The problem describes a portfolio optimization scenario with two stocks, X and Y. The expected returns and standard deviations of the two stocks are given, along with the correlation coefficient between their returns. An investor named Naya with a risk aversion coefficient of 6 wants to select the optimal portfolio. We need to find the optimal weight of stock X in the portfolio, the expected return of the optimal portfolio, and the standard deviation of the optimal portfolio.

1. Problem Description

The problem describes a portfolio optimization scenario with two stocks, X and Y. The expected returns and standard deviations of the two stocks are given, along with the correlation coefficient between their returns. An investor named Naya with a risk aversion coefficient of 6 wants to select the optimal portfolio. We need to find the optimal weight of stock X in the portfolio, the expected return of the optimal portfolio, and the standard deviation of the optimal portfolio.

2. Solution Steps

First, let's denote the following:

$E(R_X) = 0.08$, the expected return of stock X

$E(R_Y) = 0.12$, the expected return of stock Y

$\sigma_X = 0.075$, the standard deviation of stock X

$\sigma_Y = 0.16$, the standard deviation of stock Y

$\rho_{XY} = 0.20$, the correlation coefficient between stock X and Y

$A = 6$, the risk aversion coefficient

We need to find the optimal weight of stock X, $w_X$, in Naya's portfolio. The weight of stock Y will be $w_Y = 1 - w_X$.

The expected return of the portfolio is:

$E(R_p) = w_X E(R_X) + w_Y E(R_Y) = w_X(0.08) + (1-w_X)(0.12) = 0.12 - 0.04w_X$

The variance of the portfolio is:

$\sigma_p^2 = w_X^2 \sigma_X^2 + w_Y^2 \sigma_Y^2 + 2 w_X w_Y \rho_{XY} \sigma_X \sigma_Y$

$\sigma_p^2 = w_X^2 (0.075)^2 + (1-w_X)^2 (0.16)^2 + 2 w_X (1-w_X) (0.20) (0.075) (0.16)$

$\sigma_p^2 = 0.005625 w_X^2 + 0.0256 (1 - 2w_X + w_X^2) + 0.0048 w_X (1 - w_X)$

$\sigma_p^2 = 0.005625 w_X^2 + 0.0256 - 0.0512 w_X + 0.0256 w_X^2 + 0.0048 w_X - 0.0048 w_X^2$

$\sigma_p^2 = 0.026425 w_X^2 - 0.0464 w_X + 0.0256$

The standard deviation of the portfolio is:

$\sigma_p = \sqrt{\sigma_p^2} = \sqrt{0.026425 w_X^2 - 0.0464 w_X + 0.0256}$

To find the optimal weight, we need to maximize the utility function:

$U = E(R_p) - \frac{1}{2} A \sigma_p^2$

$U = (0.12 - 0.04w_X) - \frac{1}{2} (6) (0.026425 w_X^2 - 0.0464 w_X + 0.0256)$

$U = 0.12 - 0.04w_X - 3 (0.026425 w_X^2 - 0.0464 w_X + 0.0256)$

$U = 0.12 - 0.04w_X - 0.079275 w_X^2 + 0.1392 w_X - 0.0768$

$U = -0.079275 w_X^2 + 0.0992 w_X + 0.0432$

To maximize U, we take the derivative with respect to $w_X$ and set it to zero:

$\frac{dU}{dw_X} = -2(0.079275) w_X + 0.0992 = 0$

$-0.15855 w_X + 0.0992 = 0$

$0.15855 w_X = 0.0992$

$w_X = \frac{0.0992}{0.15855} \approx 0.6256$

So, the optimal weight of stock X is approximately 0.

6

2

5

6.

Now we can calculate the expected return of the optimal portfolio:

$E(R_p) = 0.12 - 0.04w_X = 0.12 - 0.04(0.6256) = 0.12 - 0.025024 = 0.094976 \approx 0.0950$ or 9.50%

And the variance of the optimal portfolio:

$\sigma_p^2 = 0.026425 w_X^2 - 0.0464 w_X + 0.0256 = 0.026425 (0.6256)^2 - 0.0464(0.6256) + 0.0256$

$\sigma_p^2 = 0.026425 (0.39137136) - 0.029038 + 0.0256 = 0.010341 - 0.029038 + 0.0256 = 0.006903$

The standard deviation of the optimal portfolio:

$\sigma_p = \sqrt{0.006903} \approx 0.0831$ or 8.31%

3. Final Answer

1. Weight of X: 0.6256

2. Expected return: 9.50%

3. Standard deviation: 8.31%