Question

Compute the expected return \(\mu_{V}\) and standard deviation \(\sigma_{V}\) of a portfolio consisting of three securities with weights \(w_{1}=40 \%, w_{2}=-20 \%\) \(w_{3}=80 \%\), given that the securities have expected returns \(\mu_{1}=8 \%\) \(\mu_{2}=10 \%, \mu_{3}=6 \%\), standard deviations \(\sigma_{1}=1.5, \sigma_{2}=0.5, \sigma_{3}=1.2\) and correlations \(\rho_{12}=0.3, \rho_{23}=0.0, \rho_{31}=-0.2\).

Short answer

\mu_V = 6\%, \sigma_V \approx 1.0587

Step-by-step solution

- Calculate the Expected Return

The expected return \(\mu_V\) of the portfolio is calculated by the weighted sum of the expected returns of the individual securities. This can be expressed as: \[ \mu_V = w_1 \mu_1 + w_2 \mu_2 + w_3 \mu_3 \] Substituting the given values: \[ \mu_V = 0.4 \times 8\% + (-0.2) \times 10\% + 0.8 \times 6\% \] Solve the equation: \[ \mu_V = 3.2\% - 2\% + 4.8\% = 6\% \]

- Calculate the Variance of the Portfolio

To calculate the variance \(\sigma_V^2\) of the portfolio, use the formula: \[ \sigma_V^2 = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + w_3^2 \sigma_3^2 + 2w_1 w_2 \sigma_1 \sigma_2 \rho_{12} + 2w_2 w_3 \sigma_2 \sigma_3 \rho_{23} + 2w_3 w_1 \sigma_3 \sigma_1 \rho_{31} \] Substituting the given values: \[ \sigma_V^2 = (0.4)^2 \times (1.5)^2 + (-0.2)^2 \times (0.5)^2 + (0.8)^2 \times (1.2)^2 + 2 \times 0.4 \times -0.2 \times 1.5 \times 0.5 \times 0.3 + 2 \times -0.2 \times 0.8 \times 0.5 \times 1.2 \times 0 + 2 \times 0.8 \times 0.4 \times 1.2 \times 1.5 \times -0.2 \] Solve the equation: \[ \sigma_V^2 = 0.36 + 0.01 + 0.9216 - 0.036 + 0 + (-0.1152) = 1.1204 \]

- Calculate the Standard Deviation

The standard deviation \(\sigma_V\) is the square root of the variance: \[ \sigma_V = \sqrt{\sigma_V^2} = \sqrt{1.1204} \approx 1.0587 \]

Key concepts

expected return calculation

Understanding the calculation of expected return is fundamental in portfolio management.
It helps investors predict the future returns of a portfolio consisting of different securities.

The expected return \(\mu_V\) of a portfolio is calculated by taking the weighted sum of the expected returns of individual securities. Here, you allocate certain weights to each security based on the proportion of your total investment.

The formula for calculating the expected return is:
\[ \mu_V = w_1 \mu_1 + w_2 \mu_2 + w_3 \mu_3 \]
This means multiplying each security's expected return \(\mu\) by its respective weight \(w\), and then summing them all up.

**Example Calculation:**
Given weights: \(w_1 = 0.4 \, w_2 = -0.2 \, w_3 = 0.8\), and expected returns: \(\mu_1 = 8\%, \mu_2 = 10\%, \mu_3 = 6\%\).
Plug these values into the formula:
\[ \mu_V = 0.4 \times 8\% + (-0.2) \times 10\% + 0.8 \times 6\% \]
Solve the equation:
\[ \mu_V = 3.2\% - 2.0\% + 4.8\% = 6.0\% \]

So, the portfolio's expected return is 6\%.

portfolio variance

Calculating the portfolio variance is essential to understand the risk associated with the portfolio. It gives an indication of the portfolio's total variability.

The variance of a portfolio, \(\sigma_V^2\), involves multiple calculations not just within each security but between pairs of securities as well.

The formula for the portfolio variance is more complex and accounts for individual variances and covariances between securities:
\[ \sigma_V^2 = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + w_3^2 \sigma_3^2 + 2w_1 w_2 \sigma_1 \sigma_2 \rho_{12} + 2w_2 w_3 \sigma_2 \sigma_3 \rho_{23} + 2w_3 w_1 \sigma_3 \sigma_1 \rho_{31} \]

In this formula, \(\sigma\) represents the standard deviation, \(w\) the weights, and \(\rho\) the correlation coefficients between pairs of securities.

**Example Calculation:**
Given values: \(w_1 = 0.4, w_2 = -0.2, w_3 = 0.8\); \(\sigma_1 = 1.5, \sigma_2 = 0.5, \sigma_3 = 1.2\); and correlations: \(\rho_{12} = 0.3, \rho_{23} = 0.0, \rho_{31} = -0.2\).
Substitute these into the variance formula:
\[ \sigma_V^2 = (0.4)^2 \times (1.5)^2 + (-0.2)^2 \times (0.5)^2 + (0.8)^2 \times (1.2)^2 + 2 \times 0.4 \times -0.2 \times 1.5 \times 0.5 \times 0.3 + 2\times-0.2\times0.8\times0.5\times1.2\times0+2\times0.8\times0.4\times1.2\times1.5\times-0.2 \]
Solve the equation step-by-step:
\[ \sigma_V^2 = 0.36 + 0.01 + 0.9216 - 0.036 + 0 + (-0.1152) = 1.1204 \]

So, the portfolio's variance is approximately 1.1204.

standard deviation

Standard deviation is a statistical measure of the dispersion or spread in a set of values. For a portfolio, it shows how much the return on the portfolio is expected to vary.
In other words, it helps investors understand the level of risk or volatility associated with the portfolio's expected return.

Calculating the standard deviation, \(\sigma_V\), involves taking the square root of the portfolio variance.
The formula is straightforward once you have the variance calculated:
\[ \sigma_V = \sqrt{\sigma_V^2} \]

**Example Calculation:**
From previous steps, if the portfolio variance \(\sigma_V^2\) is 1.1204, then the standard deviation is:
\[ \sigma_V = \sqrt{1.1204} \]
Solving for the square root gives approximately 1.0587.

So, the standard deviation of the portfolio is about 1.0587.
This value represents the average amount by which the returns of the portfolio deviate from the expected return.