**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 43 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 *Basic Ratemaking*, 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).

Some of the questions here ask for short written answers based on the reading. This is meant to give the student practice in answering questions of the format that will appear on Exam 5. Students are encouraged to type their own answers first and then to compare these answers with the solutions given here. Please note that the solutions provided here are not necessarily the only possible ones.

**Source:**

Werner, Geoff and Claudine Modlin. *Basic Ratemaking**.* Casualty Actuarial Society. 2009. Chapter 6, pp. 106-110.

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

**Problem S5-43-1.** Loss development is most often positive for a given loss (i.e., reported losses tend to increase over time). However, there are situations in which loss development can be negative (i.e., reported losses decrease as they approach their ultimate amount). Describe two reasons for which loss development can be negative.

**Solution S5-43-1.** This question is based on the discussion in Werner and Modlin, p. 106. There are three situations described in that discussion:

1. **Salvage:** The insurer pays for a total loss on the damaged property and then takes possession of the property. Thereafter, the insurer sells the damaged property and recovers some of the amount it paid. This reduces the insurer’s net payment on the loss.

2. **Subrogation:** The insurer pays for a loss for which a third party is responsible; then the insurer seeks indemnification from the responsible third party. Successful subrogation will mean that the insurer gets a reduction in net payment on the loss.

3. **Early case reserve estimate too high:** If the insurer or the insurer’s claim adjuster sets the case reserve on a claim too high, later events may lead to a downward revision of this estimate, leading to negative loss development.

**Problem S5-43-2.** For a certain kind of “long-tailed” loss that occurred in Accident Year 2052, you have the following information:

As of 12 months, total losses were $56,000.

As of 24 months, total losses were $65,700.

As of 36 months, total losses were $87,000.

As of 48 months, total losses were $90,000.

As of 60 months, total losses were $102,000.

The chain ladder method of estimating loss development is used, and it is assumed that no further changes in total losses occurred after 60 months.

From the information above, it is possible to calculate five age-to-ultimate development factors. Find the factors and specify the time periods to which each factor pertains.

**Solution S5-43-2.** Before we calculate age-to-ultimate development factors, we need to find the age-to-age development factors whose multiplication will enable us to obtain the age-to-ultimate factors. An age-to-age development factor for the time period X through Y is found via the expression

(Losses at time Y)/(Losses at time X).

The factor for months 12-24 is thus 65700/56000 = 1.173214286.

The factor for months 24-36 is 87000/65700 = 1.324200913.

The factor for months 36-48 is 90000/87000 = 1.034482759.

The factor for months 48-60 is 102000/90000 = 1.1333333333.

The five age-to-ultimate factors are from each of the times listed to ultimate (i.e., 12 months to ultimate, 24 months to ultimate, etc.). From time X to ultimate, the age-to-ultimate factor is the product of all the age-to-age factors applicable to that time range.

Thus, the **60-months-to-ultimate factor** is simply **1**, since no further loss development occurs after 60 months.

The **48-months-to-ultimate factor is** the (48-60) factor, multiplied by the (60-ultimate) factor, i.e., 1.1333333333*1 = **1.1333333333**.

The **36-months-to-ultimate factor is** the (36-48) factor, multiplied by the (48-ultimate) factor, i.e., 1.034482759*1.1333333333 = **1.172413793**.

The **24-months-to-ultimate factor is** the (24-36) factor, multiplied by the (36-ultimate) factor, i.e., 1.324200913*1.172413793= **1.552511415**.

The **12-months-to-ultimate factor is** the (12-24) factor, multiplied by the (24-ultimate) factor, i.e., 1.173214286*1.552511415= **1.821428572**.

**Problem S5-43-3. .** For a certain kind of “long-tailed” loss that occurred in Accident Year 2052, you have the following information:

As of 12 months, total losses were $56,000.

As of 24 months, total losses were $65,700.

As of 36 months, total losses were $87,000.

As of 48 months, total losses were $90,000.

As of 60 months, total losses were $102,000.

The chain ladder method of estimating loss development is used, and it is assumed that no further changes in total losses occurred after 60 months.

Age-to-ultimate factors on the basis of these data were calculated in Solution S5-43-2. These factors are then applied to subsequent years’ data, which is analyzed on accident year basis as of January 1, 2058. All that is known about the subsequent years’ losses are the reported losses as of January 1, 2058:

2053 reported losses as of January 1, 2058, are $53,000.

2054 reported losses as of January 1, 2058, are $52,000.

2055 reported losses as of January 1, 2058, are $92,000.

2056 reported losses as of January 1, 2058, are $85,000.

2057 reported losses as of January 1, 2058, are $45,000.

Using the assumptions above, find the total estimated ultimate losses for the years 2053 through 2057.

**Solution S5-43-3.** From Solution S5-43-2, we have the following age-to-ultimate development factors:

(60-ultimate): 1

(48-ultimate): 1.1333333333

(36-ultimate): 1.172413793

(24-ultimate): 1.552511415

(12-ultimate): 1.821428572

60 months have passed between the beginning of accident year (AY) 2053 and January 1, 2058, so we multiply AY 2053 reported losses by the (60-ultimate) factor of 1, resulting in estimated ultimate AY 2053 losses being 53000*1 = $53,000.

48 months have passed between the beginning of accident year (AY) 2054 and January 1, 2058, so we multiply AY 2054 reported losses by the (48-ultimate) factor of 1.1333333333, resulting in estimated ultimate AY 2054 losses being 52000*1.1333333333 = $58,933.33.

36 months have passed between the beginning of accident year (AY) 2055 and January 1, 2058, so we multiply AY 2055 reported losses by the (36-ultimate) factor of 1.172413793, resulting in estimated ultimate AY 2055 losses being 92000*1.172413793 = $107,862.07.

24 months have passed between the beginning of accident year (AY) 2056 and January 1, 2058, so we multiply AY 2056 reported losses by the (24-ultimate) factor of 1.552511415, resulting in estimated ultimate AY 2056 losses being 85000*1.552511415 = $131,963.47.

12 months have passed between the beginning of accident year (AY) 2057 and January 1, 2058, so we multiply AY 2057 reported losses by the (12-ultimate) factor of 1.821428572, resulting in estimated ultimate AY 2057 losses being 45000*1.821428572 = $81,964.29.

The sum of the estimated ultimate losses for these five years is $53,000 + $58,933.33 + $107,862.07 + $131,963.47 + $81,964.29 = **$433,723.16**.

**Problem S5-43-4.** You have the following loss data:

For the year ending during the first quarter of 2350 (Q1 2350), there were 766,000 earned exposures, 46,000 claims, and $15,331,551 in losses.

For the year ending during the second quarter of 2350 (Q2 2350), there were 341,000 earned exposures, 26,000 claims, and $6,801,800 in losses.

For the year ending during the third quarter of 2350 (Q3 2350), there were 443,138 earned exposures, 32,020 claims, and $8,889,123 in losses.

For the year ending during the fourth quarter of 2350 (Q4 2350), there were 500,000 earned exposures, 47,831 claims, and $7,000,012 in losses.

For the year ending during the first quarter of 2351 (Q1 2351), there were 871,124 earned exposures, 61,000 claims, and $15,159,012 in losses.

For the year ending during the second quarter of 2351 (Q2 2351), there were 124,000 earned exposures, 8,000 claims, and $2,442,400 in losses.

What is the annual percent change in loss frequency, measured as of the end of Q1 2351?

**Solution S5-43-4.** Here, Frequency = (Number of Claims)/(Number of Earned Exposures).

For Q1 2351, frequency is 61000/871124 = 0.0700244741. This should be compared to frequency one year ago, during the year ending during Q1 2350. This frequency is 46000/776000 = 0.0592783505. The annual percent change in frequency is therefore 100*(0.0700244741/0.0592783505 – 1) = **+18.12824331%.**

**Problem S5-43-5.** You have the following loss data on closed claims:

For the year ending during the first quarter of 2350 (Q1 2350), there were 766,000 earned exposures, 46,000 claims, and $15,331,551 in losses.

For the year ending during the second quarter of 2350 (Q2 2350), there were 341,000 earned exposures, 26,000 claims, and $6,801,800 in losses.

For the year ending during the third quarter of 2350 (Q3 2350), there were 443,138 earned exposures, 32,020 claims, and $8,889,123 in losses.

For the year ending during the fourth quarter of 2350 (Q4 2350), there were 500,000 earned exposures, 47,831 claims, and $7,000,012 in losses.

For the year ending during the first quarter of 2351 (Q1 2351), there were 871,124 earned exposures, 61,000 claims, and $15,159,012 in losses.

For the year ending during the second quarter of 2351 (Q2 2351), there were 124,000 earned exposures, 8,000 claims, and $2,442,400 in losses.

What is the annual percent change in pure premium, measured as of the end of Q2 2351?

**Solution S5-43-5.** Although pure premium can be calculated as (Frequency)*(Severity), we can also take a more direct approach and consider pure premium as

(Amount of Total Losses)/(Number of Earned Exposures). For the year ending Q2 2351, pure premium is thus 2442400/124000 = 19.69677419. For the year ending Q2 2350, pure premium is 6801800/341000 = 19.94662757. The annual percent change in pure premium is therefore 100*(19.69677419/19.94662757 – 1) = **-1.252609603%.**

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