Engineering Economy ISYE 6225
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This 0 page Class Notes was uploaded by Maryse Thiel on Monday November 2, 2015. The Class Notes belongs to ISYE 6225 at Georgia Institute of Technology - Main Campus taught by Staff in Fall. Since its upload, it has received 15 views. For similar materials see /class/234215/isye-6225-georgia-institute-of-technology-main-campus in Industrial Engineering at Georgia Institute of Technology - Main Campus.
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Date Created: 11/02/15
MeaniVariance Ef cient Frontier 1 Introduction We consider a market with n risky assets An investor wishes to invest B dollars in this market Let Bi i 1 2 n denote the allocation of the budget to asset i so that 21 Bi B and let n denote the random one period return on asset i i 1 2 n The investor s one period return on his portfolio B1 B2 B is given by B Oneperiod return 3 2mm 1 i where we let x BiB denote the weight of asset i in the portfolio We shall think of B as xed and hereafter identify a portfolio of the n assets with a vector x 12 zn such that 221 zl 1 The portfolio s random return will be denoted by Mm 2 i1 Remark 1 When z lt 0 the holder of the portfolio is short selling asset i In these notes we permit unlimited short selling In practice however there a limits to the magnitude of short selling lf short selling is not permitted ie each M is constrained to be non negative then the solution approach outlined in these notes does not directly apply though the problem can be easily solved with commercial software Let W0 denote the investor s initial wealth and let Rm 1 Mm denote the one period total random return The investor s end of period wealth is W0Rz We assume the investor is risk averse and that his utility function is sufficiently differentiable and strictly concave The investor s optimal portfolio f should maximize his expected utility of end of period wealth as follows Investor s problem mgx 1 3 i1 where for convenience we have normalized the initial wealth W0 to one The investor s problem is a non trivial one One dif culty is to obtain the distribution of Rz which depends on m In lieu of solving the investor s problem directly we shall approx imate the expected utility via a second order Taylor series expansion about Mm For each realization of the random variable Rz and ignoring higher order terms URm WNW We 7 MlUM lRM 7 Ml2UM 4 Taking expectations of both sides of 4 we have ElURml ElU l UMElR 7 WM gU LElR 7 We 5 10495 U mVarlRml7 6 where the second line follows since Uiz is a constant and uz and 7 um2 VarRz We assume that the approximation of expected utility given in 4 is accu rate Consequently maximizing the left hand side of 4 is essentially equivalent to maximizing the right hand side of Now x the value for uz say to 770 that is consider only those portfolios z for which uz 770 In words the investor is xing his desired expected portfolio return Since the sign of U is negative and since Uiz and U iz are constants it immediately follows from the right hand side of 4 that the investor s optimal portfolio subject to the expected return constraint must minimize the variance of the portfolio s return Such a portfolio is called mean variance e cient As the level of 770 is varied a collection of mean variance ef cient points will be generated that trace out what is termed the mean variance e cient frontier Using results from convex analysis calculating the mean variance efficient frontier turns out to be easily implementable 2 Portfolio Mean and Variance We recall some basic de nitionsproperties from statistics Keep in mind each r is a random variable Let E 7 02 Varm Eri 7 77 2 7 7702 8 denote respectively the mean and variance of ri For two random variables X and Y COVX Y EX 7 EXY 7 EXY 7 9 It follows directly from 9 that the variance may be expressed in terms of covariance ie VarX COVXX 10 and that covariance is symmetric ie COVXY COVY X 11 In addition covariance has the following properties COVnX Y iCOVXY 12 06YQX Y 165VX04Y aCOVX Y 04 a real number 13 In what follows we let 039 3 COVm Tj 14 and let 2 denote the n by 71 matrix Whose i j entry is given by O39l39j The matrix 2 is called the covariance matrix Due to 11 a 074 which implies that the covariance matrix is symmetric namely 2 ET here the symbol T denotes transpose The portfolio s mean return or portfolio s return77 is given by Elrltzgtl El nmi i1 i1 Using 10 and repeated application of 12 and 13 the portfolio s variance is given by Varrz Varani i1 COVani Emmi i1 j1 n E Tjij i j1 n H H M lM aijxixj l H 11 Example 1 When n 2 the portfolio s variance is given by 2 2 7 2 2 2 2 2 2017mm 7 01x1 012m1x2 021m2x1 azzz i1 j1 2 2 2 2 01x1 azzz Qalgzlmg ln matrix notation the portfolio s variance may be represented as U 012 901 Varrz lt m1 2 gt 2 03921 0392 2 mTEx Remark 2 It may be veri ed that the representation Varrz xTEz holds true for n gt 2 16 17 18 19 24 3 The MeanVariance Optimization Problem The core optimization problem is formulated as follows Pm Min Varrx 7 12 mTEm 25 subject to z1m2znl 26 mm 2222 77nn 770 27 In vector form equations 26 and 27 may be expressed as eTz 1 28 rTz i0 29 where 5T 11 1 is the n vector whose components are all ones and ET r1 r2 Wm Remark 3 A convex optimization problem is de ned by a convex objective function subject to constraints de ned by convex functions Linear programming is a special well known case The constraints in P070 are linear which are convex The objective function in P070 is convex because the variance cannot be negative The objective function is a quadratic function of the vector x and so Problem P070 is called a quadratic programming problem Quadratic programming problems are easy to solve using commercial software De ne Lx i 7 12 mTEm 7 5Tm 7 1 7 WTQE 7 70 30 Here and a are real numbers The function is called the Lagrangian for this problem Example 2 When n 2 the core optimization problem is given by P070 Min 12 am aim 21712321322 31 subject to 1 2 1 771951 902 770 33 The associated Lagrangian is given by L901 9027 7 M 3 12 Ufmi t 73953 2012901902 M901 902 i 1 071901 772902 i 70 34 Now consider the following unconstrained problem L070 VAa min Lm A a 35 I Let mAa denote an optimal solution to problem L070 The following is a fundamental theorem of convex analysis Theorem 1 1 If f is an optimal solution to P070 then values for A and a exist for which zA a 2 If mAa satis es 26 and 27 then zA a is an optimal solution to P070 Remark 4 The variables A and a are called the dual variables As in linear programming there is one dual variable for every constraint Note that there are only two dual variables regardless of the number of z variables It turns out that primal problem P070 has a dual problem given by Vr0 maxVAa 36 such that V VarRz where f is an optimal solution to problem P070 The dual problem is a concave optimization problem Essentially this duality extends the well known duality of linear programming 4 Optimality Conditions We can use Theorem 1 to obtain a closed form solution to our original problem P070 Here is the game plan Step 1 Let f be an optimal solution to P070 and suppose we were given values for A and a such that z zAa Theorem 1a guarantees that this is possible Since zAa is an optimal solution to 35 which is an unconstrained problem we know from calculus that all n partial derivatives of evaluated at zA a must vanish These n equations in n unknowns may be solved to determine zA a Step 2 It remains to pin down values for A and a which requires two additional equations Theorem 1b provides them the portfolio constraint 28 and the portfolio return constraint 29 We illustrate this game plan in the special case that n 2 When presented in matrix form the equations we derive all hold true for the general case First the partial derivatives must vanish ie 8L 7 2 7 7 7 8M 01x1 1 algzg A Ian 0 37 8L 87 U 2 717 0121 7 A 7 M72 0 38 2 To simplify notation we suppress the dependence of x on A and In In matrix form equations 37 38 may be represented as lt1gtltgtlt1gtltgt Ex A5 1 p77 40 Now by multiplying both sides of 40 by 2 1 we solve for the optimal z zA u as m A 7 7 A 2 15 2 1r 41 Since zA u must solve the two feasibility constraints 28 and 29 we have 1 1 1 m1A u 5T m1 A u 7 7 7 7T T0 T1 T2 962 71 T x2 7 5T 71 717 ltz SET M n 5712715 57127177 A lt ETE le 24 p 39 44 It follows from equation 44 that A 7 2715 271 1 1 45 p ETE le ETE W 770 39 It may be veri ed that equations 40 41 44 and 45 all hold for n gt 2 risky assets or equivalently as MM Remark 5 If m were a single variable then the Lagrangian in 30 would be more commonly written as 1 MmAM75m 7Mm7DMM7L since 2 e and 77 would be all constants The derivative of with respect to the variable x would be easily computed as Em 7 A5 7 p77 which when set equal to zero yields the equation Em A5 In and z E 1Ae 1 pi The analogous operations when x is a vector are given in 40 and 41 Setting the collection of partial derivatives of a quadratic function of several variables to zero yields a system of linear equations 5 Algorithm to Compute MeanVariance Ef cient Points Here is how you could solve P070 step by step STEP 1 STEP 2 STEP 3 STEP 4 STEP 5 Compute the n x 2 matrix B lt 2 15 2 1 gt T 71 T 717 i 7 e E e e E r 7 CBS Ce Compute the 2 x 2 matriXC 7 lt FTEAE FTEA gt 7 lt CB 07 i 1 1 Compute the optimal values for and u as u C F 0 Compute the optimal solution 070 as Blt gt BC l lt 1 70 39 Compute the minimum variance Var o as zFoTEmF0 Remark 6 STEP 3 is unnecessary for the purpose of solving the problem7 but it does give the values for the dual variables Remark 7 The computation in STEP 5 can be simpli ed In what follows we write x in lieu of 070 Using 407 Var o zTEz zTe 1 p77 mTe umT 1 p770 ltwgtltgt 1 70 gtC 1lt 710 by STEP 3 Thus7 the minimum variance is a quadratic function of 770 6 Example 46 47 48 Consider 3 risky assets whose expected returns are 771 187 772 10 and 773 8 The covariance matrix 2 is 216 70 7324 70 25 7150 7324 7150 1596 49 The inverse of the covariance matrix is 01480 705367 700204 705367 20388 00827 50 700204 00827 00043 You should verify that 704092 728673 15847 113878 51 00665 04932 12420 90136 C 90136 662109 52 669280 79112 gt 53 79112 12555 712608 01283 23039 701417 54 700431 00133 Thus the optimal portfolio vector is 712608 01283 712608 01283770 5a 23039 701417 lt gt 2303970141750 55 700431 00133 T0 7004310013350 and the minimum variance is Var0 1 50 C 1lt 71 gt 66912807182224770 1255553 56 0 Note that 070 does indeed satisfy the feasibility constraints 26 and 27 7 Minimum Variance Portfolio It is easy to nd the mean variance efficient portfolio that minimizes the portfolio variance7 otherwise called the minimum variance portfolio There are two approaches Approach 1 Minimize the Vmiance Formula Here7 one simply minimizes Var0 given in 567 which is achieved by taking its derivative and setting it equal to zero For the example in the previous section7 dVWO 7 dig 0 7182224 2511150 57 which means that the portfolio that achieves the minimum variance has an expected mean return equal to 7257 Once the value for 770 for the minimum variance portfolio is known plug its value into 55 to obtain the minimum variance portfolio f 7032971275600534 and then plug this value into 56 to obtain the minimum variance of 08078 Approach 2 Solve the Problem Directly The problem of nding the minimum variance portfolio is the same as our original problem P070 except that the expected return constraint 27 is not there Eliminating a constraint is equivalent to setting its dual variable a to zero An inspection of 41 shows that the minimum variance portfolio must be proportional to the vector E le To obtain the minimum variance portfolio compute the vector E le and then normalize this vector so that the sum of its components is one This is achieved by merely dividing each of this vector s components by the sum of all the components That is 1 tfl 724 58 minimum variance por OlO 612715 e One may further verify that the minimum variance is given by 71 minimum variance eTE le 59 In our example E le 7040811597100661T The sum of the components of this vector is 12379 which equals the reciprocal of the minimum variance Dividing 7040811597100661 by 12379 yields 7032971275600534 as in Approach 1 Remark 8 Two observations yields computational savings First the minimum variance portfolio is proportional to the vector E le which is given in the rst column of the matrix 3 Second in words eTE le merely represents the sum of all elements of the matrix 2 1 8 Computing the Tangent Market Portfolio In this section we assume the market includes a risk free security that returns rf in all states The tangent or market portfolio solves the following optimization problem V zTEz where rf denotes the risk free rate The objective function is called the Sharpe ratio TP Maxm eTx1 60 It can be shown that all mean variance e cient portfolios must be a combination of the risk free security and the tangent portfolio That is each investor should allocate a portion of their budget to the risk free security and the remaining portion to the tangent portfolio The sub allocations to the individual risky securities are determined by the weights f that de ne
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