Mean-Variance Optimization Corporate Finance and Incentives Lars - PowerPoint PPT Presentation

Mean-Variance Optimization Corporate Finance and Incentives Lars Jul Overby Department of Economics University of Copenhagen September 2010 Lars Jul Overby (D of Economics - UoC) Mean-Variance Optimization 09/10 1 / 21

Practical stuff Slides from exercise classes External presenters? Lars Jul Overby (D of Economics - UoC) 09/10 2 / 21

Risk and Return The key objective when choosing to invest in financial assets is to maximize the expected return of the investment for a given level of risk to minimize the amount of risk of the investment for a given expected return In order to achieve these objectives we need to compute the expected return of an investment quantify the risk of the investment Lars Jul Overby (D of Economics - UoC) 09/10 3 / 21

Harry Markowitz The basis of modern portfolio theory is Harry Markowitz’ mean-variance optimization theory 1 The theory assumes that individuals minimize the return variance of an investment for any given level of expected return i.e. risk is quantified as the return variance (dispersion of return outcomes) of the investment To work with this theory, we first need to look at some mathematics of portfolios 1 Markowitz, Harry, 1952; ”Portfolio Selection”, Journal of Finance 7, 77-99. Lars Jul Overby (D of Economics - UoC) 09/10 4 / 21

Asset i Return on asset i : R i = P i ,1 + D i ,1 − P i ,0 ∼ � r i P i ,0 S Expected return of asset i : E ( � r i ) = ∑ q s ∗ � r i , s ∼ r i s = 1 where q s is the probability of scenario s and � r i , s is the return on asset i is scenario s occurs. Expected value of a constant times a return: E ( x � r i ) = xE ( � r i ) Return variance of asset i : � ( R i , s − E ( R i )) 2 � � r i − r i ) 2 � S = σ 2 var ( � r i ) = = E ( � ∑ q s ∗ i s = 1 r i ) = x 2 var ( � Variance of constant times a return: var ( x � r i ) Standard deviation: σ ( x � r i ) = x σ ( � r i ) Lars Jul Overby (D of Economics - UoC) 09/10 5 / 21

Expected return and variance of asset i State � r i q s A 0.01 0.25 0.04 0.50 B C 0.08 0.25 Lars Jul Overby (D of Economics - UoC) 09/10 6 / 21

Portfolio returns Portfolio weights: x j = value of holding of asset j total value of portfolio N Portfolio returns: � R p = x 1 � r 1 + x 2 � r 2 + ... + x N � r N = ∑ x i � r i i = 1 Expected value of the sum or difference of two returns: E ( � r i + � r j ) = E ( � r i ) + E ( � r j ) and E ( � r j ) = E ( � r i ) − E ( � r j ) r i − � Expected return of 2 asset portfolio: � � � = E ( x i � r i + x j � r j ) = x i E ( � r i ) + x j E ( � r j ) E R p � � N � Expected return of N asset portfolio: E = R p ∑ x i r i i = 1 Lars Jul Overby (D of Economics - UoC) 09/10 7 / 21

Portfolio returns State q s � r 1 � r 2 � r 3 0.25 0.01 0.10 0.01 A B 0.50 0.04 0.03 0.06 C 0.25 0.08 0.07 0.09 E ( � r i ) 0.0425 0.0575 0.055 var ( � r i ) 0.000619 0.000869 0.000825 std . dev . 0.0249 0.0295 0.0287 x i 0.30 0.45 0.25 Two ways to compute expected portfolio return Compute portfolio returns in each state and take expectation Compute weighted average of expected asset returns Lars Jul Overby (D of Economics - UoC) 09/10 8 / 21

Covariances and correlations Covariance of two returns: σ ij = E [( � r i − r i ) ( � r j − r j )] = cov ( � r i , � r j ) r j ) = cov ( � r i , � r j ) Correlation between two returns: ρ ( � = ρ ij r i , � σ i σ j If ρ ij = 1, return i and j are perfectly correlated. If ρ ij = − 1, return i and j are perfectly negatively correlated. Lars Jul Overby (D of Economics - UoC) 09/10 9 / 21

Variance of a 2-asset portfolio � r j − ( x i r i + x j r j )] 2 � var ( x i � r i + x j � r j ) = E [ x i � r i + x j � � r j − r j )] 2 � = [ x i ( � r i − r i ) + x j ( � E � r i − r i ) 2 + x 2 r j − r j ) 2 + 2 x i x j ( � x 2 = i ( � j ( � r i − r i ) ( � r j − r j ) E � r i − r i ) 2 � � r j − r j ) 2 � x 2 + x 2 = i E ( � j E ( � + 2 x i x j E [( � r i − r i ) ( � r j − r j )] x 2 r i ) + x 2 = i var ( � j var ( � r j ) + 2 x i x j cov ( � r j ) r i , � x 2 i σ 2 i + x 2 j σ 2 = j + 2 x i x j σ ij x 2 i σ 2 i + x 2 j σ 2 = j + 2 x i x j ρ ij σ i σ j Given positive portfolio weights on two assets, the lower the correlation between returns, the lower the variance of the portfolio. Lars Jul Overby (D of Economics - UoC) 09/10 10 / 21

Variance-covariance-correlation matrix 2 Asset 1 2 3 1 0.000619 − 0.00019 0.000688 2 − 0.264 0.000869 − 0.00044 3 0.962 − 0.517 0.000825 2 Variance on diagonal, covariances above diagonal, correlations below diagonal. Lars Jul Overby (D of Economics - UoC) 09/10 11 / 21

Variance of an N-asset portfolio � � N N N ∑ ∑ ∑ = var x i � r i x i x j σ ij j = 1 i = 1 j = 1 N N ∑ ∑ = x i x j ρ ij σ i σ j i = 1 j = 1 Lars Jul Overby (D of Economics - UoC) 09/10 12 / 21

Markowitz’ mean-variance optimization We now have the tools to compute portfolio returns and variances. Markowitz said, that the only thing investors care about is the relation between these two. Investors wish to achieve the highest possible expected return for a given level of return variance. Lars Jul Overby (D of Economics - UoC) 09/10 13 / 21

Two asset portfolio � � � = x i E ( � r i ) + x j E ( � r j ) E R p σ 2 x 2 i σ 2 i + x 2 j σ 2 = j + 2 x i x j ρ ij σ i σ j p Assume we have a risk-free asset and a risky asset. � � � = x f r f + x j E ( � E R p r j ) σ 2 x 2 f ∗ 0 + x 2 j σ 2 j + 2 x f x j ∗ 0 ∗ 0 ∗ σ j = x 2 j σ 2 = p j = x j σ j if x j ≥ 0 σ p = − x j σ j if x j < 0 σ p � � r f + E ( � r j ) − r f � E R p = σ p if x j ≥ 0 σ j � � r f − E ( � r j ) − r f � = E R p σ p if x j < 0 σ j Lars Jul Overby (D of Economics - UoC) 09/10 14 / 21

Two risky assets Two perfectly negatively correlated assets: ρ = − 1 � � � = x 1 E ( � r 1 ) + x 2 E ( � r 2 ) E R p σ 2 x 2 1 ∗ σ 2 1 + x 2 2 σ 2 = 2 + 2 x 1 x 2 ∗ ( − 1 ) ∗ σ 1 ∗ σ 2 p ( x 1 σ 1 − x 2 σ 2 ) 2 = = x 1 σ 1 − x 2 σ 2 if x 1 σ 1 − x 2 σ 2 ≥ 0 σ p = − x 1 σ 1 + x 2 σ 2 if x 1 σ 1 − x 2 σ 2 < 0 σ p Lars Jul Overby (D of Economics - UoC) 09/10 15 / 21

Two risky assets if x 1 σ 1 − x 2 σ 2 ≥ 0 = ( 1 − x 2 ) σ 1 − x 2 σ 2 = σ 1 − x 2 ( σ 2 + σ 1 ) ⇔ σ p − σ p σ 1 = + x 2 σ 2 + σ 1 σ 2 + σ 1 � � � � σ p σ 1 � = 1 + E ( � r 1 ) E R p − σ 2 + σ 1 σ 2 + σ 1 � − σ p � σ 1 + + E ( � r 2 ) σ 2 + σ 1 σ 2 + σ 1 − E ( � r 2 ) − E ( � r 1 ) σ 1 = E ( � r 1 ) + ( E ( � r 2 ) − E ( � r 1 )) σ p σ 2 + σ 1 σ 2 + σ 1 Lars Jul Overby (D of Economics - UoC) 09/10 16 / 21

Two risky assets if x 1 σ 1 − x 2 σ 2 < 0 = ( x 2 − 1 ) σ 1 + x 2 σ 2 = − σ 1 + x 2 ( σ 2 + σ 1 ) ⇔ σ p σ p σ 1 = + x 2 σ 2 + σ 1 σ 2 + σ 1 � � � � σ p σ 1 � = 1 − E ( � r 1 ) E R p − σ 2 + σ 1 σ 2 + σ 1 � � σ p σ 1 + + E ( � r 2 ) σ 2 + σ 1 σ 2 + σ 1 + E ( � r 2 ) − E ( � r 1 ) σ 1 = E ( � r 1 ) + ( E ( � r 2 ) − E ( � r 1 )) σ p σ 2 + σ 1 σ 2 + σ 1 Lars Jul Overby (D of Economics - UoC) 09/10 17 / 21

Two risky assets Lars Jul Overby (D of Economics - UoC) 09/10 18 / 21

Four risky assets Lars Jul Overby (D of Economics - UoC) 09/10 19 / 21

Minimum variance portfolio (without a risk-free asset) The portfolio of a group of stocks that minimizes return variance is the portfolio with a return that has an equal covariance with every stock return 3 . 3 See proof in section 4.8. Lars Jul Overby (D of Economics - UoC) 09/10 20 / 21

Finding the Minimum Variance Portfolio with N stocks 1) Construct N equations consisting of the covariance of the N stocks’ return with the MV-Portfolio (containing N unknown weights) � � � � N r 1 , � ∑ = = k cov � R p cov � r 1 , w i � r i i = 1 � � N ∑ r 2 , = cov � w i � r i k i = 1 ... ... � � N ∑ r N , = cov � w i � r i k i = 1 w i 2) Rescale the weights so that they sum to 1 x i = ∑ w i See examples in section 4.9 Lars Jul Overby (D of Economics - UoC) 09/10 21 / 21