Question

Suppose that T-bills currently have a rate of return of 2%. As- sume that borrowing is possible at the risk free rate. You are risk averse and you are considering constructing a portfolio consisting of T-bills and one of the two risky assets: Stock A or Stock B. You did the following scenario analysis on stocks A and B

Events Bull Market Normal Market Bear Market

Probability Stock Aís return 0.3 50%
0.5 18%
0.2 -20%

Stock Bís return 10%
20%
-15%

1. (a) Compute the expected rate of return and the standard deviation for Stock A and Stock B.

2. (b) Based on the information you have so far, which of the two risky assets, Stock A or Stock B, would you choose to be included in your portfolio with T-bills? Explain.

3. (c) Your friend is considering the same problem but she is more risk averse than you. Should she arrive at a di§erent conclusion than you? Ex- plain.

4. (d) Suppose that you start your portfolio with $1 million and also your portfolio target risk (std dev) is 10% (this is your portfolio from part (b)), how many dollars will you invest in T-bills? What is your port- folioís expected return? Show your work.

Answer 1

(a) Calculation if  expected rate of return and the standard deviation for Stock A and Stock B.

For Stock A = 0.3 X 50% + 0.5 X 18% + 0.2 X -20% = 20%

For Stock B = 0.3 X 10% + 0.5 X 20% + 0.2 X -15% = 10%

Standard deviation : -

possible returns (x)	probability	deviation	deviation squard	product
50	0.3	30	900	270
18	0.5	-2	4	2
-20	0.2	-40	1600	320
			variance	592

variance = 592 % so standard deviation is square root of 592 = 24.331% for stock A.

possible returns (x)	probability	deviation	deviation squard	product
10	0.3	0	0	0
20	0.5	10	100	50
-15	0.2	-25	625	125
				175

variance = 175 % so standard deviation is square root of 592 = 13.228% for stock B.

(b) Based on the information you have so far, which of the two risky assets, Stock A or Stock B, would you choose to be included in your portfolio with T-bills? Explain.

For stock A the standard deviation will be 24.331% and for stock B the standard deviation will be 13.228%. since for stock A has more risk compared with stock B since he is risk averse investor so he will buy stock B since standard deviation is less ( risk is low ) but the return is less compared with stock A.yes the investor has include the T-bills in his portfolio for certain proportion so that for T bills the risk will be zero so he got in addition to the % return.

c) Your friend is considering the same problem but she is more risk averse than you. Should she arrive at a di§erent conclusion than you? Ex- plain.

even though she is the risk averse she may consider the return given by Stock A since the return is more for Stock A she may preferred to buy Stock A.

(d) Suppose that you start your portfolio with $1 million and also your portfolio target risk (std dev) is 10% (this is your portfolio from part (b)), how many dollars will you invest in T-bills? What is your port- folioís expected return? Show your work.

if he invest in stock B : -

  since the target risk is (std dev ) is 10% the proportion of investment will be if i invest in stock B since he is the risk averse investor so the proportion will be stock B will be 76% approx and in T bills the weight will be balance i,e 24% so the risk will be @ 10 %

Calculation = 13.288 X 76% + 0 X 24% = 10% , return = 10 X 76% + 2 X 24% = 8.08 %

if he purchase stock B then the investment will be stock B = $ 1000000 X 76 % = $ 760000 and in T bills the amount will be balance $ 240000.

if he invest in stock A : -

if the investor is purchasing the stock A then the proportion will be stock A 41% and in T bills the proportion will be balance i,e 59%

calculation = 24.331 X 41% + 0 X 59% = 10% approx , return = 20 X 41% + 2 X 59% = 9.38 %

if he purchase stock A then the investment will be stock A = $ 1000000 X 41 % = $ 410000 in stock A and

$ 590000 in T bills .