Avoiding Underestimation of the Risk and Overestimation of Average Profit by a Meaningful Interpretation of the Chebyshev Sum Inequality

Chebyshev’s sum inequality. Letand b_u....b_n be two sets of similarly ordered positive numbers (for example, a, <,...,< a„ and b_{ <,...,< b„). Then

If a, >,...,> a„ and b_{ <,...,< b„ are oppositely ordered positive numbers, the inequality is reversed:

Chebyshev’s sum inequality possesses a great advantage. It provides the opportunity to segment an initial complex expression into simpler expressions. Thus, the complex terms а,Д in (7.10) and (7.11) are segmented into simpler terms involving a, and b,.

The segmentation capability offered by the Chebyshev inequality will be illustrated with an example related to the danger of underestimating risk by using average values.

Suppose that a new meaning for the variables in the Chebyshev inequality (7.10) is created. The variables a, (/' = 1,...,/;) stand for the probabilities of a loss corresponding to each of n investments while the variables b, (i = l,...,n) stand for the sizes of the investments and correspond to the probabilities a,. The right-hand side of inequality 7.10 can now be interpreted as an estimate of the risk (expected potential loss) ab, made by using the average size of the investment b = (/;, + b₂ + ... + b„)/n and the average probability of a loss a = (1 //?)(<:/, + ... + «„). The left side of inequality (7.10) can be interpreted as the average of the expected potential loss from all investments.

From inequality (7.10), it can be seen that estimating the risk (expected potential loss) by using the average size of the investment and the average probability of the loss of an investment results in an underestimation of the expected potential loss.

The underestimation of the expected potential loss (the risk) is dangerous and will be illustrated by a very simple numerical example involving only two investments: an investment of size c, =$800 and an investment of size c₂ =$15000. The probability of losing the separate investments are p =0.08 and p₂ = 0.16, correspondingly. If the average probability of losing an investment and the average size of the investment are used for calculating the expected potential loss (the risk), the value pxc =0.5x(0.08 + 0.16)x0.5x(800 + 15000) = 948 for the expected potential loss will be predicted. The actual expected potential loss from the two investments is

which is significantly greater than the prediction of 948, based on the average value of the probability of losing an investment and the average size of the investment.

An equally dangerous situation is present in estimating profits with average values.

Suppose that a new meaning for the variables in the Chebyshev sum inequality

(7.11) is created: the variables a, (i = 1,...,/;). ranked in descending order: a_t >,...,> a„ now stand for profit margins corresponding to each of n customer balances represented by the variables b (i = l,...,/i), ranked in ascending order: <,...,< b„ and corresponding to the defined profit margins a_h

Consider a lending institution making profit from lending to customers. The profit margin from charging interest rates on loans to customers is negatively correlated with the size of the customers’ balance. Customers with a small balance have poorer credit ratings, the charged interest is higher and the profit margin is higher. Customers with large balances have a better credit rating, the charged interest is lower and the profit margin is lower.

The right-hand side of inequality (7.11) can now be i nterpreted as the projected expected profit ab, made by using the average size of the balances b = (b_t + b₂ +... + b„)/n and the average profit margin ci = (l/n)(«i + ... + «„). The left-hand side of inequality

(7.11) can be interpreted as the actual average profit received from all n customers. As can be seen from the Chebyshev inequality (7.11), the projected expected profit, estimated by using the average balance b = (ft, + b₂ +... + b„) / n and the average profit margin a = (/n)(ai +... + a„) is overestimated.

The overestimation of the projected expected profit can be dangerous for the lending institution and will be illustrated by a simple numerical example involving only two customers: a customer with balance b_{ = $600 and a customer with balance b₂ =$40000. The profit margins are a, =0.12 and a₂ =0.04 correspondingly. If the average profit margin and the average balance are used for calculating the average profit, the value axb =0.5x(0.12 + 0.04)x0.5x(600 + 40000) = $1624 will be predicted for the average profit. The actual average profit from the two customers is ₌ ".i: ■ <■»» ^ ши ■ 4"»"o _ j₈3_6- which is significantly smaller than the predicted

value.

The conclusion is that for negatively correlated variables, calculating the average of a product by multiplying the averages of the two variables leads to overestimation.

Conversely, according to inequality (7.10), for positively correlated variables, calculating the average of a product by multiplying the averages of the two variables leads to underestimation.