**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 64 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.

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

**Problem S6-64-1.** Pinto and Gogol (pp. 230-233) analyzed excess loss development factors using various treatments of ALAE: (1) adding ALAE to the loss amount, (2) assigning ALAE to the excess layer on a pro rata basis, and (3) not including ALAE in the loss amount. What did Pinto and Gogol observe regarding the effect of the treatment of ALAE on excess loss development factors?

**Solution S6-64-1.** Pinto and Gogol observed that, after 39 months, the treatment of ALAE has no material effect on the excess loss development factors (pp. 230-233).

Most of the factors displayed on pp. 231-233 are roughly the same after 39 months, irrespective of how ALAE is treated.

**Problem S6-64-2.** Fill in the blanks regarding the curve-fitting approach used by Pinto and Gogol. (See p. 234.)

**(a)** For each development interval, a curve was fitted to the excess loss development factors as a function of _______.

**(b)** Separate curves were fitted for each ________.

**(c)** The curve formula selected was y = ________. (Give the variables on other side of the equation, with definitions as appropriate.)

**(d)** The distribution whose qualities motivated the selection of the function in (c) was the __________ distribution.

**Solution S6-64-2.** **(a)** For each development interval, a curve was fitted to the excess loss development factors as a function of **retention**.

**(b)** Separate curves were fitted for each **line of business**.

**(c)** The curve formula selected was y = **axb**, where x is the retention divided by $10,000, and a is the value for development in excess of $10,000.

**(d)** The distribution whose qualities motivated the selection of the function in (c) was the **single-parameter Pareto** distribution. (In the function, b acts as the parameter.)

**Problem S6-64-3.** For any one of the curves y = an*xb_n used by Pinto and Gogol, explain in general terms the procedure by which the values for an and bn were determined. (See p. 234.)

**Solution S6-64-3.** Pinto and Gogol applied natural logarithms to each side of the equation y = an*xb_n to get ln(y) = ln(an) + bn*ln(x). This is a linear function, where the values ln(an) and bn can be determined using least-squares linear regression.

**Problem S6-64-4.** Pinto and Gogol compare development factors for excess losses between those based on data from the Reinsurance Association of America (RAA) and those based on data from the Insurance Services Office (ISO), fitted using the methods of Pinto and Gogol. It is observed that, after 99 months, the RAA data show materially higher development factors than the fitted ISO data. What are the possible explanations offered by Pinto and Gogol for this observation? (See pp. 239-240.)

**Solution S6-64-4.** Pinto and Gogol offer the following explanations for this observation:

1. Aggregate-basis reinsurance coverage may show its effects later on in the loss development. When reinsurance treaties are written on an aggregate basis, it takes a longer period for losses to accumulate so as to breach the aggregate retention.

2. There may be unidentified longer-tailed medical malpractice losses present in the RAA data.

3. The ISO data have a fixed retention of $250,000. The RAA data have various retentions and limits, and the distribution of those may be more conducive to larger excess loss development factors later on.

4. The RAA data include information on umbrella, excess, and surplus lines business that the ISO data exclude.

5. RAA development factors are based on actual empirical data, with no fitting. The ISO factors beyond 99 months are based on a curve fitted to the data through 99 months.

**Problem S6-64-5.** Pinto and Gogol also analyzed excess *paid* loss and ALAE development and compared it to excess *reported* loss and ALAE development (pp. 241-242), as a function of the retention. What did they observe?

**Solution S6-64-5.** Pinto and Gogol observed that the ratios of paid excess losses to reported excess losses at each age of maturity do not materially vary as a function of the retention (p. 242).

**Problem S6-64-6.** **(a)** In a single sentence, summarize the main theoretical insight of the Pinto and Gogol paper.

**(b)** What are the two principal influences on excess loss development, according to Pinto and Gogol, and which one of these influences is indispensable to the main theoretical insight in part (a) above?

**Solution S6-64-6.** **(a)** Excess loss development increases as the retention increases (p. 245).

**(b)** The two principal influences on excess loss development are

1. The change in the reporting pattern of claims over time; and

2. Change in the characteristics of the distribution for loss size at successive reports (p. 245).

Influence 2 above is the one indispensable to Pinto and Gogol’s observation. The lack of this influence would produce excess loss development factors that do not vary by retention.

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

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