Close
About
FAQ
Home
Collections
Login
USC Login
Register
0
Selected
Invert selection
Deselect all
Deselect all
Click here to refresh results
Click here to refresh results
USC
/
Digital Library
/
University of Southern California Dissertations and Theses
/
Optimal debt allocation using a dynamic programming approach
(USC Thesis Other)
Optimal debt allocation using a dynamic programming approach
PDF
Download
Share
Open document
Flip pages
Contact Us
Contact Us
Copy asset link
Request this asset
Transcript (if available)
Content
OPTIMAL DEBT ALLOCATION USING A DYNAMIC PROGRAMMING APPROACH by Melissa Maisch A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (APPLIED MATHEMATICS) August 2009 Copyright 2009 Melissa Maisch Acknowledgements I cherish each contribution to my development as a scholar and professional: To my advisor Fernando Zapatero, a gracious mentor who believed in the mathemati- cian's ability to make a contribution in nance. To my committee members Remigijus Mikulevicius, Jin Ma, Jianfeng Zhang, and Peter Baxendale for their encouraging words, thoughtful criticism, and time and attention during busy semesters. To my colleagues Aleksey Polunchenko and Johnny (Youngyun) Yun for sharing their enthusiasm for and comments on my work. To my parents, Chuck and Kay, for their support throughout the years. To my invaluable network of supportive, forgiving, generous and loving friends with- out whom I could not have survived the process: Cymra Haskel, Erin Quillen, and, of course, Mathias Knape. ii Table of Contents Acknowledgements ii List of Tables iv List of Figures v Abstract vi Chapter 1: Introduction 1 Chapter 2: The General Setting 4 2.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 The Objective Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Dynamic Programming Approach and Technical Results . . . . . . . . . 8 2.3.1 Transition Density . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3.2 The Value Function . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3.3 Lower Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Chapter 3: An Example: Ornstein-Uhlenbeck Process 14 3.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.1.1 Technical Result . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2 Analytic Solution in One Period . . . . . . . . . . . . . . . . . . . . . . 17 3.3 Numerical Solution in Multiple Periods . . . . . . . . . . . . . . . . . . . 19 3.3.1 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.4 Eect of Parameter Values on Optimal Solution . . . . . . . . . . . . . . 21 Chapter 4: Results for Various Economic Environments 26 4.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Chapter 5: Conclusions 30 5.1 The Cost of the Suboptimal Solution . . . . . . . . . . . . . . . . . . . . 30 5.2 Limitations and Further Extensions . . . . . . . . . . . . . . . . . . . . 31 References 32 Appendices 33 A.1 Key Computations for Random Components . . . . . . . . . . . . . . . 33 A.2 Optimal Allocation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 36 A.3 Calibrating the Ornstein-Uhlenbeck Process . . . . . . . . . . . . . . . . 40 A.4 Quantifying the Cost of the Suboptimal Solution . . . . . . . . . . . . . 41 iii List of Tables 4.1 Calibrating the O-U Process . . . . . . . . . . . . . . . . . . . . . . . . . 28 A.1 Calibrated to U.S. Interest Rates 1983-1988 . . . . . . . . . . . . . . . . 42 A.2 Calibrated to U.S. Interest Rates 1984-1989 . . . . . . . . . . . . . . . . 42 iv List of Figures 3.1 Parameterization: r 0 = 0:047, = 0:0139, = 0:53, = 0:01 . . . . . . 22 3.2 Parameterization: r = 0:056, = 0:0139, = 0:53, = 0:01 . . . . . . . 23 3.3 Parameterization: r = 0:056, r 0 = 0:047, = 0:53, = 0:01 . . . . . . . 24 3.4 Parameterization: r = 0:056, = 0:0139, r 0 = 0:047, = 0:01 . . . . . . 25 3.5 Parameterization: r = 0:056, = 0:0139, = 0:53, r 0 = 0:047 . . . . . . 25 4.1 Optimal Allocation of U.S. Debt as a Function of the Current Short Rate 28 4.2 Optimal Allocation of South Korean Debt as a Function of the Current Short Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.3 Optimal Allocation of Debt in Pakistan as a Function of the Current Short Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 v Abstract In this dissertation the optimal maturity structure of debt is determined. According to the 2007 Financial Report of the U.S. Government, the public debt is expected to triple by 2040. It is therefore ever more urgent to minimize this liability by optimiz- ing the structure of debt. A partial equilibrium approach is used to determine the optimal allocation between two representative debt instruments: a short-term Bill and a long-term Note. The objective function is solved numerically through the principle of dynamic programming. Calibrating the Ornstein-Uhlenbeck process to the U.S. T- Bill rates gives the optimal debt allocation as a function of the current interest rate. This method can also be extended to policymakers worldwide as analogous calibrations and conclusions are made for the developing economy of South Korea and the frontier economy of Pakistan. vi Chapter 1 Introduction A budget decit occurs when a government's expenditure exceeds its federal revenue. In the U.S., this decit is payed for through debt instruments issued by the Bureau of the Public Debt, an agency of the U.S. Department of the Treasury. As of 2008, the public held about $6 trillion in U.S. debt instruments a . These marketable securities range over a broad spectrum of maturities from 28 days to 30 years. The objective of this paper is to deduce the optimal allocation, from the standpoint of the risk-averse Treasury, of debt instruments across dierent maturities for a given exogenous scal policy and a given exogenous model of the term structure. An asset pricing approach is used to solve this nancial problem with the goal of producing practical results useful to policymakers. Many others have considered the question of debt maturity from a macroeconomic scal perspective by solving a general equilibrium problem. To the best of my knowledge, this paper is the rst to examine debt maturity from a partial equilibrium perspective. The existing debt management literature is heavily focused on macroeconomic scal and monetary policies. Of particular interest have been decisions between taxes, expenditure, and the type of debt issued. Ramsey (1927) observes that nancing government expenditure with a sudden large increase in sales tax can backre by de-incentivizing sales. Lucas and Stokey (1983) and Barro (2003) begin with the objective of smoothing taxes to maximize social welfare in accordance with the a This data can be found in the Treasury Bulletin viewable online at http://fms.treas.gov/bulletin/backissues.html 1 Ramsey (1927) objective. They then proceed to conjecture the optimal maturity structure of debt in an inter-temporal and stochastic general equilibrium economy. Lucas and Stokey (1983) determine the optimal policy subject to a time-consistent composition of debt. Barro (2003) extends this by assuming policymakers can make eective commitments about the form of future scal actions, and in turn concludes that the solution to the equilibrium is issuing long-term debt contingent on levels of expenditure and the tax base. Angeletos (2002) contributes by noting that public debt is, in reality, mostly not contingent. Angeletos (2002) demonstrates that actively managing the maturity of non-contingent debt (by increasing maturity before a large government expenditure, for instance) can almost perfectly replicate contingent debt. Angeletos (2002) concludes that any allocation which is implementable in an Arrow-Debreu economy can be replicated (or approximated) by portfolios of non-contingent bonds. One issue is that this result relies on the existence of as many bonds of dierent maturities as there are possible states. Shin (2006) recognizes that in reality, governments issue a few bonds of dierent maturities and actively manage their maturity structure. Shin (2006) replicates a complete market setting using fewer bonds by assuming the government can dynamically re-balance its portfolio. In summary, the literature to date is concerned with the maturity structure of debt as it relates to monetary and scal policies. They consider equilibrium models where current scal decisions aect future decits, which in turn aect their choice of debt maturity. One short-coming is that even in the most general setting, their equilibrium models deal with closed economies, that is, without international partners. For that reason, their predictive power is likely to be very poor. 2 Asset pricing reveals a practical approach that may produce results that are more useful to policymakers. Assuming scal policy and the interest rate are given exogenously, the problem of the treasury is to issue debt to pay for a decit. Adapted from Merton's (1969) portfolio problem, the risk-averse treasurer must choose between short and long maturities with the objective of minimizing a concave function of the nal liability. Intuitively, the choice is between paying a xed rate or paying a short rate with rollover risk (caused by the possibility that the future short-term rate may be higher). An exogenous interest rate model is used because it is easy to estimate empirically and exible enough to t the historical data. Despite the forecasting limitations imposed by the exogenous nature of the term structure, the proposed model is likely to perform competitively against the equilibrium models used previously in the literature. This paper is organized into ve sections. Section 2 introduces the general set- ting and some technical results. Section 3 considers a special case of the model and describes the algorithm for computing the exact numerical solution in multiple periods. Also included in this section are sensitivity tests to the parameter values. In Section 4, three dierent economic environments are calibrated and the resulting parameter values are used to deduce the optimal allocation of debt as a function of the current interest rate. Concluding remarks, including a quantication of the eect of choosing the optimal allocation, are given in Section 5. 3 Chapter 2 The General Setting The generalized model, objective function, and some technical results are derived in this section. 2.1 The Model Let ( ;F;fF s g s0 ;P ) be a ltered probability space. Suppose m2 N is a natural number, T2mN is a multiple of m, andS = [0;T )R. 2.1.1 Denition (Continuous-Time Markov Process). AnF s -adapted process y(s) is said to be a continuous-time Markov process if the following condition holds: For h 0 and s2S P [y(s +h)y s+h jF s ] =P [y(s +h)y s+h jy(s)]: The risk-averse treasurer is faced with an exogenously given decit and must decide how to optimally allocate this liability between two types of debt: short-term Bills and long-term Notes. The interest rate y(s) of the Bills is assumed to be a continuous-time Markov process. Let m be the maturity of the long-term Note, in years. At each time t2 T =f0;m; 2m;:::;Tmg the treasurer may re-balance his allocation of debt between the two securities. Over each period [t;t + m) the Notes pay the constant rate Y (t) given by the rate prevailing in the market at timet, the moment they are issued. It therefore makes 4 sense to model the rate Y (s) of the Notes as a step function which takes, at times t2T, the value of the short-term rate y(t) plus a constant risk premium . Y (s) = X t (y(t) +) ts<t+m t2T;s2S: (2.1.1) The rate, Y (s), of the Note changes stochastically at times t2 T in synchrony with the short-term rate and therefore is alsoF s -adapted. These assumptions characterize the Note akin to a m-year Note. The initial value of the liability is L(0) = L 0 2 R + . Let p(0) 2 [0; 1] denote the initial proportion of liability in Notes. At time 0 the treasurer chooses to allocate p(0)L(0) of the initial liability in Notes and the remaining (1p(0))L(0) in Bills. The F s -adapted stochastic process for the liability when s2 [0;m) is L(s) =L(0) p(0) exp (Y (0)s) + (1p(0)) exp Z s 0 y(u)du : The value of the liability after t =m years becomes L(m) =L(m ) +X(m) where X(m) N( X L 0 ; 2 X L 0 ) represents the government expenditure in the time period (0;m]. Let the control process p(t) = p(t;!) : T ! [0; 1] denote the proportion of liability in Notes at time t. At each time t2 T the treasurer updates the allocation of the current liability: p(t)L(t) in Notes and (1p(t))L(t) in Bills. The stochastic process for liability when s2 [t;t +m) is L(s) = L(t) p(t) exp(Y (t)(st)) + (1p(t)) exp Z s t y(u)du : 5 At time t +m the value of the liability is given by L(t +m) = L(t) p(t) exp (Y (t)10) + (1p(t)) exp Z t+10 t y(u)du +X(t +m); (2.1.2) whereX(t+m) represents the government expenditure during the time period (t;t+m]. X(t +m)N( X L 0 ; 2 X L 0 ) are i.i.d8t2T and furthermore X(t +m) is assumed to be independent of all other random processes for all t2T. Note that the model allows for a negative expenditure,X(t +m)< 0, which represents a capital gain in the period (t;t +m]. 2.2 The Objective Function The treasurer is interested in maximizing the expected utility from liability. Utility as a function of liability is rarely considered in nance literature; it is much more common to use utility from wealth. Let U :R!R denote a utility function of liability and let U : R! R denote a utility function of wealth. Notice that liability is equivalent to negative wealth. More directly, if the treasurer has a liability in the amount of x then it would be equivalent to say the treasurer has a wealth ofx. Therefore the following relation holds U(x) =U(x): (2.2.3) A more rigorous explanation of this relation follows. The properties for a utility function from liability U(x) for x2R are (i) U(x) is twice dierentiable8x2 R. This ensures the required optimization can be performed mathematically. (ii) U 0 (x)< 0,8x2R. The utility decreases as a function of x. 6 (iii) U 00 (x) < 0,8x2 R. Utility is concave to account for the risk-aversion of the treasury. This property also guarantees that the critical point found through an optimization is indeed a maximum. Notice that properties (i) and (iii) are standard properties for utility functions of wealth. Property (ii) is specic to utility functions of liability since the treasury will derive less utility from additional liability. Now consider the properties of a utility function U from wealth x. (a) U(x) is twice dierentiable8x2R. (b) U 0 (x)> 0,8x2R. (c) U 00 (x)< 0,8x2R. Straightforward calculations show that U can be chosen as a simple transformation of the familiar utility functions from wealth U. U(L) =U(L): In many cases it may be necessary to ensure the argument of this function is positive. Therefore a horizontal shift by a value of b will sometimes be considered U(L +b); for a constant b chosen so that b>L. 7 For j2T the vector ofF-adapted debt allocations p [j] :=fp(j);p(j + 10);:::;p(T 10)g2 [0; 1] (Tj)=10 refers to the allocation policy for the period [j;T ). The liability L at terminal time T depends on the chosen policy p [0] . Hence write L(T ) =L p [0] (T ): The treasurer's objective is to choose the optimal policy ^ p [0] to maximize the expec- tation of the utility derived from the liability at terminal time T given information at time 0, sup p [0] E h U(L p [0] (T )) L(0) =L 0 ;r(0) =r 0 i : (2.2.4) 2.3 Dynamic Programming Approach and Technical Results The objective function derived in Section 2.2 is sup p [0] E h U(L p [0] (T )) L(0) =L 0 ;y(0) =y 0 i : This multi-variable maximization problem cannot be solved directly because of the expectation. Hence the principle of dynamic programming is used to rewrite this as a sequence of one-period maximizations. The following lemma is necessary to validate the use of dynamic programming for the case in question. 8 2.3.1 Lemma (Discrete-Time Markov Property). If y(t) is a continuous-time Markov process then the pair (L(t);y(t)) satises the discrete-time Markov Property8t2T: P [(L(t +m);y(t +m)) (L t+m ;y t+m )j (L(t);y(t)) = (L t ;y t ); (L(tm);y(tm)) = (L tm ;y tm );:::; (L(0);y(0)) = (L 0 ;y 0 )] =P [(L(t +m);y(t +m)) (L t+m ;y t+m )j (L(t);y(t)) = (L t ;y t )]: Proof The idea of this proof is to show that y(t +m) and L(t +m) depend only on the values of y(t) and L(t) and not on the values of y(s) and L(s) for s<t. First, it is clear from the denition of a continuous-time Markov process that y(t +m) does not depend on values of y(s) for s<t. Secondly, equation (3.1.3) indicates that L(t +m) can be written as L(t +m) = L(t) p(t) exp (m(y(t) +)) + (1p(t)) exp Z t+m t y(u)du +X(t +m): Recall that p(t) is a Markov control process, X(t +m) is independent of all other processes, and y(t) is a continuous-time Markov process. This indicates that L(t +m) does not depend on values of L(s) and y(s) for s<t. Together, this implies that the pair (L(t);y(t)) satises the discrete-time Markov property. 9 2.3.1 Transition Density Since the pair (L(t);y(t)) is a Markov process, dene the transition density p (t;l;y;l 0 ;y 0 ) = @ 2 @l@y P (L(t +m)l;y(t +m)yjL(t) =l 0 ;y(t) =y 0 ): (2.3.5) 2.3.2 The Value Function The following notation will be used E t [Z] =E t;L;y [Z] =E[Zjt;L(t) =L;y(t) =y]: (2.3.6) Now dene the value function V (t;L t ;y t ) = sup p [t] E h U(L p [t] (T )) t;L(t) =L t ;y(t) =y t i to indicate the maximum value (in the expected utility sense) the treasurer can achieve at time T knowing all the information up until time t. Since V (t;L t ;y t ) depends only on the liability and interest rate at time t, it makes sense to suppress the time subscripts and dene V (t;L;y) :=V (t;L t ;y t ): 2.3.1 Theorem (Generalized Principle of Dynamic Programming). Let V (t;L;y) = sup p [t] E t;L;y [U(L p [t] (T ))]: Then the following relation holds8t2T, V (t;L;y) = sup p(t) E t;L;y h V (t +m;L p(t) (t +m);y(t +m)) i : (2.3.7) 10 Proof Using the law of iterated expectations gives V (t;L;y) = sup p [t] E t;L;y h E t+m h U(L p [t] (T )) ii Using the fact that (L(t);y(t)) is a Markov process indicates V (t;L;r) = sup p(t) E t;L;y " sup p [t+m] E t+m h U(L p [t] (T )) i # = sup p(t) E t;L;y h V (t +m;L p(t) (t +m);y(t +m)) i : In light of the transition density dened in equation (2.3.5), V (t;L;y) can also be expressed V (t;L;y) = sup p(t) Z V (t +m;L 0 ;y 0 ) p (t;L;y;L 0 ;y 0 )dL 0 dy 0 : (2.3.8) The relation given in equation (2.3.7) reduces the multi-period optimization problem to a sequence of single-period problems. At terminal time T the solution of the value function is known V (T;L;y) =U(L) because there is nothing to expect or maximize over. The relation in equation (2.3.7) is used to computeV (Tm;L;y), and then to recursively continue backwards in time until V (0;L;y) is obtained. The value ^ p(0) chosen to obtain V (0;L;y) is the optimal proportion of initial liability the treasurer should issue in the form of Notes. 11 2.3.3 Lower Bound In this section a lower bound for the objective function is determined. The following corollary will be needed for the lower bound as well as for the algorithm of the numerical solution. 2.3.1 Corollary. sup p [0] E 0 h U L p [0] (T ) i = sup p(0) E 0 " sup p(m) E m " ::: sup p(Tm) E Tm h U L p [0] (T ) i ::: ## (2.3.9) Proof This follows immediately from Theorem 2.3.1 by applying equation 2.3.7 itera- tively T=mm times. 2.3.1 Proposition. The objective function V (0;L;y) is bounded below by sup [0] E h U(L [0] (T )) 0;L;y i ; (2.3.10) where [0] := ((0);(m);:::;(Tm))2 [0; 1] T=m isF 0 -measurable. Further clarication may be necessary to understand the dierence between equation (2.3.10) and the original objective function equation (2.2.4). In the case of the lower bound, notice the vector of allocations [0] is non-random. This represents the idea that the treasurer must make all decisions about future debt allocations at time 0. In the case of the original objective function, on the other hand, the treasurer is permitted to decide the optimal debt allocation ^ p(t) at time t. Since the treasurer has less exibility and cannot use future information in the case where [0] is non-random, it trivially follows that the value derived from such a strategy should 12 be less than in the case of the original objective. A rigorous proof of this intuitive concept follows. Proof V (0;L;y) = E h U(L p [0] (T )) 0;L;y i = sup p(0) E 0 " sup p(m) E m " ::: sup p(Tm) E Tm h U L p [0] (T ) i ::: ## = sup (0) E 0 " sup p(m) E m " ::: sup p(Tm) E Tm h U L p [0] (T ) i ::: ## sup (0);(m) E 0 " E m " ::: sup p(Tm) E Tm h U L p [0] (T ) i ::: ## . . . sup [0] E 0 h E m h :::E Tm h U L [0] (T ) i ::: ii = sup [0] E 0 h U L [0] (T ) i ; where the last equality uses the law of iterated expectations. 13 Chapter 3 An Example: Ornstein-Uhlenbeck Process The Ornstein-Uhlenbeck process is a model of the term structure of interest rates which is commonly used in the nancial industry. The theory developed for the general model is applied to this very interesting case, producing tractable results. An analytic solution in one period is derived for the case of quadratic utility. A numerical solution is obtained in multiple periods using an algorithm dependent on the principle of dynamic programming. The algorithm is tested for consistency by investigating the sensitivity to input parameters. 3.1 The Model LetS = [0;T )R where T2 10N denotes the terminal time (in years). At each time t2 T =f0; 10; 20;:::;T 10g the risk-averse treasurer is faced with an exogenously given decit and must decide how to optimally allocate this liability between two types of debt: short-term 'Bills' and long-term 'Notes.' The interest rate r(s) of the Bills is assumed to follow a stochastic F s -adapted mean-reverting Ornstein-Uhlenbeck process, dr(s) = ( rr(s))ds +dW (s); s2S (3.1.1) with initial valuer(0) =r 0 , wherer 0 ;; r, and2R + nf0g are constant. W (s) denotes a standard Brownian motion process. Over each period [t;t + 10) the Notes pay the constant rate R(t) given by the rate prevailing in the market at timet, the moment they are issued. It therefore makes 14 sense to model the rate R(s) of the Notes as a step function which takes, at times t2T, the value of the short-term rate r(t) plus a constant risk premium . R(s) = X t (r(t) +) ts<t+10 t2T;s2S: (3.1.2) The rate, R(s), of the Note changes stochastically at times t2 T in synchrony with the short-term rate and therefore is alsoF s -adapted. These assumptions characterize the Note akin to a 10-year Note. The initial value of the liability is L(0) = L 0 2 R + . Let p(0) 2 [0; 1] denote the initial proportion of liability in Notes. At time 0 the treasurer chooses to put p(0)L(0) of the initial liability in Notes and the remaining (1p(0))L(0) in Bills. The F s -adapted stochastic process for the liability when s2 [0; 10) is L(s) =L(0) p(0) exp (R(0)s) + (1p(0)) exp Z s 0 r(u)du : The value of the liability after t = 10 years becomes L(10) =L(10 ) +X(10) where X(10) N( X L 0 ; 2 X L 0 ) represents the government expenditure in the time period (0; 10]. Let the control process p(t) = p(t;!) : T ! [0; 1] denote the proportion of liability in Notes at time t. At each time t2 T the treasurer updates the allocation of the current liability: p(t)L(t) in Notes and (1p(t))L(t) in Bills. The stochastic process for liability when s2 [t;t + 10) is L(s) = L(t) p(t) exp(R(t)(st)) + (1p(t)) exp Z s t r(u)du : 15 At time t + 10 the value of the liability is given by L(t + 10) = L(t) p(t) exp (R(t)10) + (1p(t)) exp Z t+10 t r(u)du +X(t + 10); (3.1.3) whereX(t+10) represents the government expenditure during the time period (t;t+10]. X(t + 10)N( X L 0 ; 2 X L 0 ) are i.i.d8t2T and furthermore X(t + 10) is assumed to be independent of all other random processes for all t2T. Note that the model allows for a negative expenditure,X(t+10)< 0, which represents a capital gain in the period (t;t + 10]. 3.1.1 Technical Result The principle of dynamic programming applies to the special case described in this section. 3.1.1 Lemma. The value function V (t;L;r) satises the Generalized Principle of Dynamic Programming with m = 10. Mathematically,8t2T V (t;L;r) = sup p(t) E t;L;r h V (t + 10;L p(t) (t + 10);r(t + 10)) i : Proof The rst step is to show that r(s) is a continuous-time Markov process. The following explicit formula for r(s) is derived in equation A.1.3 of Appendix A.1, r(s +h) = r(s)e (h) + r(1e (s1) ) + Z s+h s e (ush) dW (u): This expression indicates that r(s +h) does not depend on values of r() or W () for < s and therefore satises the denition of a continuous-time Markov process introduced in section 2.1. 16 Using this fact, Lemma 2.3.1 implies the pair (L(t);r(t)) satises the discrete- time Markov property. Applying Theorem 2.3.1 with m = 10 gives the required result. This validates the solution method of section 3.3. 3.2 Analytic Solution in One Period Consider the T = 10 case of the treasurer's problem V (0;L;r) = sup p(0) E h U L p(0) (10) L(0);r(0) i : An analytic solution is obtained when the treasurer is assumed to have quadratic utility U(x) =ax 2 +bx: The value function can be expressed as V (0;L;r) = sup p(0) E a L p(0) (10) 2 bL p(0) (10) L(0);r(0) : The rst order condition is 0 =E 0 2aL p(0) (10)b L 0 exp (R(0)) exp Z 10 0 r(u)du ; 17 where equation (3.1.3) is used to rewriteL p(0) (10). Dividing both sides byL 0 and using equation (3.1.3) once again gives 0 = E 0 2aL 0 p(0) exp(R(0)) + (1p(0)) exp Z 10 0 r(u)du + 2aX(10)b ::: exp(R(0)) exp Z 10 0 r(u)du : Denote m j i (k) :=E 0 exp k Z j i r(u)du : Then the rst order condition can be rewritten as 0 = 2aL 0 p(0) exp(2R(0)) 2aL 0 p(0) exp(R(0))m 10 0 (1)::: + 2aL 0 (1p(0)) exp(R(0)m 10 0 (1) 2aL 0 (1p(0))m 10 0 (2)::: b exp(R(0)) +bm 10 0 (1): Solving for p(0) gives ^ p(0) = 2aL 0 m 10 0 (2) 2aL 0 exp(R(0))m 10 0 (1) +b exp(R(0))bm 10 0 (1) 2aL 0 exp(2R(0)) 4aL 0 exp(R(0))m 10 0 (1) + 2aL 0 m 10 0 (2) : According to calculations found in Appendix A.1, R 10 0 r(u)du is normally distributed with mean 1 (0) and variance 2 1 (0). Therefore m 10 0 (k) is simply the moment generating function for a normal random variable with mean 1 (0) and variance 2 1 (0) and hence is known explicitly. The optimal proportion of liability to be assigned to Notes is ^ p(0) = 0:48 for the following parameter values: L(0) = 1; = 0:10; r = 0:10; = 0:20;r(0) = 0:06; = 0:02;a =0:50; and b =0:50. Notice for this choice of risk aversion parameters a 18 and b, U(L(10)) = a(L(10)) 2 bL(10) is decreasing and concave down for liability L(10)2 [0:5; 1:5]; as required. 3.3 Numerical Solution in Multiple Periods The primary problem with computing the numerical solution to a stochastic dynamic programming problem is the symbolic nature of the algorithm. At each time t2 T the computation of the expected value of a symbolic expression is required. Monte Carlo simulation computes the expected value by simulating a large number of paths for the random components and taking the mean. Taking the mean of many symbolic expressions is very cumbersome numerically and can lead to memory overload or an impractical run time. The algorithm proposed in this section circumvents this problem. 3.3.1 The Algorithm The C++ program for this algorithm is given in Appendix A.2. LetN be the number of Monte Carlo paths to compute an accurate expected value and for eacht2T letM be the number of choices forp(t) in the interval [0; 1] to ensure an accurate maximization. Then the algorithm relies on the simulation of (NM) T=10 random outcomes. According to computer science experts, this technique is original however its power is limited by its exponential runtime. To explain the idea of the algorithm, consider theT = 20 case of the treasurer's problem using a horizontal shift to the utility function as described in section 2.2 V (0;L;r) = sup p [0] E h U(L p [0] (20)b) L(0);r(0) i = sup p(0) E 0 " sup p(10) E 10 h U(L p [0] (20)b) i # ; 19 where equation (2.3.9) is used. Now using equation (3.1.3) twice with t = 10 and then t = 0 gives L p [0] (20) = L(0) p(0) exp (R(0)) + (1p(0)) exp Z 10 0 r(u)du +X(10) p(10) exp (10R(10)) + (1p(10)) exp Z 20 10 r(u)du +X(20): In the spirit of Monte Carlo simulation,N paths of the random components are needed. LetZ j t for j = 1;:::;N represent the j th set of outcomes for the random components in the interval [t;t + 10]. In this case Z j 10 = exp Z 20 10 r j (u)du ;X j (20) 2R 2 and Z j 0 = exp Z 10 0 r j (u)du ;R j (10);X j (10) 2R 3 : Z 0 is anN 3 matrix andZ 10 is anN 2 matrix. Recall thatX is independent from all other random processes, but R(10) and R 10 0 r(u)du are correlated. The maximization step requires evaluating the objective function for a discrete spectrum of allocations p(t)2 [0; 1] M . For i = 1;:::;M let p(t) i represent the i th allocation. Then p(t) i = i 1 M 1 ; t2f0; 10g: Denote J =U(L p [0] (20)b): Then J depends on p(0) and p(10)2 [0; 1] M ,Z 0 2R N3 , andZ 10 2R N2 . 20 Fix an outcome of Z 0 and p(0). Notice that there are a total of NM combina- tions. Then J(i;j) depends on p(10) i andZ j 10 . Taking the mean over each row of the matrix eectively gives E 10 []2R M . Identifying the maximum of this vector gives the sup p(10) E 10 []2R for the xed values ofZ 0 and p(0). Repeat this process for each of the pairs p(0) i and Z j 0 and put the results in a matrix A. Then taking the mean across the rows of A gives E 0 sup p(10) E 10 []2R M : The maximum is sup p(0) E 0 sup p(10) E 10 []2R: The value of p(0) which achieves this maximum is the required ^ p(0). 3.4 Eect of Parameter Values on Optimal Solution In this section, the C++ program written to implement the algorithm of section 3.3.1 is tested for consistency. The eect of changing each parameter value is documented and analyzed to ensure the results are consistent with economic intuition. In each of the following cases, N = 1000 Monte Carlo simulations were used with the parameter set indicated in the caption of each graph. The results of 30 runs were used to compute a 90% condence interval, as indicated by the blue triangles. 21 Figure 3.1: Parameterization: r 0 = 0:047, = 0:0139, = 0:53, = 0:01 Figure 3.1 shows the eect of changing the mean-reversion parameter r while keeping the remaining parameters constant. Figure 3.1 indicates the optimal propor- tion of liability assigned to long-term Notes, p 0 , increases as r increases. This makes sense because as the expected value of the interest rate increases, it is in the best interest of the treasurer to "lock in" the lower rate in the form of long-term Notes at the initial rate of 0.047. Figure 3.2 shows the eect of changing the initial rater 0 while keeping the remaining parameters constant. Figure 3.2 indicates the optimal proportion of liability assigned to long-term Notes, p 0 , decreases asr 0 increases. This makes sense because the higher the initial rate, the more willing the treasurer is to take the chance that the rate will decrease in the future by issuing more short-term debt. Treasurer hopes to roll-over the debt at a lower rate in the future. 22 Figure 3.2: Parameterization: r = 0:056, = 0:0139, = 0:53, = 0:01 Figure 3.3 shows the eect of changing the volatility of the interest rate while keeping the remaining parameters constant. Figure 3.3 indicates that the optimal proportion of liability assigned to long-terms Notes, p 0 , increases as increases. This makes sense because the risk-averse treasurer prefers to avoid the potential that high volatility could increase the rate signicantly. The treasurer therefore prefers to lock in the xed rate of the long-term Note as the volatility increases. Figure 3.4 shows the eect of changing the speed at which the interest rate mean reverts while keeping the remaining parameters constant. Figure 3.4 indicates that the optimal proportion of liability assigned to long-term Notes, p 0 , decreases as increases. This makes sense because as increases, the interest rate process is quicker to revert to the mean, hence reducing the variance of the interest rate. As the variance is reduced, it is less necessary for the risk-averse treasurer to lock in the rate of the long-term Note. Hence, p 0 decreases. 23 Figure 3.3: Parameterization: r = 0:056, r 0 = 0:047, = 0:53, = 0:01 Figure 3.5 shows the eect of changing the risk premium while keeping the remain- ing parameters constant. Figure 3.5 indicates that the optimal proportion of liability assigned to long-term Notes, p 0 , decreases as increases. The risk premium is the spread the treasurer would have to pay on top of the initial rate so that he may lock in the initial rate. It therefore makes sense that the larger the risk premium, the less incentive the treasurer will have to issue long-term Notes. The results of each of these analyses indicate the algorithm and corresponding C++ program are functioning as expected. 24 Figure 3.4: Parameterization: r = 0:056, = 0:0139, r 0 = 0:047, = 0:01 Figure 3.5: Parameterization: r = 0:056, = 0:0139, = 0:53, r 0 = 0:047 25 Chapter 4 Results for Various Economic Environments This section examines how dierent economic environments should optimally allocate debt, as dictated by the algorithm. First, the Ornstein-Uhlenbeck process was cali- brated to the interest rates of three economic environments: a developed economy, an emerging economy, and a frontier economy. Then the C++ program in Appendix A.2 was used to determine the optimal allocation for each of the parameter sets found in the calibration. 4.1 Calibration The most liquid short-term debt instrument is the 3-month T-Bill. Therefore 3-month T-Bills were used for the calibration of the Ornstein-Uhlenbeck process. As described in section 3.1 the Note is assumed to be comparable to the 10-year Note. These Notes also represent the most liquid long-term debt instruments. To calibrate the risk premium parameter, the average dierence between the 10-year government Note and the 3-month T-Bill was used. The U.S. is the representative for the developed economies. South Korea is taken to be the emerging economy since it is a member of the Big Emerging Market (BEM) as classied by the U.S. Department of Commerce. Pakistan is the delegated frontier economy as it is included in the S&P Frontier 150 a . Approximately 5 years of daily yields (12/5/2003-8/1/2008) were gathered on Bloomberg for the 3-month T-Bills and 10-year government Notes issued by each of these countries. Using the a As of August 2008 26 calibration method described in detail in Appendix A.3, the parameter sets in Table 4.1 were deduced. 4.2 Results Assuming the parameter sets described in section 4.1 the following conclusions were deduced. The United States should issue all liability in the form of long-term debt. South Korea and Pakistan, on the other hand, should issue all liability in the form of short-term debt. These conclusions are based on the assumption that the current short-term rate r 0 is the 3-month T-Bill yield on August 1, 2008. The following analysis examines the eect of a change in the current short-term rate in each of the three environments. The short-term rate in the United States as of August 1, 2008 was 0.017. Fig- ure 4.1 indicates that on that date it was optimal for the U.S. to issue all of its liability in the form of long-term debt. As the short-term rate increases, it is optimal to reallocate the liability in short-term debt. The short-term rate in South Korea as of August 1, 2008 was 0.053. Figure 4.2 indicates that on that date it was optimal for South Korea to issue all of its liability in the form of short-term debt. As the short-term rate decreases, it is optimal to reallocate the liability in long-term debt. The short-term rate in Pakistan as of August 1, 2008 was 0.12. Figure 4.3 indicates that on that date it was optimal for Pakistan to issue all of its liability in the form of short-term debt. As the short-term rate increases, it is optimal to reallocate the liability in long-term debt as well. 27 Table 4.1: Calibrating the O-U Process Parameter United States South Korea Pakistan 0.01 0.0074 0.013 r 0.036 0.048 0.25 0.38 0.45 0.12 0.012 0.016 0.021 r 0 0.017 0.053 0.12 Figure 4.1: Optimal Allocation of U.S. Debt as a Function of the Current Short Rate 28 Figure 4.2: Optimal Allocation of South Korean Debt as a Function of the Current Short Rate Figure 4.3: Optimal Allocation of Debt in Pakistan as a Function of the Current Short Rate 29 Chapter 5 Conclusions In this paper debt management is considered from a revolutionary perspective - a partial equilibrium perspective. This opened the door for the development of an innovative tool for nding the optimal allocation of debt instruments across dierent maturities. 5.1 The Cost of the Suboptimal Solution As of March 2008, approximately 50% of the U.S. public debt was held in Notes, 20% in Bills, and the rest in other securities including Bonds and in ation-protected instruments. Using the fact that the short term rate in the same month was 0.014 and assuming all other parameter values were the same, the theory developed in this article indicates it was optimal to issue the majority of U.S. debt in the form of instruments with long-term maturity. The eect of choosing the optimal allocation can be quantied in the following way. Take an initial liability L 0 and compute the expected utility using Monte Carlo for the optimal strategy. Then nd such that the expected utility using the suboptimal strategy with an initial liability of L 0 + gives the same utility as for initial liability L 0 using the optimal strategy. In other words, nd such that E 0 h U L ~ p [0] (T ) i =E 0 h U L ^ p [0] (T ) i ; (5.1.1) where ~ p [0] is the actual allocation, ^ p [0] is the optimal allocation strategy and L is computed using initial liabilityL 0 +. UsingL 0 = 100 gives the cost of the suboptimal strategy as a percentage. 30 Calibrating the U.S. interest rates to the model and applying the algorithm gives the optimal allocation ^ p. Then iteratively computing the expected utility with the suboptimal policy for the period August 1988 - August 2008 gives =23. This means the Treasury would have needed 23% less liability in August 1988 using the suboptimal policy to achieve the same expected utility as if they had used the optimal policy. In other words, the cost of the suboptimal strategy for the U.S. over this 20-year time horizon is 23%. In a separate trial, calibrating to the period May 1989 - May 2009 gives the cost of the suboptimal policy as 20%. Additional information about both of these computations can be found in the Appendix. 5.2 Limitations and Further Extensions One limitation of this method is that the algorithm is exponential time, limiting the future time horizon. This obstacle can likely be overcome through the computing power of government treasuries. The second limitation is that only two representative instruments are chosen. This means that the algorithm should be used cautiously to choose the relative weights of short-term versus long-term instruments. This work may be extended to include additional debt instruments such as longer term Bonds and Treasury In ation Protected Securities (TIPS). Another idea is to consider expenditure,X as a consumption(income) process and use existing techniques to determine the optimal expenditure process ^ X t . 31 References [1] Alfaro, L. and F. Kanczuk, 2007. Debt Maturity: Is Long-Term Debt Optimal? NBER Working Paper No. W13119. [2] Angeletos, G.M., 2002. Fiscal Policy with Non-Continent Debt and the Optimal Maturity Structure. Quarterly Journal of Economics 117, 1105-1131. [3] Barro, R., 2003. Optimal Management of Indexed and Nominal Debt. Annals of Economics and Finance 4, 1-15. [4] Bertsekas, D.P. and S.E. Shreve, 1978. Stochastic Optimal Control: The Discrete Time Case. Mathematics in Science and Engineering, Volume 139. [5] Cvitani c, J. and F. Zapatero, 2004. Introduction to the Economics and Mathe- matics of Financial Markets. MIT Press. [6] Lucas, Jr., R.E. and N.L. Stokey, 1983. Optimal Fiscal and Monetary Policy in an Economy without Capital. Journal of Monetary Economics 12, 55-93. [7] Merton, Robert C, 1969. Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case. The Review of Economics and Statistics, MIT Press, vol. 51(3), pages 247-57, August. [8] Ramsey, F.P., 1927. A Contribution of the Theory of Taxation. Economic Journal 37, 47-61. [9] Shin, Y., 2006. Managing the Maturity Structure of Government Debt. Journal of Monetary Economics Volume 54, Issue 6, 1565-1571. 32 Appendices A.1 Key Computations for Random Components Consider the Ornstein-Uhlenbeck process r(s) which satises dr(s) =( rr(s))ds +dW (s); (A.1.2) with initial value r(0); where r(0);; r; and 2R + n 0: Applying ^ Ito to r(s)e s gives d(r(s)e s ) = r(s)e s ds +e s dr(s) d(r(s)e s ) = r(s)e s ds +e s ( rr(s))ds +e s dW (s) Z t n d(r(s)e s ) = Z t n (e s r)ds + Z t n e s dW (s) r(t)e t = r(n)e n +e t re n r + Z t n e s dW (s) r(t) = r(n)e (nt) + r(1e (n1) ) + Z t n e (st) dW (s) (A.1.3) Z n+10 n r(t)dt = Z n+10 n (r(n)e (nt) + r(1e (nt) ))dt (A.1.4) + Z n+10 n Z t n (e (st) )dW (s)dt = 1 1e 10 (r(n) r) + 10 r (A.1.5) + Z n+10 n Z n+10 s e (st) dtdW (s) = 1 1e 10 (r(n) r) + 10 r + Z n+10 n 1e (sn10) dW (s) (A.1.6) 33 Dene 1 (n) =E Z n n + 10r(t)dt r(n) : Then using equation (A.1.6) gives 1 (n) = 1 1e 10 (r(n) r) + 10 r : Dene 2 1 (n) =Var Z n+10 n r(t)dt r(n) : Then 2 1 (n) =E " Z n+10 n r(t)dt 2 r(n) # 2 1 : Using equation (A.1.6) gives 2 1 (n) = E 1 2 1e 10 (r(n) r) + 10 r 2 ::: + E 2 1e 10 (r(n) r) + 10 r Z n+10 n 1e (sn10) dW (s) ::: + E Z n+10 n 1e (sn10) dW (s) 2 2 1 : Notice 2 1 cancels with the rst term and the second term is 0. So using ^ Ito Isometry for the remaining term gives 2 1 (n) = Z n+10 n 2 2 1e (sn10) 2 ds = 2 2 3 20 3 + 4e 10 e 20 : 34 Recall R(s) is a step function dened as R(s) = X t (r(t) +) ts<t+10 t2T;s2S: (A.1.7) Putting t =n + 10 in equation (A.1.3) gives r(n + 10) =r(n)e 10 + r 1e 10 + Z n+10 n e (sn10) dW (s): (A.1.8) Dene 2 (n) =E [R(n + 10)jr(n)]: 2 (n) =r(n)e 10 + r 1e 10 +: Dene 2 2 (n) =Var [R(n + 10)jr(n)]: Then using equation (A.1.8) gives 2 2 (n) =E Z n+10 n e (sn10) dW (s) 2 : Using ^ Ito Isometry 2 2 (n) = Z n+10 n e (sn10) 2 ds = 2 2 1e 20 : 35 Denote cov =Cov R(n + 10); Z n+10 n r(s)ds r(n) : Then cov =E R(n + 10) Z n+10 n r(s)ds r(n) 1 (n) 2 (n): Using equations (A.1.8) and (A.1.6) for r(n + 10) and R n+10 n r(s)ds, respectively cov =E Z n+10 n e (sn10)dW(s) Z n+10 n 1e (sn10) dW (s) ; where many terms cancel and drop out from taking the expectation of the stochastic integral. cov = Z n+10 n e (sn10) 1e (sn10) ds = 2 2 2 1e 10 2 : A.2 Optimal Allocation Algorithm The following C++ program computes the optimal allocation ^ p(0) in the T = 20 discrete case where the utility function can be specied by the user. 1 #include <iostream> 2 #include <algorithm> 3 #include <numeric> 4 #include <stdio.h> 5 #include <time.h> 6 #include <vector> 7 #include <cmath> 8 #include "rng.cpp" //comment if using Visual C++ 9 #include "rng.h" 10 using std::cout; 11 using std::cin; 12 using std::endl; 36 13 14 int main () 15 f 16 17 const int RN = 20; // Number of runs 18 const int T = 20; // Time horizon 19 const int N = 1000; // Number of MC simulations 20 const int M 1 = 11; // Number of values for p 1 21 const int M 0 = 11; // Number of values for p 0 22 const double pleft 1 = 0; // Left endpoint for p 1 23 const double pright 1 = 1; // Right endpoint for p 1 24 const double pleft 0 = 0; // Left endpoint for p 0 25 const double pright 0 = 1; // Right endpoint for p 0 26 27 //inital values and parameters for OrnsteinUhlenbeck Process 28 //dr(t)=v * (rbarr(t)) * dt+sigma * dW(t) initial value r 0; 29 const double r 0 = .024; // Initial rate of Bill 30 const double L 0 = 1; // Initial liability 31 const double lambda =0.012; // Positive constant risk premium 32 const double R 0 = r 0 + lambda; // Initial rate of Note 33 const double v = .38; // Rate of mean reversion 34 const double rbar = .036; // Mean reversion parameter 35 const double sigma = .01; // Volatility of interest rate process 36 const double gamma = 3 ; // Parameter for power utility 37 38 //Vector of values for proportion p to maximize over 39 std::vector<double> prop 1(M 1); 40 prop 1[0] = pleft 1; 41 for (int i=0; i<M 1; i++) f 42 prop 1[i] = pleft 1 + i * (pright 1pleft 1)/(M 11); 43 g 44 std::vector<double> prop 0(M 0); 45 prop 0[0] = pleft 0; 46 for (int i=0; i<M 0; i++) f 47 prop 0[i] = pleft 0 + i * (pright 0pleft 0)/(M 01); 48 g 49 50 //Specify Cholesky decomposition of covariance matrix between R 10 and exp0 10 51 const double theta 1 = ((pow(sigma,2))/(2 * v)) * (1exp(20 * v)); // Var(r 10) 52 const double theta 2 = ((pow(sigma,2))/(2 * (pow(v,3)))) * (20 * v3+4 * exp(10 * v)exp(20 * v)); // Var(exp0 10) 53 const double cov = ((pow(sigma,2))/(2 * (pow(v,2)))) * (pow((1exp(10 * v)),2) ); // Cov(R 10,exp0 10) 54 const double sd 1 = sqrt(theta 1); 55 const double sd 2 = sqrt(theta 2); 56 const double rho = cov / (sd 1 * sd 2); 57 double Chol[2][2]; 58 Chol[0][0] = sd 1; 59 Chol[0][1] = 0; 60 Chol[1][0] = rho * sd 2; 61 Chol[1][1] = sd 2 * sqrt( 1 pow(rho,2)); 62 63 ////Specify distribution of Government Expenditure 37 64 const double X1 mu = 0 * L 0; 65 const double X1 sd = .05 * L 0; 66 std::vector<double> X1(N); 67 const double X2 mu = 0 * L 0; 68 const double X2 sd = .05 * L 0; 69 std::vector<double> X2(N); 70 71 ////Specify distribution of Bills and Notes 72 const double mu 1=r 0 * exp(10 * v)+rbar * (1exp(10 * v))+lambda; // Mean of R 10 73 const double mu 2=(1/v) * ((1exp(10 * v)) * (r 0rbar)+10 * rbar * v); // Mean of int0 10 74 std::vector<double> Z(2); 75 double B[N][2]; // Matrix of bivariate normal generates with joint distribution of R 10 and int0 10 76 77 ////Initializing the matrices 78 double A[M 0][N]; 79 std::vector<double> Aavg( M 0 ); 80 std::vector<double> Asum(M 0); 81 double L 2[M 1][N]; 82 double V[M 1][N]; 83 double muj 2; 84 std::vector<double> Vsum(M 1); 85 std::vector<double> Vavg(M 1); 86 std::vector<double> V1(N); 87 88 //Repeat the simulation RN times 89 for (int number = 0; number < RN; ++number) f 90 91 //Government expenditure is normally distributed and assumed independent from period to period 92 93 RNG x; // Not seeded explicitly, so it will be given a random seed. 94 double testsum = 0; 95 for (int i = 0; i < 2; ++i) f 96 testsum += x.normal(); 97 g 98 99 //Filling a vector with N Normal(X mu, X sd) variates 100 x.normal(X1, X1 mu, X1 sd); 101 x.normal(X2, X2 mu, X2 sd); 102 103 //Generate multivariate normal random variable representatives with joint distribution of R 1 and int0 1 given time 0 104 for (int i=0; i<N/2; ++i) f 105 x.normal(Z, 0, 1); 106 B[i][0] = mu 1 + (Chol[0][0] * Z[0]); 107 B[N1i][0] = mu 1 (Chol[0][0] * Z[0]); // antithetic variates 108 B[i][1] = mu 2 + (Chol[1][0] * Z[0]) + (Chol[1][1] * Z[1]); 109 B[N1i][1] = mu 2 (Chol[1][0] * Z[0]) (Chol[1][1] * Z[1]); 110 g 111 //Ensure rates are nonnegative 112 for (int count1=0; count1<N; ++count1) f 38 113 for ( int count2=0; count2<2; ++count2) f 114 if (B[count1][count2] < 0) f 115 B[count1][count2] = 0; 116 g 117 g 118 g 119 120 //With respect the explanation of the algorithm given in the paper, A is the big outside MxN matrix and V is the small inside MxN value matrix 121 122 for (int i=0; i<M 0; i++) f 123 for (int j=0; j<N; j++) f 124 muj 2 = (1/v) * ((1exp(10 * v)) * (B[j][0]lambdarbar)+10 * rbar * v); // mean of int1 2 125 x.normal(V1, muj 2, sd 2); //V1 are random variables representing the conditional distribution of int1 2 given time 1 126 for (int k=0; k<M 1; k++) f 127 for (int n=0; n<N; n++) f 128 L 2[k][n]=(((L 0 * (prop 0[i] * exp(10 * R 0)+(1prop 0[i]) * exp(B[ j][1]))+X1[j]) * (prop 1[k] * exp(10 * B[j][0])+(1prop 1[k]) * exp(V1[n]))+X2[n])); 129 V[k][n]=(1/(1gamma)) * pow((L 2[k][n]5),(1gamma)); // power utility 130 g 131 g 132 for (int k=0; k<M 1; k++) f 133 Vsum[k] = V[k][0]; 134 for (int n=1; n<N; n++) f 135 Vsum[k] = Vsum[k] + V[k][n]; 136 g 137 g 138 for (int k=0; k<M 1; k++) f 139 Vavg[k] = Vsum[k] / N; 140 g 141 double Vmax = 999999999; 142 for (int q=0; q<M 1; q++) f 143 if (Vmax < Vavg.at(q)) Vmax = Vavg.at(q); 144 g 145 A[i][j] = Vmax; 146 g 147 g 148 for (int i=0; i<M 0; i++) f 149 Asum[i] = A[i][0]; 150 for (int j=1; j<N; j++) f 151 Asum[i] = Asum[i] + A[i][j]; 152 g 153 g 154 for (int i=0; i<M 0; i++) f 155 Aavg[i] = Asum[i] / N; 156 g 157 double Amax = 999999999; 158 for (int q=0; q<M 0; q++) f 159 if (Amax < Aavg.at(q)) Amax = Aavg.at(q); 160 g 39 161 double V 0 = Amax; 162 for (int i=0; i<M 0; i++) f 163 if (V 0 == Aavg[i])f 164 cout << "x(" << number + 1 << ") = " << i * ((pright 0pleft 0)/(M 01) )+pleft 0 << ";" << endl; 165 g 166 g 167 g 168 return 0;g A.3 Calibrating the Ornstein-Uhlenbeck Process The stochastic dierential equation for the Ornstein-Uhlenbeck process is dr(s) = ( rr(s))ds +dW (s); s2S: (A.3.9) The exact solution can be written in the form r(s +) =ar(s) +b +; (A.3.10) where are i.i.d. normally distributed with mean 0 and the constants a and b are determined by the parameters ; r; and . Making two changes of variables allows equation (A.3.9) to be written in the form of (A.3.10). The rst change of variable is Y (s) =r(s) r, so that dY (s) =Y (s)ds +dW (s): The second change of variable is Z(s) = exp(s)Y (s). Using ^ Ito's product rule gives dZ(s) = exp(s)dW (s): Integrating from s to s + gives Z(s +) =Z(s) + Z s+ s exp(u)dW (u): (A.3.11) 40 Substituting the change of variables gives Y (s +) = exp()Y (s) + Z s+ s exp ((us))dW (u) r(s +) = exp()r(s) + r(1 exp()) + Z s+ s exp((us))dW (u): Matching coecients to equation (A.3.10) implies a = exp() b = r(1 exp()) sd() = r 1 exp(2) 2 ; where sd() is computed using the ^ Ito isometry. Rewriting these equations gives = ln(a) (A.3.12) r = b 1a (A.3.13) = sd() s 2 ln(a) (1a 2 ) : (A.3.14) A linear regression between consecutive values is t using the method of least squares to deduce the values of a;b, and c that solve equation (A.3.10). The parameters of the Ornstein-Uhlenbeck process can then be recovered via the relationships given in equations (A.3.12), (A.3.13), and (A.3.14). The risk premium is computed by taking the mean dierence between the 10-year Note and the 3-month T-Bill. A.4 Quantifying the Cost of the Suboptimal Solution The following tables show the calibrated parameters for use in the section on the quantication of the cost of the suboptimal solution. 41 Table A.1: Calibrated to U.S. Interest Rates 1983-1988 0.0128 r 0.0643 0.4955 0.0216 r 0 0.0733 1988 1998 Actual: ~ p 0.78 0.80 Optimal: ^ p 0.0025 0.70 Table A.2: Calibrated to U.S. Interest Rates 1984-1989 0.0128 r 0.0668 0.6202 0.0058 r 0 0.0862 1989 1999 Actual: ~ p 0.78 0.80 Optimal: ^ p 0.00 0.005 42
Abstract (if available)
Abstract
In this dissertation the optimal maturity structure of debt is determined. According to the 2007 Financial Report of the U.S. Government, the public debt is expected to triple by 2040. It is therefore ever more urgent to minimize this liability by optimizing the structure of debt. A partial equilibrium approach is used to determine the optimal allocation between two representative debt instruments: a short-term Bill and a long-term Note. The objective function is solved numerically through the principle of dynamic programming. Calibrating the Ornstein-Uhlenbeck process to the U.S. T-Bill rates gives the optimal debt allocation as a function of the current interest rate. This method can also be extended to policymakers worldwide as analogous calibrations and conclusions are made for the developing economy of South Korea and the frontier economy of Pakistan.
Linked assets
University of Southern California Dissertations and Theses
Conceptually similar
PDF
Optimal decisions under recursive utility
PDF
A general equilibrium model for exchange rates and asset prices in an economy subject to jump-diffusion uncertainty
PDF
Credit risk of a leveraged firm in a controlled optimal stopping framework
PDF
Large deviations rates in a Gaussian setting and related topics
PDF
Essays on delegated portfolio management under market imperfections
PDF
Scheduling and resource allocation with incomplete information in wireless networks
PDF
Integrating data analytics and blended quality management to optimize higher education systems (HEES)
PDF
Nonlinear dynamical modeling of single neurons and its application to analysis of long-term potentiation (LTP)
Asset Metadata
Creator
Maisch, Melissa
(author)
Core Title
Optimal debt allocation using a dynamic programming approach
School
College of Letters, Arts and Sciences
Degree
Doctor of Philosophy
Degree Program
Applied Mathematics
Publication Date
07/29/2011
Defense Date
05/08/2009
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
bond maturity structure,debt management,dynamic programming,OAI-PMH Harvest
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Mikulevicius, Remigijus (
committee chair
), Zapatero, Fernando (
committee chair
), Ma, Jin (
committee member
)
Creator Email
melissa.maisch@gmail.com,mmaisch@usc.edu
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-m2419
Unique identifier
UC1498742
Identifier
etd-Maisch-2930 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-405175 (legacy record id),usctheses-m2419 (legacy record id)
Legacy Identifier
etd-Maisch-2930.pdf
Dmrecord
405175
Document Type
Dissertation
Rights
Maisch, Melissa
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Repository Name
Libraries, University of Southern California
Repository Location
Los Angeles, California
Repository Email
cisadmin@lib.usc.edu
Tags
bond maturity structure
debt management
dynamic programming