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Interval arithmetic and an application in finance

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Interval arithmetic and an application in finance

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Content Interval Arithmetic and an Application in Finance Zijun Liu May 2017 A thesis presented for the degree of Master of Science in Applied Mathematics Department of Mathematics Supervisor: Robert Sacker Thesis Committee: Robert Sacker (Chairman) Cymra Haskell Jianfeng Zhang UNIVERSITY OF SOUTHERN CALIFORNIA Abstract Interval Arithmetic and an Application in Finance Zijun Liu Abstract In this thesis, we explored interval arithmetic and used it as an approach to the computation of internal rate of return and net present value. We used Newton's method and secant method on intervals to handle the situation when net transfer is under uncertainty, and compared the results. We also applied interval fuzzy arithmetic to the computation of net present value. We applied the methods on a World Bank's Hunan urban development project in Changsha, China that was initiated in 2003 and will last until 2027. From the computations, we see that both the component 1 and the component 2 of this project are highly protable. i Contents Abstract i List of Figures iii List of Tables iv 1 Introduction 1 1.1 A brief history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 What is interval arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2.1 Arithmetic Operations of Intervals . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.2 Endpoint Formulas for the Arithmetic Operations . . . . . . . . . . . . . . . 3 1.2.3 Algebra laws of intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Interval iterative methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3.1 Interval Newton's method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3.2 Interval secant method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.3 Advantages and disadvantages of interval iterative methods . . . . . . . . . . 10 1.4 Fuzzy interval arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4.1 Fuzzy sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4.2 Fuzzy numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2 An application in nance: the computation of internal rate of return 15 2.1 Internal rate of return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 The component 1 of a Hunan urban development project in Changsha, China . . . . 16 2.2.1 Using Newton's method to compute internal rate of return . . . . . . . . . . 17 2.2.2 Using secant method to compute internal rate of return . . . . . . . . . . . . 19 2.2.3 Comparison of programming eciency of Newton's method and secant method in this problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3 An application in nance: the computation of net present value 21 3.1 Net present value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2 The component 2 of a Hunan urban development project in Changsha, China . . . . 22 4 Summary and future work 28 References 29 A Python programming for PC1 of Hunan urban development project 30 A.1 Python for classical Newton's method . . . . . . . . . . . . . . . . . . . . . . . . . . 30 A.2 Python for interval Newton's method . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 A.3 Python of IRR with uncertainties, using Newton's method . . . . . . . . . . . . . . . 32 A.4 Python for classical secant method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 A.5 Python of IRR with uncertainties, using secant method . . . . . . . . . . . . . . . . 35 B Python programming for PC2 of Hunan urban development project 37 B.1 Python for NPV using fuzzy interval arithmetic . . . . . . . . . . . . . . . . . . . . . 37 List of Figures 1 Iteration steps of Interval Newton's method . . . . . . . . . . . . . . . . . . . . . . . 8 2 Fuzzy sets and -cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3 An example of triangular fuzzy interval . . . . . . . . . . . . . . . . . . . . . . . . . 14 4 Cost-benet analysis for PC1 of the World Bank's project in Changsha, China . . . 17 5 The relationship of Net Present Value and Internal Rate of Return . . . . . . . . . . 21 6 Triangular fuzzy number of NPV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 7 Cost-benet analysis for PC2 of the World Bank's project in Changsha, China . . . 23 8 Fuzzy NPV of PC2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 iii List of Tables 1 IRR PC1 of Hunan Urban Development Project in China . . . . . . . . . . . . . . . 18 2 IRR of PC1 of Hunan Urban Development Project in China, with net transfer uncer- tainty of 2% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3 IRR of PC1 of Hunan Urban Development Project in China, with net transfer uncer- tainty of 5% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4 IRR of PC1 of Hunan Urban Development Project in China, with net transfer uncer- tainty of 10% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5 IRR of PC1 of Hunan Urban Development Project in China using secant method . . 19 6 Run time for each method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 7 Triangular fuzzy number of PC2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 8 CF l for each year of PC2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 9 CF r for each year of PC2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 10 Fuzzy NPV of PC2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 iv 1 Introduction Interval arithmetic, or interval analysis, is a method based on intervals as opposed to oating-point arithmetic. It is an extension of the classical real analysis. It is originated from an academic report by R. E. Moore.[1] Later, many scientists made an improvement on Interval arithmetic, and it has become an active branch of computational mathematics. 1.1 A brief history This section is largely inspired by William (Bill) Walster's article Introduction to Interval Arithmetic.[2] It is well known that there are many kinds of numerical calculation errors in scientic computa- tion, such as measuring errors, truncating errors and rounding errors. In 1957, Ramon E. Moore conceived the idea of interval arithmetic while he was an employee of Lockheed Missiles and Space Co. Inc. in Sunnyvale, CA. In April 1997, he explained his thinking at a kick-o meeting of Sun Microsystems' university R&D program in interval arithmetic. In 1957 at LMSC, Moore was considering how scientists and engineers represent measurements and values as x, where x is the computed result and is the error tolerance. Computing with such a notation is inconvenient even for simple questions like measuring the area of a room without knowing its specic length and width. Especially in large scale computations, the complexity of computation increases the accumulation of errors. Sometimes the computing process is too complex and labor intensive that the results become meaningless. Moore came up with a better idea. He suggested using two dierent numbers to represent . That is, instead of x, use x + and x, which point out the upper end points and lower end points of the interval that contains all possible values of the result. This is the birth of interval arithmetic. As Moore stated in [3], with interval computation, we can program a computer to nd intervals that contain - with absolute certainty - the exact answers to various mathematical problems. 1.2 What is interval arithmetic To begin with, Let's recall the denition of a closed interval. Denition 1 A closed interval X is a set of real numbers given by X = [x;x] =fx2R :xxxg (1) 1 where x denotes the left endpoint of X and x denotes the right endpoint of X. Denition 2 The width of an interval X is dened by w(x) =XX: (2) The absolute value of an interval X is dened by jXj =maxfjXj;jXjg: (3) The midpoint of an interval X is dened by m(X) = 1 2 (X +X): (4) The intersection of two intervals X and Y is dened by X\Y = [maxfX;Yg;minfX;Yg] (5) The union of two intervals X and Y is dened by X[Y = [minfX;Yg;maxfX;Yg] (6) 1.2.1 Arithmetic Operations of Intervals Denition 3 LetX andY be closed intervals, then their sum, dierence and product are dened to be the following sets: X +Y =fx +yjx2X \ y2Yg (7) XY =fxyjx2X \ y2Yg (8) XY =fxyjx2X \ y2Yg (9) Interval division has to be a compromise between information gain, computational eciency, and program complexity. In the past, many mathematicians and engineers have given their denition of interval division. From the denition of the standard interval division with 0 = 2Y , we have that the quotient X=Y =X [1=Y; 1=Y ] is the set 2 X=Y =fx=yjx2X \ y2Yg (10) But if we allow Y to contain the point zero, we cannot use this denition because X=0 is not dened. Therefore, in order to allow 02Y , in this paper, we apply Ratz's denition. Denition 4 Let X, Y be closed intervals. X = [X;X], Y = [Y;Y ]. Then Ratz's division is dened by X=Y =fz2Rj9x;y:x2X; y2Y;x =yzg (11) Note that this denes the quotient of two intervals to be a set which may not itself be an interval. For example, f2g=fxjx1g =fxjx2g[fx 0g We will provide explicit formulas for this quotient later in this paper. 1.2.2 Endpoint Formulas for the Arithmetic Operations Now let's dene the basic arithmetic operations of intervals. Addition Let's nd an operational way to add intervals. Since x2X means that x2 [X;X] and y2Y means that y2 [Y;Y ], it is obvious that X +Y 2 [X +Y;X +Y ]: Example Let X = [0; 1] and Y = [1; 2], then X +Y = [0 + 1; 1 + 2] = [1; 3]: Note that this is not the same as X [ Y = [0; 2]. Subtraction The operational way of subtraction is similar as addition. For subtraction we add the intervals 3 x2 [X;X] andy2 [Y;Y ] and we get xy2 [XY;XY ]: Therefore, we get the formula of interval subtraction XY = [XY;XY ]: Example Let X = [0; 1] and Y = [1; 2], then XY = [0 2; 1 1] = [2; 0]: Multiplication If X and Y are bounded, real intervals, then XY = [minS;maxS];whereS = [XY;XY;XY;XY ]: Although this formula may seem more complicated that the formulas of addtion and subtraction, this is an obvious conclusion and easy to prove. For the complete proof, please see [3]. Example Let X = [0; 1] and Y = [1; 2], then XY = [minf0 1; 0 2; 1 1; 1 2g;maxf0 1; 0 2; 1 1; 1 2g] = [0; 2]: Division The Ratz formula is given by the following theorem: Theorem 5 Let X and Y be two closed intervals. Then X=Y = 8 > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > : [X;X] [1=Y; 1=Y ] if 0 = 2Y (1;1) if 02X \ 02Y X=Y;1 if X < 0 \ Y Y = 0 1;X=Y [ [X=Y; +1) if X < 0 \ Y < 0<Y 1;X=Y if X < 0 \ 0 =Y <Y (1;X=Y ] if 0<X \ Y <Y = 0 (1;X=Y ] [ [X=Y; +1) if 0<X \ Y < 0Y X=Y; +1 if 0<X \ 0 =Y <Y ; if 0 = 2X \ 0 =Y 4 The proof of this theorem is in [4]. Since Ratz's interval division denes division among bounded intervals and the zero point is allowed to be included in the intervals, it can be used in the context of the Newton's Method. 1.2.3 Algebra laws of intervals Commutative Property X +Y =Y +X XY =YX Associative Property (X +Y ) +Z =X + (Y +Z) (XY )Z =X(YZ) The Sub-distributive Property X(Y +Z)XY +XZ X(Y +Z) =XY +XZ; if X is a real number or YZ > 0 Additive Identity X + 0 =X Denition 6 The distance of two intervals is dened by d(X;Y ) =maxfjXYj;jXYjg (12) and that d(X;Y ) 0, and d(X;Y ) = 0 i X =Y d(X;Y ) =d(Y;X) d(X;Z)d(Y;X) +d(Y;Z) Denition 7 A sequence of intervalsfX (k) g is convergent, if lim k!1 d(X (k) ;X) = 0 (13) and X is the limit of thefX (k) g. 1.3 Interval iterative methods As apposed to oating point root-nding algorithms, such as bisection method, Newton's method and secant method, we can try to apply them to the interval arithmetic in order to nd zeros in the given interval. We will introduce interval Newton's method and interval secant method in this paper. 5 1.3.1 Interval Newton's method Newton's method is a way to approximate the roots for a real-valued function by iterations. Moore created interval Newton's method in 1966 in his rst book about interval arithmetic, Interval Analysis [5], to nd zeros in an interval X. In this paper, we introduce interval Newton's method in one dimension. Letf be a real-valued function with variablex, and suppose thatf is continuously dierentiable. By the mean value theorem, we can write f(x)f(x ) = (xx )f 0 (); (14) where is some point between x and x . If x is a root of f, then x =x f(x) f 0 () : (15) If = x, then we get the ordinary Newton's method. If we let X be an interval that contains both x and x , then it is obvious that 2X. For convenience in computation, we set x to be the midpoint of X, i.e. x =m(X) = (X +X) 2 : Let F 0 (X) be an monotonic interval extension of f 0 (x). Consider the algorithm X n+1 =X n \N(X n ) (16) where N(X) =m(X) f(m(X)) F 0 (X) (17) It follows from equation 15 that x 2 N(X), and therefore 2 N(X): Then the theorem below is followed: Theorem 8 If an interval X 0 contains a root of f(x), then so does X n for n = 0; 1; 2; . The intervals X n form a nested sequence that nally converges to x , if 0 = 2F 0 (X 0 ). Proof. If 0 = 2F 0 (X 0 ), then 0 = 2F 0 (X n ) for all n. Since N(X n ) =m(X n ) f(m(X n )) F 0 (X n ) ; 6 m(X n ) = 2N(X n ) unless f(m(X n )) = 0. Therefore, w(X n+1 )< 1 2 (w(X n )); and the sequence is convergent. Theorem 9 If the intervals X n form a nested sequence that nally converges to x , 0 = 2 F 0 (X 0 ), then it converges quadratically, i.e.9 a positive real number K such that w(X n+1 )K(w(X n )) 2 Proof. Since we have that F (x )6= 0, x 2 X, and for all real numbers x2 X 0 , f(x) = F (x), f 0 (x) =F 0 (x), we can write equation 17 as N(X) =m(X) F (m(x)) F 0 (X) : (18) Recall equation 16. Without loss of generality, we assume F 0 (X) > 0, from the mean value theorem, we have F 0 ([x 1 ;x 2 ]) =F 0 (X ) +K 1 [(x 2 x 1 ); (x 2 x 1 )] (19) For any [x1;x2] containing x , [x 1 ;x 2 ]2X. Substitute equation 19 in equation 18, we get N([x 1 ;x 2 ]) x 1 +x 2 2 F ( x1+x2 2 ) F 0 (x ) +K 1 [(x 2 x 1 ); (x 2 x 1 )] ; (20) then for small enough x 2 x 1 , we have w(N([x 1 ;x 2 ])) 2K 1 jF ( x1+x2 2 )j(x 2 x 1 ) (F 0 (x )) 2 K 2 1 (x 2 x 1 ) 2 ; (21) Since x 2 [x 1 ;x 2 ] and F (x ) = 0, we have jF ( x 1 +x 2 2 )jjF 0 ([x 1 ;x 2 ])j( x 1 x 2 2 ) (22) substituting equation 19 in equation 22, we get jF ( x 1 +x 2 2 )j (F 0 (x ) +K 1 (x 2 x 1 ))( x 1 x 2 2 ) (23) 7 then substituting the above function in equation 21 we get w(N([x 1 ;x 2 ])) K 1 (F 0 (x ) +K 1 (x 2 x 1 )) (F 0 (x )) 2 K 2 1 (x 2 x 1 ) 2 (x 2 x 1 ) 2 For small enough x 2 x 1 , we have w(N([x 1 ;x 2 ])) K 1 F 0 (x ) + 1 (x 2 x 1 ) 2 Therefore9 K ( K1 F 0 (x ) ) + 1, K is a positive real number, such that w(N([x 1 ;x 2 ]))K(w([x 1 ;x 2 ])) 2 (24) By equation 16, letting X n = [x 1 ;x 2 ], we have w(N(X n ))K(w(X n )) 2 From equation 16, we know that w(X n+1 )w(N(X n )), therefore w(X n+1 )K(w(X n )) 2 and the sequence is quadratically convergent. Figure 1: iteration steps of Interval Newton's method Figure 1 gives a geometric explanation of how the iteration steps converging to the root in Interval Newton's method. Example Let f(x) = x 2 2, and X 0 = [1; 2]. Then F 0 (X) = 2X and m = 1:5, therefore we have the rst Newton's step as 8 [1; 2] \ ( 1:5 1:5 2 2 2 [1; 2] ) = [1; 2] \ [1:375; 1:4375] = [1:375; 1:4375]: The next iterations will be performed on x2 [1:375; 1:4375], and we will nally get a very small interval containing p 2. Example Using Ratz's division, 02 F 0 (X) is allowed. Let f(x) = x 2 2, and X 0 = [2; 2]. Then F 0 (X) = 2X and m = 0, therefore we have the rst Newton's step as [2; 2] \ ( 0 0 2 2 2 [2; 2] ) = [2; 2] \ ((1;0:5][ [0:5;1)) = [2;0:5][ [0:5; 2]: The next iteration will be performed separately onx2 [2;0:5] andx2 [0:5; 2], and we will nally get very small intervals containing p 2 and p 2, respectively. For interval Newton's method in higher dimensions, we can replace the derivative with the Jacobian, J(X). When using the computer to do interval arithmetic, as oating-points are stored in the computer with a certain length, which depends on the accuracy of the variable, directed-rounding is widely used to return the computed results in order to ensure the accuracy. Usually, scientists round downwards for the left endpoints and round upward for the right endpoints, both on the last digit carried, and this is outward rounding. 1.3.2 Interval secant method The secant method is also used to approximate the roots for a real-valued function by iterations. It is usually viewed as a variant of the Newton's method, although the secant method was developed 3,000 years earlier than Newton's method.[6] Compared to Newton's method, the secant method saves the time of evaluating the derivative of f(x), f 0 (x), by replacing it with f(x n )f(x n1 ) x n x n1 : In interval arithmetic, it is also applicable when it is possible to nd an interval F 0 (X) that contains the range of f 0 (x): 9 f 0 (x)2F 0 (X), for all x2X We shall assume that 0 = 2F 0 (X), which is a natural assumption since we required F 0 (X) to be monotone. The most eective way of nding F 0 (X) is highly depended on the specic situation of the problem. Here, we leave it to further discussion. Also, we don't need to calculate the midpoint of X, we can just set m =x n . Therefore, we get the interval secant method: X n+1 =X n \S(X n ); where S(X n ) =x n+1 f(x n+1 ) F 0 (X n ) ; 0 = 2F 0 (X n ) and x n+1 =x n x n x n1 f(x n )f(x n1 ) f(x n ): Note that both Newton's method and the secant method are guaranteed to converge under appropriate conditions only locally: there is the assumption that the initial iterate x 0 is already \close enough" to an isolated root.[7] Since f(x n )f(x n1 ) is a denominator, f(x n )6= f(x n1 ), and also f(x n )f(x n1 ) cannot be too small. Similar to Theorem 8, the interval secant method is guaranteed to converge. The proof of convergence and convergence rate (superlinear convergence) appears in [8]. Let's look at the example of f(x) =x 2 2, with initial interval X 0 = [1; 2]. Since the derivative is obvious in this example, we setF 0 (X) = 2X andx 0 = 1,x 1 = 2, therefore we have the rst secant step as [1; 2] \ ( 2 2 2 2 2 [1; 2] ) = [1; 2] \ [1; 1:5] = [1; 1:5]; whereas the root p 2 lies in this interval. 1.3.3 Advantages and disadvantages of interval iterative methods Advantages: 10 (1) For classical (real or complex) methods of solving nonlinear equations, sometimes it is dicult to nd the solutions and it is almost impossible to determine whether all solutions are found; but using interval Newton's method, we can nd all the solutions in the initial interval and it is easy to tell whether all the solutions are found. (2) By interval iteration method, we can know how and when to stop the iteration steps, and the accuracy and the bound of errors. (3) By interval iteration method, we can determine whether the initial interval would render a convergent sequence, therefore determine whether the solution exists or not. Disadvantages: (1) Interval algorithm requires a large amount of computation, and resulting in too long com- puting time. (2) Interval arithmetic has its built-in advantage of over-conservation. For example, we get xx = 0 from the classical oating point arithmetic, but when it comes to interval arithmetic, we get [2; 3] [2; 3] = [1; 1] instead of 0, as we regard the two intervals as uncorrelated. However, in scientic computing, it is necessary to consider this correlation. Especially in a complex interval computations, if the correlation of interval is not taken into account, the result will be far beyond the range of true value, resulting in "error explosion". (3) The interval secant method, unlike the classical secant method, has many limitations in computing and is thus not as widely used as interval Newton's method. 1.4 Fuzzy interval arithmetic Fuzzy mathematics, including fuzzy set theory, fuzzy logic theory, fuzzy algorithms, fuzzy semantics, etc, was introduced by Lot Aliasker Zadeh. He proposed the idea of fuzzy set in 1965.[9] 1.4.1 Fuzzy sets Let's dene a characteristic function I A of a set A by I A(x) = 8 < : 1 if x2A 0 if x = 2A ; as an indicator of the membership of set A. Fuzzy sets are generalizations of the classical sets represented by their characteristic functionsI A . It is not uncommon that sometimes the membership 11 of a object is clearly dened. For example, the class of trees clearly includes oaks, pine, willows, etc. as its members. However, let A be the set of all trees, and A(x) =fx2Xjx is tallg Then A is a "fuzzy subset" of X. For a tree that is 10 feet, it is not clear if it belongs to the set A. Because the criterion \tall" is not well dened. Let g A(x) be a characteristic function that represents the "grade of membership" of x in A, to which real numbers g A(x) 2 [0; 1] are assigned. g A(x) = 1 corresponds to denite membership while g A(x) = 0 is non-membership. A tree of 10 feet can be considered to be \tall" with \degree of 0.5", and a tree of 300 feet can be considered to be \tall" with \degree of 1". Let A be a fuzzy subset dened in a universe set X, with its characteristic function g A(x) . Denition 10 An -cut of a fuzzy set A is dened by A =fx2Xjg A(x) g 2 (0; 1] It is also called the level-set of A. A 1 =fx2Xjg A(x) 1g is called the core of the fuzzy set A, and suppA =fx2Xjg A(x) > 0g is called the support of the fuzzy set A. Denition 11 A fuzzy subset A of X =R is convex if and only if every -cut is convex, i.e. for any x 1 , x 2 2X, and any 2 [0; 1], g A(x) (x 1 + (1)x 2 )minfg A(x1) ;g A(x2) g Figure 2 [10] gives a geometric explanation of convex fuzzy subsets, where A 0:8 is convex and A 0:6 is not. 12 Figure 2: an example of -cuts 1.4.2 Fuzzy numbers Fuzzy numbers are generalized real numbers and are very useful in fuzzy analysis and interval arithmetic. Denition 12 A fuzzy set A is called a fuzzy number if 1. A is normal, i.e.9x 0 2R; where A(x 0 ) = 1; 2. A is convex; 3. A is upper semicontinuous onR, i.e.8> 0,9> 0, such thatA(x)A(x 0 )<,jxx 0 j<; 4. A is compactly supported, i.e. clfx2R;A(x)> 0g is compact, where clfAg denotes the closure of set A. From the denition above, we can see that any real number and closed intervals are fuzzy numbers, and fuzzy set theory are closely related with interval arithmetic. Denition 13 A triangular fuzzy number is a fuzzy number represented with three points as: A = (a 1 ;a 2 ;a 3 ), where 1. a 1 to a 2 is an increasing function 2. a 2 to a 3 is a decreasing function 3. a 1 a 2 a 3 4. g A(x) = 8 > > > > > > > > > < > > > > > > > > > : 0 if x<a 1 xa1 a2a1 if a 1 xa 2 a3x a3a2 if a 2 xa 3 0 if xa 3 13 Figure 3 shows an example of the triangular fuzzy interval. Figure 3: An example of triangular fuzzy interval 14 2 An application in nance: the computation of internal rate of return 2.1 Internal rate of return The internal rate of return (IRR) is a method of calculating rate of return on an investment or a project. The term \internal" refers to the fact that its calculation does not incorporate environmental factors (e.g., the interest rate or in ation). It is the \annualized eective compounded return rate" that makes the net present value (NPV) of all cash ows (positive or negative) from a particular investment equal to zero. It can also refer to the annual interest rate paid on a savings account. For an proposed project, given V i as the investment for year i and R i as the return received at year i, we have V 0 +V 1 (1 +r) 1 + +V N (1 +r) N =R 0 +R 1 (1 +r) 1 + +R N (1 +r) N (25) where IRR, r, is the solution of the equation. The net transfer, n i = V i R i , as dened in [Moore, 1995], is the investment made in year i minus the return received in year i. Let p i be the project value at year i, then the p i is a recursion polynomial p i (r) =n i + (1 +r)p i1 (r) for i = 1; 2; ;N: The Present value (PV) is dened as the current worth of a future sum of money or stream of cash ows given a specied rate of return. Then we can deduce that the PV of the project at year i given p i is p i (r)=(1 +r) i : We will assume thatp 0 (r) =n 0 > 0 for a positive initial investment andn N < 0 for a nal return of the total project. If n i > 0, it means that there is a positive net transfer, i.e. the investment made at year i is larger than the return received at year i, and if n i < 0, then there is a negative net transfer, i.e. the return received at year i is larger than the investment made at year i. We can re-write the equation in (1) as (V 0 R 0 ) + (V 1 R 1 )(1 +r ) 1 + + (V N R N )(1 +r ) N = 0 15 which is n 0 +n 1 (1 +r ) 1 + +n N (1 +r ) N = 0 Now let's look at a real world example of the dierent approaches to the computation of internal rate of return. 2.2 The component 1 of a Hunan urban development project in Changsha, China A World Bank's project, \On a loan in the amount of US$171 million to the People's Republic of China for a Hunan urban development project" was initiated in 2003, and the project will sustain for 23 years from 2005 to 2027. The Project Development Objective (PDO) was to foster greater integration in the Changsha - Zhuzhou-Xiangtan (CZT) Region of Hunan Province through support for a carefully selected set of priority investments to address specic regional needs in a sustainable manner. The CZT Region had a population of 13.8 million, of whom 3 million lived in the core urban areas of the three cities. Most of this population was to benet to some extent from the project through improved economic development, improved transportation, cleaner air, a better landscape and cleaner rivers. The project has two main components. One is Project Component 1: Corridor Component (US$253.8 million or 66% of total project costs), the other is Project Component 2: Changsha Wastewater Component (US$101.6 million or 26% of total project costs). In this paper, we will evaluate Project Component 1 by internal rate of return (IRR) and Project Component 2 by net present value (NPV). The cost-benet analysis of Project Component 1 for each year is as follows: 16 Figure 4: Benet and Investment for PC1 corridor component Figure 4 shows the investment, return and the net benet for each year of Project Component 1. 2.2.1 Using Newton's method to compute internal rate of return 2.2.1.1 Using the classical Newton's method Now, let's apply classical Newton's method to the computation of IRR in this project. We can deduce from the gure that this is a highly protable project and p 9 (0) < 0. Let's try that r 0 = 0:5 as our initial guess of the IRR, setting stopping criterion as r n r n1 0:001, using Newton's method, we can program in Python as in Appendix A.1. The sequence converges to 0.24318428063916714 after 7 steps of iteration and 0.0005194660043343902 seconds, as we can see from the Python output below in Table 1. 17 Table 1: IRR for PC1: Corridor Component Iteration Internal Rate of Return k: 0 r*: 0.43159390716941837 k: 1 r*: 0.36985812667066725 k: 2 r*: 0.3168922587050794 k: 3 r*: 0.27619662342341506 k: 4 r*: 0.25195757216209574 k: 5 r*: 0.24394606075046174 k: 6 r*: 0.24318428063916714 2.2.1.2 Using Interval Newton's method Another approach to solve this problem would be using internal Newton's method as we discussed in section 1.3.1. Here we use the built-in package for interval arithmetic, PyInterval, in Python as in Appendix A.2. The output is interval [0:24317790249912383; 0:24317790249912438], with run time 0.0972037059837021 seconds. 2.2.1.3 Using Newton's method under uncertainties In the previous two methods, the classic Newton's method and the interval Newton's method, we get the oating point results from precise net transfer values of each year of the project. However, in real world problems, it is very common that the actual cost and benet vary a little from the assumed value. What if the net transfer has uncertainty? We can use interval arithmetic to address this issue. For the PC1 as previously showed, let's discuss the situation when the net transfer oats by 2%, 5%, or 10%, and how will IRR oat accordingly. For the net transfer in year i, n i , the interval extension is n i = [n i ;n i ] where n i =n i ajn i jandn i =n i +ajn i j for i = 0; 1; 2; ;N. As the net transfer oat upwards and downwards for 2%, 5%, or 10%, we set a as 0.02, 0.05, and 0.10. Again, we can apply the above method to the IRR problem. Using Newton's method, setting stopping criterion as r n r n1 0:001, we can program in Python as Appendix A.3. The run time is 0.0009633370209485292 and the results with net transfer of 2%, 5%, and 10% 18 variation are shown below in Table 2, Table 3, and Table 4, respectively. Table 2: When a=0.02 Iteration left endpoint of IRR right endpoint of IRR k: 0 rl: 0.24 ru: 0.24 k: 1 rl: 0.23784737557133087 ru: 0.2495130905918041 k: 2 rl: 0.23779562506100244 ru: 0.24863052020519769 Table 3: When a=0.05 Iteration left endpoint of IRR right endpoint of IRR k: 0 rl: 0.24 ru: 0.24 k: 1 rl: 0.230846291298838 ru: 0.26070893392855393 k: 2 rl: 0.22983806117503985 ru: 0.2570658387873536 Table 4: When a=0.10 Iteration left endpoint of IRR right endpoint of IRR k: 0 rl: 0.24 ru: 0.24 k: 1 rl: 0.22150958044682878 ru: 0.2868202214433979 k: 2 rl: 0.217017005066925 ru: 0.2734139570555571 k: 3 rl: 0.21678513852928355 ru: 0.2711801423183424 2.2.2 Using secant method to compute internal rate of return Now let's approach the problem using secant method and see if there will be slight dierence. 2.2.2.1 The classic secant method Now let's use the classic secant method to solve IRR as in paragraph 2.2.1.1. We also program in Python and set stopping criterion as r n r n1 0:001 as in Appendix A.4. The run time is 8.702278137207031e-05 seconds and the results are shown below in Table 5. Table 5: IRR of PC1 using secant method Iteration left endpoint of IRR right endpoint of IRR k: 0 rl: 0.2462716388565179 ru: 0.3 k: 1 rl: 0.24459769730063088 ru: 0.2462716388565179 k: 2 rl: 0.24322464565953886 ru: 0.24459769730063088 k: 3 rl: 0.24317862010844482 ru: 0.24322464565953886 2.2.2.2 Using secant method under uncertainties In this paragraph, we apply secant method to the same problem as in paragraph 2.2.1.3. Setting stopping criterion as r n r n1 0:001, we program in Python as Appendix A.5. The run time is 0.00045800209045410156 and the results are listed below. 19 2.2.3 Comparison of programming eciency of Newton's method and secant method in this problem After the computation with each method, we get the run time as in Table 6. Table 6: run time for each method Newton's method secant method classical 0.0005194660043343902 0.0009633370209485292 with uncertainty 8.702278137207031e-05 0.00045800209045410156 We can see that although Newton's method has quadratic convergence rate, it is costs more time than the secant method. The main reason is that Newton's method needs the computation of the derivative of the function in each step, which is very time consuming, while the secant method need not. Although the dierence of run time between the two methods is not signicant in our case, it can be crucial when the complexity of the problem rises. 20 3 An application in nance: the computation of net present value 3.1 Net present value Net Present Value (NPV) is the dierence between the present value of cash in ows and the present value of cash out ows. Similar as IRR, NPV is used in capital budgeting to analyze the protability of a projected investment or project. Denition 14 For an proposed project, given R i as the net cash in ow, i.e. return, during year i, V i as the net cash out ow, i.e. investment, during year i, r as the discount rate, and n i as the net transfer in year i, CF i =n i as the cash ow, we have: NPV = N X i=0 R i (1 +r) i N X i=0 V i (1 +r) i (26) = N X i=0 n i (1 +r) i (27) = N X i=0 CF i (1 +r) i (28) If the Net Present Value of a project or investment is positive, then the projected returns exceed the anticipated costs. An investment with a positive NPV will be protable and one with a negative NPV will result in a net loss. Therefore, the only investments that should be made are those with positive NPV values. Figure 5: The relationship of NPV and IRR 21 Figure 5 shows that when NPV = 0, IRR=discount rate. However, in real world, it is not easy to determine the precise value of cash in ow and out ow, and unforeseen variations may cause mistakes in decision making. In this section, we use fuzzy interval arithmetic to approach NPV under uncertainty. Consider a triangular fuzzy number A = (a 1 ;a 2 ;a 3 ) with an -cut A = [a l();a r() ] as gure 6 [11]. Figure 6: Triangular fuzzy number of NPV We can see from gure 6 that a l() a 1 a 2 a 1 = a 3 a r() a 3 a 2 =; therefore a l() =a 1 + (a 2 a 1 ); a r() =a 3 + (a 2 a 3 ); and A = [a l() ;a r() ] = [a 1 + (a 2 a 1 ); a 3 + (a 2 a 3 )] (29) 3.2 The component 2 of a Hunan urban development project in Changsha, China Let's look at the Project Component 2 of the World Bank's project of Hunan urban development in Changsha. The cost-benet analysis of Project Component 2 for each year is shown in gure 7: 22 Figure 7: Cost-benet analysis for PC2 of the World Bank's project in Changsha, China Assume that the cash ows (net benet) are given by the following triangular fuzzy number where A = (a 1 ;a 2 ;a 3 ): 23 Table 7: triangular fuzzy number of PC2 According to equation 29, we have table 8 and table 9 24 Table 8: CF l for each year 25 Table 9: CF r for each year By equation 28, we set discount rate = 6% and program in Python as in Appendix B. Table 10: fuzzy NPV of PC2 NPV l() NPV r() 0 1854.094909 2268.880444 0.1 1874.834186 2248.141167 0.2 1895.573462 2227.401891 0.3 1916.312739 2206.662614 0.4 1937.052016 2185.923337 0.5 1957.791293 2165.18406 0.6 1978.530569 2144.444784 0.7 1999.269846 2123.705507 0.8 2020.009123 2102.96623 0.9 2040.7484 2082.226953 1.0 2061.487676 2061.487676 The values ofNPV l() andNPV r() are listed in Table 10. A graphical representation is shown 26 in Figure 8. Figure 8: fuzzy NPV of PC2 Figure 8 shows that PC2 of the project is highly protable with NPV from 1854.094909 to 2268.880444, and run time 0.000165939331055 seconds. The application of fuzzy interval arithmetic on net present value provides an alternative for computing internal rates of return. The gure of the result as in Figure 8 is more intuitive and the the program runs faster than computing IRR, which is very important for data analysts and decision makers. 27 4 Summary and future work Interval arithmetic is a relatively new method that is very useful in measuring errors and bounding uncertainties. Today it is widely used for a variety of purposes. It also helps nd reliable and guaranteed solutions to equations and optimization problems. In this paper we provided applications in interval iterative method and fuzzy interval arithmetic. Chapter 1 is an introduction to the basics and concepts of interval arithmetic as well as the theoretical basics of the applications. In Chapter 2 we explained the concept of internal rate of return, which is an important gauge of project prot in nance. Then we used the cost-benet analysis of Project Component 1 of a World Bank's Hunan urban development project in Changsha, China to further demonstrate our theories. After comparing the classical and interval's Newton's method and secant method, we nd that secant method runs faster in this problem since it does not need to compute the derivative of the function in every step as in Newton's method. In Chapter 2 we explained the concept of net present value, an alternative for internal rate of return. Then we applied fuzzy interval arithmetic on the Project Component 2 of the Hunan urban development project. Then output image of net present value is more user-friendly than interval rate of return, which is very important for data analysts and decision makers. This thesis has concentrated on the basic concepts of interval arithmetic and interval iterative methods. The programming method of interval secant method has not been developed or shown in this paper. Future work can be done to address this method since it can be more eciency than interval Newton's method. 28 References [1] R. E. Moore, \Automatic error analysis in digital computation," in Technical Report Space Div. Report, ser. LMSD84821. Lockheed Missiles and Space Co., 1959. [2] W. Walster, \Introduction to interval arithmetic," J3-Fortran Standards Committee, May 1997. [3] M. J. C. Ramon E. Moore, R Baker Kearfott, Introduction to Interval Analysis. Society for Industrial and Applied Mathematics, 2009. [4] D. Ratz, \On extended interval arithmetic and inclusion isotonicity," Institut f. Angrew. Math- ematik, Universitat Karlsruhe, 1996. [5] R. E. Moore, Interval Analysis. Prentice-Hall, 1966. [6] J. Papakonstantinou, \The historical development of the secant method in 1-d," August 2007, paper presented at the annual meeting of the Mathematical Association of America, The Fair- mont Hotel, San Jose, CA. [7] C. G. Uri M. Ascher, A First Course in Numerical Methods. Society for Industrial and Applied Mathematics, 2001. [8] A. Neumaier, \An interval version of the secant method," BIT Numerical Mathematics, vol. 24, no. 3, pp. 366{372, 1984. [9] L. A. Zadeh, \Fuzzy sets," Information and control, vol. 8, no. 3, pp. 338{353, 1965. [10] G. Chen and T. T. Pham, Introduction to fuzzy sets, fuzzy logic, and fuzzy control systems. CRC press, 2000. [11] A. C. Sampaio Filho, M. Vellasco, and R. Tanscheit, \Modied net present value under un- certainties: An approach based on fuzzy numbers and interval arithmetic," in International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems. Springer, 2012, pp. 10{19. 29 A Python programming for PC1 of Hunan urban development project A.1 Python for classical Newton's method def addData (x , n ) : i=0 while i < n : data . append ( x ) i+=1 def s e t I p ut ( ) : data . extend ([11915 , 26975 , 24173 , 57795 , 39806 , 42148 , 22600, 41123, nn 79177, 87846, 95587, 103560, 111835 ,120475 , 129537, 139078, nn 149150, 159806, 171097, 183078, 195801, 209095, 268591]) def newton ( ) : s e t I p ut ( ) s t a r t = t i m e i t . d e f a u l t t i m e r ( ) n = len ( data ) k = 0 r = 0.6 e = 0.001 while True : p = data [ 0 ] pp = 0.0 for i in range (1 , n ) : pp = p + ( 1 . 0 + r ) pp p = data [ i ] + ( 1 . 0 + r ) p r1 = r p / pp print (k , r1 , r , p , pp) i f abs ( r1 r ) < e : break 30 r = r1 k+=1 stop = t i m e i t . d e f a u l t t i m e r ( ) print ( "Newton method run time i s : " , stop s t a r t ) newton ( ) A.2 Python for interval Newton's method import t i m e i t from i n t e r v a l import i n t e r v a l , inf , imath from sympy import x=Symbol ( "x" ) def f ( x ) : return 11915 + 26975((1+x)(1)) + 24173((1+x)(2))n + 57795((1+x)(3)) + 39806((1+x)(4)) + 42148((1+x)(5))n 22600((1+x)(6)) 41123((1+x)(7))79177((1+x)(8))n 87846((1+x)(9)) 95587((1+x)(10)) 103560((1+x)(11))n 111835((1+x)(12))120475((1+x)(13)) 129537((1+x)(14))n 139078((1+x)(15)) 149150((1+x)(16)) 159806((1+x)(17))n 171097((1+x)(18)) 183078((1+x)(19)) 195801((1+x)(20))n 209095((1+x)(21)) 268591((1+x)(22)) def fp ( x ) : return 26975/(x + 1)2n 48346/(x + 1)3 173385/(x + 1)4 159224/(x + 1)5n 210740/(x + 1)6 +135600/(x + 1)7 + 287861/(x + 1)8n +633416/(x + 1)9 + 790614/(x + 1)10 +955870/(x + 1)11n +1139160/(x + 1)12 + 1342020/(x + 1)13 + 1566175/(x + 1)14n +1813518/(x + 1)15 + 2086170/(x + 1)16 + 2386400/(x + 1)17n +2716702/(x + 1)18 +3079746/(x + 1)19 + 3478482/(x + 1)20n +3916020/(x + 1)21 + 4390995/(x + 1)22 + 5909002/(x + 1)23 s t a r t = t i m e i t . d e f a u l t t i m e r ( ) 31 print ( i n t e r v a l [ 0 , 1 ] . newton (lambda x : f ( x ) , lambda x : fp ( x ) ) ) stop = t i m e i t . d e f a u l t t i m e r ( ) print ( " I n t e r v a l newton method run time i s : " , stop s t a r t ) A.3 Python of IRR with uncertainties, using Newton's method import t i m e i t data = [ ] datal = [ ] datau = [ ] def addData (x , n , data ) : i=0 while i < n : data . append ( x ) i+=1 def s e t I p ut ( a ) : global data , datal , datau data = [ ] data . extend ([11915 , 26975 , 24173 , 57795 , 39806 , 42148 , 22600, 41123, 79177, 87846, 95587, 103560, 111835 ,120475 , 129537, 139078, 149150, 159806, 171097, 183078, 195801, 209095, 268591]) n = len ( data ) addData ( 0 . 0 , n , datal ) addData ( 0 . 0 , n , datau ) for i in range (n ) : datal [ i ] = data [ i ] a abs ( data [ i ] ) datau [ i ] = data [ i ] + a abs ( data [ i ] ) def newton ( ) : A = [ 0 . 0 2 , 0.05 , 0 . 1 0 ] s t a r t = t i m e i t . d e f a u l t t i m e r ( ) for a in A: print ( "a=" , a , "nn" ) 32 s e t I p ut ( a ) n = len ( data ) k = 0 r l = 0.24 ru = 0.24 e = 0.001 while True : pl = datal [ 0 ] pu = datau [ 0 ] ppl = 0.0 ppu = 0.0 for i in range (1 , n ) : ppl = pl + ( 1 . 0 + ru ) ppl ppu = pu + ( 1 . 0 + r l ) ppu pl = datal [ i ] + ( 1 . 0 + ru ) pl pu = datau [ i ] + ( 1 . 0 + r l ) pu r1 = r l pu / ppu r2 = ru pl / ppl print ( "k : " , k , " r l : " , rl , "ru : " , ru , "nn" ) i f ( k >= 2 and abs ( r2 ru ) < e ) : break r l = r1 ru = r2 k+=1 stop = t i m e i t . d e f a u l t t i m e r ( ) print ( "Newton i n t e r v a l method run time i s : " , stop s t a r t ) newton ( ) A.4 Python for classical secant method import t i m e i t 33 data = [ ] def addData (x , n ) : i=0 while i < n : data . append ( x ) i+=1 def s e t I p ut ( ) : data . extend ([11915 , 26975 , 24173 , 57795 , 39806 , 42148 , 22600, 41123, nn 79177, 87846, 95587, 103560, 111835 ,120475 , 129537, 139078, nn 149150, 159806, 171097, 183078, 195801, 209095, 268591]) def f ( x ) : n=len ( data ) p = data [ 0 ] for i in range (1 , n ) : p = data [ i ] + ( 1 . 0 + x ) p return p def secant ( ) : s e t I p ut ( ) s t a r t = t i m e i t . d e f a u l t t i m e r ( ) k=0 rk0 = 0.25 rk1 = 0.30 eps = 0.001 e p s f = 0.001 M = 100 frk1 = f ( rk0 ) frk2 = f ( rk1 ) while abs ( rk1 rk0 ) >= eps and k <= M1: k += 1 frk0 = frk1 34 frk1 = frk2 rk2 = rk1 frk1 ( rk1 rk0 ) / ( frk1 frk0 ) rk1 , rk0 = rk2 , rk1 print ( "k : " , k , " rk1 : " , rk1 , " f ( rk1 ) : " , frk1 , " rk0 : " , rk0 ) frk2 = f ( rk2 ) i f abs ( frk2 ) <= e p s f : break stop = t i m e i t . d e f a u l t t i m e r ( ) print ( " Secant method run time i s : " , stop s t a r t ) secant ( ) A.5 Python of IRR with uncertainties, using secant method import t i m e i t data = [ ] datal = [ ] datau = [ ] def addData (x , n , data ) : i=0 while i < n : data . append ( x ) i+=1 def s e t I p ut ( a ) : global data , datal , datau data = [ ] data . extend ([11915 , 26975 , 24173 , 57795 , 39806 , 42148 , 22600, 41123,n 79177, 87846, 95587, 103560, 111835 ,120475 , 129537, 139078, n 149150, 159806, 171097, 183078, 195801, 209095, 268591]) n = len ( data ) addData ( 0 . 0 , n , datal ) addData ( 0 . 0 , n , datau ) 35 for i in range (n ) : datal [ i ] = data [ i ] a abs ( data [ i ] ) datau [ i ] = data [ i ] + a abs ( data [ i ] ) def f l ( x ) : n=len ( data ) p = data [ 0 ] for i in range (1 , n ) : p = datal [ i ] + ( 1 . 0 + x ) p return p def fu ( x ) : n=len ( data ) p = data [ 0 ] for i in range (1 , n ) : p = datau [ i ] + ( 1 . 0 + x ) p return p def f ( rl , ru ) : n=len ( datal ) pl = datal [ 0 ] pu = datau [ 0 ] for i in range (1 , n ) : pl = datal [ i ] + ( 1 . 0 + ru ) pl pu = datau [ i ] + ( 1 . 0 + r l ) pu return pl , pu def secant ( ) : A = [ 0 . 0 2 , 0.05 , 0 . 1 0 ] s t a r t = t i m e i t . d e f a u l t t i m e r ( ) for a in A: print ( "a=" , a , "nn" ) s e t I p ut ( a ) k=0 36 #rk0 = 0.217 r k 0 l = 0.216 rk0u = 0.218 #rk1 = 0.269 r k 1 l = 0.268 rk1u = 0.270 eps = 0.001 e p s f = 0.001 M = 100 f r k 1 l , frk1u = f ( rk0l , rk0u ) f r k 2 l , frk2u = f ( rk1l , rk1u ) while abs ( r k 1 l r k 0 l ) >= eps and k <= M1: k += 1 f r k 0 l , frk0u = f r k 1 l , frk1u f r k 1 l , frk1u = f r k 2 l , frk2u #rk2 = rk1 f ( rk1 ) ( rk1 rk0 ) / ( f ( rk1 ) f ( rk0 ) ) r k 2 l = r k 1 l frk1u ( r k 1 l r k 0 l ) / ( frk1u frk0u ) rk2u = rk1u f r k 1 l ( rk1u rk0u ) / ( f r k 1 l f r k 0 l ) rk1l , r k 0 l = rk2l , r k 1 l rk1u , rk0u = rk2u , rk1u f r k 2 l , frk2u = f ( rk1l , rk1u ) print ( "k : " , k , " r k 1 l : " , rk1l , "rk1u : " , rk1u ) i f abs ( f r k 1 l ) <= e p s f : break stop = t i m e i t . d e f a u l t t i m e r ( ) print ( " Secant i n t e r v a l method run time i s : " , stop s t a r t ) secant ( ) B Python programming for PC2 of Hunan urban development project B.1 Python for NPV using fuzzy interval arithmetic 37 data = [ ] def s e t I p ut ( ) : data . extend ([12.44 , 150.27 ,95.65 , 179.35 , 129.31 , 48.69 , 90.73 , n 145.82 , 190.34 , 229.43 , 272.8 , 292.8 , 313.98 , 336.44 , 360.23 , 385.44 , n 412.17 , 440.5 , 470.52 , 502.36 , 536.11 , 571.24 , 6 0 8 . 5 2 ] ) def f l ( r ) : n=len ( data ) alpha=0 while ( alpha <= 1 ) : npvtemp=data [ n1] for i in range (1 , n1): npvtemp=data [ n1i ]+npvtemp / ( 1 . 0 + r ) npvtemp=data [0]+ npvtemp /( 1 . 0 + r ) ( 0 . 9 + alpha /10.0) npv=npvtemp print npv alpha += 0.1 def f r ( r ) : n=len ( data ) alpha=0 while ( alpha <= 1 ) : npvtemp=data [ n1] for i in range (1 , n1): npvtemp=data [ n1i ] + npvtemp / ( 1 . 0 + r ) npvtemp=data [ 0 ] + npvtemp /( 1 . 0 + r ) ( 1 . 1 alpha /10.0) npv=npvtemp print npv alpha += 0.1 s e t I p ut ( ) print " Left : " print "" f l ( 0 . 0 6 ) print " Right : " 38 print "" f r ( 0 . 0 6 ) 39

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Creator Liu, Zijun (author)

Core Title Interval arithmetic and an application in finance

School College of Letters, Arts and Sciences

Degree Master of Science

Degree Program Applied Mathematics

Publication Date 02/08/2017

Defense Date 01/09/2017

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag internal rate of return,interval arithmetic,interval iterative methods,net present value,OAI-PMH Harvest

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Sacker, Robert (committee chair), Haskell, Cymra (committee member), Zhang, Jianfeng (committee member)

Creator Email zijunliu@usc.edu,zijunliu2014@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c40-330561

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Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA