# Level 1 CFA® Exam:

Properties of Random Variables

What is an expected value?

An expected value is the expected result of an event. In simple terms, it's the average value for an event.

The expected value of a random variable, for example, a rate of return, is the probability-weighted average of the possible outcomes of the random variable.

Expected value is denoted as \(E\), for example, \(E(X)\) will mean the expected value of a random variable \(X\). If, for instance, a rate of return can take three equally likely values, say 5%, 10%, and 15%, we can intuitively tell that the expected value of the rate of return is 10%. Because one-third of 5% plus one-third of 10% plus one-third of 15% gives us 10%:

\(E(X)=\frac{1}{3}\times5\%+\frac{1}{3}\times5\%+\frac{1}{3}\times5\%=10\%\)

The generalized formula for an expected value of a random variable to be used in your level 1 CFA exam looks as follows:

\(E(X)=\sum_{i=1}^nP(X_i)\times X_i\)

- \(E(X)\) - expected value
- \(n\) - number of possible outcomes
- \(X_i\) - possible outcome i
- \(P(X_i)\) - probability of outcome i

The expected value of a random variable equals the sum of the products of the possible values of the variable multiplied by their probabilities. Note that the sum of all probabilities must equal 1.

### Conditional Expected Value

Let's consider a random variable X and a scenario S. Suppose the random variable is the return on investment, and the assumed scenario is an increase in GDP of over 4% in a period of time. Conditional expected value can tell us the expected value of the random variable X given scenario S.

\(E(X|S) = P(X_{1}|S)\times X_{1} + \\+P(X_{2}|S)\times X_{2} + \ldots + P(X_{n}|S)\times X_{n}\)

- \(E(X|S)\) - expected value of a random variable \(X\) given scenario \(S\)
- \(P(X_{i}|S)\) - probability of an outcome \(X_{i}\) given scenario \(S\)
- \(X_i\) - one of possible outcomes
- \(n\) - number of possible outcomes

If we now define scenarios that are mutually exclusive and exhaustive, we'll be able to compute the expected value of the variable using the total probability rule for expected value.

Let’s take two scenarios, namely:

- \(S_1\) – increase in GDP in the analyzed period will exceed 4%, and
- \(S_2\) – increase in GDP in the analyzed period will be equal to or less than 4%.

The expected value of a random variable equals:

\(E(X)=E(X|S_1)\times{P(S_1)}+ E(X|S_2)\times{P(S_2)}\)

The generalized formula (for 'n' scenarios) looks as follows:

\(E(X)=\sum_{i=1}^nE(X|S_i)\times P(S_i)\)

- \(E(X)\) - expected value
- \(E(X|S_i)\) - expected value of \(X\) given scenario \(S_i\)
- \(P(S_i)\) - probability of scenario \(S_i\), \(\sum_{i=1}^nP(S_i)=1\)

Scenarios \(S_i\) are mutually exclusive and exhaustive.

To get a better grasp of the total probability rule for expected value, we can use a tree diagram.

Suppose we want to find out the expected value of the stock of a U.S. company. With a probability of 40%, we assume that demand for the company's products and services will increase. If this scenario occurs, the stock price will eventually reach USD 100 with a probability of 80%, and USD 90 with a probability of 20%.

Another scenario involves a decrease in the sales of the company's products and services or no change in the sales. In this scenario, the price will be USD 70 with a probability of 0.3 or USD 50 with a probability of 0.7.

(...)

Note: if you want to use the total probability rule for expected value, certain conditions must be met. First, the scenarios have to be mutually exclusive and exhaustive. Second, the values of a random variable given a scenario should cover the entire possible set of values and be exclusive.

Let's go back to the example we discussed when dealing with the expected value. Recall that for 3 equally probable returns of 5%, 10%, and 15%, the expected value was 10%. What is worth noticing is that the expected value tells us nothing about the dispersion of returns. If we take another example of 3 equally likely rates of return, let’s say -30%, 10%, and 50%, the expected value will again equal 10%, but returns will be more dispersed. The more dispersed returns, the riskier the investment. The risk is measured by variance and standard deviation. Here’s how to compute these measures for a random variable.

The variance of a random variable is the expected value of squared deviations from the random variable’s expected value:

\(\sigma^2(X)=E\left\{[X-E(X)]^2\right\}\)

- \(\sigma^2(X)\) - variance of a random variable \(X\)
- \(E(X)\) - expected value of a random variable \(X\)

Standard deviation is the square root of variance.

The formula for the variance of a random variable that will be more useful for you in your CFA exam looks as follows:

\(\sigma^2(X)=\sum_{i=1}^nP(X_i)\times [X_i-E(X)]^2\)

- \(\sigma^2(X)\) - variance of a random variable \(X\)
- \(P(X_i)\) - probability of one of possible outcomes
- \(E(X)\) - expected value of a random variable \(X\)

Compute the variance and standard deviation for two random variables:

- a random variable B with equally probable returns of 5%, 10%, and 15%,
- a random variable C with equally probable returns of -30%, 10%, and 50%.

(...)

The expected return on a portfolio is the sum of the products of the expected returns on assets included in the portfolio and their weights:

\(E(R_p)=E(w_{1}\times R_{1} + w_{2}\times R_{2} + \ldots + w_{n}\times R_{n}) =\\= w_{1}\times E(R_{1}) + w_{2}\times E(R_{2}) + \ldots + w_{n}\times E(R_{n})\)

- \(w_{1}, w_{2},\ldots, w_{n}\) - securities' weights in portfolio
- \(R_{1}, R_{2},\ldots, R_{n}\) - returns of portfolio securities
- \(R_p\) - portfolio expected return

Unlike expected return, the variance of a portfolio is not simply a weighted average of the variances of the assets. This is because of the covariance (or correlation) between returns on assets.

The portfolio variance can be expressed with the following formula:

\(\sigma^2(R_P)=\sum_{i=1}^n\sum_{j=1}^n{w_i}\times {w_j}\times Cov(R_i,R_j)\)

- \(\sigma^2(R_P)\) - variance of a portfolio return
- \(w_i\) - weight of asset \(i\) in the portfolio
- \(R_i\) - return of asset \(i\)
- \(Cov(R_i,R_j)\) - covariance between return of assets \(i\) and \(j\)

In the case of 3 assets, the above formula takes the following form:

\(\sigma^{2}(R_{p}) = 2\times{w_{1}}\times w_{2}\times Cov(R_{1},R_{2}) +\\+ 2\times{w_{1}}\times w_{3}\times Cov(R_{1},R_{3}) +\\ + 2\times{w_{2}}\times w_{3}\times Cov(R_{2},R_{3}) +\\+ w^{2}_{1}\times \sigma^{2}(R_{1}) + w^{2}_{2}\times \sigma^{2}(R_{2}) + w^{2}_{3}\times \sigma^{2}(R_{3})\)

- \(R\) - return (random variable)
- \(w\) - weight of security
- \(Cov\) - covariance
- \(\sigma^{2}(R)\) - variance

Covariance is a measure of the linear association between two variables and is given by the following formula:

\(Cov(R_i,R_j)=E[(R_i-E(R_i))\times (R_j-E(R_j))]\)

- \(Cov(R_i,R_j)\) - covariance between random variables \(R_i\) and \(R_j\)
- \(E(R)\) - expected value of a random variable \(R\)

Covariance shows us if deviations from expected values are associated. If they are associated and if both variables deviate above the expected value, or if they simultaneously deviate below the expected value, covariance is positive. If the variables deviate in opposite directions, covariance is negative. The greater the deviations in the same direction (both positive and negative), the greater the covariance. A covariance of zero shows that there is no linear association between the variables. Note, however, that the absence of linear association does not mean that the variables are not associated. There may be a non-linear association between them.

Covariance can take any value from minus infinity to plus infinity and it's an intermediate step in computing the correlation coefficient, which is easier to interpret. Correlation coefficient can take on values ranging from (-1) to (+1). The relationship between covariance and the correlation coefficient can be expressed as follows:

\(Cov(R_{i},R_{j}) = \rho_{i,j}\times \sigma_{i} \times \sigma_{j}\)

- \(Cov(R_{i},R_{j})\) - covariance between the returns on asset classes "i" and "j"
- \(\rho_{i,j}\) - correlation between the returns of asset classes "i" and "j"
- \(\sigma_{i}\) - risk of assets class "i"
- \(\sigma_{j}\) - risk of assets class "j"

(...)

The table shows the values of two random variables, namely the return on stock A and the return on stock B, and the associated probabilities. We also have two mutually exclusive and exhaustive scenarios and their probabilities.

Return on stock A (%) | Return on stock B (%) | Probability of scenario | |
---|---|---|---|

Scenario 1 | 30 | 20 | 0.75 |

Scenario 2 | 10 | 5 | 0.25 |

Expected return | 25 | 16.25 |

What is the value of the covariance?

To calculate covariance, let's put the necessary steps into another table.

Scenario 1 | Scenario 2 | COV | |
---|---|---|---|

Deviations of stock A returns (%) | \(30-25=5\) | \(10-25=-15\) | |

Deviations of stock B returns (%) | \(20-16.25=3.75\) | \(5-16.25=-11.25\) | |

Product of deviations | \(5\times3.75=18.75\) | \((-15)\times{(-11.25)}=168.75\) | |

Probability of scenario | \(0.75\) | \(0.25\) | |

Probability-weighted product of deviations | \(0.75\times18.75=14.0625\) | \(0.25\times168.75=42.1875\) | |

\(56.25\) |

To arrive at covariance, we first need to compute the deviation of the variable from the expected value for both variables in both scenarios. So, in total, we need to find 4 deviations. What we have to do next is multiply the relevant deviations by one another and by the probability of the given scenario. What we get are the so-called probability-weighted products of deviations. Finally, we need to add these products to arrive at covariance.

In our example covariance is therefore:

\(Cov(R_{A},R_{B})=\\=P(\text{Scenario 1})\times{(30-25)\times(20-16.25)}+\\+P(\text{Scenario 2})\times{(10-25)\times(5-16.25)}=\\=0.75\times18.75+0.25\times168.75=56.25\)

Remember! To use the joint probability function, the probabilities for the scenarios must be the same for each variable.

- The expected value of a random variable is the probability-weighted average of the possible outcomes of the random variable.
- The variance of a random variable is the expected value of squared deviations from the random variable’s expected value.
- Standard deviation is the square root of variance.
- The expected return on a portfolio is the sum of the products of the expected returns on assets included in the portfolio and their weights.
- Covariance is a measure of the linear association between two variables.
- A correlation coefficient of +1 means that there is a perfect positive correlation and a linear association between the variables.
- A correlation coefficient of -1 means that there is a perfect negative correlation and a linear association between the variables.
- If the correlation coefficient equals zero, the variables are not correlated, so there is no linear association between them.
- The lower the correlation between pairs of assets in a portfolio, the lower the variance of the portfolio.