INVESTMENTS Fourth Edition Risk and Risk Aversion Chapter 2 INVESTMENTS Fourth Edition W 1 = 150 Profit = 50 W 2 = 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 Investment INVESTMENTS Fourth Edition Asset Returns ? Asset returns over a given period are often uncertain. 1 0 11 0 011 ? + = ?+ = P PD P PPD R P0 is the price at the beginning of period P1 is the price at the end of the period--Uncertain D1 is the dividend at the end of the period—Uncertain So return on a asset is a random variable, characterized by: ? All possible outcomes ? Probability of each outcome INVESTMENTS Fourth Edition Expected rate of return on a investment 1 )( 0 1010 0 0110 0 ? + = ?+ = P PEDE P PPDE RE Expected rate of return compensates for time- value and risk premiumRiskRRE f +=)( 0 For example: %221 100 122 )( 0 0110 0 =?= ?+ = P PPDE RE INVESTMENTS Fourth Edition Variance on a investment ? Fro example 1176.0 %)22%20(4.0%)22%50(6.0)( 222 = ??×+?×=Rσ 3429.0 %)22%20(4.0%)22%50(6.0)( 22 = ??×+?×=Rσ INVESTMENTS Fourth Edition Measuring expected return and risk-- example ? Moments of return distributions 5 5 5 5 5 20 15 14.5 5 5 5 6 5 -10 -5 -5.5 R0(%) R1 R2 R3 Asset 0 Asset 1 Asset 2 Asset 3 1/31/31/3probabilityAsset Mean321state Between asset 0 and 1,which one would you choose? Between asset 1 and 2…… Between asset 2 and 3…… INVESTMENTS Fourth Edition These returns have the following moments -0.578.165R3 08.165R2 012.255R1 00.005R0(%) SkewnessSt DMean { } xofDStExxESkewness ./][( 3/1 3 ?= INVESTMENTS Fourth Edition ? Investor’s view of risk - Risk Averse - Risk Neutral - Risk Seeking ? Utility ? Utility Function U = E ( R) - .005 A σ 2 A measures the degree of risk aversion Risk Aversion & Utility INVESTMENTS Fourth Edition To measure risk, we need to know how investors react to uncertainty—Key assumption of most model ? Higher mean in return is preferred: ? ER ? Higher standard deviation in return is disliked ? Investor care only about mean and st. D ? Investor don’t care about higher moments, such as skewness ? Under the assumption 1-4. Std give a measure of risk ? In general, other moments may matter. 2 )( ERRE ?=σ INVESTMENTS Fourth Edition Risk Aversion and Value: Using the Sample Investment U = E ( R ) - .005 A σ 2 = .22 - .005 A (34%) 2 Risk Aversion A Value High 5 -6.90 34.6 Low 1 16.22 T-bill = 5% INVESTMENTS Fourth Edition Dominance Principle 1 23 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 INVESTMENTS Fourth Edition Utility and Indifference Curves ? Represent an investor’s willingness to trade- off return and risk ? Example Exp Ret St Deviation U=E ( R ) - .005Aσ 2 10 20.0 2 15 25.5 2 20 30.0 2 25 33.9 2 INVESTMENTS Fourth Edition Indifference Curves Expected Return Standard Deviation Increasing Utility INVESTMENTS Fourth Edition Expected Return Rule 1 : The return for an asset is the probability weighted average return in all scenarios. ∑ = s sRsRE )()Pr()( INVESTMENTS Fourth Edition Variance of Return Rule 2: The variance of an asset’s return is the expected value of the squared deviations from the expected return. ])()()[Pr( 2 2 ∑ ? = s REsRs σ INVESTMENTS Fourth Edition 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. R p = W 1 R 1 +W 2 R 2 W 1 = Proportion of funds in Security 1 W 2 = Proportion of funds in Security 2 R 1 =( Expected) return on Security 1 R 2 =( Expected) return on Security 2 2211 ERWERWER += INVESTMENTS Fourth Edition 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. σσ riskyassetriskyassetp w ×= INVESTMENTS Fourth Edition Rule 5: When two risky assets with variances σ 1 2 and σ 2 2 , respectively, are combined into a portfolio with portfolio weights w 1 and w 2 , respectively, the portfolio variance is given by σ p 2 = w 1 2 σ 1 2 + w 2 2 σ 2 2 + 2W 1 W 2 Cov(R 1 R 2 ) Cov(R 1 R 2 ) = Covariance of returns for Security 1 and Security 2 Portfolio Risk