**This section of sample problems and solutions is a part of** **The Actuary’s Free Study Guide for Exam 6, authored by Mr. Stolyarov. This is Section 66 of the Study Guide. See an index of all sections by following the link in this paragraph.**

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:**

Pinto, E.; and Gogol, D.F., “An Analysis of Excess Loss Development,” *PCAS* LXXIV, 1987, pp. 227-255. Including discussions of paper: Levine, G.M., *PCAS* LXXIV, 1987, pp. 256-271; and Bear, R.A., *PCAS* LXXIX, 1992, pp. 134-148.

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

**Problem S6-66-1.** In what ways does Bear generalize/extrapolate upon the Pinto-Gogol formula for computation of industry loss development factors for unbounded layers as a function of the retention? Give a qualitative answer. (See Bear, pp. 136-137.)

**Solution S6-66-1.** Bear generalizes the Pinto-Gogol formula to apply to large-account primary pricing and account-specific reinsurance pricing. It is possible, using Bear’s approach, to estimate the account-specific development pattern for a higher layer based on the pattern for a lower layer, if the two layers have the same ratios of gross limit to retention (Bear, pp. 136-137).

**Problem S6-66-2.** You are given the following definitions:

qi = estimated value of single-parameter Pareto distribution parameter during the valuation at i

qj = estimated value of single-parameter Pareto distribution parameter during the valuation at j

e = qi – qj

k1 = Lower bound of excess layer k

k2 = Upper bound of excess layer k

x1 = Lower bound of excess layer x

x2 = Upper bound of excess layer x

c = x1/k1 ≥ 1

d = Incurred loss development factor from valuation at i to valuation at j for losses in layer k

Then Bear’s generalized formula for the incurred loss development factor from valuation i to valuation j for losses in the layer x is as follows (fill in the blanks):

________, where ______.

**Solution S6-66-2.** Bear’s generalized formula for the incurred loss development factor from valuation i to valuation j for losses in the layer x is **dce,** where **x2/x1 = k2/k1**.

**Problem S6-66-3.** You are using Bear’s generalized formula to compare the layer from $400,000 to $600,000 with the layer from $500,000 to $750,000. The estimated value of the single-parameter Pareto distribution parameter during the valuation at 12 months is 1.56. The estimated value of this parameter during the valuation at ultimate is 1.05. These parameter estimations were made on the basis of losses in the layer in excess of $400,000.

The incurred loss development factor from 12 months to ultimate, for the layer from $400,000 to $600,000, is estimated to be 2.45. Find the incurred loss development factor from 12 months to ultimate, for the layer from $500,000 to $750,000.

**Solution S6-66-3.** Our desired factor is dce, where x2/x1 = k2/k1.Here, k1 = 400000, k2 = 600000, x1 = 500000, x2 = 750000, so 750000/500000 = 600000/400000 = 1.5 → x2/x1 = k2/k1.

Here, d = 2.45, the LDF for the layer from $400,000 to $600,000. Also, c = x1/k1 = 500000/400000 = 1.25. Finally, e = qi – qj = q12 months – qultimate = 1.56 – 1.05 = 0.51.

Thus, dce = 2.45*1.250.51 = the desired LDF = **2.745302409**.

**Problem S6-66-4.** Levine used the Pinto-Gogol technique to fit development factors to empirical reinsurance data. What did he find regarding the error between the actual excess loss development factors and the fitted excess loss development factors? (See Levine, pp. 258-263.)

**Solution S6-66-4.** Levine found that, for retentions under $250,000, the error between the actual and fitted excess loss development factors is generally small (under ± 2%), but the error becomes considerably larger for retentions in excess of $250,000. Levine concludes that the Pinto-Gogol technique provides a poor fit to actual data for retentions in excess of $250,000.

**Problem S6-66-5.** Explain the “catch-up” theory described by Levine on pp. 263-264. Why does this theory suggest that, in using the Pinto-Gogol technique, it would be desirable to exclude very high retentions?

**Solution S6-66-5.** The “catch-up” theory attempts to explain the reversal observed by Pinto and Gogol for very high retentions ($500,000 and $1,000,000) of the tendency for excess loss development to increase as retention increases. The theory posits that, many times, the initial estimate of a large claim is held at the same level for many years, until an actual jury trial is held, after which there is a dramatic revision. Because of this dramatic revision, considering development at late maturities would actually *introduce* error into one’s estimate. The “catch-up” theory suggests that different reserving practices are associated with claims that have very high retentions, so applying the Pinto-Gogol technique to those claims may not be optimal.

**Problem S6-66-6.** What did Levine observe regarding the application of the Pinto-Gogol technique to *primary* loss development? What could explain this observation? (See Levine, pp. 265-267.)

**Solution S6-66-6****.** Levine observed “disappointing” results regarding the application of the Pinto-Gogol technique to primaryloss development. Using curves fitted to a single-parameter Pareto distribution has been known to underestimate trends for small sizes of losses, and this tendency may apply to loss development as well. The Pinto-Gogol techniques works better with excess losses than with primary losses because excess losses, by definition, exclude small sizes of losses.

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

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