The expected value is a powerful tool that can be used in many spheres, not in only in gambling, but also in business or investments. Let us follow an example that will enable us to make a choice between: which of two products would be better to produce and sell in the market with respect to the expected return and risk.
Suppose that we have got limited funds and therefore we can produce only one of the two products: either the Product A or the Product B. However the same procedure can be used for any number of products. The main criterion for the product choice in our case is maximum expected return. Since we have only two products, we will choose the one with higher expected return.
We assume three possible scenarios of the product success in the market: (very) successful, normal, unsuccessful. Each of the scenarios may occur with some probability that is estimated by us based on our subjective opinion, experience and/or market research (the total of the probabilities must equal 1). The returns per cent related to their respective scenarios are estimated in the same fashion.
The initial data (the scenarios, their probabilities and returns) and the calculations of the expected return and risk are summarized in the table below. Which of the two products A or B will be preferred and why?
|Product A||Probability pn||Return xn||pnxn||xn-E(x)||[xn-E(x)]2||pn[xn-E(x)]2|
|Expected return E(x)||10%||Variance σ2||112|
|Risk measured as a standard deviation σ (sigma)||10.58 %|
|Product B||Probability pn||Return xn||pnxn||xn-E(x)||[xn-E(x)]2||pn[xn-E(x)]2|
|Expected return E(x)||10%||Variance σ2||288.0|
|Risk measured as a standard deviation σ (sigma)||16.97%|
How can these table data be interpreted? If the product A is successful, it will reach the return 20% p.a. However the probability of this scenario is 0.3 (or 30% if you like). If the product A is normally successful – there is 0.5 or 50% likelihood – the return will be 12%. If the product is unsuccessful, the return will be negative –10%, but the likelihood is 0.2 or 20% only.
We can see that the probabilities are weights for the scenarios to happen (again the sum of the weights must equal 1 or 100%). Thus the expected return E(x) is a weighted average of the returns, whereas the weights are the probabilities. But let us not jump ahead.
Just a brief look at the characteristics of the product B (its probabilities and returns) indicates that it deals with a much more aggressive product. It will deliver very high return 42% if the product is successful, but the probability is 0.1 or 10% only. On the other side the probability of failure is quite big 0.5, or 50%, and the product's return would be –6% in that case.
Now let us try to "judge" the products and choose the one with the higher expected return. The expected return of the product A =
0.3 ˟ 20 + 0.5 ˟ 12 + 0.2 ˟ (–10) = 6 + 6 + (–2) = 10%. But we can find out easily that the return of the product B is 10% as well:
0.1 ˟ 42 + 0.4 ˟ 22 + 0.5 ˟ (–6) = 4 + 9 + (–3) = 10%. Are the products equally good?
Let us assess the risk as well. The risk can be defined as a possible deviation from the expected outcome. The risk can be quantified by means of probability and it can be measured by a standard deviation. We can arrive at it and explain it by the following three formulas.
This says how much are the returns of individual scenarios deviated from the expected return. For instance in case of the product A: successful scenario: 20 – 10 = 10, normal: 12 – 10 = 2, unsuccessful: (–10) – 10 = –20.
The preceding formula, i.e. the deviations from the expected return must be squared first (if we added them up now, we would receive a classic variance). Why do we square the deviations? The answer is: In order to prevent positive and negative values (returns) from eliminating each other and thus distorting the real variance or deviation. This will be clear by the following sub-example.
Imagine that we have only three values:
–1, 0, 1. The mean value is 0 and the deviations as follows:
–1 – 0 = –1,
0 – 0 = 0 and
1 – 0 = 1. If we sum up the deviations
–1 + 0 + 1 the result is
0, but the variance around the mean value (0) is not null! Therefore the deviations must be squared, so that the negative deviations become positive and only after that they can be added up:
(–1)2 + 02 + 12 = 1 + 0 + 1 = 2, thus this is a classic and true variance!
At last we extract the root of the variance (i.e. the sum of the deviations) and receive the standard deviation
1.4142 – this is how much are the values "in average" spread around the mean value.
Finally the squared values in the previous step must be multiplied (weighted) by the probabilities and added up afterwards. By this we receive the (weighted) variance and, by extracting the root of the variance, the (weighted) standard deviation that is the desired representative of the risk. The risk of the product A is
10.58%, while the risk of the product B is
16.67%. The higher the standard deviation is, the greater the risk is (of deviation from the expected return).
Both products have the same expected return
10%. On the basis of the "maximum return" criterion we cannot determine, which of the products is more favorable. Therefore we apply an extra rule that is valid generally: if the returns of the products (or investments) are equal, we choose the one with lower risk. The standard variation or the risk of the product A
10.58% < 16.97% (the product B), therefore we prefer the product A to the product B.
In the world of investments, and production is a part of that, it holds true that a high return usually bears a high risk and vice versa. If you are not willing to risk a lot, you have to content yourself with a lower return. Whether you prefer high returns at the expense of high risks or low returns at the expense of low risks, is dependable on your aversion to risk.
By means of the standard deviation, marked as sigma, the Greek letter, so called rule "one sigma" and assuming normal probability distribution, we can determine the interval, within which the returns of the products will occur with the likelihood of
0.68 or 68%. The lower interval is the expected return minus the standard validation, while the upper interval is the expected return plus the standard deviation.
Therefore we can say that, with the likelihood 0.68 or 68%, the per cent return of the product A will lie in the interval
<10 – 10.58; 10 + 10.58>, that is