The document summarizes a portfolio optimization problem involving allocating funds between a risky fund and treasury bills. It calculates that for a portfolio with 70% in the risky fund and 30% in treasury bills, the expected return is 14% and the standard deviation is 18.9%. Both the risky fund and the overall portfolio have a Sharpe ratio of 0.370. A capital allocation line is drawn on a risk-return graph showing the efficient frontier between the risky fund and treasury bills, with the client's portfolio appearing at 18.9% risk and 14% return.

1. Essentials of Investments
BODIE, KANE, MARCUS, 8TH EDITION
Problem + Solution for Chapter 5, Problem 12

2. Preamble
Assume you manage a risky portfolio
Expected Return of 17%
Standard Deviation of 27%
T-Bill rate is 7%

3. E(r) = 17% σ = 27% T-bill=7%
Client chooses to place:
70% of their portfolio in your fund
30% in Tbill money market
What is the E(r) and σ of your client’s portfolio?

4. 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 not
lose value, they are considered “risk-free”.
T-Bill 7 0 30%

5. Expected Return
70% will have an E(r) of 17
30% will have an E(r) of 7
.7 (17) + .3 (7) =
11.9 + 2.1 = 14
Expected Return of the Portfolio is 14

6. Standard Deviation
First, calculate the variance.
This is easy, since standard deviation of a t-bill is 0.

7. Quick Solve
Ignore the T-bill, since that is 0
Weight the Standard Deviation of the Risky Portfolio

8. Standard Deviation
70% of 27%
.7 * 27% = 18.9

9. Answers to 12(a)
Mean Expected Return = 14%
Standard Deviation = 18.9%

10. Part B
Stock A 27%
Suppose your risky portfolio includes
the following investments in the
given proportions.
Stock B 33%
What are the investment proportions
of your client’s overall portfolio, including
the position in T-bills.
Stock C 40%

11. A Portfolio Partition? My Risky Portfolio
27%
40%
33%
Remember, each stock in My Risky Portfolio will Stock A Stock B Stock C
only take up a portion of 70% of the client’s
portfolio.
Client’s Portfolio
30%
70%
My Risky Portfolio T-Bills

12. Compute!
Stock A: 27% *.7 = 18.9
Stock B: 33% * .7 = 23.1
Stock C: 40% * .7=28.0

13. 12(b) Solution
Stock A Stock B Stock C T-Bills
T-Bills = 30%
18.9%
Stock A = 23.1 30.0%
Stock B = 18.9
Stock C= 28.0 23.1%
28.0%

14. 12(c) Sharpe Ratio
What are the reward-to-volatility ratios (S) of your Risky
Portfolio and your client’s portfolio?

15. How to find Sharpe
Portfolio Risk Premium
Sharpe = S =
Standard Deviation of Portfolio Excess Return

16. 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.

17. Sharpe Ratios
My Risky Portfolio has a Portfolio Risk Premium of 17% - 7% = 10%
My Client’s Portolio has a PRP of 14%-7% = 7%

18. 12(c) Sharpe Ratios
My Risky My Client’s
Portfolio Portfolio
Portfolio Risk
What! The Sharpe Ratios are the same! 10 7
Premium
How can that be, since My Risky Portfolio Standard
is riskier than My Client’s Portfolio? 27 18.9
Deviation
0.370 0.370 Sharpe Ratios

19. What The Sharpe Ratio Says
The Sharpe Ratio compares the reward, per unit of volatility
A higher Sharpe Ratio indicates a higher reward.
Higher is better.

20. Sharpe Ratio Equation
E(rp) - rf
S=
σp

21. 12(d) Draw the CAL
Draw 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.

22. What is a CAL
CAL stands for Capital Allocation Line
When graphed, The Y axis is the E(r) return, and the X axis is risk/standard deviation

23. CAL for My Risky Portfolio
CAL CLIENT
The CAL starts at 0 risk + 7% return (TBills)20
17
The CAL ends at 17% return and 27% risk
15 14
Rise/Run = Slope
10
10/27 =.3704 7
5
My Client appears at X-axis18.9% risk
with a 14% return.
0
0 18.9 27 40