Question

Stocks A and B have the following returns in each of the states given below.

	Good	Bad	Ugly
Stock A return	25%	10%	-25%
Stock B return	1%	-5%	35%

The probability of the good state is 0.5, the probability of the bad state is 0.3 and the probability of the ugly state is 0.2. What is the covariance between the returns of A and B? (Hint: The expected return of A is 10.5% and the expected return of B is 6%) What is the covariance between the return of A and the return of a risk-free asset which has an expected return of 5%?

Answer 1

1]	Stock A
	State of Economy	Probability of state of the economy [p]	Rate of return [%] if the state occurs [r]	E[r] = p*r	d = r-E[r]	d^2	p*d^2
	Good	0.5	25	12.50	$          14.50	210.25	105.13
	Bad	0.3	10	3.00	$           -0.50	0.25	0.08
	Ugly	0.2	-25	-5.00	$         -35.50	1260.25	252.05
				10.50			357.25
	Expected return			10.50
	Variance			357.25
	SD = 357.25^0.5 =			18.90
	Stock B:
	State of Economy	Probability of state of the economy [p]	Rate of return [%] if the state occurs [r]	E[r] = p*r	d = r-E[r]	d^2	p*d^2
	Good	0.5	1	0.50	$           -5.00	25.00	12.50
	Bad	0.3	-5	-1.50	$         -11.00	121.00	36.30
	Ugly	0.2	35	7.00	$          29.00	841.00	168.20
				6.00			217.00
	Expected return			6.00
	Variance			217.00
	SD = 217^0.5 =			14.73
	Covariance and Correlation [A,B]:
	State of Economy	Probability of state of the economy	dk*dm	dk*dm*p
	Boom	0.2	-72.5	-14.50
	Normal	0.5	5.5	2.75
	Recession	0.3	-1029.5	-308.85
	Covariance [A,B]			-320.60
2]	Covariance between return of A and a risk free rate of 5%, will be 0
	as there is Variance of a risk free asset is 0.