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5.5 Asset Allocation Across Risky and Risk Free Portfolios

5.5 Asset Allocation Across Risky and Risk Free Portfolios. 5- 1. Allocating Capital Between Risky & Risk-Free Assets. T-bills or money market fund. risky portfolio. $2,500. $3,000. $2,000. $7,500. Possible to split investment funds between safe and risky assets

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5.5 Asset Allocation Across Risky and Risk Free Portfolios

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

  2. Allocating Capital Between Risky & Risk-Free Assets T-bills or money market fund risky portfolio $2,500 $3,000 $2,000 $7,500 • Possible to split investment funds between safe and risky assets • Risk free asset rf : proxy; ________________________ • Risky asset or portfolio rp: _______________________ • 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 ______ 5-2

  3. Allocating Capital Between Risky & Risk-Free Assets $2,500 / $7,500 = 33.33% $3,000 / $7,500 = 40.00% $2,000 / $7,500 = 26.67% 100.00% 25% Wrf = ; Wrp = In the complete portfolio WA = 0.75 x 33.33% = 25%; 75% WB = 0.75 x 40.00% = 30% Wrf = 25% WC = 0.75 x 26.67% = 20%; Weights in rp • WA = • WB = • WC = The complete portfolio includes the riskless investment and rp. 5-3

  4. Example rf = 5% srf = 0% E(rp) = 14% srp = 22% y = % in rp (1-y) = % in rf 5-4

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

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

  7. E(rp) = 11.75% 16.5% Possible Combinations E(r) CAL (Capital Allocation Line) E(rp) = 14% P y = 1 y =.75 rf = 5% F y = 0 0 s 22% 5-7

  8. Combinations Without Leverage E(rc) = yE(rp) + (1 - y)rf y = .75 E(rc) = y = 1 E(rc) = y = 0 E(rc) = (.75)(.14) + (.25)(.05) = 11.75% 75(.22) = 16.5% (1)(.14) + (0)(.05) = 14.00% 1(.22) = 22% 0(.22) = 0% (0)(.14) + (1)(.05) = 5.00% Since σrf = 0 σc= y σp If y = .75, then σc= If y = 1 σc= If y = 0 σc= 5-8

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

  10. Risk Premium & Risk Aversion E[rasset] – rf • 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 Premiumasset = 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. 5-10

  11. Risk Aversion and Allocation 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 = 1.5 y = 0 • Greater levels of risk aversion lead investors to choose larger proportions of the risk free rate 5-11

  12. E(rp) - rf = 9% P or combinations of P & Rf offer a return per unit of risk of 9/22. CAL (Capital Allocation Line) E(r) P E(rp) = 14% ) Slope = 9/22 rf = 5% F s 0 rp = 22% 5-12

  13. Quantifying Risk Aversion E(rp) = Expected return on portfolio p rf = the risk free rate 0.5 = Scale factor A x p2 = Proportional risk premium The larger A is, the larger will be the _________________________________________ investor’s added return required to bear risk 5-13

  14. 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

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

  16. 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

  17. 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%.

  18. 5.6 Passive Strategies and the Capital Market Line 5-18

  19. 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

  20. 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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