Question

An investor gathers the following information regarding three stocks, which are not in the market portfolio: Stocks Expected Return Standard Deviation Beta A 20% 39% 1.7 B 23.5% 34% 1.4 C 22.5% 31% 1.2 Given that the return on the market portfolio is 15% with a standard deviation of 21%, and the risk-free rate is 5%, answer the following questions: a) Calculate Jensen's alpha for stocks A, B, and C. (6 points) b) Calculate non-systematic variance for stocks A, B and C. (6 points) c) If an investor holds the market portfolio, should she add any of these stocks to her portfolio? If so, which stock should have the highest weight in the portfolio?

Answer 1

Answer - a

Jensen's Alpha can be calculated using the below mentioned formula :

Alpha = Expected Return - CAPM Return

Where -

Expected Return is given for stocks A, B and C

CAPM Return = Risk Free rate + Beta (Market Return - Risk Free rate)

Statement showing calculation of CAPM Return

Stock	Risk Free Rate (A)	Beta (B)	Market Return (C )	CAPM Return [A + B * (C - A)]
A	5%	1.7	15%	22%
B	5%	1.4	15%	19%
C	5%	1.2	15%	17%

Statement showing calculation of Alpha

Stock	Expected Return (A)	CAPM Return (B)	Alpha (A - B)
A	20%	22%	-2%
B	23.5%	19%	4.5%
C	22.5%	17%	5.5%

Answer - b

Non-systematic variance can be calculated using the below mentioned formula :

Non-systematic Variance = Total Variance of Security - Systematic Variance of Security

Where -

Total Variance of Security is given for stocks A, B and C as (Standard Deviation)2

Systematic Variance of Security = (Beta of Security)2 * (Standard Deviation of Market)2

Statement showing calculation of Systematic Variance of Security

Stock	Beta (A)	Standard Deviation of Market (B)	Systematic Variance of Security [(A)2 * (B)2]
A	1.7	21%	1274.49%
B	1.4	21%	864.36%
C	1.2	21%	635.04%

Statement showing calculation of Non-systematic Variance of Security

Stock	Standard Deviation of Security (A)	Total Variance of Security [B = (A)2 ]	Systematic Variance of Security (C )	Non-systematic variance (B - C)
A	39%	1521%	1274.49%	246.51%
B	34%	1156%	864.36%	291.64%
C	31%	961%	635.04%	325.96%

Answer - c

The Investor should add Stock B and C to her portfolio since the Jensen's alpha of these stocks are positive (calculated in Answer-a above) which means that the investor would get more than CAPM return from these stocks.

Stock B should have the highest weight in the portfolio although stock B has lower Jensen's alpha than that of stock C but it also have a lower non-systematic risk (calculated in Answer-b above) which is controllable by the company.