Early probability developers asked how we could combine the various probabilities and outcome values together to obtain a single number reflecting the “value” of the multitude of different outcomes and different consequences of these outcomes. They wanted a single number that summarized in some way the entire probability distribution. In the context of the gambling games of the time when the outcomes were the amount you won in each potential uncertain state of the world, they asserted that this value was the “fair value” of the gamble. We define fair value as the numerical average of the experience of all possible outcomes if you played the game over and over. This is also called the “expected value.” Expected value is calculated by multiplying each probability (or relative frequency) by its respective gain or
loss. [4] It is also referred to as the mean value, or the average value. If X denotes the value that results in an uncertain situation, then the expected value (or average value or mean value) is often denoted by E(X), sometimes also referred to by economists as E(U)—expected utility—and E(G)—expected gain. In the long run, the total experienced loss or gain divided by the number of repeated trials would be the sum of the probabilities times the experience in each state. In Table 2.3 "Opportunity and Loss Assessment
Consequences of New Product Market Entry" the expected value is (.01)×(–10.2) + (.1) × ( −.50) + (.4) × (.1) + (.4) × (1) + (.09) × (8) = 1.008. Thus, we would say the expected outcome of the uncertain situation described in Table 2.3 "Opportunity and Loss Assessment Consequences of New Product Market
Entry" was $1.008 million, or $1,008,000.00. Similarly, the expected value of the number of points on the toss of a pair of dice calculated from example in Figure 2.2 "Possible Outcomes for a Roll of Two Dice with the Probability of Having a Particular Number of Dots Facing Up" is 2 × (1/36) + 3 × (2/36) + 4 × (3/36) + 5 × (4/36) + 6 × (5/36) + 7 × (6/36) + 8 × (5/36) + 9 × (4/36) + 10 × (3/36) + 11 × (2/36) + 12 × (1/36) = 7. In uncertain economic situations involving possible financial gains or losses, the mean value or average value or expected value is often used to express the expected returns. [5] It represents the expected return from an endeavor; however, it does not express the risk involved in the uncertain scenario. We turn to this now.
Relating back to Table 2.1 "Claims and Fire Losses for Group of Homes in Location A" and Table 2.2
"Claims and Fire Losses ($) for Homes in Location B", for locations A and B of fire claim losses, the
expected value of losses is the severity of fire claims, $6,166.67, and the expected number of claims is the frequency of occurrence, 10.2 claims per year.
KEY TAKEAWAYS
In this section you learned about the quantification of uncertain outcomes via probability models. More specifically, you delved into methods of computing:
• Severity as a measure of the consequence of uncertainty—it is the expected value or average value of the loss that arises in different states of the world. Severity can be obtained by adding all the loss values in a sample and dividing by the total sample size.
• If we take a table of probabilities (probability distribution), the expected value is obtained by multiplying the probability of a particular loss occurring times the size of the loss and summing over all possibilities.
• Frequency is the expected number of occurrences of the loss that arises in different states of the world.
• Likelihood and probability distribution represent relative frequency of occurrence (frequency of occurrence of the event divided by the total frequency of all events) of different events in uncertain situations.
DISCUSSION QUESTIONS
1. A study of data losses incurred by companies due to hackers penetrating the Internet security of the firm found that 60 percent of the firms in the industry studied had experienced security breaches and that the average loss per security breach was $15,000.
a. What is the probability that a firm will not have a security breach?
b. One firm had two breaches in one year and is contemplating spending
money to decrease the likelihood of a breach. Assuming that the next
year would be the same as this year in terms of security breaches, how
much should the firm be willing to pay to eliminate security breaches (i.e., what is the expected value of their loss)?
The following is the experience of Insurer A for the last three years:
Year Number of Exposures Number of Collision Claims Collision Losses ($)
1 10,000 375 350,000
2 10,000 330 250,000
3 10,000 420 400,000
a. What is the frequency of losses in year 1?
b. Calculate the probability of a loss in year 1.
c. Calculate the mean losses per year for the collision claims and losses.
d. Calculate the mean losses per exposure.
e. Calculate the mean losses per claim.
f. What is the frequency of the losses?
g. What is the severity of the losses?
The following is the experience of Insurer B for the last three years:
Year Number of Exposures Number of Collision Claims Collision Losses ($)
1 20,000 975 650,000
2 20,000 730 850,000
3 20,000 820 900,000