**This section of sample problems and solutions is a part of** **The Actuary’s Free Study Guide for Exam 5, authored by Mr. Stolyarov. This is Section 22 of the Study Guide. See an index of all sections by following the link in this paragraph.**

This section of the study guide is intended to provide practice problems and solutions to accompany the pages of *Foundations of Risk Management and Insurance*, cited below. Students are encouraged to read these pages before attempting the problems. This study guide is entirely an independent effort by Mr. Stolyarov and is not affiliated with any organization(s) to whose textbooks it refers, nor does it represent such organization(s).

When there exists a pooling arrangement with n members who each have an identical expected value of losses and standard deviation of losses and for whom the losses are completely independent and uncorrelated, then the following formula holds:

**Standard deviation of pool =**

**√(n)*(Standard deviation for a single member without the pooling arrangement).**

The standard deviation of losses *per member* of the pool is

√(n)*(Standard deviation for a single member without the pooling arrangement)/n =

**Standard deviation per member of pool =**

**(Standard deviation for a single member without the pooling arrangement)/√(n)**

Pooling in which the members are identical does not change the expected value of losses per member.

**Source:**

Nyce, C.M. *Foundations of Risk Management and Insurance* (Second Edition). 2006. American Institute for Chartered Property Casualty Underwriters. Chapter 7, pp. 7.34-7.39.

**Original Problems and Solutions from The Actuary’s Free Study Guide**

**Problem S5-22-1.** Joe, Jim, and Claudius decide to pool their losses. Each of them on his own has a standard deviation of losses of $670. All losses are independent and uncorrelated. What is the standard deviation of losses for the pool?

**Solution S5-22-1.** Here, n, the number of members of the pool, is 3.

We use the formula Standard deviation of pool =

√(n)*(Standard deviation for a single member without the pooling arrangement) =

√(3)*670 = 1160.474041 = approximately **$1160.47**.

**Problem S5-22-2.** Joe, Jim, and Claudius decide to pool their losses. Each of them on his own has a standard deviation of losses of $670. All losses are independent and uncorrelated. With the pooling arrangement, what is the standard deviation of losses faced by each of them?

**Solution S5-22-2.** We use the formula Standard deviation per member of pool =

(Standard deviation for a single member without the pooling arrangement)/√(n) =

670/√(3) = 386.8246804 = approximately **$386.82**.

**Problem S5-22-3.** Cuauhtémoc and Tim decide to pool their losses, which are completely independent and uncorrelated. Each of them has a 0.6 probability of no loss occurring, a 0.3 probability of a $10,000 loss occurring, and a 0.1 probability of a $100,000 loss occurring. Each person contributes an equal amount of money in the event of loss. What is the difference between the probability that Cuauhtémoc will not have to pay anything under the pooling arrangement and the probability that he would not have had to pay anything if he decided not to pool?

**Solution S5-22-3.** If Cuauhtémoc decided not to pool, he would pay nothing with probability 0.6, as given in the problem. Under the pooling arrangement, Cuauhtémoc would have to pay half of any of his own losses and half of any of Tim’s losses. Thus, in order for him to pay nothing, both he and Tim would need to not suffer any losses. The probability of this happening is 0.62 = 0.36. Thus, the desired difference is 0.36 – 0.6 = **-0.24.** Note that Cuauhtémoc’s probability of paying nothing is *reduced considerably* via the pooling arrangement.

**Problem S5-22-4.** Cuauhtémoc and Tim decide to pool their losses, which are completely independent and uncorrelated. Each of them has a 0.6 probability of no loss occurring, a 0.3 probability of a $10,000 loss occurring, and a 0.1 probability of a $100,000 loss occurring. Each person contributes an equal amount of money in the event of loss. What is the difference between the probability that Cuauhtémoc will have to pay $50,000 or more under the pooling arrangement and the probability that he would have had to pay $50,000 or more if he decided not to pool?

**Solution S5-22-4.** If Cuauhtémoc decided not to pool, the only way for him to pay $50,000 or more would be if he suffered a $100,000 loss – which would have a probability of 0.1. Under the pooling arrangement, Cuauhtémoc would have to pay half of any of his own losses and half of any of Tim’s losses. Thus, if total losses are $100,000 or more, Cuauhtémoc would have to pay at least $50,000. The only way for total losses to be $100,000 or more is if *either* member of the pool or *both* of them have a $100,000 loss. This is

Pr(Tim loses $100,000) + Pr(Cuauhtémoc loses $100,000) – Pr(Both lose $100,000) =

0.1 + 0.1 – 0.12 = 0.19. Thus, the desired difference is 0.19 – 0.1 = **0.09**, meaning that Cuauhtémoc is actually at *greater* risk for a large payout with this pooling arrangement.

**Problem S5-22-5.** Which of the following statements about insurance and pooling are true? More than one answer may be correct.

(a) Pooling functions by reducing the expected frequency and severity of individual loss exposures.

(b) As the number of members in a pool increases, the standard deviation of losses per member decreases, unless the loss exposures are perfectly correlated.

(c) Pooling can reduce the expected cost of loss for every member.

(d) One advantage of pooling is that it tends to make losses per member more consistent and more predictable.

(e) Pooling does not work with correlated loss exposures.

(f) Insurance is just another name for pooling of losses. There are no substantive differences beyond the names.

(g) Insurance is a risk sharing mechanism, whereas pooling is a risk transfer mechanism.

(h) Unlike a pool, insurers typically cannot collect additional payments from the insured if actual losses happen to exceed the losses which were estimated in calculating the premium.

(i) One advantage of insurance over a pool is that an insurer typically has other resources to draw on – such as the initial capital from its investors and its retained earnings.

**Solution S5-22-5.** The ideas in this question are addressed in Nyce 2006, pp. 7.36-7.39.

The following statements are correct: **(b)** As the number of members in a pool increases, the standard deviation of losses per member decreases, unless the loss exposures are perfectly correlated. **(d)** One advantage of pooling is that it tends to make losses per member more consistent and more predictable.

**(h)** Unlike a pool, insurers typically cannot collect additional payments from the insured if actual losses happen to exceed the losses which were estimated in calculating the premium. **(i)** One advantage of insurance over a pool is that an insurer typically has other resources to draw on – such as the initial capital from its investors and its retained earnings.

Choice (a) is incorrect, because pooling cannot reduce frequency and severity of individual losses.

Choice (c) is incorrect, because pooling does not reduce total expected loss cost, so it cannot reduce the expected loss cost for every member.

Choice (e) is incorrect; pooling works best with completely uncorrelated loss exposures, but it can also work to a lesser extent with correlated loss exposures, provided that the correlation is not perfect.

Choice (f) is incorrect; there are numerous important differences between pooling and insurance – including those discussed under the correct answers (h) and (i).

Choice (g) is incorrect; pooling is a risk sharing mechanism, whereas insurance is a risk transfer mechanism. This is one of the major differences between pooling and insurance.

**See other sections of The Actuary’s Free Study Guide for Exam 5.**