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1 Risk, Returns, and Risk Aversion Return and Risk Measures Real versus Nominal Rates EAR versus APR Holding Period Returns Excess Return and Risk Premium Variance Sharpe Ratio Risk Aversion and Capital Allocations Risk Aversion and Utility Function Capital allocation line Optimal Allocations Risk, Return and Risk Aversion

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2 Road map in this and the next lecture Risk and return Optimal allocation given risk and return tradeoff Two-asset allocation Efficient frontier Multiple asset allocation Capital asset pricing models (CAPM) Arbitrage pricing theory (APT) Fama-French three-factor model Risk, Return and Risk Aversion

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3 Nominal and Real Rates Nominal rate Real rate Risk, Return and Risk Aversion

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4 Example 5.2 Annualized Rates of Return Risk, Return and Risk Aversion

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5 Formula for EARs and APRs See page 128-129 Risk, Return and Risk Aversion

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6 HPR = Holding Period Return P 0 = Beginning price P 1 = Ending price D 1 = Dividend during period one Rates of Return: Single Period Risk, Return and Risk Aversion

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7 Ending Price =48 Beginning Price = 40 Dividend = 2 HPR = (48 - 40 + 2 )/ (40) = Rates of Return: Single Period Example Risk, Return and Risk Aversion

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8 Excess Return Risk free rate Excess return Also known as risk premium Risk, Return and Risk Aversion

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9 StateProb. of Stater in State.1 -.05.10 2.2.05 3.4.15 4.2.25 5.1.35 E(r) = Scenario or Subjective Returns: Example Risk, Return and Risk Aversion

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10 Standard deviation = [variance] 1/2 Var= Using Our Example : Variance or Dispersion of Returns Risk, Return and Risk Aversion

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11 Mean and Variance of Historical Returns Arithmetic average or rates of return Risk, Return and Risk Aversion

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12 Geometric Average Returns TV = Terminal Value of the Investment g= geometric average rate of return Risk, Return and Risk Aversion

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13 Sharpe Ratio Sharpe Ratio for Portfolios= Measure of risk-return tradeoff Other concepts: Skewness and Kurtosis – page 142-143 Check out the statistics from page 147-151 Risk, Return and Risk Aversion

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14 Page 187 Risk, Return and Risk Aversion

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15 Figure 5.4 The Normal Distribution Risk, Return and Risk Aversion

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16 Figure 5.4 The Normal Distribution Risk, Return and Risk Aversion

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17 Normality and Risk Measures What if excess returns are not normally distributed? Standard deviation is no longer a complete measure of risk Sharpe ratio is not a complete measure of portfolio performance Need to consider skew and kurtosis Risk, Return and Risk Aversion

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18 Skew and Kurtosis Skew Equation 5.19 Kurtosis Equation 5.20 Risk, Return and Risk Aversion

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19 Figure 5.5A Normal and Skewed Distributions Risk, Return and Risk Aversion

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20 Figure 5.5B Normal and Fat-Tailed Distributions (mean =.1, SD =.2) Risk, Return and Risk Aversion

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21 Value at Risk (VaR) A measure of loss most frequently associated with extreme negative returns VaR is the quantile of a distribution below which lies q % of the possible values of that distribution The 5% VaR, commonly estimated in practice, is the return at the 5 th percentile when returns are sorted from high to low. Risk, Return and Risk Aversion

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22 Expected Shortfall (ES) Also called conditional tail expectation (CTE) More conservative measure of downside risk than VaR VaR takes the highest return from the worst cases ES takes an average return of the worst cases Risk, Return and Risk Aversion

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23 Lower Partial Standard Deviation (LPSD) and the Sortino Ratio Issues: Need to consider negative deviations separately Need to consider deviations of returns from the risk-free rate. LPSD: similar to usual standard deviation, but uses only negative deviations from r f Sortino Ratio replaces Sharpe Ratio Risk, Return and Risk Aversion

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24 Historic Returns on Risky Portfolios Returns appear normally distributed Returns are lower over the most recent half of the period (1986-2009) SD for small stocks became smaller; SD for long- term bonds got bigger Risk, Return and Risk Aversion

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25 Historic Returns on Risky Portfolios Better diversified portfolios have higher Sharpe Ratios Negative skew Risk, Return and Risk Aversion

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26 Figure 5.10 Annually Compounded, 25-Year HPRs Risk, Return and Risk Aversion

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27 Risk Aversion Risk Love Risk Neutral Utility function Risk, Return and Risk Aversion

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28 Utility Function Utility Function: U = E ( r ) – 1/2 A σ 2 Where U = utility E ( r ) = expected return on the asset or portfolio A = coefficient of risk aversion σ 2 = variance of returns Risk, Return and Risk Aversion

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29 Computing Utility Scores If A=2, then See page 168 Risk, Return and Risk Aversion

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30 Figure 6.2 The Indifference Curve Page 166, Table 6.3 Risk, Return and Risk Aversion

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31 It’s possible to split investment funds between safe and risky assets. Risk free asset: proxy; T-bills Risky asset: stock (or a portfolio) Allocating Capital: Risky & Risk Free Assets Risk, Return and Risk Aversion

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32 Example Using Chapter 6.4 Numbers The total market value of an initial portfolio is $300,000, of which $90,000 is invested in the Ready Asset money market fund, a risk-free asset. The remaining $210,000 is invested in risky securities – $113,400 in equity and $96,600 in long-term bonds. Find the distribution of this portfolio. Risk, Return and Risk Aversion

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33 E(r c ) = yE(r p ) + (1 - y)r f r c = complete or combined portfolio For example, y =.75 E(r c ) = Expected Returns for Combinations Risk, Return and Risk Aversion

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34 c = If y =.75, then c = If y = 1 c = If y = 0 Combinations Without Leverage Risk, Return and Risk Aversion

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35 Borrow at the Risk-Free Rate and invest in stock. Using 50% Leverage, r c = (-.5) (.07) + (1.5) (.15) =.19 c = (1.5) (.22) =.33 Capital Allocation Line with Leverage Risk, Return and Risk Aversion

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36 Risk, Return and Risk Aversion

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37 Table 6.5 Utility Levels Risk, Return and Risk Aversion

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38 Optimal Portfolio Maximize the mean-variance utility function U=E(R)-1/2Aσ 2 Based on the expressions for expected return and σ, we have the expression for optimal allocation y*: Example 6.4 (page 175): Risk, Return and Risk Aversion

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39 Figure 6.6 Utility as a Function of Allocation to the Risky Asset, y Risk, Return and Risk Aversion

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40 Figure 6.7 Indifference Curves for U =.05 and U =.09 with A = 2 and A = 4 Risk, Return and Risk Aversion

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41 Optimal Complete Portfolio on Indifference Curves Risk, Return and Risk Aversion

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