CSCI 1951-G Optimization Methods in Finance Part 07: Portfolio Optimization

Transcription

1 CSCI 1951-G Optimization Methods in Finance Part 07: Portfolio Optimization March 9 16, / 19

2 The portfolio optimization problem How to best allocate our money to n risky assets S 1,..., S n with random returns? µ i : expected return of asset i in a time interval; Stocks Bonds Money Market µ i Σ: variance-covariance n n matrix of returns, with: σ ii : variance of the return of asset i; σ ij : covariance of the returns of assets i and j. Covariance Stocks Bonds MM Stocks Bonds MM / 19

3 The portfolio optimization problem Portfolio x = (x 1,..., x n ), where x i : proportion of money invested in asset i. Expected return: E[x] = µ 1 x µ n x n = µ T x Variance: Var[x] = i,j σ ijx i x j = x T Σx Var[x] 0, so Σ...is positive semidefinite (we assume positive definite) Feasible portfolios: set X = {x : Ax = b, Cx d} One constraint is n x i = 1 i=1 3 / 19

4 The portfolio optimization problem Efficient portfolio w.r.t. R > 0: the portfolio with minimum variance among all those with expected return at least R. (variants possible, e.g., ) Markowitz mean-variance optimization: find the efficient portfolio: min x T Σx s.t. µ T x R Ax = b Cx d This optimization problem is... convex. We assumed Σ 0, so the optimal solution is... unique. 4 / 19

5 The portfolio optimization problem Stocks Bonds MM µ i Covariance Stocks Bonds MM Stocks Bonds MM min x 2 S x S x B x S x M x 2 B x B x M x 2 M s.t x S x B x M R x S + x B + x M = 1 x S 0, x B 0, x M 0 5 / 19

6 The portfolio optimization problem Rate of Return R Variance Stocks Bonds MM Table 8.1: Efficient Portfolios Expected Return (%) Percent invested in different asset classes Stocks Bonds 7 10 MM Standard Deviation (%) Expected return of efficient portfolios (%) Figure 8.1: Efficient Frontier and the Composition of Efficient Portfolios 6 / 19

7 The efficient frontier R min, R max : minimum and maximum expected returns for efficient portfolios. σ(r) : [R min, R max ] R, σ(r) = ( x T RΣx R ) 1/2 where x R is the efficient portfolio w.r.t. R [R min, R max ]. The efficient frontier is the graph E = {(R, σ(r)) : R [R min, R max ]} 7 / 19

8 Maximizing the Sharpe ratio Consider a riskless asset with deterministic return r f R min (why does it make sense?) Consider convex combinations between a risky portfolio x with the riskless asset x θ = [(1 θ)x θ] T As θ varies, (for fixed x) the combinations form a line on the stdev/mean plot: Mean CAL r f V Figure 8.4: Capital Allocation Line For different choices of x, the slope of the line changes. 8 / 19

9 Maximizing the Sharpe ratio Which Capital Allocation Line (CAL) is the best? Mean CAL r f V Figure 8.4: Capital Allocation Line The CAL with the largest slope: the corresponding portfolio will have the lowest stdev for any given value of R r f. 9 / 19

10 Maximizing the Sharpe ratio To which portfolio x does the optimal CAL corresponds to? The feasible x that maximizes the slope: h(x) = µt x r f (x T Σx) 1/2 The quantity h(x) is known as the Sharpe ratio or the reward-to-volatility ratio. 10 / 19

11 Maximizing the Sharpe ratio To find the optimal risky portfolio we solve max µt x r f (x T Σx) 1/2 s.t. Ax = b Cx d The feasible region is polyhedral, but the objective function may be non-concave. Let s build an equivalent convex quadratic program. 11 / 19

12 Maximizing the Sharpe ratio X = {x : Ax = b, Cx d} (includes full alloc. const., assumes ˆx X s.t. µ Tˆx > r f ) X + = {(x, k) : x R n, k R ++, x k X } {(0, 0} The optimal risky portfolio is x = y /k where (y, k ) is the optimal solution of: min y T Σy s.t. (y, k) X + (µ r f 1) T y = 1 This is a quadratic convex program (why?) 12 / 19

13 Returns-based style analysis You are a portfolio manager (!) and would like to understand how your portfolio manager friend Sally, the style of her portfolio i.e., the mix of stocks in it; Sally is secretive on the mix, but publish the returns of her portfolio over time; You also have access to the returns of index funds tracking different sectors of the market; Definition (Return-Based Style Analysis (RBSA)) A technique using constrained optimization to determine the style of a portfolio using the return time series of the portfolio and of a number of other asset classes (factors). 13 / 19

14 RBSA Mathematical Model Fundamentally a linear model for regression. Data R t, t = 1,..., T : the return of Sally s portfolio over T fixed time intervals (e.g., R t is the monthly return, and T = 12 months); F it, i = 1,..., n, t = 1,..., T : the returns of factor i over T fixed time intervals (same intervals as R t ); Model R t = w 1t F 1t + w 2t F 2t + + w nt F nt + ε t = F T t w t + ε t w it : sensitivity of R t to factor i; ε t : non-factor return. 14 / 19

15 Interpretation R t = w 1t F 1t + w 2t F 2t + + w nt F nt + ε t = F T t w t + ε t Assume the F it are returns of passive investments (e.g., index funds); Then: F T t w t is the return of a benchmark portfolio of passive investments; ε t is the difference between the passive benchmark and the active strategy followed by Sally. If the passive investments considered together are representative of the market, then ε t measures the additional (or negative) return due to Sally s ability as a portfolio manager. 15 / 19

16 Optimization problem R t = w 1t F 1t + w 2t F 2t + + w nt F nt + ε t = F T t w t + ε t Additional assumptions/constraints: w it = w i, i.e., the weights do not change over time. w i > 0, n i=1 w t = 1. Constraints of our optimization problem: min??? n s.t. w i = 1 i=1 w i 0, i = 1,..., n What about the objective function? Any idea? 16 / 19

17 Optimization problem (cont.) If ε t measures Sally s ability as a portfolio manager, we can assume that it is approximately constant over time. I.e., we want the plots of the returns of Sally s portfolios and of the benchmark portfolios to be curves with approximately constant distance. I.e., we want ε t to have the smallest possible variance over time. Formulation min Var(e t) = Var(R t F T w R n t w) n s.t. w i = 1 i=1 w i 0, i = 1,..., n The objective function is convex. 17 / 19

18 Objective function Let We have R = R 1. R T, and F = F T 1. F T T, and e = 1. 1 ( Var(R t Ft T w) = 1 T T (R t Ft T w) 2 t=1 (R t Ft T w T T i=1 = 1 ( e T ) 2 T R F (R F w) w 2 T = R 2 2R T F w + w T F T F w T (et R) 2 2e T R w T F T ee T F w T 2 ) 2 18 / 19

19 Objective function Var(R t Ft T w) = R 2 2R T F w + w T F T F w T (et R) 2 2e T R w T F T ee T F w T 2 Reorganizing the terms as function of w: ( R Var(R t Ft T 2 w) = et R) 2 ) ( R T ) F T T 2 2 et R T T 2 et F w ( 1 + w T T F T F 1 ) T F T ee T F w We have 1 T F T F 1 T F T ee T F = 1 T F T ) (I eet F T The matrix M = I ee T /T is symmetric and positive semidefinite (eigenvalues: 0 and 1), and so is F T MF. Hence the objective function is convex. 19 / 19