The appendix to the Proceedings paper by Almagro and Ghezzi shows the IRS loss reserve discounting procedure, with illustrations for various lines of business. The the IASA textbook chapter describes the IRS loss reserve discounting rules.

Examination committee members are familiar with these procedures. Moreover, the syllabus readings are complex, making them excellent fodder for difficult but "fair" examination questions.

The following practice problems provide a review of the more complicated examination material that candidates should master. Most candidates find this material to be quite difficult. This study aid provides an efficient way to review the tax readings in preparation for the examination.

Practice Problem: Personal Auto Liability

You are given the following information from your 1999 Schedule P, Part 1B, for your Personal Automobile liability book of business: Accident Year Losses Paid Losses Incurred 1989 & Prior 250,000 250,000 1990 270,000 275,500 1991 300,000 316,000 1992 320,000 348,000 1993 340,000 386,500 1994 350,000 421,500 1995 370,000 480,500 1996 380,000 550,500 1997 360,000 610,000 1998 330,000 687,500 1999 200,000 571,500 The 60 month rolling average of the federal mid-term rate, from January 1994 through December 1998, is 7% per annum. Your company has elected to use its own Schedule P data to determine the IRS loss reserve discount factors. What is the loss reserve discount factor for accident year 1999 for the Personal Automobile liability book of business?

Solution: Personal Auto Liability

The solutions to the loss reserve discount problems can be divided into three components:

• Calculate the nominal amounts of
  
  
  • Cumulative percentage paid,

• Incremental percentage paid, and

• Percentage unpaid.
  
  

• Calculate the adjustments for long-tailed lines of business, showing
  
  
  • Adjusted incremental percentage paid,

• Long-tail extension of payments, and

• Adjusted percentage paid.
  
  

• Apply the appropriate discount factor to obtain the
  
  
  • Discounted percentage unpaid, and

• Loss reserve discount factor.
  
  

The column captions for the exhibits in this study aid follow the terms used by Almagro and Ghezzi. These are the terms most likely to be used on the examination, so candidates should be familiar with them.

Nominal Amounts

Exhibit A shows the completed entries for the data given in the problem. Schedule P, Part 1B, provides cumulative losses paid and losses incurred by accident year. The "cumulative percentage paid" in column (4) is the former divided by the latter. For instance, for accident year 1999, the cumulative percentage paid is $200,000 ÷ $571,500 = 35%.

The "incremental percentage paid" in column (5) is the difference between the cumulative percentages paid in adjoining rows. For instance, the incremental percentage paid for accident year 1998 is the difference between the cumulative percentages paid for accident years 1998 and 1999, or 48% - 35% = 13%. For the current accident year, the incremental percentage paid equals the cumulative percentage paid (35% in this example).

The cumulative percentages paid are determined separately for each accident year. It is possible for Schedule P to show a higher cumulative percentage paid for a more recent accident year, leading to a negative incremental percentage paid. This is most likely to occur in older accident years, where the cumulative percentages paid for adjoining accident years are often close to each other. The IRS has provided special rules to deal with negative incremental percentages paid; see the exhibits in the Almagro and Ghezzi appendix.

The "percentage unpaid" in column (6) is unity minus the cumulative percentage paid. For instance, for accident year 1998, the percentage unpaid is 1 - 48% = 52%.

Adjusted Amounts

In this problem, all losses are paid by the tenth year. Note that for accident years 1989 and prior, both the incurred losses and the cumulative paid losses are $250,000.

There are no "adjustments" for payments beyond the tenth year in this example. The entries for the "11th and prior" through the "All Prior" rows are zero in the "long-tail extension of payments" column and in the "adjusted percentage unpaid" column. For the accident years listed in Schedule P, the adjusted percentage unpaid entries equal the (unadjusted) percentage unpaid entries. (See columns (8), (9), and (10).]

Discount Factors

The "discounted percentages unpaid" are determined from the "adjusted percentage unpaid" entries and the applicable discount rate. The applicable discount rate is the 60 month moving average of the federal midterm rate. The problem specifies that this figure is 7% per annum.

The discount factor is determined at each year end and is then frozen for the accident year to which it applies. In other words, the discount factor of 7% per annum is applicable to the 1999 accident year. Another discount factor, based on the average of the federal mid-term rates from January 1993 through December 1997 would be used for the 1998 accident year. Although the average mid-term rate changes between December 1998 and December 1999, the discount rate used for accident year 1998 loss reserves does not change between December 31, 1998, and December 31, 1999.

Nevertheless, the accompanying exhibit shows the "discounted percentage unpaid" and the "loss reserve discount factors" for all accident years using the 7% per annum discount rate, for two reasons:

• The complete set of entries serves a heuristic purpose, in that it shows the candidate the type of calculations required. Note, however, that for each entry a different discount rate is needed.
  
  

• This is the form of the Almagro and Ghezzi exhibits. The Almagro and Ghezzi calculations are correct, since they are determining loss reserve discount factors when the 1986 tax reform was introduced, and all accident years used the same discount rate. The examination committee may infer from this that the same is true for subsequent years, and it may ask the candidate to reproduce the Almagro and Ghezzi exhibits. So be prepared.
  
  

Present Values

Loss payments are assumed to occur in the middle of the year. The "discounted percentage unpaid" is the "adjusted incremental percentage paid" and the "long-tail extension of payments" discounted at the 7% per annum rate. Thus, the accident year 1999 entry is

(13.00% ÷ 1.070.5) + (11.02% ÷ 1.071.5) + . . . + (3.07% ÷ 1.078.5) + (2.00% ÷ 1.079.5) = 52.26%.

The "loss reserve discount factor" is the quotient of the "discounted percentage unpaid" and the "adjusted percentage unpaid." For example, the accident year 1999 loss reserve discount factor is 52.26% ÷ 65.00% = 0.803944.

Question Formats

The CAS examination questions may be worded in various ways. The solution to this practice problem gives the loss reserve discount factor. Alternative question format are equally likely. For instance, the question may ask, "What are the discounted reserves for accident year 1999?"

The statement reserves are $571,500 - $200,000 = $371,500. The discounted reserves are $371,500 * 0.803944 = $298,665.

A Complete Tax Problem

A complete federal income tax question would provide the following information:

• The applicable discount rate for each accident year, and

• The losses paid and incurred at the current statement date and at the previous statement date.
  
  

The question would ask: "What is the addition to statutory income resulting from the IRS loss reserve discounting procedure?"

This question is real: it reflects the work actually required of the insurance company's tax accountant. However, it takes too long to perform the mathematics.

Nevertheless, the candidate should know what is required, since the examination committee may test any part of this procedure. The following items are required:

• The discounted percentage unpaid entries are determined for each accident year, using the discount rate applicable to that accident year.
  
  

• The loss reserve discount factor is the quotient of the discounted percentage unpaid and the adjusted percentage unpaid.
  
  

• These calculations are performed for the current statement date and for the previous statement date.
  
  

• For each statement date, the total undiscounted reserves and the total discounted reserves are determined. The undiscounted reserves are the losses incurred minus the losses paid. The discounted reserves are the undiscounted reserves times the loss reserve discount factor (by accident year).
  
  

• The difference between the undiscounted reserve and the discounted reserve is the amount of loss reserve discount.
  
  

• The change in the loss reserve discount from the previous statement date to the current statement date is the addition to regular taxable income. For instance, if the December 1998 loss reserve discount is $600,000, and the December 1999 loss reserve discount is $675,000, then the 1999 addition to regular taxable income is $75,000.
  
  
  
  

Practice Problem: Products Liability

You are given the following information from your 1994 Schedule P, Part 1R, for your Products Liability book of business, occurrence forms (you do not sell claims-made Products Liability business):

Accident Losses Losses

Year Paid Incurred

1984 & Prior 235,000 250,000

1985 50,000 55,500

1986 55,000 62,000

1987 60,000 70,000

1988 65,000 80,000

1989 70,000 96,000

1990 65,000 103,000

1991 60,000 115,000

1992 50,000 125,000

1993 35,000 140,000

1994 15,000 180,000

The 60 month rolling average of the federal mid-term rate, from January 1989 through December 1993, is 7% per annum. Your company has elected to use its own Schedule P data to determine the IRS loss reserve discount factors.

What is the loss reserve discount factor for accident year 1994 for the Products Liability book of business?