Assorted Exam-Style Questions for Actuarial Exam 6 -- Part 16

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.

Some of the problems in this section were designed to be similar to problems from past versions of Exam 6, offered by the Casualty Actuarial Society. They use original exam questions as their inspiration - and the specific inspiration is cited to give students an opportunity to see the original. All of the original problems are publicly available, and students are encouraged to refer to them. But all of the values, names, conditions, and calculations in the problems here are the original work of Mr. Stolyarov.

Sources:

Friedland, Jacqueline F. Estimating Unpaid Claims Using Basic Techniques. Casualty Actuarial Society. July 2009.

Past Casualty Actuarial Society exams: 2007 Exam 6 and 2008 Exam 6.

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

Problem S6-38-1. Similar to Question 47 from the 2007 CAS Exam 6. You know the following about paid defense and cost containment (DCC) expenses as of December 31, 2047, and ultimate losses for an insurer by accident year (AY):

AY 2044: Ultimate loss: 5550; Paid DCC: 200
AY 2045: Ultimate loss: 5200; Paid DCC: 150
AY 2046: Ultimate loss: 5100; Paid DCC: 110
AY 2047: Ultimate loss: 6000; Paid DCC: 20

Ratio of Cumulative Paid DCC to Cumulative Paid Loss, expressed in the format
(Ratio at 12 months, Ratio at 24 months, Ratio at 36 months, Ratio at Ultimate).

AY 2044: (0.24%, 2.22%, 3.20%, 4.25%)
AY 2045: (0.28%, 2.24%, 3.25%)
AY 2046: (0.25%, 2.30%)
AY 2047: (0.22%)

For AY 2044 through AY 2047, calculate the total DCC reserve. Show all intermediate steps contributing to the result.

Solution S6-38-1. First, we want to calculate age-to-age factors for ratio of cumulative DCC to cumulative paid loss.

Age-to-Age Factors for Ratio of Cumulative Paid DCC to Cumulative Paid Loss, expressed in the format
(Factor for 12-24 months, Factor for 24-36 months, Factor for 36 months to Ultimate).

AY 2044: (2.22%/0.24%, 3.20%/2.22%, 4.25%/3.20%)
AY 2045: (2.24%/0.28%, 3.25%/2.24%)
AY 2046: (2.30%/0.25%)

Age-to-Age Factors for Ratio of Cumulative Paid DCC to Cumulative Paid Loss

AY 2044: (9.25, 1.441441441, 1.328125)
AY 2045: (8.00, 1.450892857)
AY 2046: (9.20)

We select the simple arithmetic means of the available age-to-age factors for a given age to maturity:

12-24 months: 8.8166666667
24-36 months: 1.446167149
36 months to ultimate: 1.328125

We can now select factors to ultimate:

12-24 months: 8.8166666667*1.446167149*1.328125 = 16.93409007
24-36 months: 1.446167149*1.328125 = 1.920690745
36 months to ultimate: 1.328125

Now, for each accident year, we can project to ultimate the ratios of cumulative paid DCC to cumulative paid loss:

AY 2044: 4.25% -- Already at ultimate
AY 2045: 3.25%*1.328125 = 4.31640625%
AY 2046: 2.30%*1.920690745 = 4.417588714%
AY 2047: 0.22%*16.93409007 = 3.725499815%

This allows us to find the ultimate DCC for each accident year:

AY 2044: 5550*4.25% = 235.875
AY 2045: 5200*4.31640625% = 224.452125
AY 2046: 5100*4.417588714% = 225.2970244
AY 2047: 6000*3.725499815% = 223.5299889

Now we can find the DCC reserve for each accident year by subtracting paid DCC from ultimate DCC:

AY 2044: 235.875 - 200 = 35.875
AY 2045: 224.452125 - 150 = 74.452125
AY 2046: 225.2970244 - 110 = 115.2970244
AY 2047: 223.5299889 - 20 = 203.5299889

Total: 35.875 + 74.452125 + 115.2970244 + 203.5299889 = 429.1541383 = circa 429.15.

Problem S6-38-2. Similar to Question 3 from the 2008 CAS Exam 6. The annual projected severity trend is +2%. All data are at ultimate at 48 months. You also know the following information by accident year (AY):

Incremental Loss and ALAE Payments on Closed Claims, expressed in thousands of dollars and in the format

(Amount at 12 months, Amount at 24 months, Amount at 36 months, Amount at 48 months).

AY 2030: (200, 250, 180, 80)
AY 2031: (250, 290, 200)
AY 2032: (300, 300)
AY 2033: (350)

Incremental Number of Claims Closed, expressed in the format

(Number at 12 months, Number at 24 months, Number at 36 months, Number at 48 months).

AY 2030: (30, 60, 50, 40) Ultimate claims: 180
AY 2031: (36, 72, 60) Ultimate claims: 216
AY 2032: (24, 48) Ultimate claims: 144
AY 2033: (42) Ultimate claims: 252

(a) Use Adler and Kline's claim closure projection method to find the projected reserve as of December 31, 2033.

(b) Describe two aspects of the calculation you performed in part (a) that would recommend it as a reserve estimation technique.

Solution S6-38-2.

(a) First we note the incremental claim closure pattern, which appears to be the same for every accident year. Let N be the number of claims closed at 12 months. Then the pattern is

(N, 2N, (5/3)N, (4/3)N). Using this formula, we can extrapolate the numbers of closed claims:

AY 2030: (30, 60, 50, 40) Ultimate claims: 180
AY 2031: (36, 72, 60, 48) Ultimate claims: 216
AY 2032: (24, 48, 40, 32) Ultimate claims: 144
AY 2033: (42, 84, 70, 56) Ultimate claims: 252

We can also figure out the incremental paid severities (Paid Amounts/Closed Claims):

Incremental Paid Severities on Closed Claims, expressed in thousands of dollars and in the format

(Amount at 12 months, Amount at 24 months, Amount at 36 months, Amount at 48 months).

AY 2030: (200/30, 250/60, 180/50, 80/40)
AY 2031: (250/36, 290/72, 200/60)
AY 2032: (300/24, 300/48)
AY 2033: (350/42)

Really, we are just interested in the outermost diagonal:
Incremental Paid Severities on Closed Claims

AY 2030: (200/30, 250/60, 180/50, 2)
AY 2031: (250/36, 290/72, 3.3333)
AY 2032: (300/24, 6.25)
AY 2033: (8.3333)

Now we can apply our annual multiplicative severity trend of 1.02:
Incremental Paid Severities on Closed Claims

AY 2030: (200/30, 250/60, 180/50, 2)
AY 2031: (250/36, 290/72, 3.3333, 2.04)
AY 2032: (300/24, 6.25, 3.40, 2.0808)
AY 2033: (8.3333, 6.375, 3.468, 2.122416)

Now we can calculate the reserve amounts for each year as the sum of 1000*(Number of Closed Claims*Closed Claim Severity) for each age to maturity. There is no reserve for AY 2030, since losses are already at ultimate.

AY 2031: 1000*(2.04*48) = 97920
AY 2032: 1000*(3.40*40 + 2.0808*32) = 202585.6
AY 2033: 1000*(6.375*84 + 3.468*70 + 2.122416*56) = 897115.296
Total: 97920 + 202585.6 + 897115.296 = 1197620.896 = circa $1,197,620.90.

(b) Two advantages of the calculation in part (a) are 1) explicit incorporation of claim severity trends, which could be accounted for by economic or social inflation, and 2) no reliance on incurred losses and independence from the accuracy or lack thereof of case reserve estimates.

Problem S6-38-3. Similar to Question 4 from the 2008 CAS Exam 6. You have the following information as of December 31, 2066, all expressed in the format

(Number at 12 months, Number at 24 months, Number at 36 months, Number at 48 months), where applicable.

Cumulative Reported Loss ($000)

AY 2063: (3030, 4506, 4990, 5200)
AY 2064: (3133, 4666, 5000)
AY 2065: (3002, 4556)
AY 2066: (3000)

Cumulative Paid Loss ($000)

AY 2063: (1525, 2344, 2990, 4560)
AY 2064: (1498, 2200, 3000)
AY 2065: (1555, 2660)
AY 2066: (1500)

Average Case Reserve Per Open Claim ($000)

AY 2063: (10.03, 27.15, 66.6667, 64)
AY 2064: (11.68, 35.55, 50)
AY 2065: (11.13, 24)
AY 2066: (12)

Number of Open Claims ($000)

AY 2063: (150, 80, 30, 10)
AY 2064: (140, 75, 40)
AY 2065: (130, 79)
AY 2066: (125)

(a) Assume an annual severity trend of -4% and use the Berquist-Sherman method to create an adjusted cumulative reported loss triangle, based on a severity-adjusted case reserve triangle. Round your answers to the nearest whole number.

(b) Using the adjusted cumulative reported loss triangle from part (a) and loss development factors calculated as weighted averages of all relevant years' experience, estimate the ultimate loss for AY 2066. Use a 48-month-to-ultimate factor of 1.05.

Solution S6-38-3.

(a) We take the average case reserve triangle and project backward from the outermost diagonal using the annual severity trend of -4%. We divide each subsequent vertical entry by 0.96 to get the preceding entry.

Adjusted Average Case Reserve Per Open Claim ($000)

AY 2063: (13.56336806, 26.0416667, 52.083333, 64)
AY 2064: (13.02083333, 25, 50)
AY 2065: (12.5, 24)
AY 2066: (12)

Now, to get the adjusted reported claim triangle, for each entry except the outermost diagonal (where reported claims are unchanged from what is given), we calculate

Cumulative Paid Loss + (Number of Open Claims)*(Adjusted Average Case Reserve Per Open Claim).

Sample calculation, for AY 2063 at 12 months:

1525 + 150*13.56336806 = 3559.505209 = circa 3560.

Adjusted Cumulative Reported Loss ($000)

AY 2063: (3560, 4427, 4553, 5200)
AY 2064: (3321, 4075, 5000)
AY 2065: (3180, 4556)
AY 2066: (3000)

(b) We can calculate weighted-average age-to-age factors as follows:

For 12-24 months: (4427 + 4075 + 4556)/(3560 + 3321 + 3180) = 1.297882914.
For 24-36 months: (4553 + 5000)/(4427 + 4075) = 1.123617972
For 36-48 months: 5200/4553 = 1.142104107
12-month-to-ultimate factor: 1.297882914*1.123617972*1.142104107*1.05 = 1.748836402.
Ultimate loss for AY 2066: 1000*3000*1.748836402 = $5,246,509.21.

Problem S6-38-4. Similar to Question 11 from the 2008 CAS Exam 6. You have the following IBNR estimates from three different methods:

Loss development method: $6000
Bornhuetter-Ferguson method: $5000
Percent of premium method: $5300

The insurer's book of business has been showing a deteriorating loss ratio, with no changes in case reserve adequacy or loss emergence patterns.

(a) Rank these methods in order of accuracy in this situation. Justify your answer.

(b) For any of these methods that are inaccurate, which are self-correcting in the long term? Why?

Solution S6-38-4.

(a) The most accurate method here is the development method. Since there are no changes in case reserve adequacy or loss emergence patterns, the loss development pattern has not altered at all, and the loss development factors based on historical losses will still fully reflect the current situation. Less accurate is the percent of premium method, where the IBNR estimate is based on the premiums and losses during the time periods in question, and only part of the experience will be based on the more recent time periods of deteriorating loss ratios. The percent of premium method would thus underestimate the true IBNR. The Bornhuetter-Ferguson method would produce an even greater underestimate, as the IBNR component of ultimate losses is based on an expected loss ratio that is determined a priori. This expected loss ratio would be lower than warranted by the more recent experience. The ranking in terms of accuracy would thus be

Loss development method > Percent of premium method > Bornhuetter-Ferguson method - with the ">" sign denoting greater accuracy.

(b) The percent of premium method would be self-correcting over time, as the earlier time periods' experience falls outside the time period being analyzed and new experience, based on more recent loss ratio behavior, would replace it. The Bornhuetter-Ferguson method would require deliberate adjustment of the expected loss ratio to reflect more current conditions.

Problem S6-38-5. Similar to Question 13(a) from the 2008 CAS Exam 6. You have the following information:

Ratios of ultimate excess loss to ground-up loss
For a retention of $100,000: 0.55
For a retention of $600,000: 0.22

Excess loss development factors, 12 months to ultimate
For a retention of $100,000: 2.24
For a retention of $600,000: 5.56

What is the 12-month-to-ultimate excess loss development factor for the layer $500,000 in excess of $100,000?

Solution S6-38-5. We consider the definition of the 12-month-to-ultimate excess loss development factor. It is (Excess loss at ultimate)/(Excess loss at 12 months).

For the layer in question, the factor is

(Loss in excess of $100,000 at ultimate - Loss in excess of $600,000 at ultimate)/
((Loss in excess of $100,000 at 12 months - Loss in excess of $600,000 at12 months).

Since we are given ratios to ultimate ground-up loss, we can express the desired factor as follows:

(Ultimate ground-up loss)*((Loss in excess of $100,000 at ultimate - Loss in excess of $600,000 at ultimate)/(Ultimate ground-up loss)/
((Loss in excess of $100,000 at 12 months - Loss in excess of $600,000 at12 months) =

((Ratio of loss in excess of $100,000 at ultimate to ultimate ground-up loss) -
(Ratio of loss in excess of $600,000 at ultimate to ultimate ground-up loss))/
((Loss in excess of $100,000 at 12 months - Loss in excess of $600,000 at12 months)/(Ultimate ground-up loss)).

We can replace ((Ratio of loss in excess of $100,000 at ultimate to ultimate ground-up loss) -
(Ratio of loss in excess of $600,000 at ultimate to ultimate ground-up loss)) by 0.55 - 0.22 = 0.33, leading to

0.33/((Loss in excess of $100,000 at 12 months - Loss in excess of $600,000 at 12 months)/(Ultimate ground-up loss)).

We again consider the formula for the excess loss development factor: (Excess loss at ultimate)/(Excess loss at 12 months).

It follows that (Loss in excess of $100,000 at 12 months - Loss in excess of $600,000 at 12 months) = (Ultimate loss in excess of $100,000)/(Loss development factor in excess of $100,000) - (Ultimate loss in excess of $600,000)/(Loss development factor in excess of $600,000).

Thus we have our desired factor as

0.33/((Ultimate loss in excess of $100,000)/(Loss development factor in excess of $100,000) - (Ultimate loss in excess of $600,000)/(Loss development factor in excess of $600,000))/ (Ultimate ground-up loss) =

0.33/((Ratio of loss in excess of $100,000 at ultimate to ultimate ground-up loss)/(Loss development factor in excess of $100,000) - ((Ratio of loss in excess of $600,000 at ultimate to ultimate ground-up loss)/(Loss development factor in excess of $600,000)) =

0.33/(0.55/2.24 - 0.22/5.56) = 1.602196554.

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