Tabulate and draw the investment opportunity set

Management Finance Questions

INVESTMENT MANAGEMENT / FALL 2015

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Homework Assignment 3

Due on 11/4/2015 (Wed) at class.

You can work as a group up to THREE people. On your assignment, state clearly the name

and UB number of the members. Please read the instructions (or hints) first and show your

work/or explain your answer.

The following data apply to problems 1-6.

A pension fund manager is considering three mutual funds. The first is a stock fund, the

second is a long-term government and corporate bond fund, and the third is a T-bill

money market fund that yields an interest rate of 5.5%. The probability distributions of

the risky funds are as following table. The correlation between stock fund and bond fund

returns is 0.15 ( ρS,B=0.15)

1. Tabulate and draw the investment opportunity set of the two risky funds. Use investment

proportions for the stock fund of -20% to 120% in increments of 20%. (Hint: You can

refer to lecture notes 6 p.23-26).

% in

Stock

% in Bond Exp. Returns of

risky portfolio

Std. of the risky

portfolio σp

Sharpe Ratio

-20%

0

20%

40%

60%

80%

100%

120%

MGF 402 INVESTMENT MANAGEMENT / FALL 2015

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2. What expected return and standard deviation does your graph show for the minimum-

variance portfolio? What is the Sharpe Ratio of the minimum-variance portfolio? (Hint:

The WMin-var(S) can be found as 𝜎𝐵 2−𝐶𝑜𝑣(𝑟𝑆,𝑟𝐵)

𝜎𝑆 2+𝜎𝐵

2−2𝐶𝑜𝑣(𝑟𝑆,𝑟𝐵) , WMin-var(B)=1-WMin(S))

3. Draw a tangent from the risk-free rate to the opportunity set (CALo). What does your

graph show for the expected return and standard deviation of the optimal risky portfolio?

(Hint: You will need to use the formula on notes p.42 to first find the weight on stock and

bond funds)

4. What is the reward-to-volatility ratio of the best feasible CAL? What is the equation of

the best feasible CAL?

5. Suppose now that your client seek an complete portfolio must yield an expected return of

12% and be efficient, that is, on the best feasible CAL,

(I). What is the standard deviation of your portfolio? (Hint: Use the CAL equation

you get from question 4 and plug the expected return in).

(II). What is the proportion invested in the T-bill fund and each of the two risky funds?

(Hint: the expected returns of a completed portfolio is a weighted average returns

of the assets in the portfolio, i.e. E(rC) = (1- y)*rf + y*E(rP) ).

0

2

4

6

8

10

12

14

16

18

20

0 5 10 15 20 25 30 35 40

E x p

e c te

d R

e tu

rn (

% )

Standard Deviation (%)

Investment Opportunity Set

MGF 402 INVESTMENT MANAGEMENT / FALL 2015

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6. If you were to use only the two risky funds and still require an expected return of 12%,

what would be the investment proportions of your portfolio. Compare its standard

deviation to that of the optimal portfolio in the previous problem. What do you find?

7. Use the spreadsheet posted on UBLearns, calculate the expected returns and standard

deviation of stock and bond funds and the covariance and correlation coefficient between

the stock and bond funds.

Stock fund data

(A) (B) (C) (D) (E) (F) (G)

Scenario Probability

Stock

Rate of

Return

Col. B Deviation from

Expected

Return

Squared

Deviation

Col. B

X X

Col. C Col. F

Severe

recession 0.10 -0.37

Mild

recession 0.20 -0.11

Normal

growth 0.35 0.14

Boom 0.35 0.30

Expected Return = Variance =

Standard Deviation =

Bond fund data

(H) (I) (J) (K) (L)

Bond

Rate of

Return

Col. B Deviation from

Expected

Return

Squared

Deviation

Col. B

X X

Col. H Col. K

-0.09

0.15

0.08

-0.05

Expected Return= Variance =

Standard Deviation =

MGF 402 INVESTMENT MANAGEMENT / FALL 2015

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Covariance

(A) (B) (M) (N) (O) (P)

Scenario Probability

Stock

dev from

mean

Bond

dev from

mean M x N B x O Severe

recession 0.10

Mild

recession 0.20

Normal

growth 0.35

Boom 0.35

Expected Return =

Covariance =

Correlation Coefficient=

8. You are considering investing $1,000 in a T-bill that pays 5% and a risky portfolio, P,

constructed with two risky securities, X and Y. The weights of X and Y in P are 60% and

40%, respectively. X has an expected rate of return of 14% and variance of 0.01, and Y

has an expected rate of return of 10% and a variance of 0.0081.

If you want to form a portfolio with an expected rate of return of 10%, what percentages

of your money must you invest in the T-bill, X, and Y, respectively, if you keep X and Y

in the same proportions to each other as in portfolio P?

9. Consider a T-bill with a rate of return of 5% and the following risky securities:

Security A: E(r) = 0.15; Variance = 0.04

Security B: E(r) = 0.10; Variance = 0.0225

Security C: E(r) = 0.12; Variance = 0.01

Security D: E(r) = 0.13; Variance = 0.0625

From which set of portfolios, formed with the T-bill and any one of the four risky

securities, would a risk-averse investor always choose this portfolio? (Hint: consider the

reward-to-volatility ratio.)