Scenario analysis and probability distributions

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When we attempt to quantify risk, we begin with the question: What HPRs are possible, and how likely are they? A good way to approach this question is to devise a list of possible economic outcomes, or scenarios, and specify both the likelihood (probability) of each scenario and the HPR the asset will realize in that scenario. Therefore, this approach is called scenario analysis. The list of possible HPRs with associated probabilities is the probability distribution of HPRs. Consider an investment in a broad portfolio of stocks, say, an index fund, which we will refer to as the “stock market.” A very simple scenario analysis for the stock market (assuming only four possible scenarios) is illustrated in Spreadsheet 5.1 . The probability distribution lets us derive measurements for both the reward and the risk of the investment. The reward from the investment is its expected return, which you can think of as the average HPR you would earn if you were to repeat an investment in the asset many times. The expected return also is called the mean of the distribution of HPRs and often is referred to as the mean return. To compute the expected return from the data provided, we label scenarios by s and denote the HPR in each scenario as r ( s ), with probability p ( s ). The expected return, denoted E ( r ), is then the weighted average of returns in all possible scenarios, s 5 1, . . . , S, with weights equal to the probability of that particular scenario.

Each entry in column D of Spreadsheet 5.1 corresponds to one of the products in the summation in Equation 5.6 . The value in cell D7, which is the sum of these products, is therefore the expected return. Therefore, E ( r ) = 10%. Because there is risk to the investment, the actual return may be (a lot) more or less than 10%. If a “boom” materializes, the return will be better, 30%, but in a severe recession the return will be a disappointing - 37%. How can we quantify this uncertainty?

The “surprise” return in any scenario is the difference between the actual return and the expected return. For example, in a boom (scenario 4) the surprise is r (4) - E ( r ) 5=30% -10% 5 20%. In a severe recession (scenario 1), the surprise is r (1) - E ( r ) =  - 37% - 10% = - 47%.

Uncertainty surrounding the investment is a function of both the magnitudes and the  probabilities of the possible surprises. To summarize risk with a single number, we define the variance as the expected value of the squared deviation from the mean (the expected squared “surprise” across scenarios).

We square the deviations because negative deviations would offset positive deviations otherwise, with the result that the expected deviation from the mean return would necessarily be zero. Squared deviations are necessarily positive. Squaring (a nonlinear -transformation) exaggerates large (positive or negative) deviations and deemphasizes small deviations. Another result of squaring deviations is that the variance has a dimension of percent squared. To give the measure of risk the same dimension as expected return (%), we use the standard deviation, defined as the square root of the variance:

Column F of Spreadsheet 5.1 replicates these calculations. Each entry in that column is the squared
deviation from the mean multiplied by the probability of that scenario. The sum of the probabilityweighted squared deviations that appears in cell F7 is the variance, and the square root of that value is the standard deviation (in cell F8).

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