Portfolio Return and Risk | CFA Level I Quantitative Methods

We’re diving into the exciting world of portfolio return and risk, covering topics like variance, covariance, and correlation. Buckle up, because this lesson is a wild ride!

Expectation of Multiple Variables

Previously, we learned how to calculate the expectation of a single random variable X. Now, we’re going to extend this concept to multiple variables, each with their own weights. Here’s a useful property: the expected value of a weighted sum of random variables equals the weighted sum of the expected values. Let’s see how this applies to expected portfolio returns.

In financial markets, returns on different assets are likely not independent of each other. So, we have to evaluate assets on a joint probability basis. Let’s assume we have the following joint probabilities:

By calculating the expected returns for X and Y, we can then compute the expected return of the entire portfolio using the weighted sum of expected values:

E(X) = 6.5%
E(Y) = 3%
wx = 0.7
wy = 0.3
Expected Portfolio Return = (0.7 * 6.5%) + (0.3 * 3%) = 4.55% + 0.9% = 5.45%

Portfolio Variance and Covariance

Analysts are also concerned with the variance of the portfolio, which indicates risk. Since the returns of the two assets cannot be assumed independent, we have to consider covariance. Covariance is a measure of how two assets move together. Using the joint probabilities, we can calculate the covariance between X and Y:

There are three important properties of covariance:

1. Covariance is the general representation of variance.
2. Covariance of a random variable with itself is equal to its variance.
3. Covariance can range from negative infinity to positive infinity.

To make covariance easier to interpret, we can calculate the correlation coefficient. A correlation coefficient ranges from -1 to 1, with 0 indicating no linear relationship between the random variables. In our example:

This strong negative correlation indicates that when one asset outperforms, the other is likely to underperform.

Calculating Portfolio Variance

Now that we have the covariance, variance of X, and variance of Y, how do we compute the variance of the portfolio? The formula for a 2-asset portfolio is as follows:

Variance(Portfolio) = w_x² * Variance(X) + w_y² * Variance(Y) + 2 * w_x * w_y * Cov(X, Y)

Using this formula, we can determine the variance of our example portfolio. Let’s say we have the following data:

Now that we have the portfolio variance, we can calculate the portfolio standard deviation (risk), which is the square root of the variance:

Std. Deviation(Portfolio) = √55.04 = 7.4

This tells us that our portfolio’s risk is 7.4%.

Working with Covariance and Correlation Matrices

In the exam, you might encounter covariance and correlation matrices. They can help you calculate the variance of the portfolio without going through each individual covariance calculation. They’re especially helpful when dealing with multi-asset portfolios. To calculate the portfolio variance using these matrices, follow these steps:

Create a covariance matrix or correlation matrix. Multiply the transpose of the weights matrix by the covariance matrix or the correlation matrix. Multiply the result by the original weights matrix.

Let’s use the same example with assets X and Y, weights w_x = 0.7 and w_y = 0.3, and their covariance matrix:

140.25	-34.5
-34.5	9

Follow the steps above:

Multiply the transpose of the weights matrix (0.7, 0.3) by the covariance matrix. Multiply the result by the original weights matrix (0.7, 0.3).

After these calculations, you’ll find that the portfolio variance is 55.04, which matches our previous result.

A correlation matrix is a useful tool to understand the relationships between multiple assets in a portfolio. It is a square matrix that provides the correlation coefficients between each pair of assets, showing the extent to which they move together. Correlation coefficients range from -1 to 1, with -1 indicating perfect negative correlation, 1 indicating perfect positive correlation, and 0 indicating no correlation.

The correlation matrix is related to the covariance matrix, as both provide insight into the co-movement of assets. The main difference is that the covariance matrix provides the actual covariance values, while the correlation matrix standardizes these values by dividing the covariance of each pair of assets by the product of their standard deviations.

Key Takeaways

In this lesson, we learned the following important concepts:

1. Calculating the expected return of a portfolio using the weighted sum of expected values.
2. Covariance and correlation coefficients, which help measure how two assets move together.
3. Calculating the variance and standard deviation of a portfolio using the weights, variances, and covariance of the assets.

With this knowledge, you can better analyze and manage the risk and return of investment portfolios. Remember, diversification is key, and understanding the relationships between assets helps achieve optimal risk-adjusted returns.