Question

Portfolio expected return. You own a portfolio that is invested 35% in stock X, 20% in stock Y, and 45% in stock Z. The expected returns on these three stocks are 9%, 15% and 12%, respectively. What is the expected return, variance and standard deviation on the portfolio?

Answer 1

To calculate the expected return on the portfolio, we use the following formula:

Expected return = (weight of stock X * expected return of stock X) + (weight of stock Y * expected return of stock Y) + (weight of stock Z * expected return of stock Z)

Expected return = (0.35 * 0.09) + (0.2 * 0.15) + (0.45 * 0.12) = 0.0321 or 3.21%

To calculate the variance of the portfolio, we use the following formula:

Variance = (weight of stock X)^2 * variance of stock X + (weight of stock Y)^2 * variance of stock Y + (weight of stock Z)^2 * variance of stock Z + 2 * weight of stock X * weight of stock Y * covariance of stocks XY + 2 * weight of stock X * weight of stock Z * covariance of stocks XZ + 2 * weight of stock Y * weight of stock Z * covariance of stocks YZ

Assuming that the stocks are uncorrelated, the covariance terms will be zero. Also, we assume that the variances of the stocks are equal to the square of their standard deviations. Therefore, we can simplify the formula to:

Variance = (weight of stock X)^2 * standard deviation of stock X^2 + (weight of stock Y)^2 * standard deviation of stock Y^2 + (weight of stock Z)^2 * standard deviation of stock Z^2

Variance = (0.35)^2 * (0.09)^2 + (0.2)^2 * (0.15)^2 + (0.45)^2 * (0.12)^2 = 0.00060167 or 0.060167%

To calculate the standard deviation of the portfolio, we take the square root of the variance:

Standard deviation = sqrt(0.00060167) = 0.0245 or 2.45%

Therefore, the expected return of the portfolio is 3.21%, the variance is 0.060167% and the standard deviation is 2.45%.

Answer 2

Explanation:

To calculate the expected return on the portfolio, we need to take the weighted average of the individual stock returns based on their proportions in the portfolio. The expected return on the portfolio can be calculated as follows:

Expected return on portfolio = (weight of stock X × expected return on stock X) + (weight of stock Y × expected return on stock Y) + (weight of stock Z × expected return on stock Z)

Expected return on portfolio = (0.35 × 9%) + (0.20 × 15%) + (0.45 × 12%)

Expected return on portfolio = 3.15% + 3.00% + 5.40%

Expected return on portfolio = 11.55%

To calculate the variance and standard deviation of the portfolio, we need to use the formula that takes into account the individual stock variances, covariances, and the weights of the stocks in the portfolio. Assuming the covariances between the stocks are zero, we can use the following formulas:

Portfolio variance = (wX^2 * σX^2) + (wY^2 * σY^2) + (wZ^2 * σZ^2) + 2(wX * wY * σXY) + 2(wX * wZ * σXZ) + 2(wY * wZ * σYZ)

Portfolio standard deviation = sqrt(portfolio variance)

where wX, wY, and wZ are the weights of stocks X, Y, and Z in the portfolio respectively, σX, σY, and σZ are the standard deviations of returns on stocks X, Y, and Z respectively, and σXY, σXZ, and σYZ are the covariances between the returns on stocks X and Y, X and Z, and Y and Z respectively.

Since we assume the covariances between the stocks are zero, we can simplify the formulas as follows:

Portfolio variance = (wX^2 * σX^2) + (wY^2 * σY^2) + (wZ^2 * σZ^2)

Portfolio standard deviation = sqrt[(wX^2 * σX^2) + (wY^2 * σY^2) + (wZ^2 * σZ^2)]

Substituting the values, we get:

Portfolio variance = (0.35^2 * 0.09) + (0.20^2 * 0.15) + (0.45^2 * 0.12)

Portfolio variance = 0.0036475

Portfolio standard deviation = sqrt[(0.35^2 * 0.09) + (0.20^2 * 0.15) + (0.45^2 * 0.12)]

Portfolio standard deviation = sqrt(0.0036475)

Portfolio standard deviation = 0.06035

Therefore, the expected return on the portfolio is 11.55%, the portfolio variance is 0.0036475, and the portfolio standard deviation is 0.06035.