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CHAPTER 3
Risk Assessment and Pooling
I. SUGGESTED CLASSROOM TIME: 60–75 MINUTES
II. CHAPTER OVERVIEW
This chapter examines the risk assessment step of the risk-management process.
Economic entities have an interest in making informed estimates of the likely impact of
losses in a budget year. The key pieces of data for these informed estimates are the mean
frequency and severity of losses.
Because losses are random variables, some basic statistical concepts can be employed
III. LECTURE OUTLINE
A. Introduction: The key task is to estimate the financial impact of losses.
1. Trying to gauge loss experience, so that budget decisions can be made
2. Estimating frequency and severity
B. Basic Statistical Concepts (See illustrations in text and review questions)
1. Random variables
C. Risk Assessment: Estimating Loss Frequency and Loss Severity
1. Average values come from historical data.
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E. Diversification of Risk Using Pooling of Exposure Units
1. Change in risk brought about by risk pooling; establishment of the mean loss
distribution
2. Normal distribution and its uses
IV. ANSWERS TO REVIEW QUESTIONS
1. Explain why flipping a coin is a random variable, and show the probability
distribution of outcomes that can result from a single flip of a coin. A coin
2. Show the physical damage loss distribution for Scary Airline’s planes.
Loss Outcome Probability Size of loss
3. Calculate the expected value of the physical damage loss.
4. Show the calculations for the variance and the standard deviation, as
described in footnote 3.
Loss
outcome
Probability
Loss outcome
– expected loss
(Loss outcome –
expected loss)2
(Loss outcome – expected
loss)2 * Probability
Chapter 3/Risk Assessment and Pooling 17
5. Describe the difference between loss severity and loss frequency. In your
answer, give an example of a loss control technique (as described in
Chapter 2, “Risk Identification”) that can be used by the owner of an
apartment complex to reduce the severity and the frequency of fire losses.
6. Use the information in the table to find the average frequency of losses per
worker.
7. Use the information in the table to estimate a probability distribution for
the frequency distribution of losses per worker in a year.
8. Use the information in the table to find the average severity per claim.
9. Use the information in the table to estimate a probability distribution for
the loss severity per claim.
10. Using your answers from Question 7 and Question 9, use convolution to
find the average loss.
Loss 1
Loss 2
Total loss
Probability
Total loss *
probability
Joint
probabilities
—
—
0
0.8500
0.0
1,000
—
1,000
0.0900
90.0
0.1*0.9
—
0.0100
60.0
1,000
0.0405
1,000
0.0045
0.0045
0.0005
6.0
1.0000
300.0
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11. Al’s Toy Store faces the following probability distribution of fire losses in
its store over the next year:
Calculate the expected value and standard deviation of Al’s losses for the
year.
prob loss prob*loss
prob loss loss-mean (loss – exp. value)2 (loss– exp. value)2*prob
0.85 $0 -4,000 16,000,000 13,600,000
12. Refer to Question 11. Assume that Al pools his losses with Ed’s store, which
has an identical loss distribution. Ed’s losses are independent of Al’s. Al
and Ed agree to split the total losses in the pool equally. Show the revised
probability distribution for the mean loss from the pool.
prob loss
0.7225 $0
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13. Refer to Questions 11 and 12. Calculate the expected value and standard
deviation of the pooled mean losses.
prob loss prob*loss
0.7225 $0 0
prob loss (loss – exp. value)2 (loss– exp. value)2*prob
14. Insurers combine a large number of exposure units in the process of risk
pooling. Describe the effect of increasing the size of the risk pool on the
mean loss of the pool and on the standard deviation of the mean loss in the
pool. In your answer, assume that the losses of all the exposure units in
the pool are independent and homogeneous. Increasing the size of the risk pool
will have little effect on the mean loss of the pool. If losses are random in a group of
15. Describe the shape of the normal distribution, indicating the statistical
relationship between its mean and its standard deviation. The normal
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16. Describe the concept of a risk charge as the term is used in the calculation
of insurance premiums. Is the risk charge affected by an increase in the
number of exposure units in a risk pool? Because there is deviation from the
17. List and briefly describe two examples of loss exposures that are not well
suited to risk pooling, and explain the reasons why risk pooling is not
effective for these loss exposures. Catastrophic exposures such as floods and
V. ANSWERS TO OBJECTIVE QUESTIONS
1. A(n) ______ is a table or graph that shows all possible outcomes for a random
variable, as well as their respective probabilities of occurring.
2. Which of the following statements about probability distributions is not correct?
3. Joe faces a 25 percent chance of a $1 million loss and a 75 percent chance of no loss.
Joe’s expected loss equals
4. Which of the following is NOT a statistical calculation used to measure risk?
Chapter 3/Risk Assessment and Pooling 21
5. The process of convolution
I. uses the probability distribution for loss frequency in the process of forecasting the
average loss.
II. uses the probability distribution for loss severity in the process of forecasting the
average loss.
d. Neither I nor II
6. Which of the following statements best describes the calculation of a confidence
interval (CI) for the estimate of the value of the mean loss?
7. Which of the following statements about risk pooling is correct?
I. By joining an insurance risk pool, the insured becomes financially responsible for
the mean loss of the pool instead of his or her own personal risk exposure.
II. By joining an insurance risk pool, the insured can decrease the expected loss that
he or she must pay to finance her exposure to loss.
8. Assume that Fred faces a potential liability exposure. The standard deviation of the
probability distribution for the loss equals $12,000. Assume that Fred joins a risk pool
with 899 other people with the same loss distribution whose losses are independent of
each other. As one of 900 pool members, Fred’s risk equals which of the following?
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9. Which of the following statements regarding the use of risk pooling is correct?
I. Insurers can use risk pooling to diversify risk in products like auto or life insurance.
II. Risk managers cannot use risk pooling to reduce risk because they do not have
sufficient number of homogeneous and independent exposure units.
10. Which of the following statements about the normal curve is correct?
VI. IDEAS FOR INSTRUCTORS AND TEACHING METHODS
1. Discuss with your class the notion of “confidence intervals.” Is it intuitively obvious to
them that larger sample sizes will bring smaller confidence intervals?
2. Ask your class to name some things that represent as a normal distribution, i.e., the
bell curve. Do they believe that human IQs are normally distributed, with a mean of