Question

Consider the following data about the expected returns, standard deviations, and correlation between two assets:

	Asset 1	Asset 2
Expected return	5.3%	6.8%
Standard deviation	4.5%	7.8%
Correlation coefficient	-0.6

Calculate the expected return and standard deviation of a portfolio consisting of a 20% weight in asset 1 and an 80% weight in asset 2. What happens to the expected return and standard deviation of the portfolio when the weight combination changes to 50% in asset 1 and 50% in asset 2?

Answer 1

	Asset 1	Asset 2
Expected return	5.3%	6.8%
Standard deviation	4.5%	7.8%

Correlation coefficient between asset 1 and asset 2 = ρ = -0.6

Expected return of asset 1 = E[R1] = 5.3%, Standard deviation of asset 1 = σ1 = 4.5%

Expected return of asset 2 = E[R2] = 6.8%, Standard deviation of asset 2 = σ2 = 7.8%

Part -1

	Weight	Expected return	Standard deviation
Asset 1	20%	5.3%	4.5%
Asset 2	80%	6.8%	7.8%

Weight of asset 1 in the portfolio = W1 = 20%

Weight of asset 2 in the portfolio = W2 = 80%

Expected return of the portfolio is calculated using the formula:

Expected return of portfolio = E[RP] = W1*E[R1] + W2*E[R2] = 20%*5.3% + 80%*6.8% = 6.5%

Variance of the portfolio is calculated using the formula:

Portfolio variance = σP2 = W12*σ12 + W22*σ22 + 2*ρ*W1*W2*σ1*σ2 = (20%)2*(4.5%)2 + (80%)2*(7.8%)2 + 2*(-0.6)*20%*80%*4.5%*7.8% = 0.000081 + 0.00389376 + (-0.00067392) = 0.00330084

Standard deviation of the portfolio is the square root of variance

Standard deviation of portfolio = σP = (0.00330084)1/2 = 5.74529372617275% ~ 5.75% (Rounded to two decimals)

Expected return of the portfolio consisting of a 20% weight in asset 1 and 80% weight in asset 2 = 6.5%

Standard deviation of the portfolio consisting of a 20% weight in asset 1 and 80% weight in asset 2 = 5.75%

Part - 2

	Weight	Expected return	Standard deviation
Asset 1	50%	5.3%	4.5%
Asset 2	50%	6.8%	7.8%

Weight of asset 1 in the portfolio = W1 = 50%

Weight of asset 2 in the portfolio = W2 = 50%

Expected return of the portfolio is calculated using the formula:

Expected return of portfolio = E[RP] = W1*E[R1] + W2*E[R2] = 50%*5.3% + 50%*6.8% = 6.05%

Variance of the portfolio is calculated using the formula:

Portfolio variance = σP2 = W12*σ12 + W22*σ22 + 2*ρ*W1*W2*σ1*σ2 = (50%)2*(4.5%)2 + (50%)2*(7.8%)2 + 2*(-0.6)*50%*50%*4.5%*7.8% = 0.00050625 + 0.001521 + (0.001521) = 0.00097425

Standard deviation of the portfolio is the square root of variance

Standard deviation of portfolio = σP = (0.00097425)1/2 = 3.12129780700272% ~ 3.12% (Rounded to two decimals)

Expected return of the portfolio consisting of a 50% weight in asset 1 and 50% weight in asset 2 = 6.05%

Standard deviation of the portfolio consisting of a 50% weight in asset 1 and 50% weight in asset 2 = 3.12%