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

CSCI 1951-G – Optimization Methods in Finance Part 07: Portfolio Optimization March 9–16, 2018 1 / 19

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 0.1073 0.0737 0.0627 • 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 0.02778 0.00387 0.00021 Bonds 0.00387 0.01112 -0.00020 MM 0.00021 -0.00020 0.00115 2 / 19

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 1 + · · · + µ n x n = µ T x Variance: Var[ x ] = � i,j σ ij x i x j = x T Σ x 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

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 Σ x 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

The portfolio optimization problem Covariance Stocks Bonds MM Stocks Bonds MM Stocks 0.02778 0.00387 0.00021 0.1073 0.0737 0.0627 Bonds 0.00387 0.01112 -0.00020 µ i MM 0.00021 -0.00020 0.00115 min 0 . 02778 x 2 S + 2 · 0 . 00387 x S x B + 2 · 0 . 00021 x S x M + 0 . 01112 x 2 B − 2 · 0 . 00020 x B x M + 0 . 00115 x 2 M s.t. 0 . 1073 x S + 0 . 0737 x B + 0 . 0627 x M ≥ R x S + x B + x M = 1 x S ≥ 0 , x B ≥ 0 , x M ≥ 0 5 / 19

The portfolio optimization problem Rate of Return R Variance Stocks Bonds MM 0.065 0.0010 0.03 0.10 0.87 0.070 0.0014 0.13 0.12 0.75 0.075 0.0026 0.24 0.14 0.62 0.080 0.0044 0.35 0.16 0.49 0.085 0.0070 0.45 0.18 0.37 0.090 0.0102 0.56 0.20 0.24 0.095 0.0142 0.67 0.22 0.11 0.100 0.0189 0.78 0.22 0 0.105 0.0246 0.93 0.07 0 Table 8.1: E ffi cient Portfolios 10.5 100 90 10 Percent invested in different asset classes 80 9.5 70 Expected Return (%) 9 60 8.5 50 40 8 30 7.5 Stocks 20 Bonds 7 MM 10 6.5 0 2 4 6 8 10 12 14 16 6.5 7 7.5 8 8.5 9 9.5 10 10.5 Standard Deviation (%) Expected return of efficient portfolios (%) Figure 8.1: E ffi cient Frontier and the Composition of E ffi cient Portfolios 6 / 19

The efficient frontier R min , R max : minimum and maximum expected returns for efficient portfolios. � 1 / 2 x T � σ ( R ) : [ R min , R max ] → R , σ ( R ) = R Σ x R 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

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 θ ] T x θ = [(1 − θ ) x As θ varies, (for fixed x ) the combinations form a line on the stdev/mean plot: Mean CAL r f Variance Figure 8.4: Capital Allocation Line For different choices of x , the slope of the line changes. 8 / 19

Maximizing the Sharpe ratio Which Capital Allocation Line (CAL) is the best ? Mean CAL r f Variance 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

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

Maximizing the Sharpe ratio To find the optimal risky portfolio we solve max µ T x − r f ( x T Σ x ) 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

Maximizing the Sharpe ratio X = { x : Ax = b, Cx ≥ d } x ∈ X s.t. µ T ˆ (includes full alloc. const., assumes ∃ ˆ x > 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 Σ y s.t. ( y, k ) ∈ X + ( µ − r f � 1) T y = 1 This is a quadratic convex program (why?) 12 / 19

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

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 1 t F 1 t + w 2 t F 2 t + · · · + 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

Interpretation R t = w 1 t F 1 t + w 2 t F 2 t + · · · + 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

Optimization problem R t = w 1 t F 1 t + w 2 t F 2 t + · · · + 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

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 w ∈ R n Var( e t ) = Var( R t − F T min t w ) n � s.t. w i = 1 i =1 w i ≥ 0 , i = 1 , . . . , n The objective function is convex . 17 / 19

Objective function Let    F T    R 1 1 1 . . . . . . R =  , and F =  , and e =       . . .     F T R T 1 T We have � 2 T �� T t =1 ( R t − F T t w ) = 1 t w t w ) 2 − � Var( R t − F T ( R t − F T T T i =1 � 2 � e T ( R − Fw ) = 1 T � R − Fw � 2 − T = � R � 2 − 2 R T Fw + w T F T Fw T − ( e T R ) 2 − 2 e T R − w T F T ee T Fw T 2 18 / 19

Objective function t w ) = � R � 2 − 2 R T Fw + w T F T Fw Var( R t − F T T − ( e T R ) 2 − 2 e T R − w T F T ee T Fw T 2 Reorganizing the terms as function of w : � � R � 2 − e T R ) 2 � R T F − e T R � � Var( R t − F T T 2 e T F t w ) = − 2 w T T 2 T � 1 � T F T F − 1 + w T T F T ee T F w We have I − ee T T F T F − 1 1 T F T ee T F = 1 � � T F T 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