BU353 Chapter 4 – Pooling Arrangements and Diversification of Risk Week 3
Risk Reduction through Pooling Independent Losses
-Diversification is an essential aspect of insurance and financial markets
Two Person Pooling Arrangement
-This involves two people and they agree to share losses equally, each paying the average loss
-This is referred to as a pooling arrangement, because the two are pooling their resources
together to pay the accident costs that my occur
-The cost paid by each person is the average loss of the two people
-Since the losses incurred by Emily are independent of the losses incurred by Samantha, the
probability that neither woman has an accident is simply the probability that Emily does not have
an accident times the probability that Samantha does not have an accident
-Ex. When flipping two coins, the result of flipping the second coin is independent of the first
coin
-Given the accidents are independent, the probability that both Samantha and Emily will have
an accident is lower than the probability that only Emily (or Samantha) will have an accident
-Because the pooling arrangement reduces the probabilities of the extreme outcomes, the
standard deviation (risk) of accident costs paid by both Emily and Samantha is reduced
-The pooling arrangement does not change either person’s expected cost, but it reduces the
standard deviation
-Accident costs have become more predictable
-The pooling arrangement reduces risk for each individual
-The average loss is much more predictable than each individual’s loss
Pooling Arrangement with Many People or Businesses
-Additional risk reduction can be obtained from pooling by adding people (or businesses) to the
arrangement
-The addition of another person whose losses are independent of the other two cause an
additional reduction in the probability of the extreme outcomes
-The standard deviation for each individual decreases with the addition of another participant
-The probability distribution of each person’s accident cost will continue to change as more
people are added
-As the number of participants in the pooling arrangement increases, the probability of the
extreme outcomes goes down
-As the number of participants increases, the probability distribution of each person’s cost
becomes more bell shaped, that is, less skewed
-Pooling makes the amount of accident losses that each person must pay less risky, because
pooling reduces the standard deviation of the average loss for all the participants and thus the
standard deviation of the payment by each participant
-Risk can be reduced substantially for the participants
-Pooling arrangements reduce the amount of risk that each participant has to bear
-The standard deviation of average loss tends to decline when more and more participants with
different loss distributions are added to a pooling arrangement
-Risk in principle still becomes negligible as the number of participants becomes infinitely large
-Law of Large Numbers – As N (number insured) gets large, the average outcome is likely to get
very close to the expected value
-Central Limit Theorem – As N gets large, the distribution of the average outcome becomes
more symmetric and bell shaped
Pooling Arrangements and the Probability of Ruin
-Pooling reduces the risk that each participant faces. As the number of individuals in the pool
gets larger, the standard deviation of the average loss becomes smaller, and the probability
distribution of the average loss becomes more bell-shaped
-Using this information, we can use probability values from the normal distribution to estimate BU353 Chapter 4 – Pooling Arrangements and Diversification of Risk Week 3
the probability that the average outcome will exceed some critical value that represents the
insurer’s ability to pay claims
-By increasing the number of similar drivers who are insured, the insurer improves its ability to
estimate the average loss
-As the number of insured gets larger, the standard deviation becomes smaller
-By pooling independent, similar exposure insurers experience a reduction in the standard
deviation of average losses
-As N increases, the insurer’s actual average loss will become closer to the expecte
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