2.1 Analytically – Part 1

2.1.1 Discrete Random Variable

A random variable is a function, which assigns a real value to each possible outcome of a random event. Random variables can be either discrete or continuous. In the examples below, we distinguish between discrete and continuous random variables.

In this example, the stock return is a random variable that has two possible outcomes: a 30% return if the research effort is successful, and a 5% return if it is not. A random variable, which can take on only a countable number of distinct values, is called discrete random variable. The stock return in our hypothetical example is a discrete random variable as it has only two possible outcomes. In reality, however, stock returns, as well as many economic variables such as insurance losses, profits, inflation, and interest rates are continuous random variables. A continuous random variable has an uncountable set of outcomes.

Probability is the likelihood that a given outcome will occur out of all possible outcomes. The probability can be deduced or computed using a controlled experiment. Consider, for example, the outcomes from a coin toss: you will either land heads or tails. If it is a fair coin then these outcomes are equally likely, and the probability of getting heads (or tails) on a given flip will be ½. Alternatively, you can toss the coin 1,000 times and, assuming it is fair, about half of the time you will observe heads (tails).

Probabilities are a number between 0 and 1. A probability of 0 indicates that the outcome cannot occur while a probability of 1 indicates that the outcome will occur with certainty.

A random variable has a probability distribution. The probability distribution of a discrete random variable lists all the possible outcomes and their corresponding, objectively known, probabilities. The outcomes must be mutually exclusive meaning that only one of these outcomes can occur and exhaustive in the sense that no other outcome is possible. If we know all the possible states of the world, their probabilities, and the values that the random variable will take in each state, we can calculate ex ante (before the event) its expected return and variance.

Table 1 shows the probability distribution of the RedPill-M rate of return. There are two possible states of the world: success and failure. We do not know which outcome will occur, but we do know that one of them will occur because the outcomes are mutually exclusive and exhaustive.

Event tree diagrams are often used to visualize and make decision in uncertain situations. Figure 1 depicts the event tree for Example 2. The number of branches that radiate from the initial node correspond to the possible outcomes of buying the stock, with the probability of each outcome shown along the branch. The end nodes show all possible outcomes.

Two major statistics describing the probability distribution of a random variable are the expected value and standard deviation. If [latex]p_{1}[/latex] denotes the probability of success and [latex]p_{2}[/latex] the probability of failure, the expected return in our example, [latex]\overline{R}[/latex], can be calculated as the probability weighted average of the two possible values that the stock return can take on:

$$\overline{R}=p_1\times 30\%+p_2\times 5\%=0.6\times 30\%+0.4\times 5\%=18\%+2\%=20\%$$

Notice that the expected value need not be one of the possible outcomes of the discrete random variable. We will never realize a rate of return of 20% – our actual rate of return will be either 30% or 5%.

As the outcomes are mutually exclusive and exhaustive, if you add up the probabilities of all possible scenarios, they should sum to 1. More generally, if [latex]X[/latex] is a random variable with two possible outcomes, [latex]X_{1}[/latex] and [latex]X_{2}[/latex], and their probabilities are
[latex]p_{1}[/latex] and [latex]p_{2}[/latex], respectively, the expected value of [latex]X[/latex] is

$E\left(X\right)=\overline{X}=p_1\times X_1+p_2\times X_2$

In the general case, if we denote the state of the world with [latex]s[/latex] and we have [latex]n[/latex] possible states of the world, the expected value of the random variable is the probability-weighted average of all the values the random variable can take on

where

$p_1+p_2+...+p_n=\sum _{s=1}^n{p}_s=1$

where [latex]p_{s}[/latex] is the probability of state [latex]s[/latex] occurring. One useful way to think about risk is in terms of how dispersed the values of a random variable are around its mean. The more spread out the values, the more uncertain the outcome. The variance and standard deviation of a random variable are measures of dispersion around the mean, and the standard deviation is commonly used as a measure of risk. The variance of a discrete random variable [latex]X[/latex], [latex]Var(X)[/latex], is given by the probability-weighted sum of the squared deviations of [latex]X[/latex] from its expected value:

And so, we have,

$Var\left(X\right)=p_1\times {\left(X_1–\overline{X}\right)}^2+\cdots +p_n\times {\left(X_n–\overline{X}\right)}^2=\sum _{s=1}^n{p}_s{\left(X_s–\overline{X}\right)}^2$

The standard deviation is then calculated as the square root of the variance

If we denote rates of return during success and failure with [latex]R_{1}[/latex] and [latex]R_{2}[/latex], respectively, the variance of the RedPill-M stock return can be computed as the probability-weighted average of the squared deviations of returns, [latex]R[/latex], from their mean value of 20%:

${\sigma }^2=p_1\times {\left(R_1–\overline{R}\right)}^2+p_2\times {\left(R_2–\overline{R}\right)}^2$

So, in this example we have,

${\sigma }^2=0.6{\times \left(30\%–20\%\right)}^2+0.4\times {(5\%–20\%)}^2=1.14\%$

Then, the standard deviation is

$\sigma =\sqrt{1.14\%}\approx 10.68$

Table 2 summarizes these calculations.

We said that the standard deviation of a random variable is conventionally used to measure the degree of risk. The higher the standard deviation, the riskier your position is considered to be. Notice, however, that the standard deviation considers both positive and negative deviations from the mean. While we welcome positive deviations from the mean, it is negative deviations or downside risk that we are concerned about in this context. In a different context, e.g., if we are looking at insurance losses, it will be the positive deviations from the mean that will be of concern.