Question

Assume that there are two assets (A and B) and there are four possible future scenarios. The four scenarios and their probabilities are shown in the following table. The last two columns show the returns on assets A and B in the four possible scenarios.

Scenario	Probability	RA	RB
Boom	0.3	0.15	-0.02
Normal	0.3	0.08	-0.01
Recession	0.3	-0.05	0.03
Disaster	0.1	-0.20	0.05

What is the covariance and correlation between the two asset returns? Please show work.

Answer 1

First, Expected return -

Expected Return_RA = 0.3 x 15% + 0.3 x 8% + 0.3 x -5% + 0.1 x -20% = 3.4%

Expected Return_RB = 0.3 x -2% + 0.3 x -1% + 0.3 x 3% + 0.1 x 5% = 0.5%

Next, Standard deviation (σ) -

$\sigma_{Asset} = \sqrt{\sum (Return_{Scenario}-Expected Return)^{2}*Probability_{Scenario}}$

$\sigma_{RA} = \sqrt{(15-3.4)^{2}*0.3+(8-3.4)^{2}*0.3+(-5-3.4)^{2}*0.3+(-20-3.4)^{2}*0.1}=11.07$ %

$\sigma_{RB} = \sqrt{(-2-0.5)^{2}*0.3+(-1-0.5)^{2}*0.3+(3-0.5)^{2}*0.3+(5-0.5)^{2}*0.1}=2.54$ %

Next, Covariance -

$Covariance = \sum (Return_{Asset1}-Expected Return_{Asset1})*(Return_{Asset2}-Expected Return_{Asset2})*Probability_{Scenario}$

%²

Finally, Correlation (We needed standard deviation for this formula)

Correlation = Covariance / ( σ_RA x σ_RB ) = (-)27.6%² / (11.07% x 2.54%) = (-)0.9816 or (-)0.98