Question

There are only two possible states of the economy. State 1 has a 49% chance of occurring. In State 1, Asset A returns 8.75% and Asset B returns 11.75%. In State 2, Asset A returns -4.50% and Asset B returns -7.50%. A portfolio of just these two assets is invested 61% in Asset A (with Asset B comprising the remainder without any negative weights). What is the standard deviation of the portfolio's returns?

a) 7.01%

b) 7.21%

c) 7.40%

d) 7.60%

e) 7.79%

Answer 1

Mena = sum of probability * returns

variance of stock = sum of (Probability * ( return - means of returns )^2

covariance = sum of [ Probability * ( return from asset A - Mean of return from A )^2 * ( Return on asset B - means of return from B)^2

Asset A :

State of Economy	Probability (P)	Return (x)	Px	x - sum of Px	(x - sum of Px)^2	P*(x - sum of Px)^2
State 1	0.49	8.75	4.2875	6.7575	45.66380625	22.37526506
State 2	0.51	-4.5	-2.295	-6.4925	42.15255625	21.49780369
Mean	1.9925	Variance	43.87306875

Asset B :

State of Economy	Probability (P)	Return (y)	Py	y - sum of Py	(y - sum of Py)^2	P*(y - sum of Py)^2
State 1	0.49	11.75	5.7575	9.8175	96.38330625	47.22782006
State 2	0.51	-7.5	-3.825	-9.4325	88.97205625	45.37574869
Mean	1.9325	Variance	92.60356875

Covariance between Asset A & Asset B :

State of Economy	Probability (P)	x - sum of Px	y - sum of Py	P*(x - sum of Px * )(y - sum of Py)
Steady Growth	0.49	6.7575	9.8175	32.50746056
Recession	0.51	-6.4925	-9.4325	31.23265819
Covariance between Asset A & Asset B	63.74011875

Standard deviation of portfolio return = [ weight of asset A^2 * Variance of asset A + Weight of Asset B^2 * Variance of asset B + 2 weright of asset A * weigh of asset B * Covariance of asset A with asset B ] ^ 0.50

= (0.61^0* 43.87 + 0.39^2* 92.60 + 2*0.61*0.39*63.74 ) ^0.50

= ( 16.33+ 64.14 + 30.33 )^0.50

= 10.53%