Question

Assume that the following market model adequately describes the return generating behavior of risky assets:

  

Rit=αi+βiRMt+ϵitRit=αi⁢+βiRMt+ϵit

  

Here:

  

Rit = The return on the ith asset at Time t.

RMt = The return on a portfolio containing all risky assets in some proportion at Time t.

RMt and ϵitϵit are statistically independent.

  

Short selling (i.e., negative positions) is allowed in the market. You are given the following information:

  

Asset	βi	E(Ri )		Var(€i)
A	.65	7.91	%	.0200
B	1.15	11.56		.0145
C	1.51	13.25		.0226

  

The variance of the market is .0122, and there are no transaction costs.

  

a.	Calculate the standard deviation of returns for each asset. (Do not round intermediate calculations and enter your answers as a percent rounded to 2 decimal places, e.g., 32.16.)

  

Asset	Standard deviation
A	%
B	%
C	%

  

b.	Calculate the variance of return of three portfolios containing an infinite number of asset types A, B, or C, respectively. (Do not round intermediate calculations and round your answers to 6 decimal places, e.g., 32.161616.)

  

Asset	Variance of return
A
B
C

  

c-1.	Assume the risk-free rate is 3 percent and the expected return on the market is 9 percent. What is the expected return of each asset? (Do not round intermediate calculations and enter your answers as a percent rounded to 2 decimal places, e.g., 32.16.)

  

Asset	Expected returns
A	6.9%
B	9.9 %
C	12.06 %

  

c- 2.	Which asset will not be held by rational investors?
	X Asset C Asset A Asset B

Answer 1

a.

Variance= Variance of (a_i+Betai*RMT+€_i) a_i and Betai are constants

Therefore Variance= βiai² * Variance of (RMT)+ Variance of (€_i)

Var(A)= (0.65)² * 0.0122 + 0.0200

= 0.4225 * 0.0122 + 0.0200

= 0.2515

Var(B)= (1.15)² * 0.0122 + 0.0145

= 1.3225 * 0.0122 + 0.0145

= 0.03063

Var(C)= (1.51)² * 0.0122 + 0.0226

= 2.28 * 0.0122 + 0.0226

= 0.05042

Assets	Standard Deviation= Var(ai)1/2
A	15.86%
B	17.5%
C	22.45%

b. Variance of a portfolio= Beta² * Var(RMT)

Variance of (portfolio of N number of asset type A) = (0.65)² * 0.0122

= 0.4225 * 0.0122

= 0.00515

Variance of (portfolio of N number of asset type B) = (1.15)² * 0.0122

= 1.3225 * 0.0122

=0.01613

Variance of (portfolio of N number of asset type C) = (1.51)² * 0.0122

= 2.28 * 0.0122

= 0.02782

c-1 Expected Return on each asset as per CAPM

E(R_i)= rf + E(RMT-rf) * Betai

E(R_A)= 3 + (9-3) * 0.65

= 4.95%

E(R_B)= 3 + (9-3) * 1.15

= 6.45%

E(R_C)= 3 + (9-3) * 1.51

= 7.53%

c-2

Assets	Risk	Return
A	15.86%	4.95%
B	17.5%	6.45%
C	22.45%	7.53%

Investor will not hold security B because its expected return is low relative to the risk associated with it.