In this chapter, we begin by introducing two themes in portfolio theory that are centered on risk. The first is the tenet that investors will avoid risk unless they can anticipate a reward for engaging in risky investments. The second theme allows us to quantify investors’ personal trade-offs between portfolio risk and expected return. | Chapter 6 Risk and Risk Aversion W = 100 W1 = 150 Profit = 50 W2 = 80 Profit = -20 p = .6 1-p = .4 E(W) = pW1 + (1-p)W2 = 6 (150) + .4(80) = 122 s2 = p[W1 - E(W)]2 + (1-p) [W2 - E(W)]2 = .6 (150-122)2 + .4(80=122)2 = 1,176,000 s = Risk - Uncertain Outcomes W1 = 150 Profit = 50 W2 = 80 Profit = -20 p = .6 1-p = .4 100 Risky Inv. Risk Free T-bills Profit = 5 Risk Premium = 17 Risky Investments with Risk-Free Investor’s view of risk Risk Averse Risk Neutral Risk Seeking Utility Utility Function U = E ( r ) - .005 A s 2 A measures the degree of risk aversion Risk Aversion & Utility Risk Aversion and Value: U = E ( r ) - .005 A s 2 = .22 - .005 A (34%) 2 Risk Aversion A Value High 5 3 Low 1 T-bill = 5% Dominance Principle 1 2 3 4 Expected Return Variance or Standard Deviation • 2 dominates 1; has a higher return • 2 dominates 3; has a lower risk • 4 dominates 3; has a higher return Utility and Indifference Curves Represent an investor’s willingness to trade-off return and risk. Example Exp Ret St Deviation U=E ( r ) - .005As2 10 2 15 2 20 2 25 2 Indifference Curves Expected Return Standard Deviation Increasing Utility Expected Return Rule 1 : The return for an asset is the probability weighted average return in all scenarios. Variance of Return Rule 2: The variance of an asset’s return is the expected value of the squared deviations from the expected return. Return on a Portfolio Rule 3: The rate of return on a portfolio is a weighted average of the rates of return of each asset comprising the portfolio, with the portfolio proportions as weights. rp = W1r1 + W2r2 W1 = Proportion of funds in Security 1 W2 = Proportion of funds in Security 2 r1 = Expected return on Security 1 r2 = Expected return on Security 2 Portfolio Risk with Risk-Free Asset Rule 4: When a risky asset is combined with a risk-free asset, the portfolio standard deviation equals the risky asset’s standard deviation multiplied by the portfolio proportion invested in the risky asset. Rule 5: When two risky assets with variances s12 and s22, respectively, are combined into a portfolio with portfolio weights w1 and w2, respectively, the portfolio variance is given by: p2 = w12 12 + w22 22 + 2W1W2 Cov(r1r2) Cov(r1r2) = Covariance of returns for Security 1 and Security 2 Portfolio Risk | Chapter 6 Risk and Risk Aversion W = 100 W1 = 150 Profit = 50 W2 = 80 Profit = -20 p = .6 1-p = .4 E(W) = pW1 + (1-p)W2 = 6 (150) + .4(80) = 122 s2 = p[W1 - E(W)]2 + (1-p) [W2 - E(W)]2 = .6 (150-122)2 + .4(80=122)2 = 1,176,000 s = Risk - Uncertain Outcomes W1 = 150 Profit = 50 W2 = 80 Profit = -20 p = .6 1-p = .4 100 Risky Inv. Risk Free T-bills Profit = 5 Risk Premium = 17 Risky Investments with Risk-Free Investor’s view of risk Risk Averse Risk Neutral Risk Seeking Utility Utility Function U = E ( r ) - .005 A s 2 A measures the degree of risk aversion Risk Aversion & Utility Risk Aversion and Value: U = E ( r ) - .005 A s 2 = .22 - .005 A (34%) 2 Risk Aversion A Value High 5 3 Low 1 T-bill = 5% Dominance Principle 1 2 3 4 Expected Return Variance or Standard Deviation • 2 dominates 1; has a higher return • 2 dominates 3; has a lower risk • 4 dominates 3; has a higher return Utility and Indifference Curves Represent an investor’s willingness to trade-off .
