Actuarial Triangles Involving Reported Claims, Paid Claims, and Earned Premium: Practice Questions and Solutions

This section of the study guide is intended to provide practice problems and solutions to accompany the pages of Estimating Unpaid Claims Using Basic Techniques, 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. This is meant to give the student practice in answering questions of the format that will appear on Exam 6. 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.

Sources:
Friedland, Jacqueline F. Estimating Unpaid Claims Using Basic Techniques. Casualty Actuarial Society. July 2009. Chapter 6, pp. 63-69.

Original Problems and Solutions from The Actuary's Free Study Guide

Refer to the following information for Problems S6-13-1 through S6-13-5:

You are given the following information for an insurer:

In Calendar Year (CY) 2041, earned premium was $13,135.
In CY 2042, earned premium was $31,631, and the company increased its rate level by 20%.
In CY 2043, earned premium was $24,124, and the company decreased its rate level by 5%.
In CY 2044, earned premium was $26,750, and the company decreased its rate level by 7%.

Assume all rate changes occurred on January 1 of their respective years.

Assume that reported claims for Accident Year 2041 were as follows:
As of 12 months: $8,214
As of 24 months: $10,233
As of 36 months: $11,351
As of 48 months: $14,888

Assume that reported claims for Accident Year 2042 were as follows:
As of 12 months: $12,124
As of 24 months: $16,774
As of 36 months: $20,004

Assume that reported claims for Accident Year 2043 were as follows:
As of 12 months: $15,123
As of 24 months: $16,125

Assume that reported claims for Accident Year 2044 were as follows:
As of 12 months: $14,499

Assume that paid claims for Accident Year 2041 were as follows:
As of 12 months: $6,662
As of 24 months: $8,821
As of 36 months: $9,821
As of 48 months: $13,333

Assume that paid claims for Accident Year 2042 were as follows:
As of 12 months: $10,242
As of 24 months: $13,474
As of 36 months: $18,821

Assume that paid claims for Accident Year 2043 were as follows:
As of 12 months: $10,033
As of 24 months: $14,300

Assume that paid claims for Accident Year 2044 were as follows:
As of 12 months: $12,232

Problem S6-13-1. For (i) CY 2042, (ii) CY 2043, and (iii) CY 2044, find the following:

(a) The cumulative percent rate level change from CY 2041

(b) The annual percent exposure change just for the calendar year in question

Solution S6-13-1. This problem is based on the discussion in Friedland, p. 64.

(a) We calculate the cumulative rate level change from CY 2041 for each year:
(i) For CY 2042, the rate level change is just the given value of +20%.
(ii) For CY 2043, the cumulative rate level change is 1.2*(1-0.05) - 1 = 0.14 = +14%.
(iii) For CY 2044, the cumulative rate level change is 1.2*(1-0.05)*(1-0.07) = 1.0602 = +6.02%.

(b) The annual exposure change is the change in earned premium that was not accounted for by the rate level change during the calendar year in question.
(i) For CY 2042, the exposure change is
((CY 2042 earned premium)/(CY 2041 earned premium))/(1+CY 2042 rate level change) - 1 =
(31631/13135)/1.2 - 1 = 1.006788479 = +100.6788479%.
(ii) For CY 2043, the exposure change is
((CY 2043 earned premium)/(CY 2042 earned premium))/(1+CY 2042 rate level change) - 1 = (24124/31631)/0.95 - 1 = -0.1971899652 = -19.71899652%.
(iii) For CY 2044, the exposure change is
((CY 2044 earned premium)/(CY 2043 earned premium))/(1+CY 2043 rate level change) - 1 = (26750/24124)/0.93 - 1 = 0.1923164011 = +19.23164011%.

Problem S6-13-2. Use the given information to construct a ratio of reported claims to earned premium triangle, in the following format for as many ratios as are applicable:

Given AY: (12-month ratio, 24-month ratio, ...).

Solution S6-13-2. This problem is based on the discussion in Friedland, p. 66.

For each accident year, we divide the reported claim figures by the earned premium for that accident year. Thus, we get the following:

AY 2041: (8214/13135, 10233/13135, 11351/13135, 14888/13135)
AY 2042: (12124/31631, 16774/31631, 20004/31631)
AY 2043: (15123/24124, 16125/24124)
AY 2044: (14499/26750)

Our triangle will appear as follows (with factors rounded to three decimal places):

Ratio of reported claims to earned premium triangle
AY 2041: (0.625, 0.779, 0.864, 1,133)
AY 2042: (0.383, 0.530, 0.632)
AY 2043: (0.627, 0.668)
AY 2044: (0.542)

Problem S6-13-3. Use the given information to construct a ratio of reported claims to on-level earned premium triangle, in the following format for as many ratios as are applicable:

Given AY: (12-month ratio, 24-month ratio, ...).

Solution S6-13-3. This problem is based on the discussion in Friedland, p. 66.

For each accident year, we divide the reported claim figures by the earned premium for that accident year and then divide the result by the cumulative rate level change factor applicable to the period from the accident year in question through 2044. Thus, we get the following:

AY 2041: (8214/(13135*1.0602), 10233/(13135*1.0602), 11351/(13135*1.0602), 14888/(13135*1.0602))
AY 2042: (12124/(31631*0.95*0.93), 16774/(31631*0.95*0.93), 20004/(31631*0.95*0.93))
AY 2043: (15123/(24124*0.93), 16125/(24124*0.93))
AY 2044: (14499/26750)

Our triangle will appear as follows (with factors rounded to three decimal places):

Ratio of reported claims to on-level earned premium triangle
AY 2041: (0.590, 0.735, 0.815, 1.069)
AY 2042: (0.434, 0.600, 0.716)
AY 2043: (0.674, 0.719)
AY 2044: (0.542)

Problem S6-13-4. Use the given information to construct a ratio of paid claims to reported claims triangle, in the following format for as many ratios as are applicable:

Given AY: (12-month ratio, 24-month ratio, ...).

Solution S6-13-4. This problem is based on the discussion in Friedland, p. 68.

For each accident year, we divide the paid claim figures by the reported claim figures. Thus, we get the following:

AY 2041: (6662/8214, 8821/10233, 9821/11351, 13333/14888)
AY 2042: (10242/12124, 13474/16774, 18821/20004)
AY 2043: (10033/15123, 14300/16125)
AY 2044: (12232/14499)

Our triangle will appear as follows (with factors rounded to three decimal places):

Ratio of paid claims to reported claims
AY 2041: (0.811, 0.862, 0.865, 0.895)
AY 2042: (0.844, 0.803, 0.941)
AY 2043: (0.663, 0.887)
AY 2044: (0.844)

Problem S6-13-5. Use the given information to construct a ratio of paid claims to on-level earned premium triangle, in the following format for as many ratios as are applicable:

Given AY: (12-month ratio, 24-month ratio, ...).

Solution S6-13-5. This problem is based on the discussion in Friedland, p. 69.

For each accident year, we divide the paid claim figures by the earned premium for that accident year and then divide the result by the cumulative rate level change factor applicable to the period from the accident year in question through 2044. Thus, we get the following:

AY 2041: (6662/(13135*1.0602), 8821/(13135*1.0602), 9821/(13135*1.0602), 13333/(13135*1.0602))
AY 2042: (10242/(31631*0.95*0.93), 13474/(31631*0.95*0.93), 18821/(31631*0.95*0.93))
AY 2043: (10033/(24124*0.93), 14300/(24124*0.93))
AY 2044: (12232/26750)

Our triangle will appear as follows (with factors rounded to three decimal places):

Ratio of reported claims to on-level earned premium triangle
AY 2041: (0.479, 0.633, 0.705, 0.957)
AY 2042: (0.366, 0.482, 0.673)
AY 2043: (0.447, 0.637)
AY 2044: (0.457)

See other sections of The Actuary's Free Study Guide for Exam 6.