
from an ARMA(1,1)-GARCH(1,1) time series model. The row labeled "Impact of 1 SD Var
t
[g
t+1
]" is the standard deviation on the
variance of growth multiplied by the coefficient on the variance of growth. Standard errors are Newey West to using 3 lags. The top
number is the coefficient, the lower number in parentheses is the t-statistic.
30 Day 3 Month 6 Month 1 Year
E
t
[g
t+1
] -1.0190 -0.8343 0.7201 -0.5185
(-1.12) (-0.88) (1.05) (-0.30)
Var
t
[g
t+1
] -58.8798 -60.3887 -66.2152 -74.9875
(-3.08) (-3.30) (-6.64) (-2.35)
Constant 0.0400 0.0378 0.0143 0.0431
(1.89) (1.72) (0.87) (1.08)
Observations 76 76 76 64
Adjusted R
2
0.14 0.14 0.19 0.12
R
2
Excluding Var
t
[g
t+1
] 0.00 0.00 0.02 0.00
Impact of 1 SD Var
t
[g
t+1
] -0.012 -0.012 -0.013 -0.015
30 Day 3 Month 6 Month 1 Year
E
t
[g
t+1
] 0.194 0.239 0.213 0.347
(0.93) (1.14) (0.78) (0.77)
Var
t
[g
t+1
] -12.799 -12.902 -10.576 -14.331
(-10.61) (-11.22) (-5.73) (-5.18)
Constant 0.019 0.020 0.025 0.025
(3.14) (3.03) (3.45) (2.00)
Observations 76 76 76 64
Adjusted R
2
0.56 0.55 0.36 0.45
R
2
Excluding Var
t
[g
t+1
] -0.01 -0.01 -0.01 0.15
Impact of 1 SD Var
t
[g
t+1
] -0.023 -0.023 -0.019 -0.026
30 Day 3 Month 6 Month 1 Year
E
t
[g
t+1
] 0.021 0.049 0.305 0.333
(0.09) (0.20) (1.70) (1.46)
Var
t
[g
t+1
] -2.419 -2.441 -2.360 -6.018
(-3.01) (-3.04) (-3.82) (-6.59)
Constant 0.019 0.020 0.024 0.034
(3.74) (3.72) (3.86) (4.29)
Observations 76 76 76 64
Adjusted R
2
0.36 0.35 0.31 0.45
R
2
Excluding Var
t
[g
t+1
] 0.00 0.00 -0.01 -0.01
Impact of 1 SD Var
t
[g
t+1
] -0.018 -0.018 -0.018 -0.045
Panel A - Consumption
Panel B - GDP
Panel C - Industrial Production
Table 2.8
Regression of the Real Interest Rate on Uncertainty, Ex-Post Growth
and Expert Forecasts of Growth
Consumption GDP
Industrial
Production
E
t
[g
t+1
] 0.1046 0.0572 0.0406
(0.36) (1.17) (1.16)
Var
t
[g
t+1
] -56.6577 -12.8533 -2.5723
(-5.04) (-8.68) (-4.16)
Constant 0.0171 0.0234 0.0207
(1.91) (4.95) (3.90)
Observations 76 76 76
Adjusted R
2
0.12 0.54 0.35
R
2
Excluding Var
t
[g
t+1
] 0.00 0.00 0.00
Impact of 1 SD Var
t
[g
t+1
] -0.011 -0.023 -0.019
Consumption GDP
Industrial
Production
E
i,t
[g
t+1
] 0.144 -0.084 -0.002
(0.91) (-0.70) (-0.03)
Var
t
[g
t+1
] -56.619 -5.672 -5.446
(-5.74) (-0.52) (-4.30)
Constant 0.025 0.022 0.030
(4.78) (3.44) (6.76)
Observations 2,955 4,856 4,710
Number of Years 29 42 42
Adjusted R
2
0.19 0.01 0.11
R
2
Excluding Var
t
[g
t+1
] 0.03 0.00 0.00
Impact of 1 SD Var
t
[g
t+1
] -0.011 -0.010 -0.041
Panel A - Ex-Post Growth
Panel B - Expert Forecasts
95
Panel A contains regressions as described in Table 2, but the realized ex-post value for growth is used.
Panel B presents panel regressions by year and individual expert forecast. The log of one plus the nominal
3 month treasury bill rate in year t minus the expert forecast of inflation is regressed on forecasts of the log
of one plus growth and the variance of the growth in year t+1 (calculated from an ARMA(1,1)-
GARCH(1,1) model) using data from 1982 until 2010 for consumption and 1969-2010 for GDP and
industrial production. Standard errors are clustered by year and by expert ID. The top value is the
coefficient, the lower value in parentheses is the t-statistic.
Consumption GDP
Industrial
Production
E
t
[g
t+1
] 0.8737 -0.0072 0.0818
(1.80) (-0.02) (0.54)
Var
t
[g
t+1
] -2.6771 -0.4375 -0.8540
(-4.26) (-0.48) (-5.14)
Constant -0.0034 0.0162 0.0160
(-0.27) (1.70) (2.18)
Observations 60 60 60
Adjusted R
2
0.16 -0.03 0.10
R
2
Excluding Var
t
[g
t+1
] -0.01 -0.02 -0.01
Impact of 1 SD Var
t
[g
t+1
] -0.013 -0.001 -0.008
Panel C - Rolling Forecasts
96
Table 2.9
VIX Regressions of the Real Interest Rate
on Forecasts of Growth and Uncertainty
This table presents regressions of the 3 month treasury bill rate in year t on growth forecasts measured by
the variable indicated in the column header and variance measured by the VIX index using data from 1990
until 2010. Growth Measures are in logs. In Panel A newey-west regressions are run with 3 lags. In Panel B
the mARM persistence correction is used. Standard errors in Panel B for the coefficients on E
t
[g
t+1
] and
Var
t
[g
t+1
] are mARM corrected and all others are White robust errors. The top value is the coefficient, the
lower value in parentheses is the t-statistic.
Leading Index for the US Consumer Sentiment
E
t
[g
t+1
] 0.024 0.012
(0.74) (3.71)
VIX -0.014 -0.011
(-2.18) (-2.81)
Constant 0.012 0.013
(6.35) (13.46)
Observations 246 246
Adjusted R
2
0.08 0.21
R
2
Excluding VIX 0.09 0.19
Impact of 1 SD VIX -0.001 -0.001
Leading Index for the US Consumer Sentiment
E
t
[g
t+1
] 0.026 0.013
(1.17) (6.46)
VIX -0.018 -0.014
(-2.82) (-3.30)
Residual E
t
[g
t+1
] -0.166 -0.012
(-2.64) (-2.06)
Residual Var
t
[g
t+1
] 0.012 0.012
(1.30) (1.53)
Constant 0.012 0.014
(7.64) (16.11)
Observations 245 245
Adjusted R
2
0.11 0.23
Impact of 1 SD VIX -0.001 -0.001
Panel A - Newey Regression
Panel B - mARM Procedure
97
Table 2.10
Controlling for Inflation Risk in Regression of the Real Interest Rate
on Forecasts of Growth and Uncertainty
This table presents regressions of the 3 month tresury bill rate in year t on forecasts of the log of one plus
expected economic growth and the variance of the growth in year t+1 as well as inflation forecasts using
data from 1934 until 2010. Growth is measured by NIPA consumption, GDP and industrial production
respectively. Forecasts are made from an ARMA(1,1)-GARCH(1,1) time series model. The row labeled
"Impact of 1 SD Var
t
[g
t+1
]" is the standard deviation of the variance of growth multiplied by the coefficient
on the variance of growth. Standard errors are Newey West to using 3 lags. The top number is the
coefficient, the lower number in parentheses is the t-statistic.
Consumption GDP
Industrial
Production
E
t
[g
t+1
] -1.492 0.524 0.247
(-1.24) (1.44) (0.67)
Var
t
[g
t+1
] -53.782 -15.075 -2.005
(-1.75) (-4.78) (-1.90)
Var
t
[i
t+1
] -13.995 5.224 -4.097
(-3.04) (0.79) (-0.93)
Cov
t
[g
t+1
,i
t+1
] 132.277 -22.711 -14.038
(2.01) (-1.04) (-1.04)
Constant 0.082 0.009 0.016
(2.26) (0.89) (2.22)
Observations 76 76 76
Adjusted R
2
0.10 0.44 0.18
Impact of 1 SD Var
t
[g
t+1
] -0.011 -0.027 -0.015
Consumption GDP
Industrial
Production
E
t
[g
t+1
] 0.222 -0.086 0.004
(1.40) (-0.72) (0.07)
Var
t
[g
t+1
] -63.389 -4.928 -6.574
(-5.22) (-0.40) (-4.18)
Var
t
[i
t+1
] 2.350 -8.063 32.892
(3.52) (-0.21) (0.92)
Constant 0.009 0.023 0.029
(1.34) (3.70) (6.51)
Observations 2955 4856 4710
Adjusted R
2
0.29 0.01 0.13
Impact of 1 SD Var
t
[g
t+1
] -0.013 -0.009 -0.049
Panel B: Inflation Risk as Variance of Expert Inflation Forecasts
Panel A: Inflation Risk from Time Series Model
98
Table 2.11
Persistence Tests for Regressions of the Real Interest Rate
on Forecasts of Growth and Uncertainty
This table presents regressions of the 3 month treasury bill rate in year t on forecasts of the log of one plus
expected economic growth and the variance of the growth in year t+1 using data from 1934 until 2010.
Growth is measured by NIPA consumption, GDP and industrial production respectively. In Panel A one
observations is used every 5 years starting in 1935 through the end of 2010. In Panel B the mARM
persistence correction is used. Forecasts are made from an ARMA(1,1)-GARCH(1,1) time series model.
The row labeled "Impact of 1 SD Var
t
[g
t+1
]" is the standard deviation on the variance of growth multiplied
by the coefficient on the variance of growth. Standard errors in Panel B for the coefficients on E
t
[g
t+1
] and
Var
t
[g
t+1
] are mARM corrected and all others are White robust errors. The top value is the coefficient, the
lower value in parentheses is the t-statistic.
Consumption GDP
Industrial
Production
E
t
[g
t+1
] 0.334 0.807 0.894
(0.41) (1.53) (3.12)
Var
t
[g
t+1
] -80.626 -11.688 -2.589
(-3.04) (-2.73) (-4.58)
Constant 0.023 0.010 0.013
(1.03) (0.96) (2.02)
Observations 16 16 16
Adjusted R
2
0.27 0.31 0.37
R
2
Excluding Var
t
[g
t+1
] 0.10 0.00 -0.07
Impact of 1 SD Var
t
[g
t+1
] -0.016 -0.021 -0.019
Consumption GDP
Industrial
Production
E
t
[g
t+1
] -3.274 0.386 0.158
(-1.25) (0.80) (0.34)
Var
t
[g
t+1
] -80.070 -12.364 -2.897
(-4.00) (-6.22) (-5.63)
Residual E
t
[g
t+1
] 2.562 -0.206 0.074
(0.95) (-0.40) (0.14)
Residual Var
t
[g
t+1
] 34.325 -2.674 2.566
(1.87) (-0.86) (2.42)
Constant 0.094 0.016 0.022
(1.70) (1.86) (3.54)
Observations 75 75 75
Adjusted R
2
0.17 0.55 0.38
Impact of 1 SD Var
t
[g
t+1
] -0.016 -0.022 -0.022
Panel A - 5 Year Interval
Panel B - mARM Procedure
99
Table 2.12
Controlling for Surplus Consumption in Regressions of the Real Interest Rate
on Forecasts of Growth and Uncertainty
This table presents regressions of the 3 month tresury bill rate in year t on forecasts of the log of one plus
expected economic growth and the variance of the growth in year t+1 as well as surplus consumption using
data from 1934 until 2010. Growth is measured by NIPA consumption, GDP and industrial production
respectively. Forecasts are made from an ARMA(1,1)-GARCH(1,1) time series model. Panel A presents
regressions of the interest rate on surplus consumption for various measures of growth. Panel B adds
forecasts of growth, uncertainty and inflation. The row labeled "Impact of 1 SD Var
t
[g
t+1
]" is the standard
deviation of the variance of growth multiplied by the coefficient on the variance of growth. Standard errors
are Newey West using 3 lags. The top number is the coefficient, the lower number in parentheses is the t-
statistic.
Surplus Consumption
Constant
Observations
Adjusted R
2
E
t
[g
t+1
] -0.824 -0.832 0.307 0.284 0.118 0.115
(-0.54) (-0.54) (1.49) (1.24) (1.28) (1.22)
Var
t
[g
t+1
] -96.030 -102.658 -13.794 -15.346 -5.804 -6.789
(-3.17) (-1.31) (-9.66) (-4.26) (-5.08) (-3.00)
Surplus Consumption -0.195 -0.209 0.022 -0.007 -0.007 -0.052
(-1.12) (-0.90) (0.65) (-0.11) (-0.22) (-0.72)
Surplus Cons.*Var
t
[g
t+1
] 55.104 7.723 5.097
(0.10) (0.52) (0.65)
Constant 0.066 0.067 0.017 0.022 0.030 0.036
(1.88) (1.61) (2.37) (1.52) (4.49) (2.53)
Observations 71 71 71 71 71 71
Adjusted R
2
0.17 0.15 0.57 0.57 0.47 0.47
Impact of 1 SD Var
t
[g
t+1
] -0.012 -0.013 -0.025 -0.027 -0.021 -0.024
(2.87)
76
0.15
Consumption GDP Industrial Production
-0.137
GDP Industrial Production
-0.094
(-3.18)
0.021
0.05 0.17
71
(3.72)
0.026
(-4.13)
Panel A - Wachter Replication
Panel B - Regression with Consumption Surplus
Consumption
-0.267
(-1.15)
0.039
(1.47)
71
100
Table 2.13
Long-Run Risk Coefficients from Various Papers
This table presents coefficients of r
f,t
=A
0,f
+A
1,f
x
t
+A
2,f
σ
t
2
and z
t
=A
0
+A
1
x
t
+A
2
σ
t
2
based on parameter values
from Bansal and Yaron (BY) (2004), Bansal, Kiku and Yaron (BKY) (2007) and Constantinides and Gosh
(CG) (2012). Ψ is the elasticity of intertemporal substitution used in the calibration
BY (2004) BKY (2007)
BKY (2007)
with ψ=0.9
CG 2012
Table 4
CG (2012)
Table 6
A
0,f
0.004 0.018 0.003 0.036 0.023
A
1,f
0.667 0.667 1.111 1.111 1.111
A
2,f
-13.551 -11.133 150.142 450.733 224.481
A
0
6.261 6.790 6.688 3.824 4.491
A
1
14.343 12.471 -4.157 -0.183 -0.311
A
2
-449.114 -1632.138 544.046 147.290 3.817
ψ 1.5 1.5 0.9 0.9 0.9
Abstract (if available)
Abstract
In the first chapter of my dissertation “The Worst, the Best, Ignoring All the Rest: The Rank Effect and Trading Behavior” I document a new stylized fact about how investors trade assets: individuals are more likely to sell the extreme winning and extreme losing positions in their portfolio (“the rank effect”). This effect is not driven by firm‐specific information or the level of returns itself, but is associated with the salience of extreme portfolio positions. The rank effect is exhibited by both retail traders and mutual fund managers, and is large enough to induce significant price reversals in stocks of up to 160 basis points per month. The effect indicates that trades in a given stock depend on what else is in an investor’s portfolio. ❧ The second chapter of my dissertation is “Economic Uncertainty and Interest Rates.” A number of asset‐pricing models predict a positive relation between the risk‐free interest rate and expected economic growth, and a negative relation between the interest rate and the uncertainty (i.e. the conditional variance) of growth. I document that uncertainty and the interest rate have a strong negative relation. This holds when examining up to 140 years of data, using various measures of economic growth and uncertainty, and after controlling for inflation. The result has a number of implications for models such as habit and long‐run risks. A negative relation between habit and the interest rate disappears after controlling for uncertainty. Previous research presents a puzzle as to the lack of relation between the macroeconomy and the real interest rate which this paper partially resolves.
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University of Southern California Dissertations and Theses
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Asset Metadata
Creator
Hartzmark, Samuel M.
(author)
Core Title
Essays in asset pricing
School
Marshall School of Business
Degree
Doctor of Philosophy
Degree Program
Finance
Publication Date
04/29/2014
Defense Date
03/24/2014
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
finance,OAI-PMH Harvest
Format
application/pdf
(imt)
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Ferson, Wayne (
committee chair
), Ahern, Kenneth (
committee member
), Solomon, David (
committee member
), Subramanyam, K.R. (
committee member
), Zapatero, Fernando (
committee member
)
Creator Email
hartzmar@usc.edu
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https://doi.org/10.25549/usctheses-c3-404540
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UC11296649
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