Calculating Covariance Given a Joint Probability Function

Calculating Covariance Given a Joint Probability Function

Covariance between variables can be calculated in two ways. One method is the historical sample covariance between two random variables \(X_i\) and \(Y_i\). It is based on a sample of past data of size n and is given by:

$$\text{Cov}_{X_i,Y_i}=\frac{\sum_{i=1}^{n}{(X_i -\bar{X})(Y_i -\bar{Y})}}{n-1}$$

Alternatively, covariance can be defined as probability-weighted average of the cross-products of each random variable’s deviation from its own expected value. That is:

$$\text{Cov}_{X_i,Y_i}=E\left[(X_i -\bar{X})(Y_i -\bar{Y})\right]$$

Consider the following example:

Example

Assume that we want to find the variance of each asset and the covariance between the returns of ABC and XYZ, given that the amount invested in each company is $1,000.

This table is used to calculate the expected returns:

$$ \begin{array}{c|c|c|c} & \textbf{Strong Economy} & \textbf{Normal Economy} & \textbf{Week Economy} \\ \hline \text{Probability} & {15\%} & {60\%} & {25\%} \\ \hline \text{ABC Returns} & {40\%} & {20\%} & {0} \\ \hline \text{XYZ Returns} & {20\%} & {15\%} & {4\%} \\ \end{array} $$


Solution

For us to find the covariance, we must calculate the expected return of each asset as well as their variances. The assets weights are:

$$ \text W_{\text{ABC}}=\cfrac {1000}{2000} = 0.5 $$

$$ \text W_{\text{XYZ}}=\cfrac {1000}{2000} = 0.5 $$

Next, we should calculate the individual expected returns:

$$ \text E(\text R_{\text{ABC}}) = 0.15 * 0.40 + 0.60 * 0.2 + 0.25 * 0.00 = 0.18 $$

$$ \text E(\text R_{\text{XYZ}}) = 0.15 * 0.2 + 0.60 * 0.15 + 0.25 * 0.04 = 0.13 $$

Finally, we can compute the covariance between the returns of the two assets:

$$ \begin{align*}
\text{Cov}(\text R_{\text{ABC},\text{XYZ}}) &= 0.15(0.40 – 0.18)(0.20 – 0.13) \\
& + 0.6(0.20 – 0.18)(0.15 – 0.13) \\
& + 0.25(0.00 – 0.18)(0.04 – 0.13) \\
& = 0.0066
\end{align*} $$

Interpretation: since covariance is positive, the two returns show some co-movement, though it’s a weak one.

Question

The following table represents the estimated returns for two motor vehicle production brands – TY and Ford, in 3 industrial environments: strong (50% probability), average (30% probability) and weak (20% probability).

$$ \begin{array}{c|c|c|c} {} & \text{TY Returns +6%} & \text{TY Returns +3%} & \text{TY Returns -1%} \\ \hline {\text{Ford Sales }+10\%} & \text{Strong(0.5)} & {} & {} \\ \hline {\text{Ford Sales }+4\%} & {} & \text{Average(0.3)} & {} \\ \hline {\text{Ford Sales }-4\%} & {} & {} & \text{Weak(0.2)} \\ \end{array} $$

Given the above joint probability function, the covariance between TY and Ford returns is closest to:

A. 0.054.

B. 0.1542.

C. 0.1442.

Solution

The correct answer is C.

First, we must start by calculating the expected return for each brand:

$$ \text{Expected return for TY} = (0.5 * 6\%) + (0.3 * 3\%) + (0.2 * (-1\%)) = 3\% + 0.9\% – 0.2\% = 3.7\% $$

$$ \text{Expected return for Ford} = (0.5 * 10\%) + (0.3 * 4\%) + (0.2 * (-4\%)) = 5\% + 1.2\% – 0.8\% = 5.4\% $$

Next, we can now compute the covariance:

$$ \begin{align*}
\text{Covariance} & = 0.5(6\% – 3.7\%)(10\% – 5.4\%) \\
& + 0.3(3\% – 3.7\%)(4\% – 5.4\%) \\
& + 0.2(-1\% – 3.7\%)(-4\% – 5.4\%) \\
& = 5.29\% + 0.294\% + 8.836\% \\
& = 0.1442 \\
\end{align*} $$

Interpretation: the covariance is positive. This means that the returns for the two brands show some co-movement in the same direction. (This would most likely be the case in real life because the companies are in the same industry and therefore, the systematic risks affecting them are quite similar.)

Reading 8 LOS 8m

Calculate and interpret covariance given a joint probability function.

 

Shop CFA® Exam Prep

Offered by AnalystPrep

Featured Shop FRM® Exam Prep Learn with Us

    Subscribe to our newsletter and keep up with the latest and greatest tips for success
    Shop Actuarial Exams Prep Shop Graduate Admission Exam Prep


    Sergio Torrico
    Sergio Torrico
    2021-07-23
    Excelente para el FRM 2 Escribo esta revisión en español para los hispanohablantes, soy de Bolivia, y utilicé AnalystPrep para dudas y consultas sobre mi preparación para el FRM nivel 2 (lo tomé una sola vez y aprobé muy bien), siempre tuve un soporte claro, directo y rápido, el material sale rápido cuando hay cambios en el temario de GARP, y los ejercicios y exámenes son muy útiles para practicar.
    diana
    diana
    2021-07-17
    So helpful. I have been using the videos to prepare for the CFA Level II exam. The videos signpost the reading contents, explain the concepts and provide additional context for specific concepts. The fun light-hearted analogies are also a welcome break to some very dry content. I usually watch the videos before going into more in-depth reading and they are a good way to avoid being overwhelmed by the sheer volume of content when you look at the readings.
    Kriti Dhawan
    Kriti Dhawan
    2021-07-16
    A great curriculum provider. James sir explains the concept so well that rather than memorising it, you tend to intuitively understand and absorb them. Thank you ! Grateful I saw this at the right time for my CFA prep.
    nikhil kumar
    nikhil kumar
    2021-06-28
    Very well explained and gives a great insight about topics in a very short time. Glad to have found Professor Forjan's lectures.
    Marwan
    Marwan
    2021-06-22
    Great support throughout the course by the team, did not feel neglected
    Benjamin anonymous
    Benjamin anonymous
    2021-05-10
    I loved using AnalystPrep for FRM. QBank is huge, videos are great. Would recommend to a friend
    Daniel Glyn
    Daniel Glyn
    2021-03-24
    I have finished my FRM1 thanks to AnalystPrep. And now using AnalystPrep for my FRM2 preparation. Professor Forjan is brilliant. He gives such good explanations and analogies. And more than anything makes learning fun. A big thank you to Analystprep and Professor Forjan. 5 stars all the way!
    michael walshe
    michael walshe
    2021-03-18
    Professor James' videos are excellent for understanding the underlying theories behind financial engineering / financial analysis. The AnalystPrep videos were better than any of the others that I searched through on YouTube for providing a clear explanation of some concepts, such as Portfolio theory, CAPM, and Arbitrage Pricing theory. Watching these cleared up many of the unclarities I had in my head. Highly recommended.