Portfolio Optimization- Chapter 7
Measuring Portfolio Risks For individual securities, one thinks in terms of the standard deviation of the returns to that security. When we combine two or more securities into a portfolio (diversify), the risk of the portfolio is related to that of the individual securities depending upon their correlation coefficient. For two securities, say gold and Infosys, the risk of a portfolio of the two will depend upon: a) the proportion of funds invested in each (portfolio weights, W) b) the individual risks of each (s). b) the correlation between the two (rho). As we talked about earlier, the search for low correlations has what has kept money sloshing around the globe. Correlations range from -1 (perfect negative) to +1 (perfect correlation). Discuss bonds and stocks, stocks and commodities, inter- and intra-industry connections.
Naïve diversification The power of diversification Just randomly picking stocks gets rid of 60% of the risk of the typical individual security by naive diversification Most of the diversifiable risk eliminated at 25 or so stocks 6-3
E(r) 13% r = -1 r = 0 r = .3 8% r = +1 St. Dev A=12% B=20% St. Dev TWO-SECURITY PORTFOLIOS WITH DIFFERENT CORRELATIONS WA = 0% WB = 100% r = 0 r = -1 r = +1 r = .3 50%A 50%B Complicated graph, but hopefully it will help. Imagine two securities 1. Expected return is 8% and SD is 12% 2. Expected return is 13% and SD is 20% Depending on the amount of correlation in the returns when we combine them we will alter the portfolio standard deviation. If there is perfect correlation the combination of the two securities has no diversification effects. However if the assets are perfectly negatively correlated we can combine the two securities to completely eliminate variance in the combined portfolio. Generally assets will be somewhere in between where the combination can eliminate some risk but not completely remove it. WA = 100% WB = 0% 6-4
Minimum Variance Combinations -1< r < +1 Choosing weights to minimize the portfolio variance 1 2 - Cov(r1r2) W1 = + - 2Cov(r1r2) W2 = (1 - W1) s 2 One question of interest is: With a given level of correlation how can we find the optimal weights of the securities so that we can minimize the variance of the portfolio. It turns out that we can solve for those weights using these equations. Recall that Covariance(r1,r2) = 1,212 6-5
Extending to All Securities Consider all possible combinations of securities, with all possible different weightings and keep track of combinations that provide more return for less risk or the least risk for a given level of return and graph the result. The set of portfolios that provide the optimal trade-offs are described as the efficient frontier. The efficient frontier portfolios are dominant or the best diversified possible combinations. All investors should want a portfolio on the efficient frontier. Dominant means they provide the best return for the given risk level. … Until we add the riskless asset 6-6
The minimum-variance frontier of risky assets Efficient Frontier is the best diversified set of investments with the highest returns Efficient frontier Found by forming portfolios of securities with the lowest covariances at a given E(r) level. Individual assets Global minimum variance portfolio Minimum variance frontier Individual assets combining them into portfolios, considering different weights. So looking at many risky assets using the same techniques it is possible to build a minimum variance frontier. We are only concerned with the upper portion of the curve. Any minimum variance point on the bottom of the curve can be dominated by the similar point on the upper portion of the curve. St. Dev. 6-7
The EF and asset allocation E(r) EF including international & alternative investments Efficient frontier 100% Stocks 80% Stocks 20% Bonds 60% Stocks 40% Bonds 40% Stocks 60% Bonds Ex-Post 2000-2002 20% Stocks 80% Bonds Alternative investments: REITs, mortgage backed, gold, other precious metals, other commodities and then the international investments. 100% Stocks St. Dev. 6-8
Efficient frontier for international diversification 6-9
ALTERNATIVE CALS s E(rP) CAL (P) P E(rA) A CAL (A) E(r) P&F Efficient Frontier P&F E(rP&F) CAL (Global minimum variance) G P There will only be one optimal Capital Allocation Line. This line will be the line that passes through the risk free rate at 0 SD and is tangent to the efficient frontier. It will dominate all other Capital Allocation Lines. F Risk Free s A 6-10
The Capital Market Line or CML CAL (P) = CML E(r) Efficient Frontier E(rP&F) The optimal CAL is called the Capital Market Line or CML The CML dominates the EF P E(rP) E(rP&F) There will only be one optimal Capital Allocation Line. This line will be the line that passes through the risk free rate at 0 SD and is tangent to the efficient frontier. It will dominate all other Capital Allocation Lines. F Risk Free s P&F P P&F 6-11 11
The Capital Market Line or CML E(r) Efficient Frontier E(rP&F) P Both investors choose the same well diversified risky portfolio P and the risk free asset F, but they choose different proportions of each. E(rP) A=4 E(rP&F) There will only be one optimal Capital Allocation Line. This line will be the line that passes through the risk free rate at 0 SD and is tangent to the efficient frontier. It will dominate all other Capital Allocation Lines. F Risk Free s P&F P P&F 6-12 12
Risk Premium Format Let: Ri = (ri - rf) Risk premium format Rm = (rm - rf) The Model: Rewritten by substituting in these risk premium variables this formula can be written as given. Ri = ai + ßi(Rm) + ei 6-13
Estimating the Index Model Scatter Plot Excess Returns (i) Security Characteristic Line . . . . . . . . . . . . . . . . . . . . . . . . . . . Excess returns on market index . . . . . . . . Each point would represent a sample pair of returns observed for a particular holding period. A regression analysis will find the “best fit” line to fit the data. The expected return for the security when the market has zero excess return is the point where the line crosses the vertical axes. Beta is the slope of the regression line. Higher beta means higher systematic risk. Beta above 1 is riskier than the market. . . . . . . . . . . . . . . . Ri = a i + ßiRm + ei Slope of SCL = beta y-intercept = alpha 6-14
Estimating the Index Model Scatter Plot Excess Returns (i) Security Characteristic Line Ri = a i + ßiRm + ei . . . . . . . . . . . . . . . . . . . . . . . . . . . Excess returns on market index . . . . . . . . Each point would represent a sample pair of returns observed for a particular holding period. A regression analysis will find the “best fit” line to fit the data. The expected return for the security when the market has zero excess return is the point where the line crosses the vertical axes. Beta is the slope of the regression line. Higher beta means higher systematic risk. Beta above 1 is riskier than the market. . . . . . Variation in Ri explained by the line is the stock’s _____________ Variation in Ri unrelated to the market (the line) is ________________ . . . . . . . . . . systematic risk unsystematic risk 6-15 15
Advantages of the Single Index Model Reduces the number of inputs needed to account for diversification benefits If you want to know the risk of a 25 stock portfolio you would have to calculate 25 variances and (25x24) = 600 covariance terms With the index model you need only 25 betas Rather than calculating all pairwise covariances, you can calculate covariances of securities versus the index which is a lot easier. For 100 securities you would have (100 x 99 =) 990 covariances to calculate. Against the index you have only 100 covariances to calculate. This type of Beta model is extremely popular. We will be talking about the single factor CAPM model in the next chapter. Easy reference point for understanding stock risk. βM = 1, so if βi > 1 what do we know? If βi < 1? 6-16