Measures of Frequency and Severity

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" show the compilation of the number of claims and their dollar amounts for homes that were burnt during a five-year period in two different locations labeled Location A and Location B. We have information about the total number of claims per year and the amount of the fire losses in dollars for each year. Each location has the same number of homes (1,000 homes). Each location has a total of 51 claims for the five-year period, an average (or mean) of 10.2 claims per year, which is the frequency. The average dollar amount of losses per claim for the whole period is also the same for each location, $6,166.67, which is the definition of severity.

As shown in 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", the total number of fire claims for the two locations A and B is the same, as is the total dollar amount of losses shown. You might recall from earlier, the number of claims per year is called the frequency. The average frequency of claims for locations A and B is 10.2 per year. The size of the loss in terms of dollars lost per claim is called severity, as we noted previously. The average dollars lost per claim per year in each location is $6,166.67.

The most important measures for risk managers when they address potential losses that arise from uncertainty are usually those associated with frequency and severity of losses during a specified period of time. The use of frequency and severity data is very important to both insurers and firm managers concerned with judging the risk of various endeavors. Risk managers try to employ activities (physical construction, backup systems, financial hedging, insurance, etc.) to decrease the frequency or severity (or both) of potential losses. In Chapter 4 "Evolving Risk Management: Fundamental Tools", we will see frequency data and severity data represented. Typically, the risk manager will relate the number of incidents under investigation to a base, such as the number of employees if examining the frequency and severity of workplace injuries. In the examples in 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", the severity is related to the number of fire claims in the five-year period per 1,000 homes. It is important to note that in these tables the precise distribution (frequencies and dollar losses) over the years for the claims per year arising in Location A is different from distribution for Location B. This will be discussed later in this chapter. Next, we discuss the concept of frequency in terms of probability or likelihood.

Frequency and Probability

Returning back to the quantification of the notion of uncertainty, we first observe that our intuitive usage of the word probability can have two different meanings or forms as related to statements of uncertain outcomes. This is exemplified by two different statements:See Patrick Brockett and Arnold Levine Brockett, Statistics, Probability and Their Applications (W. B. Saunders Publishing Co., 1984), 62.

1. “If I sail west from Europe, I have a 50 percent chance that I will fall off the edge of the earth.”

2. “If I flip a coin, I have a 50 percent chance that it will land on heads.”

Conceptually, these represent two distinct types of probability statements. The first is a statement about probability as a degree of belief about whether an event will occur and how firmly this belief is held. The second is a statement about how often a head would be expected to show up in repeated flips of a coin. The important difference is that the first statement’s validity or truth will be stated. We can clear up the statement’s veracity for all by sailing across the globe.

The second statement, however, still remains unsettled. Even after the first coin flip, we still have a 50 percent chance that the next flip will result in a head. The second provides a different interpretation of “probability,” namely, as a relative frequency of occurrence in repeated trials. This relative frequency conceptualization of probability is most relevant for risk management. One wants to learn from past events about the likelihood of future occurrences. The discoverers of probability theory adopted the relative frequency approach to formalizing the likelihood of chance events.

Pascal and Fermat ushered in a major conceptual breakthrough: the concept that, in repeated games of chance (or in many other situations encountered in nature) involving uncertainty, fixed relative frequencies of occurrence of the individual possible outcomes arose. These relative frequencies were both stable over time and individuals could calculate them by simply counting the number of ways that the outcome could occur divided by the total number of equally likely possible outcomes. In addition, empirically the relative frequency of occurrence of events in a long sequence of repeated trials (e.g., repeated gambling games) corresponded with the theoretical calculation of the number of ways an event could occur divided by the total number of possible outcomes. This is the model of equally likely outcomes or relative frequency definition of probability. It was a very distinct departure from the previous conceptualization of uncertainty that had all events controlled by God with no humanly discernable pattern. In the Pascal-Fermat framework, prediction became a matter of counting that could be done by anyone. Probability and prediction had become a tool of the people! Figure 2.2 "Possible Outcomes for a Roll of Two Dice with the Probability of Having a Particular Number of Dots Facing Up" provides an example representing all possible outcomes in the throw of two colored dice along with their associated probabilities.

Figure 2.2 "Possible Outcomes for a Roll of Two Dice with the Probability of Having a Particular Number of Dots Facing Up" lists the probabilities for the number of dots facing upward (2, 3, 4, etc.) in a roll of two colored dice. We can calculate the probability for any one of these numbers (2, 3, 4, etc.) by adding up the number of outcomes (rolls of two dice) that result in this number of dots facing up divided by the total number of possibilities. For example, a roll of thirty-six possibilities total when we roll two dice (count them). The probability of rolling a 2 is 1/36 (we can only roll a 2 one way, namely, when both dice have a 1 facing up). The probability of rolling a 7 is 6/36 = 1/6 (since rolls can fall any of six ways to roll a 7—1 and 6 twice, 2 and 5 twice, 3 and 4 twice). For any other choice of number of dots facing upward, we can get the probability by just adding the number of ways the event can occur divided by thirty-six. The probability of rolling a 7 or an 11 (5 and 6 twice) on a throw of the dice, for instance, is (6 + 2)/36 = 2/9.

The notions of “equally likely outcomes” and the calculation of probabilities as the ratio of “the number of ways in which an event could occur, divided by the total number of equally likely outcomes” is seminal and instructive. But, it did not include situations in which the number of possible outcomes was (at least conceptually) unbounded or infinite or not equally likely.Nor was the logic of the notion of equally likely outcomes readily understood at the time. For example, the famous mathematician D’Alembert made the following mistake when calculating the probability of a head appearing in two flips of a coin (Karl Pearson, The History of Statistics in the 17th and 18th Centuries against the Changing Background of Intellectual, Scientific and Religious Thought [London: Charles Griffin & Co., 1978], 552). D’Alembert said the head could come up on the first flip, which would settle that matter, or a tail could come up on the first flip followed by either a head or a tail on the second flip. There are three outcomes, two of which have a head, and so he claimed the likelihood of getting a head in two flips is 2/3. Evidently, he did not take the time to actually flip coins to see that the probability was 3/4, since the possible equally likely outcomes are actually (H,T), (H,H), (T,H), (T,T) with three pairs of flips resulting in a head. The error is that the outcomes stated in D’Alembert’s solution are not equally likely using his outcomes H, (T,H), (T,T), so his denominator is wrong. The moral of this story is that postulated theoretical models should always be tested against empirical data whenever possible to uncover any possible errors. We needed an extension. Noticing that the probability of an event, any event, provided that extension. Further, extending the theory to nonequally likely possible outcomes arose by noticing that the probability of an event—any event—occurring could be calculated as the relative frequency of an event occurring in a long run of trials in which the event may or may not occur. Thus, different events could have different, nonequal chances of occurring in a long repetition of scenarios involving the possible occurrences of the events. Table 2.3 "Opportunity and Loss Assessment Consequences of New Product Market Entry" provides an example of this. We can extend the theory yet further to a situation in which the number of possible outcomes is potentially infinite. But what about a situation in which no easily definable bound on the number of possible outcomes can be found? We can address this situation by again using the relative frequency interpretation of probability as well. When we have a continuum of possible outcomes (e.g., if an outcome is time, we can view it as a continuous variable outcome), then a curve of relative frequency is created. Thus, the probability of an outcome falling between two numbers x and y is the area under the frequency curve between x and y. The total area under the curve is one reflecting that it’s 100 percent certain that some outcome will occur.

The so-called normal distribution or bell-shaped curve from statistics provides us with an example of such a continuous probability distribution curve. The bell-shaped curve represents a situation wherein a continuum of possible outcomes arises. Figure 2.3 "Normal Distribution of Potential Profit from a Research and Development Project" provides such a bell-shaped curve for the profitability of implementing a new research and development project. It may have profit or loss.

To find the probability of any range of profitability values for this research and development project, we find the area under the curve in Figure 2.3 "Normal Distribution of Potential Profit from a Research and Development Project" between the desired range of profitability values. For example, the distribution in Figure 2.3 "Normal Distribution of Potential Profit from a Research and Development Project" was constructed to have what is called a normal distribution with the hump over the point $30 million and a measure of spread of $23 million. This spread represents the standard deviation that we will discuss in the next section. We can calculate the area under the curve above $0, which will be the probability that we will make a profit by implementing the research and development project. We do this by reference to a normal distribution table of values available in any statistics book. The area under the curve is 0.904, meaning that we have approximately a 90 percent change (probability of 0.9) that the project will result in a profit.

In practice, we build probability distribution tables or probability curves such as those in Figure 2.2 "Possible Outcomes for a Roll of Two Dice with the Probability of Having a Particular Number of Dots Facing Up", Figure 2.3 "Normal Distribution of Potential Profit from a Research and Development Project", and Table 2.3 "Opportunity and Loss Assessment Consequences of New Product Market Entry" using estimates of the likelihood (probability) of various different states of nature based on either historical relative frequency of occurrence or theoretical data. For example, empirical data may come from repeated observations in similar situations such as with historically constructed life or mortality tables. Theoretical data may come from a physics or engineering assessment of failure likelihood for a bridge or nuclear power plant containment vessel. In some situations, however, we can determine the likelihoods subjectively or by expert opinion. For example, assessments of political overthrows of governments are used for pricing political risk insurance needed by corporations doing business in emerging markets. Regardless of the source of the likelihoods, we can obtain an assessment of the probabilities or relative frequencies of the future occurrence of each conceivable event. The resulting collection of possible events together with their respective probabilities of occurrence is called a probability distribution, an example of which is shown in Table 2.3 "Opportunity and Loss Assessment Consequences of New Product Market Entry".

Measures of Outcome Value: Severity of Loss, Value of Gain

We have developed a quantified measure of the likelihood of the various uncertain outcomes that a firm or individual might face—these are also called probabilities. We can now turn to address the consequences of the uncertainty. The consequences of uncertainty are most often a vital issue financially. The reason that uncertainty is unsettling is not the uncertainty itself but rather the various different outcomes that can impact strategic plans, profitability, quality of life, and other important aspects of our life or the viability of a company. Therefore, we need to assess how we are impacted in each state of the world. For each outcome, we associate a value reflecting how we are affected by being in this state of the world.

As an example, consider a retail firm entering a new market with a newly created product. They may make a lot of money by taking advantage of “first-mover” status. They may lose money if the product is not accepted sufficiently by the marketplace. In addition, although they have tried to anticipate any problems, they may be faced with potential product liability. While they naturally try to make their products as safe as possible, they have to regard the potential liability because of the limited experience with the product. They may be able to assess the likelihood of a lawsuit as well as the consequences (losses) that might result from having to defend such lawsuits. The uncertainty of the consequences makes this endeavor risky and the potential for gain that motivates the company’s entry into the new market. How does one calculate these gains and losses? We already demonstrated some calculations in the examples above in 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 the claims and fire losses for homes in locations A and B. These examples concentrated on the consequences of the uncertainty about fires. Another way to compute the same type of consequences is provided in the example in Table 2.3 "Opportunity and Loss Assessment Consequences of New Product Market Entry" for the probability distribution for this new market entry. We look for an assessment of the financial consequences of the entry into the market as well. This example looks at a few possible outcomes, not only the fire losses outcome. These outcomes can have positive or negative consequences. Therefore, we use the opportunity terminology here rather than only the loss possibilities.

As you can see, it’s not the uncertainty of the states themselves that causes decision makers to ponder the advisability of market entry of a new product. It’s the consequences of the different outcomes that cause deliberation. The firm could lose $10.2 million or gain $8 million. If we knew which state would materialize, the decision would be simple. We address the issue of how we combine the probability assessment with the value of the gain or loss for the purpose of assessing the risk (consequences of uncertainty) in the next section.

Combining Probability and Outcome Value Together to Get an Overall Assessment of the Impact of an Uncertain Endeavor

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 valueThe numerical average of the experience of all possible outcomes if you played a game over and over.” 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.In some ways it is a shame that the term “expected value” has been used to describe this concept. A better term is “long run average value” or “mean value” since this particular value is really not to be expected in any real sense and may not even be a possibility to occur (e.g., the value calculated from Table 2.3 "Opportunity and Loss Assessment Consequences of New Product Market Entry" is 1.008, which is not even a possibility). Nevertheless, we are stuck with this terminology, and it does convey some conception of what we mean as long as we interpreted it as being the number expected as an average value in a long series of repetitions of the scenario being evaluated. 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.Other commonly used measures of profitability in an uncertain opportunity, other than the mean or expected value, are the mode (the most likely value) and the median (the number with half the numbers above it and half the numbers below it—the 50 percent mark). 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.