Question

You believe that the likelihood that next year is a bull market year is 50%. You also believe that a bear market may occur with 20% likelihood. The following table contains information on the rates of return on various assets next year under various economic scenarios.

Security	Bull Market	Normal Market	Bear Market
Security X	-2%	18%	-10%
Security Y	25%	10%	-15%
Market	11%	8%	-2%
Risk-free asset	5%	5%	5%

a) The capital allocation line is also called the capital market line, if the risky portfolio is the market portfolio. Write down the equation for the Capital Market Line using the information on the market portfolio in the table above.

b) What are the betas of security X and security Y? According to CAPM, are securities X and Y fairly priced? Explain.

Answer 1

Calculation of Average return, Standard deviation and Covariance with Market
Security X
		Actual ret	Exp. Ret	Std Dev.		Actural retu	Exp. Ret		Covariance with X and Market
State	Prob.	AR	AR * Prob.	((AR-ER)^2)*Prob.		Ar M	AR * Prob.	(Ar M - Erm)	(Ar M-Er M)*(Ar X - Erx) * Prob.
Bull	0.5	-2%	-1.00%	0.10%		11.00%	5.50%	3.50%	-0.08%
Normal	0.3	18%	5.40%	0.73%		8.00%	2.40%	0.50%	0.02%
Bear	0.2	-10%	-2.00%	0.31%		-2.00%	-0.40%	-9.50%	0.24%
			2.40%	1.13%			7.50%	-5.50%	0.18%
ER = sum=	2.40%
Std. Dev.	1.13%
Covariance =	0.18%
Security Y
		Actual ret	Exp. Ret	Std Dev.		Actural retu	Exp. Ret		Covariance with X and Market
State	Prob.	AR	AR * Prob.	(AR-ER)^2*Prob.		Ar M	AR * Prob.	(Ar M - Erm)	(Ar M-Er M)*(Ar Y - Er Y) * Prob.
Bull	0.5	25%	12.50%	0.78%		11.00%	5.50%	3.50%	0.22%
Normal	0.3	10%	3.00%	0.02%		8.00%	2.40%	0.50%	0.00%
Bear	0.2	-15%	-3.00%	1.51%		-2.00%	-0.40%	-9.50%	0.52%
			12.50%	2.31%			7.50%	-5.50%	0.74%	R
ER = sum=	12.50%
Std. Dev.	2.31%
Covariance =	0.74%
Market
		Actual ret	Exp. Ret	Std Dev.
State	Prob.	AR	AR * Prob.	(AR-ER)^2*Prob.
Bull	0.5	11%	5.50%	0.06%
Normal	0.3	8%	2.40%	0.00%
Bear	0.2	-2%	-0.40%	0.18%
			7.50%	0.24%
ER = sum=	7.50%
Std. Dev.	0.24%
Part (a)
Capital Market Equation = Risk free rate of return + Std. De. Stock * ( Return of market - Risk free return)/ Std. Dev. Market
Std. dev. Stock is variable figure. So Capital market equation shall be
	5 + S.D. stock * (7.50 - 5) /0.24
	5 +( S.D. Stock * 10.4167)
So, Capital market equation is 5 + ( σS * 10.4167)
Part (b)
Beta of X =	Covariance of X with market / (Std. Dev. Of Market)^2
	0.18/(0.24)^2
	3.12500
CAPM equation = Risk free rate of return + (Beta * (Market return - Risk free return)
	5 + (3.125 * (7.50 - 5)
Required return	12.8125
Expected return =	2.40%
Expected return is less than required rate of return. So security X is not fairly priced. It is over priced.
Beta of Y =	Covariance of Y with Market / (std. dev. Of market)^2
	12.84722
CAPM equation = Risk free rate of return + (Beta * (Market return - Risk free return)
	5 + (12.84722 * (7.50 - 5)
Required return	37.11806
Expected return	12.50%
Expected return is less than required rate of return. So security X is not fairly priced. It is over priced.