Question

Given the following data about risky portfolios P and M and the risk-free asset (T-bills):

State	State 1 (recession)	State 2 (normal)	State 3 (boom)
State probability	0.4	0.4	0.2
Return, P	-0.15	0.15	0.45
Return, M	-0.18	0.28	0.35
Return, T-bills	0.05	0.05	0.05

1. Find the expected return, variance, standard deviation, and Sharpe ratio of P

2. Find the expected return, variance, standard deviation, and Sharpe ratio of M

3. Find the covariance between P and M

4. Assuming M is the true market portfolio, find beta of P

5. Find alpha of P. Based on alpha, and assuming M is the true market portfolio, is portfolio P priced correctly?

Answer 1

For a portfolio P, if states are denoted by i, probability of states are denoted by Pi, return in a given state i is denoted by Ri and risk-free rate is denoted by Rf then

Expected return E(r) = Sum of probability weighted returns

Variance (V) = Sum of [Pi*(Ri - E(r))^2]; Standard deviation (SD) = V^0.5; Sharpe ratio = (E(r) - rf)/SDp

Parts (a) & (b):

Formula				∑Pi*Ri where i is state	∑Pi*(Ri-E('r))^2 where i is state	V^0.5	(E('r) - rf)/SD
State	State 1 (recession)	State 2 (normal)	State 3 (boom)	Expected return (E('r))	Variance (V)	St Dev (SD)	Sharpe ratio
State probability (P)	0.4	0.4	0.2
Return, P (Rp)	15.00%	15.00%	45.00%	21.00%	0.0144	12.00%	1.3333
Return, M (Rm)	-18.00%	28.00%	35.00%	11.00%	0.0567	23.82%	0.2519
Return, T-bills (Rf)	5.00%	5.00%	5.00%

c). Covariance (P,M) = Sum of [P*(Rpi - Ep(r))*(Rmi - Em(r))] where Rmi = return in a given state i for portfolio M and Em(r) is expected return for portfolio M, so

Covariance = 0.40*(15%-21%)*(-18%-11%) + 0.4*(15%-21%)*(28%-11%) + 0.2*(45%-21%)*(35%-11%) = 0.0144

d). Correlation (P,M) = Covariance(P,M)/SDp*SDm;

Correlation (P,M) = 0.0144/(12%*23.82%) = 0.5039

Beta = Correlation (P,M)*SDp/SDm = 0.5039*12%/23.82% = 0.2539

e). Alpha (P) = Ep(r) - [Rf + Betap*(Em(r) - Rf)] = 21% - [5% + 0.2539*(11%-5%)] = 0.1448

Since portfolio P has a positive alpha, it is undervalued.