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5.5Asset Allocation Across Risky and Risk Free Portfolios 5-1

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Allocating Capital Between Risky & Risk-Free Assets Possible to split investment funds between safe and risky assets Risk free asset rf : proxy; ________________________ Risky asset or portfolio r p : _______________________ Example. Your total wealth is $10,000. You put $2,500 in risk free T-Bills and $7,500 in a stock portfolio invested as follows: –Stock A you put ______ –Stock B you put ______ –Stock C you put ______ $2,500 $3,000 $2,000 T-bills or money market fund risky portfolio $7,500 5-2

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Weights in rp –W A = –W B = –W C = The complete portfolio includes the riskless investment and r p. $2,500 / $7,500 =33.33% $3,000 / $7,500 =40.00% $2,000 / $7,500 =26.67% 100.00% W rf = ; W rp = In the complete portfolio W A = 0.75 x 33.33% = 25%; W B = 0.75 x 40.00% = 30% W C = 0.75 x 26.67% = 20%; 25% 75% W rf = 25% Allocating Capital Between Risky & Risk-Free Assets 5-3

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r f = 5% rf = 0% E(r p ) = 14% rp = 22% y = % in r p (1-y) = % in rf Example 5-4

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E(r C ) = Expected Returns for Combinations E(r C ) = For example, let y = ____ E(r C ) = E(r C ) =.1175 or 11.75% C = y rp + (1-y) rf C = (0.75 x 0.22) + (0.25 x 0) = 0.165 or 16.5% c = yE(r p ) + (1 - y)r f y rp + (1-y) rf Return for complete or combined portfolio 0.75 (.75 x.14) + (.25 x.05) 5-5

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Complete portfolio Varying y results in E[r C ] and C that are ______ ___________ of E[rp] and rf and rp and rf respectively. E(r c ) = yE(r p ) + (1 - y)rf c = y rp + (1-y) rf linear combinations This is NOT generally the case for the of combinations of two or more risky assets. 5-6

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E(r) E(r p ) = 14% r f = 5% 22% 0 P F Possible Combinations E(r p ) = 11.75% 16.5% y =.75 y = 1 y = 0 5-7 CAL(CapitalAllocationLine)

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Combinations Without Leverage Since σ rf = 0 σ c = y σ p If y =.75, then σ c = If y = 1 σ c = If y = 0 σ c = 75(.22) = 16.5% 1(.22) = 22% 0(.22) = 0% E(r c ) = yE(r p ) + (1 - y)rf y =.75 E(r c ) = y = 1 E(r c ) = y = 0 E(r c ) = (.75)(.14) + (.25)(.05) = 11.75% (1)(.14) + (0)(.05) = 14.00% (0)(.14) + (1)(.05) = 5.00% 5-8

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Using Leverage with Capital Allocation Line Borrow at the Risk-Free Rate and invest in stock Using 50% Leverage E(r c ) = c = (1.5) (.14) + (-.5) (.05) = 0.185 = 18.5% (1.5) (.22) = 0.33 or 33% E(r C ) =18.5% 33% y = 1.5 y = 0 5-9

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Risk Premium & Risk Aversion The risk free rate is the rate of return that can be earned with certainty. The risk premium is the difference between the expected return of a risky asset and the risk-free rate. Excess Return or Risk Premium asset = Risk aversion is an investor’s reluctance to accept risk. How is the aversion to accept risk overcome? By offering investors a higher risk premium. E[r asset ] – rf 5-10

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Risk Aversion and Allocation Greater levels of risk aversion lead investors to choose larger proportions of the risk free rate Lower levels of risk aversion lead investors to choose larger proportions of the portfolio of risky assets Willingness to accept high levels of risk for high levels of returns would result in leveraged combinations y = 0 y = 1.5 5-11

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E(r)E(r) E(r p ) = 14% r f = 5% = 22% = 22% 0 P FF rp ) Slope = 9/22 ) Slope = 9/22 E(r p ) - r f = 9% CAL(CapitalAllocationLine) P or combinations of P & Rf offer a return per unit of risk of 9/22. 5-12

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Quantifying Risk Aversion E(r p ) = Expected return on portfolio p r f = the risk free rate 0.5 = Scale factor A x p 2 = Proportional risk premium The larger A is, the larger will be the _________________________________________ investor’s added return required to bear risk 5-13

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Quantifying Risk Aversion Rearranging the equation and solving for A Many studies have concluded that investors’ average risk aversion is between _______ 2 and 4 5-14

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Using A What is the maximum A that an investor could have and still choose to invest in the risky portfolio P? Maximum A = 3.719 5-15

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Sharpe Ratio Risk aversion implies that investors will accept a lower reward (portfolio expected return) in exchange for a sufficient reduction in risk (std dev of portfolio return) A statistic commonly used to rank portfolios in terms of the risk-return trade-off is the Sharpe measure (also reward-to-volatility measure) The higher the Sharpe ratio the better Also the slope of the CAL

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Sharpe ratio Example: You manage an equity fund with an expected return of 16% and an expected std dev of 14%. The rate on treasury bills is 6%.

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5.6 Passive Strategies and the Capital Market Line 5-18

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A Passive Strategy Investing in a broad stock index and a risk free investment is an example of a passive strategy. –The investor makes no attempt to actively find undervalued strategies nor actively switch their asset allocations. –The CAL that employs the market (or an index that mimics overall market performance) is called the Capital Market Line or CML. 5-19

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Active versus Passive Strategies Active strategies entail more trading costs than passive strategies. Passive investor “free-rides” in a competitive investment environment. Passive involves investment in two passive portfolios –Short-term T-bills –Fund of common stocks that mimics a broad market index –Vary combinations according to investor’s risk aversion. 5-20

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