Optimal Investment Strategies: A Constrained Optimization Approach

Transcription

1 Optimal Investment Strategies: A Constrained Optimization Approach Janet L Waldrop Mississippi State University jlc3@ramsstateedu Faculty Advisor: Michael Pearson Pearson@mathmsstateedu

2 Contents Introduction 2 Lagrange Multipliers 3 Efficient Frontier 3 4 Example 6 5 Optimal Investment Strategy 7 6 Example 8 7 The Capital Asset Pricing Model 9 8 Bibliography 9 Introduction The ultimate goal of modern investment theory is to construct an optimal portfolio of investments, where a portfolio is simply a collection of risky investments The Capital Asset Pricing Model (CAPM), developed in 964 by William F Sharpe, seeks to find the relationship between the way assets are priced and their risk Sharpe s work was an extension of the work done by Harry Markowitz, who developed a method of efficient portfolio selection To understand both the CAPM and Markowitz s portfolio selection technique, we first need to develop the method of Lagrange Multipliers 2 Lagrange Multipliers Consider the problem of maximizing or minimizing a function subject to given constraints One way to solve this optimization problem is the method of Lagrange Multipliers Suppose we have a function g, andwe want to maximize (or minimize) this function subject to the constraint F ( x )= c where F :R n R m for F =(f,f 2,, f m ) We assume that there exists an open subset U of R n such that f i is differentiable on U for i =, 2,, m and that all the points that satisfy the constraint function lie in U Also, we assume that g is differentiable on U Recall that the Jacobian of F, denoted JF( x ), isgivenby JF( x )= f ( x ) f m ( x ) = f x f x n f m x fm x n Note that JF is a m x n matrix, so JF( x ) is a linear map from R n R m If the rank of JF = m n on F ( c ), we say that F has full rank on F ( c ) In other words, { f ( x ),, f m ( x )} is a set of linearly independent vectors for any x in F ( c ) By the Implicit Function Theorem, there exists ϕ :R n m R m so that if (x,, x n ) is near a local extremum p =(p,, p n ) then (x,, x n )=(x,, x n m, ϕ(x,, x n m )) F ( c ) Moreover, p = (p,, p n m, ϕ(p,, p n m )) Furthermore,ifg :R n R is constrained by F ( x )= c, then we are really just maximizing G(x,, x n m )=g(x,, x n m, ϕ(x,, x n m )) Nowif p =(p,, p n m,p n m+,, p n )= (p,, p n m, ϕ(p,, p n m )) is a maximum of G, theng(p,, p n m )=g(p,, p n m, ϕ(p,, p n m )) and

3 we need G x i (p,, p n m )=0for i =, 2,, n m Since G = g + x k x k ϕ n m+ (x,, x n m ) ϕ n (x,, x n m ) i=n m+ = Hence, p mustbeasolutiontothesystem Note that g x i,k=,, n m x i x k x n m+ x n, x i x k = ϕ i x k,i= n m +,, n G = g + x k x k i=n m+ In matrix form we can write this system as g ϕ i =0,k=,, n m x i x k g x, g x 2,, g x n x x ϕ n x ϕ n x n m ϕ n m+ ϕ n m+2 ϕ n m+ x n m ϕ n m+2 x n m =(0, 0,, 0) If we let A represent the above matrix, we have g A =(0,, 0) xm Furthermore, our constraint function, F (x,, x n m, ϕ(x,, x n m )) = c gives f i (x,,x n m, ϕ(x,, x n m )) = c i for i =, 2,, n m, which implies that f i f i ϕ + i =0for i =, 2,, n m x i x i x i i=n m+ So, f i A =(0,, 0) xm as well Now, A is a linear map from R n m R n,sotherankofa n m Furthermore, the upper portion of A forms the (n m) x (n m) identity matrix, and A has at least n m linearly independent rows Thus, the rank of A must be at least (n m) and is therefore equal to (n m) But, g( p ), f ( p ),, f m ( p ) are all solutions to the homogeneous system A T y y n = As we have already seen, the rank A =(n m), so the dimension of the kernel of A is n (n m) =m Furthermore, since we have m + solutions, the set must be linearly dependent But we assumed that 0 0 m 2

4 the set { f ( x ),, f m ( x )} is linearly independent, so g is in the span of { f ( x ),, f m ( x )} In other words, there exist m scalars λ, λ 2,, λ m such that g( p )=λ f ( p )+ + λ m f m ( p ) We call the scalars λ, λ 2,, λ m Lagrange multipliers So this gives us n equations, and the constraint F ( x )= c gives us m equations for a total of n + m (possibly nonlinear) equations with n + m unknowns, namely {x,, x n, λ,, λ m } Solving this system of equations is referred to as the method of Lagrange Multipliers The solutions to these equations supply the possible solutions to the constrained optimization problem 3 Efficient Frontier We will now use the method of Lagrange Multipliers to construct what is referred to as the efficient frontier of investment portfolios That is, the portfolios that have the least amount of risk for a given level of return Note that a portfolio is simply a collection of risky investments The underlying assumptions of the model are homogeneity of investor expectations, that is all investors agree on expected values, standard deviations, and covariances, and all investors are able to borrow and lend on equal terms at some risk free rate We know that the risk of a particular security can be measured by the variance or standard deviation of that security from its expected return Similarly, the risk of an investment portfolio can be measured by the standard deviation or variance of that portfolio from its expected return So our problem becomes that of minimizing portfolio risk for a given level of return Consider a portfolio consisting of n risky investments, and let w i represent the proportion of the total investment placed in a particular investment i Also, let E(r i ) denote the expected return of i and σ 2 p be the variance of the portfolio The return of a portfolio is given by P w i E( r i ), anditcanbeshownthatthe P variance is given by n np w i w j cov(i, j), where cov(i, j) is the covariance between returns from investments i and j i=j= Thus, our problem is to minimize subject to the constraints w i i= σ 2 p = where is the desired return of the portfolio i= j= =and w i w j cov(i, j) w i E(r i )=, i= In this case, we can let f (w,, w n )=w + + w n and f 2 (w,,w n )=E(r )w + + E(r n )w n Then f f =,, f f2 =(,, ) and f 2 =,, f 2 =(E(r ),, E(r n )), w w n w w n and using the method of Lagrange Multipliers, we seek λ and λ 2 such that σ 2 p = λ f + λ 2 f 2 order to determine σ 2 p,consider σ 2 p = w i w j cov(i, j) =2w i cov(i, i)+2 w j cov(i, j) =2 w j cov(i, j) w i w i Thus, i= j= j=,j6=i j= In σ 2 p = 2w cov(, ) + +2w n cov(,n) 2w cov(2, ) + +2w n cov(2,n) 2w cov(n, ) + +2w n cov(n, n) =2 cov(, ) cov(, n) cov(, n) cov(n, n) w w n 3

5 since cov(i, j) =cov(j, i) By the Method of Lagrange Multipliers, we seek to solve E(r ) E(r ) σ 2 p = λ + λ 2 = λ λ 2 E(r n ) E(r n ) If we absorb the 2 into our constants λ and λ 2,weobtain cov(, ) cov(, n) w = cov(, n) cov(n, n) w n E(r ) E(r n ) Hence, our system of equations consists of the two constraint functions λ λ 2 () w i = E(r i )w i =, i= i= and the following equations constructed using Lagrange Multipliers: w cov(,i) = λ + λ 2 E(r ) i= w n cov(n, i) = λ + λ 2 E(r n ) i= This system can be written in matrix form as follows: 0 0 w E(r ) E(r 2 ) E(r n ) 0 0 cov(, ) cov(, 2) cov(,n) E(r ) w n = 0 (2) λ cov(,n) cov(2,n) cov(n, n) E(r 2 ) λ 2 0 Now let the symmetric matrix of covariances be denoted by V,letR =,andlet E(r ) E(r n ) w and λ represent the n x weight vector and 2 x Lagrange multiplier vector respectively Then our system in (2) can be represented by the block matrices w λ R 02x2 V R T = (3) 0 nx and our system under the method of Lagrange Multipliers in () can be represented as Recall that σ 2 p := n P i=j= V w = R T λ or w = V R T λ (4) np w i w j cov(i, j), which can be represented in matrix form as σ 2 p = w T V w Also, note that in order for V to exist, the rows of V must be linearly independent, which is equivalent to the assumption that no single investment in our portfolio can be replaced with a linear combination of 4

6 other investments We can make this assumption, since if our portfolio contained an investment that was dependent we could replace it with the appropriate linear combination of investments without changing the characteristics of our portfolio Therefore, we assume that the rows of V are linearly independent, and thus V exists Now, substituting (4) into (3), weobtain à R 02x2 V R T! λ V R T = λ 0 nx = à RV R T λ VV R T λ R T λ = RV R T λ 0 nx! = = 0 nx = RV R T λ = 0 nx Letting H = RV R T, H λ = and λ = H (5) Note that R is 2 x n, V is n x n, andr T is n x 2, soh = RV R T is 2 x 2 symmetric, so H = RV R T = RV R T T = H T and H is symmetric Using (4), weget Furthermore, V is σ 2 p = w T VV R T λ = w T R T λ, and (5) gives σ 2 p = w T R T H = (R w ) T H = H, (6) since R w = is one of the constraints We have already noted that H is a 2 x 2 symmetric matrix, so define H as a b H = for some a, b, c R b c Then we can compute H,namely H = ac b 2 c b b a 5

7 and equation (6) becomes σ 2 p = H c b = ac b 2 b + a = a2 2b + c ac b 2 α 2 + β + γ Hence, our variance is a quadratic function of the expected return If we plot the return on the x-axis and the risk, or variance, on the y-axis, the right half of the parabola determined by this function defines our efficient frontier We only need to consider the right half of the parabola, as any point on the left half gives a lower return for the same amount of risk 4 Example Using the daily stock prices of several stocks, we can actually construct the efficient frontier for portfolios consisting of these stocks In order to calculate the needed statistics, the daily stock prices were recorded from July 3, 962 to December 3, 998 for the following stocks: S&P 500, Coca-Cola, Disney, IBM, International Paper, and Philip Morris (See Appendix) From this data, the return of each individual stock, r t, was calculated for each day so that the average daily return given by NP r t t= r =, where N is the number of observations, N could be calculated Using this data, the covariances between each pair of stocks was then calculated by NP (r A,t r A )(r B,t r B ) t= Cov(r A,r B )= N Using the construction in the previous section, we find for this particular example that the coefficients for the polynomial defining our efficient frontier are α = β = γ =

8 From this data, we can generate the following plot of the efficient frontier: x Optimal Investment Strategy In order to determine the optimal investment for a particular investor, we must first introduce the idea of utility functions A utility function for a particular investor measures the effect of the investment or portfolio on the investor s level of wealth For the CAPM, we assume that investors utility functions depend only on an investment s risk and expected return, that is U = f(e(r), σ 2 ) As investors prefer a higher expected U E(r) return versus a lower return, we can also assume that > 0 Furthermore, we assume that investors are risk averse, that is given an expected return, investors will choose the lowest possible risk Mathematically, risk averse simply means that U σ < 0 The set of level curves to a particular utility function are called 2 the indifference curves for that investor Based on our above assumptions, we know that these indifference curves will be upward sloping and that each indifference curve represents a given level of expected utility Aswemovedownand totherightonthesetofindifference curves, we reach a more desirable position Next, we introduce the idea of a risk free asset denoted r f,thatisanassetwithσ 2 r f =0and expected return equal to that of the pure interest rate, say rf Suppose we want to invest a portion of our wealth α in the risk free asset and the rest in a portfolio A Then the expected return of this investment is given by and E(r) =αe(r rf )+( α)e(r A )=α rf +( α)e(r A ) σ 2 = α 2 σ 2 r f +( α) 2 σ 2 A +2E(r A )α( α)σ A σ rf = ( α) 2 σ 2 A = σ =( α)σ A, a linear relationship Thus, the variance of any investment combining some portfolio and a risk free asset must lie along the line passing through the points in the return-risk (or return-variance) plane representing theriskfreeassetandthatportfolio 7

9 We have already shown that the optimal investments lie along the efficient frontier, so the line that we are interested in is the line tangent to the efficient frontier and passing through the risk free asset This line is referred to as the Capital Market Line Using basic calculus techniques, we can easily construct this tangent line The equation of our efficient frontier is a quadratic, namely σ 2 p = a 2 + b + c, so the slope of the tangent line at any point is 2a + b We also know that our tangent line must intersect the efficient frontier at some point of the form (bx, abx 2 + bbx + c) and that it must pass through the point (r f, 0) Now, we have the simple problem of finding the equation of a line with slope 2abx + b and passing through the points (bx, abx 2 + bbx + c) and (r f, 0) where bx is our only unknown So we have abx 2 + bbx + c 0 = 2abx + b(bx r f ) abx 2 2ar f bx r f b c = 0 q ar f ± a 2 rf 2 + abr f + ac bx = a and the equation of our line is one of the equations ³ ³ q y = 2 ar f ± a 2 rf 2 + abr f + ac + b (x r f ) Ofcourse,wewouldchoosethelinethatistangenttotherightsideoftheparabolaandignorethetangent line on the left The optimal investment is then the investment along this line that is tangent to a particular investor s indifference curves 6 Example If we go back to our previous example, where we constructed the efficient frontier, we can also construct the line on which our optimal investment must lie As stated previously, we know that this line must be tangent to the efficient frontier and must pass through the point on the x axis representing the risk free rate For our example, we will assume a risk free rate of 5% per year As our stock returns are given in terms of daily returns, we must first convert this yearly rate to a daily rate: ( + x) 252 = 05 = x = 09% Here is a plot of the line passing through the point (0009,0) and tangent to the efficient frontier together 8

10 with some typical indifference curves x The Capital Asset Pricing Model William Sharpe, in his 964 paper, extended the ideas presented thus far to argue that, under appropriate assumptions, the price of a particular asset would adjust in equilibrium so that the asset would plot along the capital market line Thus, in equilibrium, the price and risk of an asset are related linearly Recall that an underlying assumption of our model is the homogeneity of investor expectations, and therefore, the model does not capture all aspects of market behavior However, modern investment theory, in particular the study of equilibrium pricing of assets under conditions of uncertainty, is based largely on the early work of Markowitz and Sharpe 8 Bibliography References [D] Dineen, Sean, Multivariate Calculus and Geometry, Springer, Great Britain 998 [H] Haugen, Robert A, Modern Investment Theory, Prentice-Hall, Upper Saddle River 200 [M] Markowitz, Harry, 952, Portfolio Selection, Journal of Finance, December [Sh] Sharpe, William F, 964, Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk, Journal of Finance [St] Stephens, Alan A, 998, Markowitz and the Spreadsheet, Journal of Finance Education, Fall 9