**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 44 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. 110-113.

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

**Problem S5-44-1.** By fitting exponential models to observed loss data, an actuary has estimated that loss frequency changed by +6% during time period X, while loss severity changed by -8% during the same time period. What is the percentage change in pure premium during time period X?

**Solution S5-44-1.** Since Pure Premium = (Frequency)*(Severity), it follows that

(Factor by which pure premium changed) = (Factor by which frequency changed)*(Factor by which severity changed).

The factor by which frequency changed is 1.06; the factor by which severity changed is 0.92. Thus, the factor by which pure premium changed is 1.06*0.92 = 0.9752. The corresponding pure premium percentage change is 100*(0.9752 – 1) = **-2.48%**.

**Problem S5-44-2.** An actuary is attempting to trend historical losses to current levels. The losses to be trended are from calendar-accident year (CAY) 3540. The rates which the actuary is developing would take effect for all new policies written on or after January 1, 3556. The actuary determines that the annual loss trend is -2%. The insurance company in question writes annual policies. Determine the trend factor by which historical losses would need to be multiplied in order to be brought to current levels in the actuary’s ratemaking analysis.

**Solution S5-44-2.** This question is based on the discussion in Werner and Modlin, p. 112. To determine the *trend period,* we need to consider the midpoint of CAY 3540, which is where the loss data are taken from. This midpoint is June 30, 3540, i.e., halfway through the calendar year. This is the beginning of our trend period. The end of our trend period is the midpoint of policy year (PY) 3556, which occurs on December 31, 3556, since losses on PY 3556 policies will continue to occur until December 31, 3557. Between June 30, 3540, and December 31, 3556, there are 16.5 years. Thus, our trend factor is (1 – 0.02)16.5 = **0.716523187.**

**Problem S5-44-3.** An actuary is attempting to trend historical losses to current levels. The losses to be trended are from calendar-accident year (CAY) 3540. The rates which the actuary is developing would take effect for all new policies written on or after January 1, 3556. The actuary determines that the annual *linear* loss trend is +54.3 Golden Hexagons (GH). The insurance company in question writes annual policies. Determine the trend addend by which historical losses would need to be adjusted in order to be brought to current levels in the actuary’s ratemaking analysis.

**Solution S5-44-3.** This question is based on the discussion in Werner and Modlin, p. 112. To determine the *trend period,* we need to consider the midpoint of CAY 3540, which is where the loss data are taken from. This midpoint is June 30, 3540, i.e., halfway through the calendar year. This is the beginning of our trend period. The end of our trend period is the midpoint of policy year (PY) 3556, which occurs on December 31, 3556, since losses on PY 3556 policies will continue to occur until December 31, 3557. Between June 30, 3540, and December 31, 3556, there are 16.5 years. For a linear trend, the annual trend is simply multiplied by the term length and *added* to (or subtracted from) historical losses. Here, the addend is 54.3*16.5 = **+895.95 GH**.

**Problem S5-44-4.** An actuary is attempting to trend historical losses to current levels. The losses to be trended are from calendar-accident year (CAY) 3540. The rates which the actuary is developing would take effect for all new policies written on or after January 1, 3556. The actuary is using a two-step loss trending method. The latest loss data available are from CAY 3549; for this period, the actuary estimates the annual loss trend to be +6%. Thereafter, the actuary estimates the annual loss trend to be -3%. Determine the trend factor by which historical losses would need to be multiplied in order to be brought to current levels in the actuary’s ratemaking analysis.

**Solution S5-44-4.** This question is based on the discussion in Werner and Modlin, pp. 112-113. We need to determine the trend period for each of the two steps of this method. For the first step, the trend period is from the midpoint of CAY 3540 (June 30, 3540) to the midpoint of CAY 3549 (June 30, 3549) – i.e., 9 years. For the second step, the trend period is from June 30, 3549, to the midpoint of PY 3556 (December 31, 3556) – i.e., 7.5 years. Thus, our trend factor is

(1.06)9*(1-0.03)7.5 = **1.344438047**.

**Problem S5-44-5.** According to Werner and Modlin, p. 113, what is one fundamental, possibly incorrect assumption entailed in using calendar-year data for the measurement of loss trends?

**Solution S5-44-5.** The fundamental assumption is that the insurer’s book of business has not significantly changed in size during the time period for which data are considered. If the book of business size has changed, there may be an overestimation or underestimation of the loss trend if calendar year data are used. This problem arises because losses from older accident years may possibly be matched with exposures from the year in question.

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