Question

Ms. Smith wants to invest in two securities, ABC and PQR, and the relevant information is below:

State of the Economy	Probability	Return on ABC (%)	Return on PQR (%)
Bear	0.25	-15%	4%
Moderate	0.5	5%	4%
Bull	0.25	20%	4%

1. Calculate expected returns and standard deviations of two securities.

2. Ms. Smith shorts $1,000 of PQR and invests all proceeds from this short sale as well as $3,000 of her own money into ABC. What is the expected return and the standard deviation of her portfolio?

show formula and explanation

Answer 1

State of Economy	Probability	Return on ABC	Return on PQR
Bear	0.25	-15%	4%
Moderate	0.5	5%	4%
Bull	0.25	20%	4%

For ABC

Expected Return = E[R] = P_BEAR*R_BEAR + P_MODERATE*R_MODERATE + P_BULL*R_BULL

E[R_ABC] = 0.25*(-15%) + 0.5*(5%) + 0.5*20% = 3.75%

Variance of the return, when different state of economies are given, can be calculated using the below formula:

σ_ABC² = P_BEAR *( R_BEAR - E[R_ABC])² + P_MODERATE *( R_MODERATE - E[R_ABC])² + P_BULL *( R_BULL - E[R_ABC])²σ_ABC² = 0.25*(-15%-3.75%)² + 0.5*(5%-3.75%)² + 0.25*(20%-3.75%)² = 0.01546875

Standard Deviation of ABC = σ_ABC = (0.01546875)^1/2 = 0.1243734 or 12.44%

For PQR

Expected Return = E[R] = P_BEAR*R_BEAR + P_MODERATE*R_MODERATE + P_BULL*R_BULL

E[R_PQR] = 0.25*(4%) + 0.5*(4%) + 0.5*4% = 4%

Variance of the return, when different state of economies are given, can be calculated using the below formula:

σ_PQR² = P_BEAR *( R_BEAR - E[R_PQR])² + P_MODERATE *( R_MODERATE - E[R_PQR])² + P_BULL *( R_BULL - E[R_PQR])²

σ_PQR² = 0.25*(4%-4%)² + 0.5*(4%-4%)² + 0.25*(4%-4%)² = 0

Standard Deviation of PQR = σ_PQR = (0)^1/2 = 0%

	Expected Return	Standard Deviation
ABC	3.75%	12.43734%
PQR	4%	0%

b. In PQR, return is 4%

So, Ms. Smith will get 1.04*1000 = 1040, after selling PQR. She then invests 1040+3000 = 4040 into ABC.

From above calculation, return on ABC is 3.75% and standard deviation is 12.44%

Therefore, expected return = 3.75%*4040 = $151.50

Standard Deviation = 12.43734%*4040 = $502.468