Pinto and Gogol: Step-by-Step Guide to Paid Loss Excess Development Factors
(The attached PDF file has better formatting.)
We derive paid loss development factors from reported loss development factors for
ground-up losses
losses excess of a retention
losses bounded by a limit
a layer of loss
We may derive age-to-age link ratios or loss development factors to ultimate. These form 4 × 2 = 8 combinations. The exam problems on paid loss development are variations on a single theme.
Focus on the theme, which the same for all exam problems.
Learn the method first for a scenario with no limit or retention. The method is easy. The paid loss development factor is the reported loss development factor times the ending paid-to-reported ratio divided by the beginning paid-to-reported ratio.
The paid-to-reported ratio at ultimate is one.
Learn two details in the Pinto and Gogol paper which are used in many exam problems:
The paid-to-reported ratios at any maturity are the same for retentions over $25,000
The outstanding to reported ratios for general liability decline by two thirds each year after 63 months.
Step #1: Data
We use
The reported loss development factor excess of the retention.
The paid-to-reported ratios at the beginning and end of the period.
Take heed: At ultimate, the paid-to-reported ratio is 100%. To derive paid development factors to ultimate, we need the paid-to-reported ratio at the beginning of the period only.
The exam problem may give other data which you convert to the data needed.
Common Errors
If the exam problem gives a ratio using paid losses, check the denominator:
The ratio of paid losses at a valuation date to ultimate losses is the reciprocal of the paid loss development factor to ultimate.
The ratio of paid-to-reported losses at a valuation date is the ratio of the reported loss development factor to the paid loss development factor.
Both ratios are used in the formulas. The first is a ratio along maturities; the second is ratio by type of loss.
Step #2: Intuition
We start with no retention. The age-to-age link ratios and loss development factors are for ground-up losses. The solution method is the same for LDF’s limited to or excess of a retention or LDF’s for a layer of loss. We use the following abbreviations:
α
reported loss development factor or link ratio α
1 is the cumulative reported losses at the beginning of the period α
2 is the cumulative reported losses at the end of the periodβ
paid loss development factor or link ratio β
1 is the cumulative paid losses at the beginning of the period β
2 is the cumulative paid losses at the end of the periodγ
paid-to-reported ratio γ
1 is the paid-to-reported ratio at the beginning of the period γ
2 is the paid-to-reported ratio at the end of the period
The cumulative paid losses and cumulative reported losses increase from zero at policy inception to total losses at ultimate.
The paid-to-reported ratio is for cumulative losses and differs at the beginning and end of the interval.
The paid-to-reported ratio is zero at policy inception. A loss may occur as soon as the policy is written, but it will not be paid for some time.
The paid-to-reported ratio is 100% at ultimate, when all losses have been paid.
The paid-to-reported ratio increases as losses mature.
We have four relations:
α
= α2 / α1 β
= β2 / β1 β
1 = γ1 × α1 β
2 = γ2 × α2
We derive:
β = β2 / β1 = γ2 × α2 / (γ1 × α1) = α × γ2 / γ1For loss development factors to ultimate, γ2 = 100%. The relation is β = α / γ1.
The same relations hold for excess loss development factors. The paid-to-reported ratio is for losses above the retention. Solve the problems in the same fashion.
The study aid has numerical examples. For optimal exam preparation, use the step-by-step guide in conjunction with the study aid. Solving each problem by first principles and then seeing the algebraic formula is the ideal study method.
Common Errors
The exam problems deal with loss development excess of a retention or in a layer of loss. Master the basic principles first. If you understand the rules for ground-up losses, you can extend them to excess losses and layers of loss.
(1) The paid-to-reported ratio increases as losses mature, so γ2 > γ1. This implies that
β
> α: the paid loss development factor exceeds the reported loss development factor.If you get the opposite relation, you have reversed the figures. This error occurs at times under exam pressure. Paid loss development exceeds reported loss development.
(2) The paid-to-reported ratio in the exam problem may be for losses below the retention
γ– or above the retention γ+. The relation is γ– < γ < γ+. Pinto and Gogol use γ+; the exam problem may use either one. Check carefully the wording of the exam problem.Step #3: Convert Ratios to Dollars
For your pre-seminar home study, use dollars of loss. You spend an extra minute, but you save time as you work the problem and you don’t make arithmetic errors.
Illustration: Suppose the reported loss development factor from 48 months to ultimate is 1.25 and the paid-to-reported ratio at 48 months is 75%.
The formula gives the paid loss development factor from 48 months to ultimate as 1.25 / 75% = 1.667. On the exam, you may erroneously multiply 75%.
Let ultimate losses be $100,000. The reported losses at 48 months are $100,000 / 1.25 = $80,000. The paid losses at 48 months are $80,000 × 75% = $60,000. The paid loss development factor is $100,000 / $60,000 = 1.667.
This illustration is simple enough that we see the intuition even without dollars of loss. For more complex problems, always use dollars of loss.
Illustration: Suppose the loss development factors to ultimate are
| 24 Mos to Ultimate | 36 Mos to Ultimate |
Reported loss | 2.000 | 1.500 |
Paid loss | 6.500 | Z |
Suppose the paid-to-reported ratio is 40% at 36 months. The exam problem asks for the 24 to 36 month paid loss age-to-age link ratio.
Using formulas leads to arithmetic errors. Assume $100,000 of losses at ultimate:
Reported losses at 24 months are $100,000 / 2.000 = $50,000.
Paid losses at 24 months are $100,000 / 6.500 = $15,385.
Reported losses at 36 months are $100,000 / 1.500 = $66,667
Paid losses at 36 months are $66,667 × 40% = $26,667.
The 24 to 36 months paid loss age-to-age factor is $26,667 / $15,385 = 1.628
Step #4: Single Maturity
Illustration: For a retention of $75,000 at 36 months, suppose that
Reported losses are 60% below the $75,000 retention and 40% above the retention.
The paid-to-reported ratios are 30% below the retention and 20% above the retention.
The ground-up paid-to-reported ratio is 60% × 30% + 40% × 20% = 26.00%.
Verify this with first principles: Assume $100 million of total losses at 36 months.
Reported losses are $100 million × 60% = $60 million below a $75,000 retention and $100 million × 40% = $40 million above the retention.
Paid losses are $60 million × 30% = $18 million below a $75,000 retention and $40 million × 20% = $8 million above the retention.
Total paid losses are $18 million + $8 million = $26 million.
The ground-up paid-to-reported ratio is $26 million / $100 million = 26.00%.
The exam problem may give the reported loss development factor excess of the $75,000 retention and ask you to derive the paid loss development factor.
We need the paid-to-reported ratio for loss above the retention.
The exam problem may give the ground-up paid-to-reported ratio (26%), the paid-to-reported ratio for losses below the retention (30%), and the percentage of reported losses below the retention at 36 months (60%).
You first derive the paid-to-reported ratio for losses above the $75,000 retention and then derive the paid loss development factor from the reported loss development factor.
Common Errors
Take Heed: The exam problem may give the percentage of paid losses below the retention at 36 months instead of the percentage of reported losses. In this illustration, the figure is (60% × 30%) / (60% × 30% + 40% × 20%) = 18 / (18 + 8) = 69.23%.
If you derive the solution by formula, you are likely to make errors. Fill out the loss matrix.
Step #5: Two Maturities
The exam problem may test age-to-age link ratios from one maturity to another or loss development factors from one maturity to ultimate.
Illustration: The reported LDF from 36 months to ultimate above a $25,000 retention is 2.5.
At 36 months, the reported losses below $25,000 are 50% of all reported losses, and the paid losses below $25,000 are 80% of all paid losses. The paid-to-reported ratio at 36 months for losses below $25,000 is 50%.
At ultimate, 40% of losses are below $25,000. What is the paid loss development factor from 36 months to ultimate for losses excess of a $25,000 retention?
Solution: Assume $100 million of losses at ultimate. The paid-to-reported ratio at ultimate is 100% in all layers.
Reported losses above $25,000 are (1 – 40%) × $100 million = $60 million at ultimate.
Reported losses above $25,000 are $60 million / 2.5 = $24 million at 36 months.
Reported losses below $25,000 are also $24 million at 36 months (50%-50% ratio).
Paid losses below $25,000 are $24 million × 50% = $12 million at 36 months.
Paid losses above $25,000 are $12 million × 20% / 80% = $3 million at 36 months.
Paid losses above $25,000 at ultimate are $24 million (like reported losses).
The paid loss development factor excess of $25,000 from 36 months to ultimate is $24 million / $3 million = 8.000.
Common Errors
(1) The exam problem may give you the paid-to-reported ratio for losses below the retention and ask you to derive the paid excess loss development factor from the reported excess loss development factor. Common errors are to use
the same paid-to-reported ratio for losses excess of the retention
the complement of the paid-to-reported ratio for loss excess of the retention
Both errors lose full credit. The exam problem will give other information from which you derive the paid-to-reported ratio excess of the retention.
(2) Pinto and Gogol assume the paid-to-reported ratios are the same for retentions of $25,000 or greater. If the exam problem gives the paid-to-reported ratio for a retention of $25,000 and the reported loss development factor for a higher retention, use this paid-to-reported ratio for the higher retention as well.
(3) The exam problem may require you to back into the paid-to-reported ratio above a retention of $25,000 from the paid-to-reported ratio below the retention and the paid loss distribution.
Step #6: Consistency
The exam problem may give you a paid-to-reported ratio for losses below the retention along with other figures. The following ratios are related:
The percent of ultimate paid or reported at each maturity
The excess loss development factors for any retention at each maturity
The percent paid or reported above or below a retention at each maturity
The paid-to-reported ratios at each retention
The exam problem may give figures that you use to derive other figures. Memorizing formulas for each scenario is not efficient, since you will confuse the formulas on the exam.
For your pre-seminar home study, use first principles. Assume total or excess losses of $100 million at either end date: the beginning maturity, the later maturity, or ultimate.
If the exam problem tests excess loss development, assume excess losses of $100 million at one end date.
If the exam problem tests loss development in a layer of loss, assume total losses of $100 million at one end date.
The choice of beginning date or end date depends on the other data in the exam problem. Use the date for which the exam problem gives
The paid-to-reported ratio for deriving paid loss development factors.
The loss distribution for deriving the development in a layer of loss.
The paid-to-reported ratio is 100% at ultimate at all retentions. If the exam problem gives a reported loss development factor to ultimate, use the paid-to-reported ratio at ultimate.
The exam problem may give the loss distribution at ultimate, such as
60% of losses are below $50,000 and 75% of losses are below $250,000.
Derive the dollars of loss in each range. Using dollars instead of percentages keeps the intuition clear and helps avoid arithmetic errors. This example has $60 million below $50,000, $15 million between $50,000 and $250,000, $25 million above $250,000, and $40 million above $50,000.
Step #7: Retentions
For losses excess of a retention, most actuaries assume that the paid-to-reported ratios decrease with the retention. We express this as small losses settle more quickly.
Illustration: If paid losses are 50% of reported losses at 36 months:
Excess of a $100,000 retention: paid losses may be 30% of reported losses
Excess of a $200,000 retention: paid losses may be 20% of reported losses
Pinto and Gogol say this is true for ground-up losses vs losses excess of a low retention. For retentions of $25,000 or more, they say the retention does not affect the paid-to-reported ratio. For exam problems on the Pinto and Gogol paper, use their assumption.
The exam problem may give you the reported loss development factor excess of a $100,000 retention, and the paid-to-reported ratios for ground-up losses and losses excess of a $25,000 retention. Use the paid-to-reported ratios excess of a $25,000 retention for the $100,000 retention as well.
The study aid gives a possible rationale for this assumption. It is not actually correct, and it should not be in a syllabus reading.
Common Errors
(1) If you are not expecting this type of exam problem, you will waste time seeking a logical relation for the paid-to-reported ratio excess of the $100,000 retention.
The exam problem may give paid-to-reported ratios for retentions of zero, $10,000, and $25,000. Use the $25,000 retention for higher retentions as well. Do not trend, interpolate, or take averages.
(2) If the exam problem gives data from which to derive a paid-to-reported ratio, do not assume the ratio at $25,000 is the same for higher retentions.
The exam problem may derive the paid loss development factor for a layer of loss from the reported loss development factors excess of the upper and lower bounds of the layer. Suppose the LDF is from 60 months to ultimate and the layer is $100,000 to $250,000.
For each bound, the reported loss development factor divided by the paid-to-reported ratio is the paid loss development factor. Suppose the exam problem gives the paid-to-reported ratio for $100,000 but not $250,000, the paid loss distribution at ultimate at 50% below $100,000 and 20% above $250,000 and the paid loss distribution at 60 months as 70% below $100,000 and 10% above $250,000.
We are missing the paid-to-reported ratio at $250,000, so we assume it is the same as the paid-to-reported ratio at $100,000. Some exam problems expect you to assume this. But if the exam problem also gives the reported loss distribution at 60 months as 60% below $100,000 and 15% above $250,000, we derive the paid-to-reported ratio at $250,000 from the data. We don’t assume the paid-to-reported ratio is the same at all retentions.
Step #8: Late Maturities
Pinto and Gogol use a formula to derive paid-to-reported ratios at maturities where they lack data. They assume the outstanding to reported ratio declines by 66.7% each year. The outstanding-to-reported ratio is the complement of the paid-to-reported ratio.
This approximation is a minor detail in the paper, but it is used much on exam problems. It is difficult to learn this approximation from the paper itself, as no clear example is given. Rely on the step-by-step guide and the study aid.
The study aid shows the data used by Pinto and Gogol to derive this approximation.
Some candidates have trouble finding the section of the paper with this assumption.
Once you see the figures, it is easy to grasp this part of the step-by-step guide.
Pinto and Gogol use this assumption where they have no data.
The exam problem may specify this relation or it may specify another percentage.
If the exam problem says nothing, assume a 66.7% relation.
Illustration: Suppose the paid-to-reported ratio is cc4 at 63 months. (We use cc1 at 27 months, cc2 at 39 months, cc3 at 51 months, and cc4 at 63 months.) We infer
The outstanding to reported ratio is (1 – cc4) at 63 months.
The outstanding to reported ratio is
b × (1 – cc4) at 75 months. The paid-to-reported ratio is cc5 = 1 – b × (1 – cc4) at 75 months. The outstanding to reported ratio is
b × b × (1 – cc4) at 87 months. The paid-to-reported ratio is cc5 = 1 – b × b × (1 – cc4) at 87 months.
With the paid-to-reported ratios at each maturity, we derive paid loss development factors from reported loss development factors.
Pinto and Gogol assume this relation for premises and operations, which are OLT and MC.
If the exam problem uses these sublines, assume this relation.
If the exam problem uses a different line of business, such as commercial auto, mention that Pinto and Gogol assume this relation for premises and operations and you are using it for the other line of business as well. This extension may not be justified, but it is what the examiner wants.
Common Errors
(1, 2) Two common errors are
Using the paid-to-reported ratio instead of the outstanding to reported ratio to estimate the ratio at future maturities.
Forgetting to convert back to paid-to-reported ratios.
Be sure you use the proper ratio.
Do not multiply the paid-to-reported ratios by 66.7%. This loses full credit.
Convert the outstanding to reported ratios back to paid-to-reported ratios at the end.
Paid-to-reported ratios increase by maturity. You can usually catch the first error. As a check, divide the paid loss development factor by the reported loss development factor at the earlier and later maturities, if you have both. The ratio should decrease from the earlier to the later maturity.
The second error is harder to catch. At late maturities, the paid-to-reported ratio is high and the outstanding to reported ratio is low.
(3) If you are given the paid-to-reported ratio at maturities beyond 63 months, use them. Do not assume a 66.7% ratio.
Illustration: If the exam problem gives paid-to-reported ratios of 60% at 87 months and 70% and 99 months, use these ratios. Do not use the 66.7% proxy to derive new ratios.
(4) If the exam problem gives different paid-to-reported ratios by retention, use them. Do not assume the paid-to-reported ratios are the same if the problem gives different ratios.
If the exam problem gives a paid-to-reported ratio for a $25,000 retention but not for the higher retention being priced, use the same paid-to-reported ratio.
If the exam problem gives a paid-to-reported ratio for a $50,000 retention but not a $250,000 retention and it prices the paid loss development factor for the layer of loss from $50,000 to $250,000, assume the paid-to-reported ratios are the same.
(5) One past exam problem gave the actual ratio for the accident year in question, not the expected ratio. The exam problem was faulty, since the actual ratio is not relevant. The problem was not thrown out. You were expected to use the actual ratio.
If the exam problem gives the paid-to-reported ratio for other accident years, we use that ratio as the expected ratio.
Step #9: Three Maturities, Two Retentions
Expect an exam problem with three maturities and two retentions. The problem will give
The reported loss development factor excess of a retention L
N from j-1 to j months.L
N will be more than $25,000.j will be more than 63 months.
The paid-to-reported ratio at 63 months for a retention of $25,000.
The exam problem will ask for the paid loss development factor excess of a retention L
N from j-1 to j months.
Use the 66.7% relation to derive paid-to-reported ratios at j-1 and j months.
Use the paid-to-reported ratios to derive the paid loss development factor.
It might seem like you can combine the steps, cancel terms, and get a simple formula. We do not recommend this. The terms do not cancel, and the formula is not simple.
If you combine the steps and make an arithmetic error, you will lose full credit. The examiner wants to see that you understand the procedure. Create a table on your answer sheet showing the outstanding to reported ratios and the final paid-to-reported ratios.
Step #10: Three Maturities, Three Retentions
Expect an exam problem with three maturities and three retentions. The problem will give
The reported loss development factor in the layer from L
N to LNN from j-1 to j months.L
N will be more than $25,000.j will be more than 63 months.
The paid-to-reported ratio at 63 months for a retention of $25,000.
The distribution of paid or reported losses at j-1 or j months.
The exam problem will ask for the paid loss development factor in the layer from L
N to LNN from j-1 to j months. Use the same procedure as in the previous step:
Use the 66.7% relation to derive paid-to-reported ratios at j-1 and j months.
Use the paid-to-reported ratios to derive the paid loss development factor.
Step #11: Optimal Exam Preparation
Examiners take past exam problems and invert them to create new exam problems. Even examiners who intend to create new problems may run out of time and reuse old problems,
For optimal exam preparation, form all permutations of past exam problems.
Hurrying to a second problem after completing one problem lowers study efficiency.
Suppose a past exam problem in the study aid gives you
The reported loss age-to-age link ratio from 36 to 48 months.
The paid-to-reported ratios at 36 and 48 months.
You solve for the paid loss age-to-age link ratio from 36 to 48 months.
After completing the problem, form the other combinations: solving for the reported loss age-to-age link ratio, the paid-to-reported ratio at 36 months, and the paid-to-reported ratio at 36 months. For each variant of the problem, vary the figures, so you redo the calculation.
Do not leave a problem until you have formed all the permutations. Forming all the permutations helps you recall the relations among the input data.
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