Diversification Finance 100 Prof. Michael R. Roberts 1 Topic Overview How to measure risk and return» Sample risk measures for some classes of securities Brief Statistics Review» Realized and Expected Risk and Return Diversification» How to analyze the benefits from diversification» How to determine the trade-off between risk and return» Two-Asset Example Firm-Specific vs Market Risk 1 1
US Wealth Indices 196-005 3 Realized Return Characteristics 196-004 4
Empirical Distributions of Annual Returns 196-004 5 Risk-Return Tradeoff in Large Portfolios, 196-004 Returns and Risk exhibit clear positive relation: More Risk, More Return 6 3 3
Risk-Return Tradeoff for 500 Individual Stocks by Size, 196 004 7 Individual Shares and the Stock Market A Paradox? Investment Risk Premium Variability Stock market index 8-9% 0% Typical individual share 8-9% 30-40% The risk premium for individual shares is not closely related to their volatility.» Need to understand diversification» Begs the question of why one would hold an individual stock. 8 4 4
Diversification: The Basic Idea Construct portfolios of securities that offer the highest expected return for a given level of risk. The risk of a portfolio will be measured by its standard deviation (or variance, same result). Diversification plays an important role in designing efficient portfolios (I.e. portfolios whose return is maximized for a given level of risk or, equivalently, portfolios whose risk is minimized for a given level of return). 9 Fire Insurance Policies: Diversification with Two Assets Consider assets:» Your house, worth $100,000» Insurance Policy Two things can happen in the future:» Your house will burn down with probability 10% resulting in a total loss» Your house does not burn down, retaining its full value Question:» What is the risk of each of these assets held separately and together as a portfolio? 10 5 5
Insurance Policies: States and Payoffs State (Pr) Fire (0.1) No Fire (0.9) Expected Value SD Value House 0 100,000 90,000 30,000 ( ) ( ) ( ) Asset Insurance 100,000 0 10,000 30,000 E House Value = 0.1 0 + 0.9 100, 000 = 90, 000 Portfolio 100,000 100,000 100,000 ( ) ( ) ( ) SD House Value = 0.1 0 90, 000 + 0.9 100, 000 90, 000 = 30, 000 By combining the assets, we reduce risk without sacrificing return hmmm 0 1/ 11 Two Asset Case Suppose we want to form portfolios of Microsoft and IBM and assume the following Security Microsoft IBM Corr(r1,r) E( r ) 0% 1% ρ SD( r ) 40% 0% What do the risk-return pairs for portfolios of Microsoft and IBM look like for different correlations?» Just compute the E( R ) and SD( R ) for various pairs of weights (i.e., portfolios) 1 6 6
Case 1: Perfect Negative Correlation (ρ=-1) w Microsoft -0% -10% 0% 10% 0% 30% E.g., w Microsoft = 0% w IBM 10% 110% 100% 90% 80% 70% E( r ) 10% 11% 1% 13% 14% 14% SD( r ) 3% 6% 0% 14% 8% % ( ) 0.0( 0.0) 0.80( 0.1) 0.14 Er = + = ( ) ( ) ( )( )( )( )( ) SD r = + + = 0.08 ( ) 0.0 0.40 0.80 0.0 0.0 0.80 0.40 0.0 1 13 Mean Variance Frontier Perfect Negative Correlation (ρ=-1) 30% Short IBM 5% 0% MVP Microsoft E(r) 15% IBM 10% 5% Short Microsoft 0% 0% 10% 0% 30% 40% 50% 60% 70% 80% 90% 100% SD(r) 14 7 7
Minimum Variance Portfolio The portfolio with the smallest variance possible is called the global minimum-variance portfolio (MVP). For the two stock case, the global minimum variance portfolio has the following portfolio weights: σ ρ1σ 1σ x1 = x = 1 x σ + σ ρ σ σ The variance of the global MVP is: Var 1 1 1 σ σ ( ) ( 1 ρ ) = 1 1 r P σ + σ σ σ ρ 1 1 Note: we have not excluded short-selling here; x i <0 is possible! 1 1 15 Minimum Variance Portfolio Perfect Negative Correlation (ρ=-1) For our example of IBM and Microsoft, what are the portfolio weights for the MVP? What is the variance of the MVP?»or 16 8 8
Case : Perfect Positive Correlation (ρ=1) 30% 5% 0% MVP Microsoft Short IBM E(r) 15% 10% IBM 5% Short Microsoft 0% 0% 10% 0% 30% 40% 50% 60% 70% 80% 90% 100% SD(r) 17 Minimum Variance Portfolio Perfect Positive Correlation (ρ=1) What are the portfolio weights for the MVP? What is the variance of MVP? 18 9 9
Imperfect Correlation What happens in the general case where -1<ρ 1 < 1?» Diversification can reduce risk but cannot completely eliminate risk» When -1<ρ 1 < σ /σ 1 (correlation is small) there are gains from diversification MVP has positive weights in both assets and there are gains from diversification» When σ /σ 1 < ρ 1 < 1 (correlation is large) there are no gains to diversification. MVP has negative weight in one asset 19 Case 3: Weak Positive Correlation (ρ=0.5) 30% 5% 0% MVP Microsoft Short IBM E(r) 15% 10% IBM 5% Short Microsoft 0% 0% 10% 0% 30% 40% 50% 60% 70% 80% 90% 100% SD(r) 0 10 10
Minimum Variance Portfolio Weak Positive Correlation (ρ=0.5) What are the portfolio weights of the MVP? What is the variance and SD of the MVP? 1 Case 4: Strong Positive Correlation (ρ=0.75) 30% 5% 0% MVP Microsoft Short IBM E(r) 15% 10% IBM Short Microsoft 5% 0% 0% 10% 0% 30% 40% 50% 60% 70% 80% 90% 100% SD(r) 11 11
Minimum Variance Portfolio Strong Positive Correlation (ρ=0.75) What are the portfolio weights of the MVP? What is the variance and SD of the MVP? 3 Mean Variance Frontier with Multiple Assets E[r] Investors prefer Efficient Frontier Asset 1 Asset Portfolios of other Portfolios of assets Asset 1 and Asset Minimum-Variance Portfolio 0 σ 4 1 1
Efficient Portfolios with Multiple Assets II With multiple assets, the set of feasible portfolios is a hyperbola. Efficient portfolios are those on the thick part of the curve in the figure.» They offer the highest expected return for a given level of risk. Assuming investors want to maximize expected return for a given level of risk, they should hold only efficient portfolios. Common sense procedures:» Invest in stocks in different industries.» Invest in both large and small company stocks.» Diversify across asset classes. Stocks Bonds Real Estate» Diversify internationally. 5 Limits to Diversification Consider an equally-weighted portfolio. The variance of such a portfolio is: 1 j= N 1 i= N σ p = σ = j= 1 i= 1 ij N N 1 Average 1 Average + 1 N Variance N Covariance As the number of stocks gets large, the variance of the portfolio approaches: var( r p ) cov The variance of a well-diversified portfolio is equal to the average covariance between the stocks in the portfolio. 6 13 13
Limits to Diversification Example What is the expected return and standard deviation of an equally-weighted portfolio, where all stocks have E(r j ) = 15%, σ j = 30%, and ρ ij =.40? N x j =1/N E(r p ) σ p 1 1.00 15% 30.00% 10 0.10 15% 0.35% 5 0.04 15% 19.53% 50 0.0 15% 19.6% 100 0.01 15% 19.1% 1000 0.001 15% 18.99% 7 Limits to Diversification Illustration Portfolio Risk, σ Total Risk Average Covariance Firm-Specific Risk Market Risk Number of Stocks 8 14 14
Firm-Specific vs. Market Risk Firm-Specific Risk» a.k.a. Idiosyncratic Risk, Diversifiable Risk, Independent Risk, Unique Risk, Unsystematic Risk» Refers to news about an individual company or entity» What are some examples? Market Risk» a.k.a. Undiversifiable Risk, Systematic Risk» Refers to news about the market or economy» What are some examples? 9 Summary It is not possible to characterize securities in terms of risk alone» Need to understand risk Risky investments» Riskier investments have higher returns» Risk premia are not related to the risk of individual assets Diversification benefits» Depend on correlation of assets» Possibility of short sales» Cannot eliminate market risk Minimum variance portfolios» Riskless if correlation perfectly negative» Applications for hedging 30 15 15