Question

A portfolio has $200,000 invested in bonds and $300,000 invested in stocks. The bonds have an expected return of 8% a with a standard deviation of 12%. The stocks have an expected return of 12% with a standard deviation of 20%. The correlation between the stocks and bonds is 0.40.

1. What is the portfolio weight for the bonds?

2. What is the portfolio weight for the stocks?

3. What is the expected return for the portfolio? (Treat the expected values of the individual assets as whole numbers not percentages.)

4. What is the expected variance for the portfolio? (Treat the standard deviations of the individual assets as whole numbers not percentages.)

5. What is the expected standard deviation for the portfolio? (Treat the standard deviations of the individual assets as whole numbers not percentages.)

Answer 1

From the above questions we have:

Two securities say A & B

Security A invested in bonds = $200,000

Security B invested in stocks = $300,000

Total investment in the portfolio = $200,000 + $300,000 = $500,000

Expected return on security A = 8% & Standard deviation = 12%

Expected return on B = 12 & Standard deviation = 20%

Correlation between A & B = 0.40

(1). Weight of the bond in the portfolio = Amount invested in bonds / Total investment in portfolio

therefore, Weight of the bond = $200,000 / $500,000 = 40% or 0.40

(2). Weight of the stocks in the portfolio = Amount invested in stocks / Total investment in the portfolio

Therefore, Weight of stocks = $300,000 / $500,000 = 0.60 or 60%

(3). Expected return of the portfolio = Return on security A * weight of security A(bonds) + Return on security B * weight of security B(stocks).

therefore, Expected return of the portfolio = 8% * 40% + 12% + 60% or it can be written in other way in whole number terms as follows.

Expected portfolio return = 0.08 * 0.40 + 0.12 * 0.60 = 0.104 or 10.4%

(4). Now, to calculate portfolio variance we have the following formula and by using it we can easily solve this problem.

Formula for Portfolio variance = W­­₁² * σ²₁ + w₂² * σ₂² + 2 * w₁ * w₂ * ρ_1,2 * σ₁ * σ₂

where, w1 = weight of security A = 40% or 0.40

w2 = weight of security B = 60% or 0.60

σ1 = Standard deviation of security A = 12% or 0.12

σ2 = Standard deviation of security B = 20% or 0.20

ρ1,2 = Correlation between security A & B = 0.40

therefore, from the above formula:

Variance of the Portfolio = 0.40 * 0.12 + 0.60 * 0.20 + 2 * 0.40 * 0.40 * 0.60 * 0.12 * 0.20 = 0.1795 or 17.95%

(5). Standard deviation of the portfolio = Variance of the Portfolio

therefore, we have calculated the variance of the portfolio above as 0.1795

hence, Standard deviation of the Portfolio = 0.1795 = 0.4237 or 42.37%