Lecture Notes – Mean-Variance Optimization PDF

Lecture Notes – Mean-Variance Optimization Utility Rank portfolios by distribution of returns Mean-Variance rankings Other Approaches Utility Functions As Functions As Indifference Curves Rank Portfolios by Distributions of Returns Bottom line – what is the best portfoli...

Lecture Notes – Mean-Variance Optimization Utility Rank portfolios by distribution of returns Mean-Variance rankings Other Approaches Utility Functions As Functions As Indifference Curves Rank Portfolios by Distributions of Returns Bottom line – what is the best portfolio of assets for me Need to judge whether one portfolio is better than another Portfolios returns are random variables Mean-Variance Rankings Idea of diminishing marginal utility Risk aversion, Risk Tolerance Vs. risk seeking Vs. risk neutral Traditional attitudes towards random variables Likes increases in means Likes decreases in standard deviations Traditional risk-return tradeoff Other Approaches Still emphasize losses more than gains Asymmetry of distributions Normal distributions vs others Gambling vs. Insurance Downside risk vs upside risk Compared with 0 Compared with target Behavior Finance: Prospect Theory Target is where you started Or different assets for different needs: security, lottery ticket Utility Functions As Functions CFA: Utility function U() = E(R) -1/2Aσ2 A is the risk aversion coefficient. Will differ across individuals Use decimals in calculation How useful is this function? In practice, not very, since you don’t know the actual utility function Q1. Which assets would an individual with A=2 prefer? X: E(r) = 2%, σ = 5% : 0.02-.0025=.0175 1  Y: E(r) = 4%, σ = 10% : 0.04-.0100=.03 Prefer Y Q2. What if A= 8? X:.02-.01=.01 Y:.04-.04=0 Prefer X Q3. If A=2, what σ of Y would make them indifferent?.0175 =.04-x; x=.0225; σ =.15 Indifference Curves Points represent asset return distribution People prefer points to the northwest Indifference curves are assets that are equally good Expected Return Risk (Standard Deviation) Why curved? A little risk is not too bad, but each unit of additional risk makes you increasingly worse requiring increasing compensation. Increasing marginal disutility of a bad is like diminishing marginal utility of a good. Efficient Frontier Represent all possible portfolios How to calculate expected return and standard deviation Which portfolios are non-dominated? How to do in Excel Practical Issues in Mean-Variance Optimization 2 Represent all possible portfolios From the last lecture: describe portfolios by mean and variance What are our set of choices (non-dominated portfolios) Do not pick best portfolio, that’s for the next lecture. This is a technical calculation Does not depend on efficient markets or rationality Still meaningful with market inefficiency Does depend on estimation of distributions How to calculate expected returns and standard deviations Quick review of statistics Mean for single asset given distribution =1/n, arithmetic means rather than geometric means for distribution variance/standard deviation var= 1/(n-1)Σ(x-xbar)2 sample vs. population s.d. = sqrt(var) covariance/correlation cov=1/(n-1)Σ(x-xbar)(y-ybar) how they vary, magnitude corr=cov(x,y)/(sd(x)sd(y)) controls for magnitude from -1 to 1 Portfolio return: Weighted average of asset returns This is random variable Describe distribution by expected value and standard deviation 𝑅𝑅𝑝𝑝 = 𝑤𝑤1 𝑅𝑅1 + 𝑤𝑤2 𝑅𝑅2 𝑅𝑅𝑝𝑝 = ∑𝑁𝑁 𝑖𝑖=1 𝑤𝑤𝑖𝑖 𝑅𝑅𝑖𝑖 Expected (mean, average) portfolio return Weighted average of asset expected returns 𝐸𝐸 𝑅𝑅𝑝𝑝 = ∑𝑁𝑁 𝑖𝑖=1 𝑤𝑤𝑖𝑖 𝐸𝐸(𝑅𝑅𝑖𝑖 ) Variance of portfolio return Not weighted average of asset standard deviations Diversification depending on correlation of asset returns 3  𝜎𝜎𝑝𝑝2 = ∑𝑁𝑁 𝑖𝑖,𝑗𝑗=1 𝑤𝑤𝑖𝑖 𝑤𝑤𝑗𝑗 𝐶𝐶𝐶𝐶𝐶𝐶(𝑅𝑅𝑖𝑖 , 𝑅𝑅𝑗𝑗 ) Cov(Ri,Rj) = ρijσiσj Cov(Ri,Ri) = Var(Ri) ≡ σi2 Summation just adds up all possible combinations: w1σ1 w2σ2 w3σ3 w4σ4 w1σ1 ρ11 w1σ1 w1σ1 ρ12w1σ1 w2σ2 ρ13 w1σ1 w3σ3 ρ14 w1σ1 w4σ4 w2σ2 ρ21 w2σ2 w1σ1 ρ22 w2σ2 w2σ2 ρ23 w2σ2 w3σ3 ρ24 w2σ2 w4σ4 w3σ3 ρ31 w3σ3 w1σ1 ρ32 w3σ3 w2σ2 ρ33 w3σ3 w3σ3 ρ34 w3σ3 w4σ4 w4σ4 ρ41 w4σ4 w1σ1 ρ42 w4σ4 w2σ2 ρ43 w4σ4 w3σ3 ρ44 w4σ4 w4σ4 For 2 assets 𝜎𝜎𝑝𝑝2 = 𝑤𝑤12 𝜎𝜎12 + 𝑤𝑤22 𝜎𝜎22 + 2𝑤𝑤1 𝑤𝑤2 𝐶𝐶𝐶𝐶𝐶𝐶(𝑅𝑅1 , 𝑅𝑅2 ) 𝜎𝜎𝑝𝑝 = 𝜎𝜎𝑝𝑝2 Which portfolios are non-dominated? Dominated Portfolio More or equal risk Less or equal expected return Strictly worse on one or both dimensions [Graph with dots] All non-dominated portfolios are on an upward sloping line segment Expected Return Standard Deviation This is called the Efficient Frontier The best portfolio will be one of the portfolios on the frontier 4 Practical Issues in Mean-Variance Optimization Unrealistic allocations Extreme short positions -> place limits on short positions Extremely large positions -> place limits on long positions Why? Probably using atypical distribution Combine estimates with priors on distributions Too many parameters to estimate 2 assets: 2 expected returns, 2 standard deviations, 1 correlations 3 assets: 3 expected returns, 3 standard deviations, 3 correlations 4 assets: 4 expected returns, 4 standard deviations, 6 correlations N assets: N expected returns, N standard deviations, N(N-1)/2 correlations How to fix Use asset categories rather than individual securities Use return generating model Specify how returns are related to some common variable We’ll do this later Allows you to calculate some of the numbers Utility Maximization and Portfolio Choice Combine utility functions with efficient frontier Problems with this approach Combine utility function with efficient frontier Expected Return Standard Deviation 5  Tangency of the indifference curve with the efficient frontier is the optimal portfolio. The less risk averse, the flatter the indifference, the farther to the right the tangency will be Problems with this Approach Hard to find actual functions in practice What is your indifference curve? Instability If estimated distributions change (and they can) Optimal portfolio will change Need to rebalance often So put restrictions on rebalancing Asset Allocators Describe investor by range of risk aversion Describe investment choices by selected points on efficient frontier Stocks Bonds High Risk Aversion 80% 20% Moderate Risk Aversion 60% 40% Low Risk Aversion 30% 70% Three ways to be more complicated More asset classes Wider range of risk aversion categories More complicated ways of determining risk aversion 6