Question

Consider two assets (A and B), both of which have a price of $100 per share at the beginning of the year. At the end of the year, one of three possible states of the world will be realized. Prob. State Good Bad Ugly Final Price A 120 140 80 Final Price B 140 60 200 25 a) Calculate holding period returns for assets A and B associated with each state, and use these to calculate the expected holding period return for each asset. b) Calculate the Var(HPR) for A & B, and the covariance of the HPR of A & B.

Answer 1

(a) Holding Period Return = (P1-P0+D)÷P0

Where, P1 is Price at the end of the period

P0 is Price at the beginning of the period

D is the Dividend for the period

As there is no mention of Dividends or earnings in the question, Dividends are assumed to be not declared.

For Security A

P0	State	P1	R=(P1-P0)÷P0	Probability(p)	p×R
100	Good	120	20%	0.5	10%
100	Bad	140	40%	0.25	10%
100	Ugly	80	-20%	0.25	-5%
				Total( mA)	15%

Here mA means Return of the Security, while R represents return at each phase.

For Security B

P0	State	P1	R=(P1-P0)÷P0	Probability(p)	p×R
100	Good	140	40%	0.5	20%
100	Bad	60	-40%	0.25	-10%
100	Ugly	200	100%	0.25	25%
				Total (mB)	35%

Here mB means Return of security B, while R represents return at each stage

(b) Calculation of Variance and Covariance of Securities A and B

mA	mB	A-mA	B-mB	p	Variance of A =p(A-mA)2 [in %2]	Variance of B=p(B-mB)2 [ in %2]	Covariance of A and B=p(A-mA)(B-mB)
15%	35%	5%	5%	0.5	12.5%	12.5%	12.5%
15%	35%	25%	-75%	0.25	156.25%	1406.25%	-468.75%
15%	35%	-35%	65%	0.25	306.25%	1056.25%	-568.75%
				Total	(475%)2	(2475%)2	(-1025%)2