FIN 435 (Faculty- SfR)

CCHHAAPPTTEERR 66RISK AVERSION AND CAPITAL ALLOCATION TO RISKY ASSETS

Suggested Problems: 4, 5, 6, 10, 11, 13,14,15,16, 19

(4) a. The expected cash flow is: (0.5 x $70,000) + (0.5 x 200,000) = $135,000With a risk premium of 8% over the risk-free rate of 6%, the required rate of return is 14%.

Therefore, the present value of the portfolio is:

$135,000/1.14 = $118,421

b. If the portfolio is purchased for $118,421, and provides an expected cash inflow of $135,000, then the expected rate of return [E(r)] is derived as follows:

$118,421 x [1 + E(r)] = $135,000Therefore, E(r) = 14%. The portfolio price is set to equate the expected rate or return with

the required rate of return.

c. If the risk premium over T-bills is now 12%, then the required return is:6% + 12% = 18%

The present value of the portfolio is now:$135,000/1.18 = $114,407

d. For a given expected cash flow, portfolios that command greater risk premium must sell at lower prices. The extra discount from expected value is a penalty for risk.

(5) When we specify utility by U = E(r) – .005A2, the utility from bills is 7%, while that from the risky portfolio is U = 12 – .005A x 182 = 12 – 1.62A. For the portfolio to be preferred to bills, the following inequality must hold: 12 – 1.62A > 7, or,A < 5/1.62 = 3.09. A must be less than 3.09 for the risky portfolio to be preferred to bills.

(6) Points on the curve are derived as follows:

U = 5 = E(r) – .005A2 = E(r) – .0152

The necessary value of E(r), given the value of 2, is therefore:

2 E(r)

0% 0 5.0% 5 25 5.37510 100 6.515 225 8.37520 400 11.025 625 14.375

FIN 435 (Faculty- SfR)

The indifference curve is depicted by the bold line in the following graph (labeled Q3).

(10) The portfolio expected return can be computed as follows:

Portfolio Portfolio

Wbills xReturnon bills + Wmarket x

Exp. returnon market =

expectedreturn standard deviation

_______________________________________________________________(=wmarketx16.2%)___

0.0 5% 1.0 8.66% 8.66% 16.5%.2 5 .8 8.66 7.93 13.2.4 5 .6 8.66 7.20 9.9.6 5 .4 8.66 6.46 6.6.8 5 .2 8.66 5.73 3.3

1.0 5 0.0 8.66 5.00 0

E(r) U(Q4,A=4)

U(Q3,A=3)

U(Q5,A=0)

U(Q6,A<0)

5

4

FIN 435 (Faculty- SfR)

(11) Computing the utility from U = E(r) – .005 x A2 = E(r) – .0152 (because A = 3), we arrive at the following table.

Wbills Wmarket E(r) 2 U(A=3) U(A=5)___________________________________________________________________

0. 1.0 8.66% 16.5 271.92 4.58 1.86 .2 .8 7.93 13.2 174.03 5.32 3.58 .4 .6 7.20 9.9 97.89 5.73 4.75 .6 .4 6.46 6.6 43.51 5.81 5.38 .8 .2 5.73 3.3 10.88 5.57 5.46 1.0 0 5.0 0 0 5.0 5.0

The utility column implies that investors with A = 3 will prefer a position of 40% in the market and 60% in bills over any of the other positions in the table.

(13) Expected return = .3 8% + .7 18% = 15% per year.

Standard deviation = .7 28% = 19.6%

(14) Investment proportions: 30.0% in T-bills.7 27% = 18.9% in stock A.7 33% = 23.1% in stock B.7 40% = 28.0% in stock C

(15) Your reward-to-variability ratio = 18 – 8

28 = .3571

Client's reward-to-variability ratio = 15 – 819.6 = .3571

ClientP

0

5

10

15

20

25

30

0 10 20 30 40

E( r)%

CAL (Slope = .3571)

Problem 16

FIN 435 (Faculty- SfR)

(19) a.

y* = E(rP) - rf

.01 As2P

= 18 - 8

.01 3.5 282 = 10

27.44 = .3644

So the client's optimal proportions are 36.44% in the risky portfolio and 63.56% in T-bills.

b. E(rC) = 8 + 10y* = 8 + .3644 10 = 11.644%

C= .3644 28 = 10.20%

EXTRA PROBLEM

1. XYZ wish to combine two stocks: Brac and Delta, into a portfolio with an expected return of 16%. The expected return of Brac is 2% and the standard deviation of 1%. The expected return of Delta is 25% with a standard deviation of 10%. The correlation between the two stocks is 0.40. A. What is the composition (weights) of the portfolio?B. What is the portfolio standard deviation?Solution:A. What is the composition (weights) of the portfolio?Set w = weight in Brac

16% *2% (1 )*25%

16% 25%

2% 25%39.1304%

w w

w

B. What is the portfolio standard deviation?

2 2 2 22 .391304 .01 .608696 .10 2 *.4 * .391304 .608696 .01 .10

0.00391097

0.062538

6.2538%

P

P

2. An investor can design a risky portfolio with two stocks, A and B. Stock A has an expectedreturn of 18% and a standard deviation of 22.5%. Stock B has an expected return of 15% and a standard deviation of 15%. The correlation coefficient between the two stocks is 0.75 and the risk-free T-bill rate is 9%.

(a) what are the weights of stocks A and B in the optimal risky portfolio (P)?

FIN 435 (Faculty- SfR)

%6060.04.01

%404.000127.0

00051.0

75.0*15.0*225.0*]09.015.009.018.0[)225.0(*]09.015.0[)15.0(*]09.018.0[

75.0*15.0*225.0*]09.015.0[)15.0(*]09.018.0[

])()([])([])([

])([])([

22

2

22

2

B

A

A

ABBAfBfAAfBBfA

ABBAfBBfAA

w

w

w

rrErrErrErrE

rrErrEw

(b) what is the expected return on the portfolio P?

%20.1615.0*6.018.0*4.0)( PrE

(c) what is the standard deviation of the return on portfolio P?

%8375.16168375.002835.0

75.0*)225.0(*)15.0(*)6.0(*)4.0(*2)15.0(*)6.0()225.0(*)4.0( 2222

p

p

3. Assume the T-Bill rate is 10% and the expected return on the market portfolio is 18%. An investment is twice as risky as the market portfolio, in terms of its systematic risk. What is the opportunity cost of capital (OCC) for this investment?

Based on CAPM:

r= rf + β [E (Rm) – rf] where r = Required rate of return, rf = risk-free rate, β= Beta, E(Rm) = Market Index Return

OCC = 10% + 2(18% – 10%) = 26.0%