For assets A and B we know the following: E(RA) = 0.10, E(RB) = 0.10, Var(RA) = 0.18, Var(RB) = 0.36 and the correlation of the returns is 0.6. What is the variance of the return of a portfolio that is equally invested in the two assets?

To calculate the variance of the return of a portfolio equally invested in two assets (A and B), we can use the formula for the variance of a two-asset portfolio:

\[
\text{Var}(R_p) = w_A^2 \times \text{Var}(R_A) + w_B^2 \times \text{Var}(R_B) + 2 \times w_A \times w_B \times \text{Cov}(R_A, R_B)
\]

Where:
- \( \text{Var}(R_p) \) is the variance of the portfolio's return.
- \( w_A \) and \( w_B \) are the weights of assets A and B in the portfolio. Since the portfolio is equally invested, \( w_A = w_B = 0.5 \).
- \( \text{Var}(R_A) \) and \( \text{Var}(R_B) \) are the variances of returns of assets A and B, respectively.
- \( \text{Cov}(R_A, R_B) \) is the covariance between the returns of assets A and B.

### Step 1: Calculate the Covariance
The covariance \( \text{Cov}(R_A, R_B) \) can be calculated using the formula:

\[
\text{Cov}(R_A, R_B) = \rho_{A,B} \times \sqrt{\text{Var}(R_A)} \times \sqrt{\text{Var}(R_B)}
\]

Where:
- \( \rho_{A,B} \) is the correlation coefficient between the returns of A and B.
- \( \text{Var}(R_A) = 0.18 \) and \( \text{Var}(R_B) = 0.36 \).
- \( \rho_{A,B} = 0.6 \).

Let's compute the covariance:

\[
\text{Cov}(R_A, R_B) = 0.6 \times \sqrt{0.18} \times \sqrt{0.36}
\]

\[
\text{Cov}(R_A, R_B) = 0.6 \times 0.4243 \times 0.6 = 0.6 \times 0.25458 = 0.1527
\]

### Step 2: Calculate the Portfolio Variance
Now, substitute the values into the portfolio variance formula:

\[
\text{Var}(R_p) = (0.5^2 \times 0.18) + (0.5^2 \times 0.36) + 2 \times 0.5 \times 0.5 \times 0.1527
\]

\[
\text{Var}(R_p) = (0.25 \times 0.18) + (0.25 \times 0.36) + (0.5 \times 0.1527)
\]

\[
\text{Var}(R_p) = 0.045 + 0.09 + 0.07635 = 0.21135
\]

So, the variance of the return of a portfolio that is equally invested in the two assets is approximately **0.21135**.

Would you like to delve deeper into any of these steps, or do you have further questions?

Here are 8 related questions:
1. How do you calculate the expected return of a portfolio with unequal weights?
2. What impact does correlation have on portfolio variance?
3. How can diversification reduce portfolio risk?
4. What is the significance of covariance in portfolio management?
5. Can two assets with a high correlation still provide diversification benefits?
6. What is the difference between systematic and unsystematic risk?
7. How does increasing the weight of one asset affect the portfolio variance?
8. How would the portfolio variance change if the correlation were negative?

**Tip:** Diversification can reduce the overall risk of a portfolio, but the degree of risk reduction depends heavily on the correlation between the assets. Lower correlation generally leads to greater risk reduction.

Learn how to calculate the variance of a portfolio equally invested in assets A and B using their expected returns, variances, and correlation coefficient. Step-by-step solution with detailed formulas and calculations provided.

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