**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 59 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:**

Teng, M.T.S.; and Perkins, M.E., “Estimating the Premium Asset on Retrospectively Rated Policies,” *PCAS* LXXXIII, 1996, pp. 611-647, excluding Section 5. Discussion by Feldblum, S., *PCAS* LXXXV, 1998, pp. 274-315, Sections 1 and 2 only.

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

**Problem S6-59-1.** Consider the Teng and Perkins approach to estimating the premium asset for books of retrospectively rated insurance policies.

**(a)** Why is it sometimes the case that the losses emerged for a time period such as 0 to 18 months would correspond to premium booked at, for instance, 27 months for the first retro adjustment? That is, why is there a mismatch between the period for emerged losses and the period for booked premiums?

**(b)** Why is it common for the premium development to loss development (PDLD) ratio to exceed 1 at the first retro adjustment and to be less than 1 for subsequent retro adjustments?

**Solution S6-59-1.**

**(a)** It takes time to perform the retro adjustment calculations and to convert emerged losses into retrospective premium – especially if one has to do this with a large number of insureds. This is known as the booking lag. For losses emerged from 0 to 18 months, it is possible for the corresponding emerged premium to only be known and booked at 27 months. (See Teng and Perkins, p. 619.)

**(b)** It is common for the premium development to loss development (PDLD) ratio to exceed 1 at the first retro adjustment because (1) the basic premium is considered as part of premium development, and (2) fewer losses are likely to reach the amount of the loss cap, meaning that, in their conversion to retrospective premium, these losses are fully affected by the tax multiplier and the loss conversion factor. Losses emerging for subsequent retro adjustments are much likelier to be affected by the cap, meaning that there is a decreased likelihood of each incremental dollar of uncapped loss translating into premium. (See Teng and Perkins, p. 622.)

**Problem S6-59-2.** Teng and Perkins (pp. 627-628) discuss three further issues to consider when calculating the premium asset for loss-sensitive rating plans. What are these issues as they relate to the diversity possible among such plans?

**Solution S6-59-2.** Teng and Perkins mention the following issues:

1. Are allocated loss adjustment expenses (ALAE) considered part of the loss for the purposes of retrospective rating? This affects whether ALAE are also considered in calculating the PDLD ratios.

2. Have there been changes in the mix of business – e.g., by state, geographical region, or industry group? If so, this may affect both loss emergence and premium sensitivity to loss.

3. How collectible are the premiums considered to be a part of the premium asset? Are they secured, or is there a need for some consideration of the possibility of bad debt?

**Problem S6-59-3.**

**(a)** Feldblum mentions three distinct advantages that the Teng and Perkins PDLD method has over other procedures. Briefly describe these advantages. (See Feldblum, pp. 274-275.)

**(b)** Feldblum discusses a difficulty of applying a simple chain-ladder method to estimating development for premium on loss-sensitive contracts. Identify such a difficulty and how specialized methods like the PDLD method and Fitzgibbon’s method overcome it. (See Feldblum, pp. 276-277.)

**Solution S6-59-3.**

**(a)** The following are the advantages of the Teng and Perkins PDLD method mentioned by Feldblum:

1. It is based directly on the retrospective rating formula and can be easily explained to underwriters and claims personnel.

2. There is correspondence with statutory reporting of accounting information (in particular, Part 7 of Schedule P). This arises because the PDLD approach emphasizes premium sensitivity to losses.

3. Other methods’ indications may be distorted by changes in retrospective rating plans’ parameters, whereas the PDLD method would take these changes into account.

**(b)** The chain-ladder method relies on direct estimates of retrospective premiums, which take much longer to emerge than incurred losses, resulting in a lag of about 9 months in many cases. The PDLD and Fitzgibbon approaches rely on emerged losses, from which retrospective premiums can be estimated, resulting in more expeditious calculations.

**Problem S6-59-4.**

**(a)** Describe the central assumption of the Fitzgibbon approach. How does linear regression relate to this assumption? (See Feldblum, pp. 280-281.)

**(b)** Why might using regression even be desirable in addressing retrospectively rated policies?

**Solution S6-59-4.**

**(a)** The central assumption of the Fitzgibbon approach is that retrospective premium linearly relates to incurred losses, via the formula (Retrospective Premium) = C + B*Losses. This translates into the formula (Retro Adjustment) = A+B *(Standard Loss Ratio), where the constant A and the slope B are estimated via linear regression on historical data for loss ratios and retro adjustments from mature policy years (Feldblum, pp. 280-281).

**(b)** There can be much variation in the details of specific retrospective rating plans – for instance, in the basic premiums, in the maximum premiums set, and in the premium taxes and involuntary market loads by jurisdiction. It may be extremely challenging and time-consuming to collect and analyze all these data for individual retrospective rating plans. Instead, average characteristics could be estimated using regression. Moreover, because the parameters established for a particular retrospective plan by its pricing actuary might not correspond to the parameters actually used in practice, regression analysis of empirical data might give a more accurate understanding of what is actually happening than inferences from how the retrospective plans “ought” to be applied. (See Feldblum, pp. 281-282.)

**Problem S6-59-5.**

**(a)** Identify two weaknesses of the Fitzgibbon approach.

**(b)** How does the Teng and Perkins PDLD approach depart from the fundamental assumption of the Fitzgibbon method?

**(c)** Feldblum identifies two factors that correlate with the decline of premium responsiveness on loss-sensitive contracts. What are they? (See Feldblum, pp. 288-289.)

**(d)** What, according to Feldblum, are the two assumptions of the PDLD method? (See Feldblum, p. 291.)

**Solution S6-59-5.**

**(a)** The Fitzgibbon approach has the following weaknesses:

1. In assuming a linear relationship between premium development and loss development, the Fitzgibbon approach tends to overstate premium development at later retro adjustments.

2. The Fitzgibbon approach does not distinguish, in its projections, between the impact of many small losses and one large loss of the same magnitude. However, because of loss limits and maximum premiums in the retrospective rating plan, this can make a material difference in the retrospective premium. (See Feldblum, pp. 284-285.)

**(b)** The Fitzgibbon approach assumes a linear relationship between premium development and loss development. The Teng and Perkins PDLD approach, instead, posits a decreasing slope over time for premium development as compared to loss development. The slope between every two subsequent retro adjustments is less steep than the slope preceding it, which counteracts the Fitzgibbon method’s tendency to overestimate premium development at later retro adjustments.

**(c)** Premium responsiveness on loss-sensitive contracts with (1) **increase in maturity (age) of a book of business** and (2) **higher reported loss ratios** (Feldblum, pp. 288-289).

**(d)** The PDLD method assumes

1. The independence of premium development for subsequent retro adjustments from premium development for prior retro adjustments and

2. Dependence on the slope of premium development versus loss development on the time period, not on the beginning loss ratio or retrospective premium ratio (Feldblum, p. 291).

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