RESERVING FOR LOSS SENSITIVE PREMIUM ITEMS. By Brian Z. Brown and Michael C. Schmitz. Abstract
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1 RESERVING FOR LOSS SENSITIVE PREMIUM ITEMS By Brian Z. Brown and Michael C. Schmitz Abstract Many insurers and reinsurers have significant accruals for loss sensitive premium items. Examples of contract features which may require an accrual are: 1. Retrospective rated policies and dividend policies for primary insurers; and 2. Contingent profit commission and no claim bonuses for reinsurers. In estimating these accruals it is important to reflect the range of potential outcomes as illustrated by probability distributions as expected values will usually create biased results. This paper discusses a general framework for establishing these accruals and then describe the process for two hypothetical policies. H:\BRIAN\PAPERS\APR.99\abstract.doc
2 RESERVING FOR LOSS SENSITIVE PREMIUM ITEMS By Brian Z. Brown and Michael C. Schmitz Many insurers and reinsurers have significant accruals for loss sensitive premium items. Examples of contract features which may require an accrual are: 1. Retrospectively rated policies and dividend policies for primary insurers; and 2. Contingent profit commission and no claim bonus plans for reinsurers. In estimating these accruals it is important to reflect the range of potential outcomes as illustrated by probability distributions since expected values will usually create biased results. This paper discusses a general framework for establishing these accruals and then describes the process for two hypothetical policies. The general approach involves estimating a profit sharing/loss sensitive provision at policy inception and revising the provision over the life of the policy based on:. Actual loss experience; and. An estimated reporting pattern. PRIMARY INSURER RETROSPECTIVE PREMIUM ACCRUALS For a primary insurer, we will discuss the process of establishing a reserve for a retrospectively rated policy. Background A basic form of a retrospectively rated policy premium is: H < R = T(B+CL) < G where H is the minimum premium; R is the retrospectively rated premium; T is the tax multiplier; B is the basic premium; C is the loss conversion factor; L is the actual losses (reported or paid) during the policy period; and G is the maximum premium. See Retrospective Rating: Excess Factors by William R. Gillian, PCAS, LXXVIII, Part 1
3 - 2 - To make the example easier to follow we will assume that: H = 300 T = 1.00 B = 300 C = 1.1 losses = 700 G = 1,100 One may ask if the expected premium is equal to: R = 1(300+(1.1)(700)) = 1,070. The answer may be no because of the limiting effect of the minimum and maximum premium. For our example, we will assume that the loss distribution is as follows: LOSS DISTRIBUTION AT INCEPTION Probability Outcome 0.10 $ ,000 In practice, the actual loss distribution will not be discrete, but rather continuous. However, the above simple distribution will illustrate the concept. As Exhibit 1 displays, if the insured generates losses of 800, 900 or 1,000, then the retrospective premium is limited at the maximum premium of 1,100. Reflecting the maximum, the expected retrospective premium is 1,013, which is below the premium of 1,070 calculated based on expected losses alone. If the retrospective reserve is calculated as the ultimate retrospective premium less premium paid to date and the premium paid-to-date is 500, then the retrospective reserve is However, if the expected values were used in the calculation instead of reflecting the probability distribution, 2 It may be appropriate to reflect the possibility that the insured it not able to pay the premium in the calculations (e.g., non payment risk due to bankruptcy). See Brian Z. Brown, Pricing for Credit Exposure, PCAS LXXIX, 1992, page 186 for more details on a process to estimate the non-payment reserve.
4 - 3 - then the expected premium of 1,070 would be used to calculate an incorrect reserve of 570 (an overstatement of 11.1%). Premium Calculations Post Policy Inception A complicating factor regarding retrospective reserves is that the distribution outlined above and in much of actuarial literature relates to loss outcomes at the beginning of the policy period. The formulation of the loss distribution becomes more problematic after claims are a number of years old. The above concept may be illustrated by a simple example. We will use the retrospective parameters outlined above but assume that it is currently three years after policy inception. If actual incurred losses are 560 and we expect these actual losses to ultimately develop by 25% to 700, then one may ask if the ultimate premium is equal to: and the reserve equal to: R = 1(300+(1.1))(560x1.25) = 1,070 1,070 less the indicated premium with losses at 560 (i.e., 154 = 1, ). The answer again may be no since it depends upon the distribution of the development of losses. Let us assume the 560 of incurred losses will develop to an expected value of 700 based on the following distribution of ultimate losses: LOSS DISTRIBUTION 3 YEARS POST-INCEPTION Probability es Exhibit 2 displays the premium calculations based on the loss probabilities in the above table. The expected value ultimate premium reflecting the maximum premium is 1, This generates a reserve of which is 12% below the reserve of 154 calculated above based on the expected value of losses.
5 - 4 - Additional Complications/Adjustments Another difficulty with a standard projection method based on expected values is that one average loss development factor (LDF) is frequently applied to every account. While this process develops theoretically correct losses in total it will not develop losses correctly for each account. Due to differences in development patterns by account and even for an account over time, a group of losses will have their own unique pattern of development. For example, we would expect the following two claim populations for workers compensation to display different development patterns. Claim Population Number of Open Claims Paid es Case Reserves Incurred es A 25 1, ,000 B 50 1, ,000 Even if the claim populations came from the same state/classification and are at the same stage of maturity, population B will probably develop differently than population A because it has twice as many open claims with three times as much case reserves as population A. Therefore, we would normally expect that population B s claims will have more potential for unusual development and have a larger variance than population A s claims (additionally B s losses often will develop to a higher ultimate than A s due to supplemental case development). In some of our analyses, we have found that it may be more accurate (especially for accounts with limited credibility) to estimate development by account into two pieces: 1. Supplemental case development; and 2. True IBNR. This method appears to work well for coverages where true IBNR is not the major component of development. An example of a coverage with this characteristic is workers compensation. Also, recall that for workers compensation, another element of IBNR is the reserve for reopened claims. For workers compensation we could estimate supplemental development as a factor times case reserves (or a combination of limited case reserves and open claim counts). 3 3 Methods to estimate case reserve development factors can be found in Pricing for Retained Workers Compensation Exposures by Brian Z. Brown and Michael D. Price, PCAS, LXXXIV, page 128.
6 - 5 - True IBNR can then be estimated based on the following: Premium; Reported losses; Reported claims; Closed claims; and Other factors. The above factors can then be varied to determine the sensitivity of the premium reserve to changes in loss outcomes. If a small or moderate change in the assumptions results in a significant change in the premium reserve accrual, then more detailed analysis may be performed (e.g., independent review of case reserve adequacy). Alternatively, the actuary may wish to establish a lower premium reserve to reflect a margin for the uncertainty associated with estimating loss reserves by account. Another element to consider when establishing reserves for a retrospectively rated policy is the probability of default. Some insureds may file for bankruptcy and be unable to make future premium payments. One way to estimate this reserve is to apply default probabilities to the projected payments by year. This method is outlined in a previous PCAS Proceedings Paper. 4 REINSURANCE PROFIT SHARING AGREEMENTS Similar issues arise in calculating accruals for profit sharing agreements on reinsurance contracts. Background An example of one form of a reinsurance profit sharing agreement is illustrated by the following terms: Layer: $500,000 excess of $500,000 Premium: 15% of Subject Net Earned Premium (SNEP) Reinsurer Expense Factor: 40% of Reinsurance Premium (Including Margin) Profit Sharing = ½ x (Reinsurance premium losses Reinsurers expenses) Timing of Disbursements: at 42 months and annually thereafter. 4 ibid 2.
7 - 6 - Therefore, if no losses are ceded and SNEP is 10,000,000 then the profit sharing payment to the ceding company is: 0.5 x [1,500,000 0 (0.40)(1,500,000] = $450,000 The question is how should the profit sharing contract be booked on the reinsurer s and ceding company s financial statement. Inception Calculations This issue may be easiest to solve at contract inception since the cedant and reinsurer may have gone through detailed loss simulations to determine the expected losses ceded to the contract and the likely variation in the loss outcomes. For illustrative purposes, we assume that both the cedant and reinsurer have estimated the following distribution of loss outcomes in the $500,000 excess of $500,000 layer. Ceded Distribution/Profit Sharing Calculation Probability Ceded Outcome Profit Sharing Return 1 10% $0 $450, , , ,000, ,500, ,000, ,500,000 0 Value $1,250,000 $85,000 1) Profit sharing return amounts are calculated for various loss outcomes based on the formula above. The expected value profit sharing amount is calculated as the weighted average return using the given probabilities (i.e., it is not calculated from the formula using the expected value losses). We will further assume the reinsurer writes 100 such contracts for insureds with similar loss distributions. It would be incorrect to use the expected value loss outcome of $1,250,000 in computing the profit sharing contributions for each insured, as some insureds will have worse experience and some will have better experience. As the above table and Exhibit 3 illustrate, an expected value profit sharing return of $85,000 is generated based on the above loss distribution with an expected value of $1,250,000. While an actual loss outcome of $1,250,000 implies a zero profit sharing amount, the distribution of loss outcomes comprising an expected loss amount of $1,250,000 implies a positive expected profit sharing amount.
8 - 7 - At contract inception, the reinsurer should establish a loss reserve of $1,250,000 and a profit sharing reserve of $85,000 for each of the 100 contracts (assuming for illustrative purposes that all premium is earned immediately at inception). While the two reserves at first glance appear inconsistent, they are not. Calculations Post Policy Inception One difficulty that arises is determining how to change the reserve over time as experience emerges. As time progresses, some insureds will move towards a $0 or $500,000 loss outcome with profit sharing payments while other insureds will move towards high loss outcomes and no profit sharing returns. The difficulty lies in projecting accurate loss outcomes for each insured. For example, if a Bornhuetter-Ferguson (B-F) method is used to project losses by account for 100 accounts, then chances are that the analyst will project ultimate losses near $1,250,000 per contract early on (due to the fact that the reporting pattern for casualty excess of loss contracts is extended). If the analyst then uses this loss outcome to estimate a profit sharing return accrual, then a return of $0 is likely to be indicated early on. However, a profit sharing return of $85,000 may still be appropriate for the same reason it was initially. To overcome this flaw it may be prudent to estimate the expected (based on all possible loss outcomes) profit sharing return at policy inception and gradually change the expected return over time as individual account s actual experience becomes mature (and thereby reflecting a full spectrum of outcomes). Additionally, making the calculation at policy inception allows the actuary to determine the significance of the profit sharing return relative to loss reserves. In our example, the expected profit sharing reserve of $85,000 is 6.8% of the initial loss projection of $1,250,000. Some actuaries may therefore believe that total reserves (both loss and profit sharing) are appropriate if a 7% cushion exists in loss reserves with no profit sharing reserve. This assumption may be reasonable at initial evaluations but will likely be flawed as the contracts mature. On an individual account basis, maturing contracts with relatively high losses and loss reserves should be associated with lower reserves for profit sharing and vice versa. Clearly, this runs counter to tying the profit sharing reserves to a fixed percentage of the loss reserve. One method to determine a profit sharing reserve involves establishing a profit sharing reporting patterns by applying a consistent methodology to estimate ultimate losses at regular evaluation points. These ultimate losses can then be used to estimate the profit sharing return at each point. To illustrate this concept, we will use the previous example and assume a B-F method is used to estimate losses by account and losses follow the reporting pattern displayed below:
9 - 8 - Evaluation Period (years after inception) Percentage of es Reported 1 10% If we write 100 identical accounts with the parameters displayed above, our initial B-F loss selection is $1,250,000 and our B-F calculation at the end of year 1 is: (1) (2) Actual Outcome (3) es (4) (5) IBNR (6) Actual Reported (7) Estimate Reporting Probability Pattern 10% $0 $1,250 10% $1,125 $0 $1, , , , ,000 1, , , ,500 1, , , ,000 1, , , ,500 1, , ,375 Average $1,250 (5) = (3) x [1-(4)] (6) = (2) x (4) At the first evaluation, all of the loss projections exceed $900,000 {i.e., $1,500,000 x (1-40%)} so the indicated profit sharing return is $0. However, some accounts will generate a profit, sharing return based on the actual loss outcomes in column (2). $85,000 is the expected profit sharing return. Exhibits 4 through 8 display subsequent B-F calculations. We note that the B-F method produces an unbiased estimate of ultimate losses of $1,250,000. Based on the above mentioned B-F calculations, reflecting the probability distribution of losses, the estimate of ultimate losses is equal to the expected value of $1,250,000 for every calculation.
10 However, the profit sharing return is understated at early evaluations due to the fact that the B-F calculations produce loss estimates near the mean (and thus do not reflect the true variation in losses e.g., with some loss outcomes at low amounts). Thus the indicated profit sharing return is underestimated using this method. The following table displays the indicated profit sharing return by year based on our B-F calculations for losses: 4 43, , ,000 However, as discussed previously, if the reinsurer wrote 100 accounts we would expect that 30% of the accounts would ultimately generate losses low enough for a profit sharing payment. Therefore, a profit sharing return of $8.5 million is expected to be paid (i.e., $85,000 x 100). Performing a B-F calculation on each account to determine ultimate losses and then using the B- F indicated losses in the profit sharing formula produces unbiased estimates of losses but understates the profit sharing calculation at early evaluations. An alternate method is to use an a priori estimate of the profit sharing accrual and age this a priori estimate over time (based on a reporting pattern) to establish a bulk profit sharing accrual. For example if we use the profit sharing reporting pattern above, we have the following: (2) = Based on B-F Method by Account (3) = (2) + 85,000 (5) =(4)x(1-3) (6) = (5) + (2)
11 The individual contract profit sharing return is added to the bulk provision to estimate the ultimate profit sharing return (see Column (6)). Profit sharing payments to date can be subtracted to estimate the accrual for future profit sharing payments. CONCLUSION This paper has attempted to illustrate the importance of reflecting the probability distributions in estimating various non-loss reserve accruals. If the probability distribution is not reflected the non-loss reserve accruals are likely to be mis-stated. We have also outlined an approach to estimate profit sharing returns. The approach involves estimating a loss sensitive provision at policy inception and revising the provisions over the life of the policy based on actual loss experience, projected ultimate losses, and an estimated profit sharing reporting pattern.
12 Exhibit 1 (1) RETROSPECTIVE PREMIUM ESTIMATE AT INCEPTION (2) (3) (4) Retro Premium (1)x(3) Retro Probability es Premium , , * , *.10 1,000 1, * Premium 1,013 * Limited to maximum premium
13 Exhibit 2 (1) RETROSPECTIVE PREMIUM ESTIMATE POST-INCEPTION (2) (3) (4) Premium (1)x(3) Probability es Premium , , ,100* ,100* Premium 1,051.5 * Limited to maximum premium
14 Exhibit 3 If the insured writes 100 risks which are identical, the following reserves are needed: Number of Policies A Outcome Per Policy Profit Sharing Return Per Policy , , , ,500 0 Total All Policies 125,000 8,500 Average B 1, A B Based on probabilities from page 6 Total divided by 100 policies Therefore, for these 100 risks, a reserve of $125,000,000 is needed for losses and $8,500,000 for profit sharing returns or an average $1,250,000 loss reserve and $85,000 profit sharing reserve per contract.
15 Exhibit 4 (1) (2) Outcome (3) es AT THE END OF YEAR 2 (4) (5) IBNR (6) Actual Reported (7) Estimate (8)* Estimated Profit Share Probability Reporting Pattern 10% $0 $1,250 30% $875 $0 $ , , ,000 1, , ,500 1, , ,000 1, , ,500 1, ,625 0 Average $1, * (5) = (3)[1-(4)] (6) = 30% of (2) (8) = Max {(.5) x [1,500-(7)-600], 0} At this evaluation the account with $0 in ultimate losses generates a profit sharing amount and the expected amount based on the above methodology is 1,250 compared to the correct amount of $85,000. Again the ultimate loss estimate is unbiased.
16 Exhibit 5 (1) (2) Outcome (3) es AT THE END OF YEAR 3 (4) (5) IBNR (6) Actual Reported (7) Estimate (8)* Estimated Profit Share Probabilit y Reporting Pattern 10% $0 $1,250 50% $625 $0 $ , ,000 1, , ,500 1, , ,000 1, ,000 1, ,500 1, ,250 1,875 0 Average $1, (5) = (3)[1-(4)] (6) = 50% of (2) (8) = Max {(.5) x [1,500-(7)-600], 0} Profit sharing amount based on the above methodology is $16,250 compared to correct amount of $85,000.
17 Exhibit 6 (1) (2) Outcome (3) es AT THE END OF YEAR 4 (4) (5) IBNR (6) Actual Reported (7) Estimate (8)* Estimated Profit Share Probability Reporting Pattern 10% $0 $1,250 70% $375 $0 $ , ,000 1, , ,500 1, ,050 1, ,000 1, ,400 1, ,500 1, ,750 2,125 0 Average $1, (5) = (3)[1-(4)] (6) = 70% of (2) (8) = Max {(.5) x [1,500-(7)-600], 0} Profit sharing amount based on the above methodology is $43,750 compared to correct amount of $85,000.
18 Exhibit 7 (1) (2) Outcome (3) es AT THE END OF YEAR 5 (4) (5) IBNR (6) Actual Reported (7) Estimate (8)* Estimated Profit Share Probability Reporting Pattern 10% 0 1,250 90% $ , ,000 1, , ,500 1, ,350 1, ,000 1, ,800 1, ,500 1, ,250 2,375 0 Average $1, (5) = (3)[1-(4)] (6) = 90% of (2) (8) = Max {(.5) x [1,500-(7)-600], 0} Profit sharing amount based on the above methodology is $71,250 compared to correct amount of $85,000.
19 Exhibit 8 (1) (2) Outcome (3) es AT THE END OF YEAR 6 (4) (5) IBNR (6) Actual Reported (7) Estimate (8)* Estimated Profit Share Probability Reporting Pattern 10% $0 $1, % $0 $0 $ , ,000 1, ,000 1, ,500 1, ,500 1, ,000 1, ,000 2, ,500 1, ,500 2,500 0 Average $1, (5) = (3)[1-(4)] (6) = 100% of (2) (8) = Max {(.5) x [1,500-(7)-600], 0} Profit sharing amount based on the above methodology is $85,000.
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