Essentials of Investments BODIE, KANE, MARCUS, 8TH EDITION Problem + Solution for Chapter 5, Problem 12
PreambleAssume you manage a risky portfolioExpected Return of 17%Standard Deviation of 27%T-Bill rate is 7%
E(r) = 17% σ = 27% T-bill=7%Client chooses to place:70% of their portfolio in your fund30% in Tbill money marketWhat is the E(r) and σ of your client’s portfolio?
Gather the Data Security E(r) σ p(s)Standard deviation of T-bills is always 0% Risky Fund 17 27 70%This is by definition. Because they can notlose value, they are considered “risk-free”. T-Bill 7 0 30%
Expected Return70% will have an E(r) of 1730% will have an E(r) of 7.7 (17) + .3 (7) =11.9 + 2.1 = 14Expected Return of the Portfolio is 14
Standard DeviationFirst, calculate the variance.This is easy, since standard deviation of a t-bill is 0.
Quick SolveIgnore the T-bill, since that is 0Weight the Standard Deviation of the Risky Portfolio
Part B Stock A 27%Suppose your risky portfolio includesthe following investments in thegiven proportions. Stock B 33%What are the investment proportionsof your client’s overall portfolio, includingthe position in T-bills. Stock C 40%
A Portfolio Partition? My Risky Portfolio 27% 40% 33%Remember, each stock in My Risky Portfolio will Stock A Stock B Stock Conly take up a portion of 70% of the client’sportfolio. Client’s Portfolio 30% 70% My Risky Portfolio T-Bills
12(b) Solution Stock A Stock B Stock C T-BillsT-Bills = 30% 18.9%Stock A = 23.1 30.0%Stock B = 18.9Stock C= 28.0 23.1% 28.0%
12(c) Sharpe RatioWhat are the reward-to-volatility ratios (S) of your RiskyPortfolio and your client’s portfolio?
How to find Sharpe Portfolio Risk PremiumSharpe = S = Standard Deviation of Portfolio Excess Return
Risk Premium? Excess Return?The Portfolio Risk Premium is the E(r) above the risk-free rate.In this case, the risk-free rate is the T-bill rate of 7%The standard deviation is the same, since the T-bill SD is 0.
Sharpe RatiosMy Risky Portfolio has a Portfolio Risk Premium of 17% - 7% = 10%My Client’s Portolio has a PRP of 14%-7% = 7%
12(c) Sharpe Ratios My Risky My Client’s Portfolio Portfolio Portfolio RiskWhat! The Sharpe Ratios are the same! 10 7 PremiumHow can that be, since My Risky Portfolio Standardis riskier than My Client’s Portfolio? 27 18.9 Deviation 0.370 0.370 Sharpe Ratios
What The Sharpe Ratio SaysThe Sharpe Ratio compares the reward, per unit of volatilityA higher Sharpe Ratio indicates a higher reward.Higher is better.
12(d) Draw the CALDraw the CAL of your portfolio on an expected return/standard deviation diagram.What is the slope of the CAL?Show the position of your client’s fund on the CAL.
What is a CALCAL stands for Capital Allocation LineWhen graphed, The Y axis is the E(r) return, and the X axis is risk/standard deviation
CAL for My Risky Portfolio CAL CLIENTThe CAL starts at 0 risk + 7% return (TBills)20 17The CAL ends at 17% return and 27% risk 15 14Rise/Run = Slope 1010/27 =.3704 7 5My Client appears at X-axis18.9% riskwith a 14% return. 0 0 18.9 27 40