MS-E2114 Investment Science Lecture 5: Mean-variance portfolio theory A. Salo, T. Seeve Systems Analysis Laboratory Department of System Analysis and Mathematics Aalto University, School of Science
Overview Random returns Portfolio mean and variance Markowitz model Two-fund theorem One-fund theorem 2/38
This lecture In earlier lectures, we have considered practically certain cash flows Emphasis on mathematical analysis of fixed income securities Credit risks have not been thoroughly addressed Yet cash flows for most investments are uncertain Examples include stock prices, dividends, real property,... The period for which capital is tied can be uncertain, too We cover the Nobel prize winning framework of Harry Markowitz for portfolio choice under uncertainty Markowitz H. (1952). Portfolio selection, Journal of Finance 7, 77-91. Link to Markowitz on the Nobel Prize website 3/38
Overview Random returns Portfolio mean and variance Markowitz model Two-fund theorem One-fund theorem 4/38
Random returns Assume that you invest a fixed amount X 0 e now and receive the random amount X 1 e one year later (Total) return R = X 1 X 0 (Rate of) return r = X 1 X 0 X 0 Thus R = 1 + r and X 1 = (1 + r)x 0 X 1 is random r is random, too If X1 < X 0, then r will be negative 5/38
Short selling Shorting (short selling) = Selling an asset that one does not own To short an asset, one can borrow the asset from someone who owns it (=has a long position) (e.g., brokerage firm) and sell it for, say, X 0 e At end of borrowing period, one has to buy the asset from the market for X 1 e to return it (plus possible dividends the stock may have paid during the period) to the original owner If the asset value declines to X1 < X 0, the deal gives a profit X 0 X 1 > 0 If the asset value increases to X 1 > X 0, the cash flow X 0 X 1 will be negative and the deal leads to a loss of X 1 X 0 Because prices can increase arbitrarily, losses can become very large Shorting can be very risky and is therefore prohibited by some institutions 6/38
Short selling Total return for a short position: Receive X 0 and pay X 1 R = X 1 X 0 = X 1 X 0 = 1 + r This is same as for the long position Initial position X 0 < 0 of asset profit rx 0 Allows one to bet on declining asset values (r < 0) Short 100 stock and sell them for 1 000 e. If price declines by 10% and you buy the stock back for 900 e, and you obtain a profit of 100 e 900 ( 1 000) r = = 0.1 1 000 rx 0 = 0.1 ( 1 000) = 100 7/38
Portfolio return Portfolio of n assets X0i = investment in the i-th asset (negative when shorting) X0 = n X 0i = total investment Weight of the i-th asset i is wi = X 0i X 0 n w i = 1 X1i = cash flow from investment in by the end of the period Return of i-th asset R i = 1 + r i Portfolio return n R = X n 1i = R n ix 0i = R iw i X 0 = w i R i X 0 X 0 X 0 1 + r = r = w i (1 + r i ) = 1 + w i r i w i r i 8/38
Random variables Expected value E[x] is the mean (average) outcome of a random variable For a finite number of possible realizations xi with probabilities p i, i = 1, 2,..., n, E[x] = p i x i = x Variance Var[x] is the expected value of the squared deviation from the mean x σ 2 = Var[x] = E[(x x) 2 ] = E[x 2 2x x + x 2 ] = E[x 2 ] 2 E[x] x + x 2 σ 2 = Var[x] = E[x 2 ] E[x] 2 9/38
Random variables Standard deviation is the expected deviation from the mean σ = Std[x] = Var[x] Covariance Cov[x 1, x 2 ] is the expected product of deviations from the respective means of two random variables x 1, x 2 σ 12 = Cov[x 1, x 2 ] = E[(x 1 x 1 )(x 2 x 2 )] = E[x 1 x 2 x 1 x 2 x 1 x 2 + x 1 x 2 ] = E[x 1 x 2 ] E[x 1 ] x 2 x 1 E[x 2 ] + x 1 x 2 σ 12 = Cov[x 1, x 2 ] = E[x 1 x 2 ] E[x 1 ] E[x 2 ] Covariance and variance closely related σ 2 1 = Var[x 1] = Cov[x 1, x 1 ] = σ 11 10/38
Random variables (Pearson) correlation Corr[x 1, x 2 ] measures the strength of the linear relationship of two random variables ρ 12 = Corr[x 1, x 2 ] = σ 12 σ 1 σ 2 Cov[x 1, x 2 ] Var[x1 ] Var[x 2 ] No correlation ρ12 = 0 σ 12 = 0 Positive correlation ρ12 > 0 Negative correlation ρ12 < 0 Perfect correlation ρ 12 = ±1 We have ρ 12 1 σ 12 σ 1 σ 2 11/38
Random variables Variance of a linear combination of two random variables σa 2 1 x 1 +a 2 x 2 = Var[a 1 x 1 + a 2 x 2 ] = a1 2 Var[x 1] + a2 2 Var[x 2] + 2a 1 a 2 Cov[x 1, x 2 ] More generally, the variance of a linear combination of random variables x 1, x 2,..., x n is [ σ 2 n a i x i = Var = a i x i ] a i a j Cov[x i, x j ] = j=1 j=1 a i a j σ ij 12/38
Overview Random returns Portfolio mean and variance Markowitz model Two-fund theorem One-fund theorem 13/38
Random returns Consider n assets with random returns r i, i = 1, 2,..., n, such that E[r i ] = r i Expected return of the portfolio r = w i r i E[r] = w i E[r i ] = Portfolio variance [ ] σ 2 = Var w i r i σ 2 = j=1 w i w j Cov[r i, r j ] = w i r i j=1 w i w j σ ij 14/38
Diversification Investing in several assets leads to lower variance Variability in the realized returns of different assets tend to average out Consider investing an equal amount in n assets Expected return of each asset is m and variance σ 2 Uncorrelated returns (σ ij = 0, i j) 1 r = w i r i = n m = m Var[r] = w i w j σ ij = w 2 1 i σ ii = n 2 σ2 = 1 n σ2 j=1 1 lim Var[r] = lim n n n σ2 = 0 No variation, yet the expected return is the same! 15/38
Diversification Uncorrelated assets are ideal for diversification If the returns are correlated, say, σ ij = 0.3σ 2, i j, we have Var[r] = j=1 w i w j σ ij = 1 n 2 σ ii + = 1 n σ2 + n(n 1) 1 n 2 0.3σ2 = lim Var[r] = lim (0.7 1n ) + 0.3 σ 2 = 0.3σ 2 n n Hence, variance cannot be reduced to zero j=1 j i 1 n 2 σ ij (0.7 1n + 0.3 ) σ 2 16/38
Mean-standard deviation diagram Variance (or standard deviation) of returns can be used as a measure of risk If two portfolios have equal expected return, then the one with smaller variance is preferred Consider 3 assets Asset i 1 2 3 E[r i ] 10% 12% 14% σ i 8% 10% 12% 1 0.2 0.2 ρ = 0.2 1 0.3 0.2 0.3 1 ρ 12 = σ 12 σ 1 σ 2 σ 11 σ 12 σ 13 0.64% 0.16% 0.19% Σ = σ 21 σ 22 σ 23 = 0.16% 1% 0.36% σ 31 σ 32 σ 33 0.19% 0.36% 1.44% 17/38
Mean-standard deviation diagram What are the possible combinations of returns and variances? Pairs (σ, E[r]) such that { E[r] = n w i E[r i ] σ 2 = n n j=1 w iw j σ ij Portfolio A with w 1 = w 2 = w 3 = 1/3 has Expected return 3 1 r A = 3 E[r i] = 0.12 Standard deviation σ A = j=1 1 3 2 σ ij = 7.07% 18/38
Expected return E r Mean-standard deviation diagram Asset 3 Portfolio A Asset 2 Asset 1 Standard deviation σ Yellow area = Set of all possible (σ, E[r]) that can be obtained from portfolios such that w i 0, n w i = 1 Blue area = As above but with shorting allowed (w i s, i = 1, 2, 3, can be negative as well) 19/38
Expected return E r Efficient frontier Efficient points Standard deviation σ Green curve = Minimum variance set (minimum variance attainable for a given return) Green point = Minimum variance point (minimum variance attainable using assets 1,2 and 3) Efficient frontier = Curve above (and including) the point 20/38
Overview Random returns Portfolio mean and variance Markowitz model Two-fund theorem One-fund theorem 21/38
Markowitz model Portfolios of the efficient frontier r can be found by solving min w 1 2 s.t. j=1 w i w j σ ij w i r i = r w i = 1 22/38
Markowitz model Set up the Lagrangian ( L = 1 ) ( ) w i w j σ ij λ w i r i r µ w i 1 2 j=1 Equations of the efficient set are solved by setting the partial derivatives of L to zero L = w j σ ij λ r i µ = 0, i = 1, 2,..., n w i λ L = j=1 w i r i r = 0 µ L = w i 1 = 0 23/38
Example: Solving minimum variance portfolio Asset i 1 2 3 1 0 0 E[r i ] 1 2 3 ρ = Σ = 0 1 0 σ i 1 1 1 0 0 1 ( L = 1 ) ( ) w i w j σ ij λ w i r i r µ w i 1 2 j=1 L = w 1 σ1 2 λ r 1 µ = w 1 λ µ = 0 w 1 L = w 2 σ2 2 λ r 2 µ = w 2 2λ µ = 0 w 2 L = w 3 σ3 2 λ r 3 µ = w 3 3λ µ = 0 w 3 λ L = w i r i r = w 1 + 2w 2 + 3w 3 r = 0 µ L = w i 1 = w 1 + w 2 + w 3 = 0 (1a) (1b) (1c) (1d) (1e) 24/38
Example: Solving minimum variance portfolio Equations (1a)-(1c) yield w 1 = λ + µ, w 2 = 2λ + µ, w 3 = 3λ + µ Substituting these into (1d) and (1e) yields { { 14λ + 6µ = r 6λ + 3µ = 1 λ = 1 2 r 1 µ = 2 3 r, and thus w 1 = 4 3 1 2 r, w 2 = 1 3, w 3 = 1 2 r 2 3 25/38
Example: Solving minimum variance portfolio Substituting the optimal weights w 1, w 2, w 3 into the objective of minimizing the portfolio variance yields min w σ2 = min w 3 w 2 i = 1 2 r 2 2 r + 7 3 (2) The expected return r for which variance is minimized can be found by differentiating (2) with respect to r r 2 = 0 r = 2 σ 2 min = 1 3, σ min = 1 3 0.577 26/38
Example: Solving minimum variance portfolio r Asset 3 Expected return ҧ (σ min, r) ҧ Asset 2 Asset 1 Standard deviation σ Blue area = The set of all possible pairs (σ, E[r]) that a portfolio can obtain for some w 1, w 2, w 3 0, n w i = 1 27/38
Overview Random returns Portfolio mean and variance Markowitz model Two-fund theorem One-fund theorem 28/38
Two-fund theorem Theorem (Two-fund theorem) There are two efficient funds (portfolios) such that any efficient portfolio can be duplicated, in terms of mean and variance, as a combination of these two. In other words, all investors seeking efficient portfolios need invest in combinations of these two funds only. Proof : Let w 1 and w 2 be efficient portfolios with corresponding Lagrange multipliers be λ 1, µ 1 and λ 2, µ 2. Form the portfolio w α = αw 1 + (1 α)w 2, α R. Weights in w α sum to 1 The return of w α is r = αr 1 + (1 α)r 2 If r 1 r 2, then any r can be obtained by choosing a suitable α (this α may be negative) 29/38
Two-fund theorem Is w α efficient? Check optimality w i L = λ L = w j σ ij λ r i µ = 0, j=1 w i r i r = 0 µ L = w i 1 = 0 i = 1, 2,..., n (w i, λ i, µ i ), i = 1, 2 satisfies these But so does the point α(w 1, λ 1, µ 1 ) + (1 α)(w 2, λ 2, µ 2 ) Can be verified by substitution Hence w α is optimal, which completes the proof. 30/38
Overview Random returns Portfolio mean and variance Markowitz model Two-fund theorem One-fund theorem 31/38
Risk-free asset Thus, two funds suffice if: 1. Only expected return and variance matter to the investor 2. Everyone has the same estimates about expectations and variances 3. There is a single investment period What if one asset is risk-free? Return r f and variance σ 2 f = 0 Assume that unlimited lending and borrowing are possible at the risk-free rate r f 32/38
Risk-free asset Let us invest the share 1 α in a portfolio of risky assets A with expected return r A and variance σa 2, and the share α to the risk-free asset The expected return is r α = αr f + (1 α) r A Standard deviation is σ α = (1 α) 2 σa 2 = (1 α)σ A 33/38
One-fund theorem Theorem (One-fund theorem) There is a single fund F of risky assets such that any efficient portfolio can be constructed as a combination of the fund F and the risk-free asset. Proof. Unlimited lending and borrowing at rate r f (σ α, r α ) forms a line (σ α, r α ) should be selected so that the line is as steep as possible n max tan θ = w i( r i r f ) w n n j=1 w iw j σ ij 34/38
One-fund theorem Expected return ҧ r F θ r f Standard deviation σ Green point = The portfolio that maximizes the slope k of line r = r f + kσ from (σ α, r α ) through the feasible set 35/38
One-fund theorem How to determine this portfolio F? 0 = tan θ w k r k r 0 = f 1 n w i( r i r f ) n j=1 w iw j σ 2 ( n ) 3 2 n ij j=1 w iw j σ ij n r k r f = n w i( r i r f ) n n j=1 w iw j σ ij w i σ ik = λ(w) Change of variable v i = λ(w)w i r k r f = Solve for v i and retrieve the weights v k n v i = λ(w)w k λ(w) n w i = w k n w i w i σ ik v i σ ik w i σ ik = w k, k = 1, 2,..., n 36/38
Example: One-fund theorem Asset i 1 2 3 Risk-free E[r i ] 1 2 3 1/2 σ i 1 1 1 0 1 0 0 Σ = 0 1 0 0 0 1 Optimality conditions v 1 = 1 1/2 = 1/2 r k r f = v i σ ik v 2 = 2 1/2 = 3/2 v 3 = 3 1/2 = 5/2 Normalization of weights w k = v 3 w 1 = (1/2)/(9/2) = 1/9 k n v, v i = 9/2 w 2 = (3/2)/(9/2) = 3/9 i w 3 = (5/2)/(9/2) = 5/9 37/38
Overview Random returns Portfolio mean and variance Markowitz model Two-fund theorem One-fund theorem 38/38