APRENDIZAJE SIGNIFICATIVO

Even if the characteristics of the population were known with certainty, insurers do not insure pop- ulations. Rather, they select a sample from the population and insure the sample. Obviously, the relationship between population parameters and the characteristics of the sample (mean and stand- ard deviation) is important for insurers, since actual experience may vary significantly from the popula- tion parameters. The characteristics of the sampling For example, assume that an actuary estimates

the following probabilities of various losses for a certain risk: Amount of Loss (X_i ) Probability of Loss (P _i ) X _i P _i $ 0 _* .30 ₌ $ 0 $360 * .50 = $180 $600 _* .20 ₌ $120 π X _i P _i = $300

Thus, we could say that the mean or expected loss given the probability distribution is $300.

Although the mean value indicates central ten- dency, it does not tell us anything about the riskiness or dispersion of the distribution. Consider a second probability-of-loss distribution: Amount of Loss (X _i ) Probability of Loss (P _i ) X _i P _i $225 * .40 = $ 90 $350 _* .60 _{= $210} π X i P i = $300

This distribution also has a mean loss value of $300. However, the first distribution is riskier because the range of possible outcomes is from $0 to $600. With the second distribution, the range of possible outcomes is only $125 ($350 – $225), so we are more certain about the outcome with the second distribution.

Two standard measures of dispersion are employed to characterize the variability or dispersion about the mean value. These measures are the variance ( s 2 _{) and the standard deviation ( s ). The variance of}

a probability distribution is the sum of the squared differences between the possible outcomes and the expected value, weighted by the probability of the outcomes:

s2 _{= a P}

i1Xi - EV 22

So the variance is the average squared devia- tion between the possible outcomes and the mean. Because the variance is in “squared units,” it is neces- sary to take the square root of the variance so that

L A W O F L A R G E N U M B E R S 4 1

The second important implication of the Central Limit Theorem for insurers is that the standard error of the sample mean distribution declines as the sample size increases. Recall that the standard error is defined as

sx = sx> 2n

In other words, the standard error of the sample mean loss distribution is equal to the standard deviation of the population divided by the square root of the sam- ple size. Because the population standard deviation is independent of the sample size, the standard error

of the sampling distribution, sx can be reduced by simply increasing the sample size .

This result has important implications for insurers. For example, assume that an insurer would like to select a sample to insure from a population where the mean loss is $500 and the standard deviation is $350. As the insurer increases the number of units insured ( n ), the standard error of the sampling distribution _sx

will decline. The standard error for various sample sizes is summarized below: n _sx 10 110.68 100 35.00 1,000 11.07 10,000 3.50 100,000 1.11

Thus, as the sample size increases, the difference between actual results and expected results decreases. Indeed, sx approaches zero as n gets very large. This

result is shown graphically in Exhibit A2.2 . distribution help to illustrate the law of large

numbers, the mathematical foundation of insurance. It can be shown that the average losses for a ran- dom sample of n exposure units will follow a normal distribution because of the Central Limit Theorem, which states:

If you draw random samples of n observations from any population with mean μ _x and standard deviation

s x , and n is sufficiently large, the distribution of sample

means will be approximately normal, with the mean of the distribution equal to the mean of the population

mx = mx , and the standard error of the sample mean

sx equal to the standard deviation of the population

( s _x ) divided by the square root of n (sx = sx> 2n)

This approximation becomes increasingly accurate as the sample size, n , increases.

The Central Limit Theorem has two important implications for insurers. First, it is clear that the sample distribution of means does not depend on the population distribution, provided n is sufficiently large. In other words, regardless of the population

distribution (bimodal, unimodal, symmetric, skewed right, skewed left, and so on), the distribution of sample means will approach the normal distribution as the sample size increases . This result is shown in

Exhibit A2.1 .

The normal distribution is a symmetric, bell- shaped curve. It is defined by the mean and standard deviation of the distribution. About 68 percent of the distribution lies within one standard deviation of the mean, and about 95 percent of the distribution lies within two standard deviations of the mean. The nor- mal curve has many statistical applications (hypothe- sis testing, confidence intervals, and so on) and is easy to use. f(x ) E(X ) n 10,000 n 1000 x Exhibit A2.1

Sampling Distribution Versus Sample Size

sx sx n

sx

n

Exhibit A2.2

Standard Error of the Sampling Distribution Versus Sample Size

reduce their objective risk. There truly is “safety in numbers” for insurers.

NOTES

1. The number of runs scored in a baseball game is a dis- crete measure as partial runs cannot be scored. Speed and temperature are continuous measures as all values over the range of values can occur.

2. Other measures of central tendency are the median, which is the middle observation in a probability dis- tribution, and the mode, which is the observation that occurs most often.

3. Introductory statistics texts discuss several popular theoretical distributions, such as the binomial and Poisson distributions, that can be used to estimate losses. Another popular distribution, the normal distribution, is discussed next under the “Law of Large Numbers.”

Obviously, when an insurer increases the size of the sample insured, underwriting risk (maximum insured losses) increases because more insured units could suffer a loss. The underwriting risk for an insurer is equal to the number of units insured mul- tiplied by  the standard error of the average loss distribution, sx . Recalling that sx is equal to sx> 2n,

we can rewrite the expression for underwriting risk as:

n * sx = n * sx> 2n = 2n * sx

Thus, while underwriting risk increases with an increase in the sample size, it does not increase proportionately.

Insurance companies are in the loss business— they expect some losses will occur. It is the deviation between actual losses and expected losses that is the major concern. By insuring large samples, insurers

4 3

CHAPTER 3

INTRODUCTION TO

RISK MANAGEMENT

“The essence of risk management lies in maximizing the

areas where we have some control over the outcome while minimizing the areas where we have absolutely no control over the outcome …”

Peter L. Bernstein Against the Gods: The Remarkable Story of Risk

Learning Objectives

After studying this chapter, you should be able to

◆ Define risk management and explain the objectives of risk management.

◆ Describe the steps in the risk management process.

◆ Explain the major risk-control techniques, including Avoidance

Loss prevention Loss reduction

◆ Explain the major risk-financing techniques, including Retention

Noninsurance transfers Insurance

D

prepared when losses occur.

The above example shows how a business firm could benefit from a risk management program. Today, risk management is widely used by corporations, small employers, nonprofit organizations, and state and local governments. Families and students can also benefit from a personal risk management program.

In this chapter —the first of two dealing with risk management— we discuss the fundamentals of traditional risk management. The following chapter discusses the newer forms of risk management that are rapidly emerging, including enterprise risk