Actuarial Notes

Basics of Reinsurance Pricing

By David R. Clark · Casualty Actuarial Society · 2014

Study guides: Exam 9, Exam GI-301

An actuarial study note on the basic tools for pricing reinsurance treaties, first written in 1996 and revised in 2014 by David R. Clark, FCAS. It treats a reinsurance program as tailored to its buyer, so that each contract must be priced individually, and it introduces the standard tools for each type of treaty in Section A of each part, with criticisms and advanced topics in Section B. It focuses on domestic treaty covers and gives numerical examples meant to illustrate the techniques, not to recommend actual factors.

On the syllabus

  • Exam 9 — objectives A2–A3; the whole study note (the content outline lists "Clark, D. R., 'Basics of Reinsurance Pricing,' CAS Study Note, Revised 2014" with no exclusions).
  • Exam GI-301 — Outcomes 7e-h; the syllabus lists the study note with no section exclusions.

Introduction

  • Like primary insurance, reinsurance is a mechanism for spreading risk: a reinsurer takes some portion of the risk assumed by the primary insurer (or other reinsurer) for premium charged, and most of the basic concepts for pricing that assumption are the same as those underlying ratemaking for other types of insurance; the note assumes a knowledge of basic ratemaking concepts.
  • A reinsurance program is generally tailored more closely to the buyer, so there is no "average" reinsured or "average" reinsurance price and each contract must be individually priced to meet the particular needs and risk level of the reinsured.
  • The "pricing paradox": if you can precisely price a given contract, the ceding company will not want to buy it — if the historical experience is stable enough for a precise expected loss estimate, the reinsured would be willing to retain that risk.
  • The "basic" pricing tools are usually only a starting point in determining an adequate premium; the actuary proves his or her worth by knowing when the assumptions in these tools are not met and how to supplement the results with additional adjustments and judgment.
  • For each type of reinsurance, basic pricing tools are introduced in Section A (methods generally accepted and standard throughout the industry) and criticisms and advanced topics in Section B (areas which require the actuary's expertise but have not been solved to universal agreement).
  • The note focuses on domestic treaty covers; pricing for facultative covers or international (non-U.S.) treaties is not addressed explicitly, but may be viewed as variations on the same themes, with differences in accounting, loss sensitive features and the amount of judgment needed.

1 Proportional Treaties

  • Section 1A. Basic Tools
    • A proportional treaty is an agreement between a reinsurer and a ceding company (the reinsured) in which the reinsurer assumes a given percent of losses and premium.
    • In a quota share treaty, the reinsurer receives a flat percent, say 50%, of the premium for the book of business reinsured and pays that percent of losses, including allocated loss adjustment expenses, plus a ceding commission designed to reflect the differences in underwriting expenses incurred.
    • A surplus share treaty, common on property business, allows the reinsured to limit its exposure on any one risk to a given amount (the "retained line"); the reinsurer assumes a part of the risk in proportion to the amount that the insured value exceeds the retained line, up to a limit expressed as a multiple of the retained line, or "number" of lines.
    • With a retained line of 100,000 and a 1st surplus of 4 lines (400,000), a 250,000 risk has 150,000 (60%) reinsured, a 500,000 risk has 400,000 (80%) reinsured, and a 10,000,000 risk has 400,000 (4%) reinsured; this is not excess insurance, because the retained line only establishes the percent of the risk reinsured, and the reinsurer then pays that percent of any loss on the risk.
    • Other proportional treaties include fixed and variable quota share arrangements on excess business (for example, commercial umbrella policies), where the underlying business is excess of loss but the reinsurer takes a proportional share of the ceding company's book.
    • Step 1: Compile the historical experience on the treaty — five or more years of premium and incurred losses, or the gross experience adjusted "as if" the surplus share terms had been in place; a "losses occurring" treaty uses earned premium and accident year losses, a "risks attaching" treaty uses written premium and the losses on policies written during the treaty period.
    • Step 2: Exclude catastrophe and shock losses — catastrophe losses are due to a single event that may affect a large number of risks, and shock losses are any other losses, usually affecting a single policy, which may distort the overall results.
    • Step 3: Adjust experience to ultimate level and project to future period — develop losses to ultimate (loss development factors from other sources if the treaty experience is insufficient), adjust historical premiums to the future level for rate changes and average pricing factors (the parallelogram method for "losses occurring" treaties, including anticipated rate changes) and for exposure inflation if the premium base is insured value, and trend the losses to the future period.
    • Step 4: Select the expected non-catastrophe loss ratio for the treaty — if the data is reliable, the average of the historical loss ratios adjusted to the future level, compared with the ceding company's gross calendar year experience and industry averages.
    • Step 5: Load the expected non-catastrophe loss ratio for catastrophes — there is usually insufficient credibility in the historical experience to price the loading, reinsurers once priced it by "spreading" large losses over expected payback periods (a 1-in-20-year event as a 5% loading), and the most common procedure is now an engineering-based catastrophe model that incorporates the risk profile of the ceding company.
    • Step 6: Estimate the combined ratio given ceding commission and other expenses — the ceding commission (often on a "sliding scale" basis), the reinsurer's general expenses and overhead, and brokerage fees where applicable, then an evaluation of whether the projected combined ratio is acceptable given potential investment income and risk level.
    • In the worked example, a property quota share treaty effective 1/1/97 on a "losses occurring" basis has six years of experience evaluated 9/30/96, with a Hurricane Andrew loss of 1,582,758 in 1992 excluded; losses are trended at 4% a year and developed to ultimate, premium is adjusted to the 1997 rate level (including a pending +10% rate increase effective 4/1/97) and trended at 3%, and the non-catastrophe loss ratio is 54.6%.
    • The example loads a 10% catastrophe loading (Hurricane Andrew would have added about 15% to the historical loss ratio) for an expected loss ratio of approximately 65%; with a 30% ceding commission, 5.0% brokerage, 1.0% administrative expenses and 1.0% unallocated expenses, the indicated combined ratio is 102.0%, unlikely to produce an acceptable return, so a reduction in the ceding commission may be recommended.
  • Section 1B. Special Features of Proportional Treaties
    • In theory a reinsurer should "follow the fortunes" of the ceding company, but in practice results may be quite different, and disagreement over the loss ratio and the appropriate ceding commission is often resolved by building adjustable features into the treaty.
    • a) Sliding Scale Commission
      • A sliding scale commission is a percent of premium paid by the reinsurer to the ceding company which "slides" with the actual loss experience, subject to set minimum and maximum amounts; in the example, a 30% provisional commission slides 1:1 from 25% at a 65% loss ratio to 35% at a 55% loss ratio, then .5:1 to a maximum of 45% at a 35% loss ratio.
      • In a "balanced" plan it is fair to calculate the ultimate commission for the expected loss ratio, but if the expected loss ratio is towards one end of the slide (65% gives a 25% commission and a 90% technical ratio), the loss ratio should be viewed as a random variable and the expected commission as the average over all possible outcomes.
      • The simplest approach estimates the expected commission from historical loss ratios adjusted to future level, including catastrophe and shock losses; it is a good reasonability check but may be distorted and leaves out many possible outcomes.
      • A better approach is an aggregate loss distribution model, which gives the probability of each range of loss ratios and its average commission; in the example the expected commission is 31.0%, so the expected technical ratio is 96% (65% + 31%) rather than the 90% naively estimated.
      • A carryforward provision lets a past loss ratio above the one corresponding to the minimum commission be included with the current year's loss in estimating the current year's commission; the first pricing approach shifts the slide by the carryforward (a 5% carryforward moves the 65% minimum point to 60%, giving an expected commission of 29.2%), but it ignores the potential for carryforward beyond the current year.
      • The second approach models the sliding scale over the "long run" of the contract, a longer block of years, reducing the variance of the average loss ratio (in the example the standard deviation is cut by the square root of 5 for a five-year block, giving an expected commission of 28.3%); the method for reducing the variance is not obvious, and the contract may not renew the following year.
      • The issue is further complicated when a commission deficit can be carried forward but not a credit, and there is no standard method for handling these questions so far as the author is aware.
    • b) Profit Commission
      • A profit commission subtracts the actual loss ratio, ceding commission and a "margin" for expenses from the treaty premium and returns a percent of this as additional commission; in the example, a 55% loss ratio, 25% ceding commission and 10% margin leave a 10% reinsurer's profit, of which 50% is returned, a 5% profit commission.
      • Like the sliding scale commission, it should be evaluated using an aggregate distribution on the loss ratio, and there is some ambiguity concerning the handling of carryforward provisions.
    • c) Loss Corridors
      • A loss corridor provides that the ceding company will reassume a portion of the reinsurer's liability if the loss ratio exceeds a certain amount; with a corridor of 75% of the layer from an 80% to a 90% loss ratio, a 100% loss ratio before the corridor becomes 92.5% after it (80.0% below the corridor, 2.5% within it, 10.0% above it).
      • The impact should be estimated using an aggregate distribution; in the example an expected loss ratio of 75.0% before the corridor falls to 73.0% after it, so the corridor lowers the reinsurer's expected loss ratio even though 75.0% is less than the 80% attachment point.
      • Bear and Nemlick provide further background on handling loss sensitive features, and the actuary's job is not finished after the expected loss ratio is calculated.

2 Property Per Risk Excess Treaties

  • Section 2A. Experience and Exposure Rating Models
    • Property per risk excess treaties provide a limit of coverage in excess of the ceding company's retention, applying on a "per risk" basis (typically a single property location), which is narrower than a "per occurrence" property excess treaty that applies to multiple risks to provide catastrophe protection.
    • The treaty premium is set as a percent of a subject premium base — "gross net earned premium income" (GNEPI) for losses occurring policies or "gross net written premium income" (GNWPI) for risks attaching policies — net of any other reinsurance inuring to the benefit of the per risk treaty, such as a surplus share treaty, but gross of the per risk treaty being priced.
    • The main tools for pricing per risk treaties are experience and exposure rating.
    • a) Experience Rating
      • Experience rating is sometimes referred to as a "burn cost" model, though that phrase more commonly denotes just the unadjusted experience and not the projected cost; the basic idea is that the historical experience, adjusted properly, is the best predictor of future expectations.
      • Step 1: Gather the subject premium and historical losses for as many recent years as possible (ten should be sufficient, balancing credibility and responsiveness), including all losses that would pierce the layer being priced after trend.
      • Step 2: Adjust the subject premium to the future level using rate, price and exposure inflation factors.
      • Step 3: Apply loss inflation factors to the historical large losses, determine the amount in the layer and sum it for each historical period, adding allocated expense individually for each loss if ALAE applies pro-rata with losses.
      • Step 4: Apply excess development factors to the summed losses for each period, derived from the same ceding company data if possible, along with any needed frequency trend.
      • Step 5: Divide the trended and developed layer losses by the adjusted subject premium to produce loss costs by year, and average them to project the expected loss cost.
      • The projected loss costs should be randomly distributed about the average; if they increase or decrease from the earliest to latest years, the trend or development factors may be too high or low, or there may have been shifts in the types of business or sizes of risks written.
      • In the example, a treaty effective 1/1/97 with a 400,000 limit and a 100,000 attachment point trends each historical loss at 4% a year to the prospective period and takes the portion that penetrates the layer; applying the LDFs to eight accident years (1988–1995) gives trended ultimate layer losses of 1,991,789 on on-level subject premium of 16,112,389, a loss cost of 12.4%.
    • b) Exposure Rating
      • Exposure rating models the current risk profile rather than what was written years earlier, using an exposure curve first developed by Ruth Salzmann in 1963 for Homeowners business and eventually adapted for commercial property.
      • The exposure curve P(p) represents the amount of loss capped at a given percent p of the insured value (IV) relative to the total value of the loss, where f(x) is the distribution of individual loss dollar amount:

P(p) = \frac{\int_0^{p \cdot IV} x\, f(x)\, dx + \int_{p \cdot IV}^{\infty} p \cdot IV\, f(x)\, dx}{\int_0^{\infty} x\, f(x)\, dx} = \frac{\int_0^{p \cdot IV} [1 - F(x)]\, dx}{E[X]}

    - The portion of the expected loss on a risk which falls in the treaty layer is P((Retention+Limit)/Insured Value) − P(Retention/Insured Value); for a $400,000 excess of $100,000 layer on a $500,000 risk, the factors at 20% and 100% of insured value are 49% and 93%, an exposure factor of 44%.
    - The illustrative curve allows for exposure above the insured value (100% at 120%), because the limits profile often does not include business interruption coverage for commercial policies or living expenses for homeowners policies.
    - For a portfolio, the calculation is performed on a distribution of premium by ranges of insured values, the "limits profile", which must be verified to be on a per location basis, since total values for policies covering multiple locations cause distortions; in the example, a 65% expected loss ratio on 2,284,000 of subject premium gives reinsurer's losses of 351,124 and a loss cost of 15.37%.
    - The expected loss ratio is of critical importance, since the final rate moves proportionally with it, and it should be projected rigorously following the procedures for proportional treaties.
    - An implicit assumption is that the same exposure curve applies regardless of the size of the insured value (scale independence): reasonable for homeowners business, but possibly a serious problem for large commercial risks, which the Lloyds scales did not recognize.
    - Ludwig gives a more detailed description of this topic.
  • Section 2B. Other Issues on Property Per Risk Treaties
    • After loss costs are estimated by experience and exposure rating, the actuary's task is to reconcile the results and select a final expected loss cost.
    • a) Free Cover
      • "Free cover" refers to an experience rating in which no losses trend into the highest portion of the layer being priced; with a largest trended loss of 500,000 ground up, a 750,000 excess of 250,000 layer and a 250,000 excess of 250,000 layer produce the same loss cost, the top 500,000 excess of $500,000 being implicitly a "free cover".
      • One approach uses the experience rating as a basis for the lowest portion of the layer and the relativities in the exposure rating to project the higher layer: with experience loss costs of 16% and 0% and exposure loss costs of 20% and 10% for the 250k xs 250k and 500k xs 500k layers, the selected costs are 16% and 8% (= 16% × 10%/20%), and 24% for the 750k xs 250k layer.
    • b) Credibility
      • A first measure of credibility is the number of claims expected during the historical period, not the number actually observed, since credibility set on the observed number gives more credibility to experience rating projections that are fortuitously worse than average; the dollars of expected loss from the exposure rating may be used instead.
      • A second measure is the year-to-year variation in the projected loss cost from each historical period, where stability adds credibility even if the number of claims is small.
      • Assigning credibility is in part subjective, and significant credibility is often given to experience rating simply because there are too many limitations to the exposure rating alternative; Mashitz and Patrik and Clark discuss reconciling the two.
    • c) Inuring Reinsurance
      • An excess treaty may apply to the ceding company's retention after a surplus share treaty; for experience rating, the only accurate way to reflect this is to restate the historical loss experience on a basis net of the inuring reinsurance.
      • Exposure rating can be applied directly to a risk profile adjusted to the terms of the inuring surplus share; if the actuary has exposure curves varying by size of insured value, the curve is selected on the insured value before the surplus share but the exposure factor is applied to the subject premium after the surplus share.
      • In the example, a surplus share in which the ceding company retains a maximum of 200,000 on any one risk, followed by a 100,000 excess of $100,000 per-risk cover, leaves a subject premium of 1,088,333, reinsurer's losses of 61,793 and a loss cost of 5.68%.

3 Casualty Per Occurrence Excess Treaties

  • Section 3A. Experience and Exposure Rating Models
    • Like property excess, casualty lines use experience and exposure rating; the discussion refers to general liability (including products), auto liability and workers compensation, and the same techniques can be adapted for other casualty lines with some modifications.
    • Casualty per occurrence excess treaties are often separated into three categories: a Working Layer (low attachment, expected to be penetrated, often multiple times each annual period), Exposed Excess (attaching below some of the policy limits on the underlying business, so some years do not penetrate the layer) and Clash Covers (high attachment, penetrated by multiple policies in a single occurrence, or by extra-contractual obligations (ECO) or excess of policy limits (XPL) rulings).
    • The distinctions between these categories are generally soft in the pricing process, and experience rating techniques are still used as experience approaches the "exposed excess" category.
    • a) Experience Rating
      • Step 1: Gather the subject premium and historical losses for as many years as possible, capturing allocated loss adjustment expenses (ALAE) separately from losses and, for general liability and auto liability, the underlying policy limit; for auto losses on split limits rather than a combined single limit, other modifications may be needed, and workers compensation losses should be requested on a full undiscounted basis because large ones are often shown on a discounted case reserve basis.
      • Step 2: Adjust the subject premium to the future level using rate, price and exposure inflation factors, which vary for each line of business.
      • Step 3: Apply loss inflation factors to the individual historical losses, varying by line; ISO estimates basic and total limits trend factors, but theoretically what should be used is an unlimited trend factor derived from large losses only.
      • The trended losses must be capped at applicable policy limits, for which there is no generally accepted solution: applying the historical policy limit to each trended loss leaves out that insureds generally increase their limits, while applying the trend factor without a cap assumes limits "drift" upwards to match inflation and requires the subject premium to be adjusted to the level that would have been charged had the higher limits been in effect.
      • Mata and Verheyen give more advanced concepts on using exposure rating techniques to adjust for changes in the policy limit profile.
      • ALAE is included in one of two ways: pro-rata with loss (ALAE in the layer allocated in proportion to losses) or as part-of-loss, "on top" or "add-on" (ALAE added to loss and the treaty limit applies to the sum); in the examples, a 640,000 loss with 320,000 ALAE against a 600,000 limit excess of 400,000 puts 360,000 in the layer pro rata and 560,000 on top, and a 920,000 loss with 460,000 ALAE puts 780,000 in the layer pro rata and 600,000 on top, with the amount in the layer not necessarily higher or lower for either method.
      • Step 4: Apply excess development factors to the summed losses for each period — critical for casualty lines because of the very large factors needed — derived from the ceding company's data where possible, otherwise from benchmarks such as the Reinsurance Association of America (RAA) biennial loss development study, which shows a significant lag between reported losses for a primary company and a reinsurer (the 2012 edition's graphs are the note's Supplement).
      • Cautions on the RAA data: the reporting lag from occurrence to reinsurer case reserve varies by company and includes retrocessional business; the mix of attachment points and limits is not cleanly broken out; the data is requested exclusive of asbestos and environmental claims but other long-term exposure claims (medical products, mold, tobacco) are not excluded; and for workers compensation the members may not handle the tabular discount on large claims consistently, so an unwinding discount can produce a very high development factor.
      • Very slow development patterns often produce either zero or very high projected ultimate layer losses by year, and the actuary will often need smoothing techniques such as a Bornhuetter-Ferguson approach or Cape Cod (Stanard-Bühlmann).
      • Step 5: Divide the trended and developed layer losses by the adjusted subject premium to produce loss costs by year, average them and select a final expected loss cost, which may be adjusted for the time value of money, expenses and risk load, dealt with in the last section.
    • b) Exposure Rating
      • Exposure rating estimates a loss cost based on the premium and limits expected to be exposed during the treaty period, using a severity distribution based on industry statistics to calculate increased limits factors (ILF) for general liability and auto liability, and excess loss factors (ELF) for workers compensation.
      • With x the size of loss, F(x) its CDF, f(x) its density, E[x] the average unlimited loss and E[x;L] the expected value of losses capped at L:

E[x;L] = E[\min(x,L)] = \int_0^L x\, f(x)\, dx + \int_L^{\infty} L\, f(x)\, dx \qquad ILF_{L,U} = \frac{E[x;U]}{E[x;L]} \qquad ELF_L = \frac{E[x] - E[x;L]}{E[x]}

    - For general liability and auto liability, one option is the truncated [Pareto](/wiki/concept/Pareto+Distribution) distribution, whose five parameters are T (truncation point, below which losses are "small"), P (probability of a "small" loss), S (average small loss severity), B (scale parameter) and Q (shape parameter), with E[x;L] for Q ≠ 1 and L > T:

E[x;L] = P \cdot S + \left(\frac{1-P}{Q-1}\right) \cdot \left[(B + Q\cdot T) - (B + L)\left(\frac{B+T}{B+L}\right)^{Q}\right]

    - The scale is adjusted for inflation by multiplying T, S and B by the same amount; the formula applies only above T (set well below any treaty attachment point), and excess factors for higher layers become very dependent on Q, which must be watched carefully when the curves are updated.
    - When B = 0 and Q = 1 the distribution becomes a log-logistic form, E[x;L] = P·S + (1−P)·T·[1 − ln(T/L)], with the property that expected losses in layers are equal if limit and attachment point are in the same ratio, approximated when B is small and Q is close to 1; this holds for individual claim severities but not necessarily for treaty loss costs, which decrease for higher layers due to fewer policies being exposed.
    - The "BQPST" form of the Pareto is not the only choice: a discrete mixture is a weighted average of relatively simple curve forms, and a popular example is the Mixed Exponential, E[x;L] = Σ w_j·μ_j·(1 − exp(−L/μ_j)) with Σ w_j = 1.
    - Once a severity distribution is selected, the exposure factor is applied to ground-up expected losses to estimate the loss cost to the treaty layer, with PL the ceding company policy limit, AP the treaty attachment point and Lim the treaty limit:

\text{Exposure Factor} = \frac{E[x;\min(PL, AP+Lim)] - E[x;\min(PL, AP)]}{E[x;PL]}

    - If the treaty includes ALAE in proportion to losses, the factor is applied to subject premium times an expected loss and ALAE ratio; if ALAE is included with losses, the limits are divided by (1+e), where e is ALAE as a percent of loss capped at PL.
    - The key assumption in both cases is that ALAE varies directly with capped indemnity loss, which is not accurate (losses closed without an indemnity payment may still incur large expense, and ALAE as a percent of loss tends to decrease as loss size increases), and perfect correlation tends to overstate expected amounts in the higher layers.
    - Another limitation of the "with losses" formula is that an exposure factor of zero is applied to high layers which are indeed exposed: with a $1,000,000 policy limit and an ALAE loading of 1.500, a treaty attaching at $1,500,000 is not considered exposed.
    - Severity curves are also used for proportional treaties on excess business, either quota share or on a "cessions" basis (the percent depends on the attachment point and limit written on each policy, with the ceding company using increased limits factors to price the portion exposing the treaty), where the exposure factors can be compared with the ceding company's own.
    - For workers compensation, severity distributions most commonly come from the National Council on Compensation Insurance (NCCI), which publishes excess factors for retrospective rating plans in many states, varying by state and hazard group; these can be approximated by an inverse power curve ELF_L = a·L^(−b), with a and b estimable from selected excess factors, behaving in the higher layers much like the Pareto.
    - Workers compensation has no policy limits corresponding to those on liability policies, so the exposure factor is calculated using only the treaty attachment point and limit, ELF_AP − ELF_(AP+Limit), separately for each state and hazard group with its own expected loss ratio.
    - In the example, a $750,000 limit excess of $250,000 with four state and hazard group cells of 100,000 standard premium each gives treaty losses of 13,695 on 400,000, a loss cost of 3.42%.
  • Section 3B. Special Problems on Casualty Excess Treaties
    • The methods described are the author's suggestions and not the consensus opinion, but however the issues are addressed they cannot be ignored in the pricing process; credibility and "free cover" should equally be considered for casualty treaties.
    • a) Including Umbrella Policies
      • Umbrella policies are excess of an underlying retention and "drop down" if an underlying aggregate is exhausted; if they sit above primary policies written by the ceding company, the combination is best considered a single policy with a higher limit, and for experience rating the pieces are simply added together.
      • When the umbrella is above primary policies from other carriers, the experience rating trend factor is applied with the underlying limit added before trend and subtracted after: Trended Loss = (Loss + Underlying Limit) · (Trend Factor) − Underlying Limit, which still leaves out losses on the underlying policy that did not exhaust its limit historically but would have after trend.
      • For exposure rating, with UL the limit of underlying policies, PL the umbrella policy limit, AP the treaty attachment point and Lim the treaty limit:

\text{Exposure Factor} = \frac{E[x;\min(UL+PL,\, UL+AP+Lim)] - E[x;\min(UL+PL,\, UL+AP)]}{E[x;UL+PL] - E[x;UL]}

    - For a $1,000,000 excess of $1,000,000 umbrella under a $500,000 excess of $500,000 treaty the exposure factor is (E[x;2] − E[x;1.5]) / (E[x;2] − E[x;1]); an approximation including the "drop down" feature weights this and the drop-down terms with ϕ, the aggregate excess factor on the underlying policy, analogous to a Table M charge factor.
- b) Loss Sensitive Features
    - For working layer excess, the ceding company is often willing to retain more of the losses, and an annual aggregate deductible (AAD) lets it retain the first losses in the layer while maintaining protection if there are more losses than anticipated, making the treaty an excess of aggregate cover; savings can be estimated from the experience rating if set sufficiently low (say, half of the expected value), but an aggregate distribution model is better.
    - The excess charge factor for a given AAD, where g(y) is the distribution of aggregate losses in the layer, is the form underlying Table M and is analogous to the ELF calculation used for per occurrence excess; it is multiplied by the loss cost for the layer gross of the AAD to estimate the net loss cost:

\phi_{AAD} = \frac{\int_{AAD}^{\infty} (y - AAD)\, g(y)\, dy}{E[y]}

    - A "swing plan" is a type of [retrospective rating](/wiki/concept/Retrospective+Rating) program in which actual losses to the layer are loaded for expenses and charged back to the ceding company, subject to maximum and minimum constraints, and likewise requires aggregate distribution models; in the example, Retro Premium = Actual Layer Losses × 100/80, with a 15% provisional rate, a maximum premium of 30% and a minimum of 10% of subject premium, so actual layer losses of $100,000 produce $125,000.
    - The swing plan may apply to a block of years; in the example the expected loss ratio is 96.5% (22.1%/22.9%), not 80%, because the maximum and minimum are not in "balance", and the 15% provisional rate is well below the expected ultimate premium rate of 22.9%, an added cash flow advantage for the ceding company to include in the final pricing evaluation.
- c) Workers Compensation Experience Rating
    - Experience rating for [workers compensation](/wiki/concept/Workers+Compensation+Insurance) may be distorted depending on how tabular discounts are taken into account; a way to avoid this is to collect information on individual claimants — current age, sex, estimated annual indemnity cost including escalation, estimated annual medical cost and amounts paid to date.
    - For claims with the potential for penetrating the layer, all future payments are determined and an appropriate mortality factor is applied to the payments falling in the excess layer; some claims whose reported incurred is below the treaty retention will show an expected amount in the layer, so a smaller development factor is needed to include only "true IBNR" claims.

4 Aggregate Distribution Models

  • Aggregate distribution models are used for pricing a variety of treaty features; their results are very sensitive to the input assumptions, so sensitivity analysis on the parameters, or even several approaches, should be used wherever possible.
  • All the approaches are "advanced", but not optional: improper evaluation of features which vary with loss experience could lead to significant under- or over-pricing.
  • a) Empirical Distribution
    • With five or more years of loss ratios on a treaty, the impact of an adjustable feature (a sliding scale commission, for example) can be estimated by calculating it as if the current terms had been in effect over the historical period, adjusted to current rate level; it is easy to calculate and should be examined at least as a check.
    • Caveats: the experience does not take into account all possible outcomes; changing volume or mix of business may make future volatility very different; and if loss development used Bornhuetter-Ferguson or Cape Cod, historical loss ratios may be artificially smooth.
  • b) Single Distribution Model
    • The single distribution approach assumes that the aggregate of all losses to the treaty follows a known CDF form, in contrast to a "collective risk" model with explicit frequency and severity distributions; a commonly used model is the lognormal, a reasonable approximation to empirical distributions, with CDF G(y) = Φ((ln(y) − μ)/σ).
    • The parameters can be set by method of moments from an expected value and coefficient of variation (CV, standard deviation over the mean):

\sigma^2 = \ln(CV^2 + 1) \qquad \mu = \ln(\text{mean}) - \frac{\sigma^2}{2}

- The limited expected loss function is E[y;L] = exp(μ + σ²/2)·Φ((ln(L) − μ − σ²)/σ) + L·[1 − Φ((ln(L) − μ)/σ)], the excess charge function is φ_L = (E[y] − E[y;L])/E[y], and a formula gives the conditional expected value E[y | L < y < U] within a given range, with U = ∞ and L = 0 as special cases.
- Most adjustable features break down into piecewise linear functions, so only the expected value is needed within each linear range: the premium or commission is estimated for the expected value in each range and weighted by the probabilities G(P_i) − G(P_(i−1)); the table can be set up for sliding scale commissions, profit commissions, swing plans, loss corridors or many other common features.
- Other curve forms, such as transformed gamma (see Venter) or inverse Gaussian, have been recommended as also providing good fits to aggregate loss data.
- The model is relatively simple to use even when source data is limited, and gives a reasonable fit even when frequency and severity distributions are not known; its disadvantages are no allowance for the loss free scenario (the lognormal is not defined for y = 0) and no easy way to reflect the impact of changing per occurrence limits on aggregate losses, for which Bear and Nemlick offer suggestions.
  • c) Recursive Calculation of Aggregate Distribution
    • The recursive formula, introduced into the actuarial literature by Panjer, is a convenient tool for low frequency scenarios, assuming a Poisson, negative binomial or binomial frequency distribution and a severity distribution defined in equally spaced discrete steps, the largest severity possibly set at the per occurrence limit (or the limit times an ALAE loading).
    • For a Poisson frequency with mean λ, Pr(0) = e^(−λ) and Pr(n) = (λ/n)·Pr(n−1); with severities 250, 500, 750 and 1,000 of probabilities .400, .150, .100 and .350 and λ = 3, the aggregate probabilities are A0 = Pr(0) = .050, A1 = (λ/1)·S1·A0 = .060, A2 = (λ/2)(S1·A1 + 2·S2·A0) = .059, and so on:

A_k = \sum_{i=1}^{k} \frac{\lambda}{k}\, i\, S_i\, A_{k-i}

- For aggregate amounts above the largest possible individual severity the number of terms does not increase; with a Poisson frequency the mean and variance of the aggregate distribution are λ·Σ(amount × S) and λ·Σ(amount² × S).
- The generalization for other frequency distributions is A_k = Σ (a + b·i/k)·S_i·A_(k−i), with Poisson a = 0, b = λ; negative binomial a = (1 − p), b = (α − 1)(1 − p); binomial a = p/(p − 1), b = (M + 1)·p/(1 − p); negative binomial or binomial frequency allows greater flexibility.
- The advantages are simplicity and accurate handling of low frequency scenarios, with severity points expandable to closely approximate continuous curves; the disadvantages are that for higher expected frequencies all probabilities up to the desired level must be calculated, and that only a single severity distribution can be used.
  • d) Other Collective Risk Models
    • Collective risk models are distributions for which frequency and severity are explicitly recognized, the aggregate loss being the sum of n severities where n is also a random variable, and most aggregate loss models allow more than a single severity distribution; the recursive method is a straightforward example.
    • The aggregate distribution may be evaluated using simulation or numerical methods (see Heckman and Meyers, Robertson or Wang), given the severity distribution(s) and parameters for the frequency distribution; most models produce the CDF G(y) and excess charge factor ϕ(y) at requested points, and the expected aggregate loss in a range is E[y | L < y < U] = (E[y]·{ϕ_L − ϕ_U} + L·{1 − G(L)} − U·{1 − G(U)}) / (G(U) − G(L)).
    • The collective risk model is generally the best way to price "pure" excess of aggregate covers such as Stop Loss treaties, which cover losses in excess of a set loss amount or loss ratio.
    • Cautions: the complexity of the calculations can lead to a "black box" mentality, so more than one set of results should be produced and basic statistics such as the CV and percentiles compared to the empirical data; most models assume that each occurrence is independent of the others and that frequency and severity are independent; some numerical methods have a large error term for low frequency scenarios; and the aggregate distribution reflects the process variance of losses but not the full parameter variance, where "parameter variance" in the broader sense of not being sure that the right model is used could be called "model risk" (process, parameter and model risk).

5 Property Catastrophe Covers

  • Section 5A. Traditional Products and Methods
    • A property catastrophe cover provides protection for a catastrophic event, such as a hurricane or earthquake, which may often affect multiple risks and multiple policies; it typically applies to the ceding company's retained exposure net of surplus share, per risk excess treaties and facultative certificates, so other reinsurance inures to its benefit.
    • The limit is defined in excess of a total loss amount (for example, 10,000,000 in excess of 30,000,000 per occurrence), and because the limit is often a substantial dollar amount the contract provides a limited number of reinstatements, "pro-rata as to amount" and less often "pro-rata as to time".
    • Pro-rata as to amount means that if part of the limit is exhausted, it can be reinstated for premium proportional to the amount reinstated: with a 10,000,000 occurrence limit, 2,000,000 annual premium, a 110% pro-rata as to amount provision and a 4,500,000 loss, the reinstatement premium is 990,000 (= $2,000,000 × 1.10 × 4.5/10).
    • The treaty effectively has an aggregate limit equal to one plus the number of reinstatements, times the occurrence limit; pro-rata as to time further reduces the premium to reflect only the time left in the policy period, and given the seasonal nature of some catastrophes (hurricanes) relatively few contracts include it.
    • Before catastrophe models there were few tools to systematically price catastrophe covers; the most common was the payback approach, in which premium is set so the offered limit is paid back over a given period (in the example, 2,000,000 of annual premium covers a single 10,000,000 total loss every five years).
    • Catastrophe models are now the generally accepted approach for pricing of natural and some man-made events, with four main components: event sets simulating the covered hazards (based on estimates of frequency and intensity at specific locations), calculation of local event intensity for each property in a portfolio, estimation of damage for each property impacted by a given event, and insured loss estimates based on the ceding company's policies.
    • The event sets are created and stored within the model before pricing; the damage and insured loss estimates are specific to the ceding company's portfolio and require a measure of exposure (insured values, construction types, occupancies, attachment points for excess contracts), geographical information (addresses converted to latitude and longitude by a geocoding engine, with less precise results if aggregated by zip code or state), the terms of the insurance policies (deductibles and coinsurance) and details of inuring reinsurance (occurrence caps or loss corridors on a surplus share affect the catastrophe exposure).
    • The output is a distribution of possible losses on the subject business, from which the expected amount in the treaty layer, the average annual loss (AAL), and its standard deviation are calculated as a starting point for a loss cost on the cover.
    • Accumulation management uses an Occurrence Exceedance Probability (OEP) curve, the probability that at least one event during the year will exceed a given loss amount, and possibly an Annual Exceedance Probability (AEP), the probability that the total of all modeled events in a single year exceeds a given loss amount.
    • Items that may need to be included more subjectively if not explicitly modeled: workers compensation losses within the cover (an earthquake during working hours), inuring reinsurance terms the model cannot calculate, "fire following" an earthquake even if earthquake coverage is not sold, and other coverage terms such as replacement cost versus actual cash value, where post-event demand for materials and labor raises the total cost borne by the insurer.
    • Coverage may be "losses occurring" or "losses occurring on risks attaching" during the period; with risks attaching contracts the reinsurer may pay twice on the same loss event (a 10,000,000 layer on a 3/15/95 event under both the 1/1/94 treaty, for 7/1/94 policies, and the 1/1/95 treaty, for 1/1/95 policies, may produce 20,000,000), so many treaties include an "interlocking clause" to equitably apportion losses covered under more than one contract.
  • Section 5B. Alternative Risk Products
    • A great variety of products are grouped under "financial reinsurance" or "finite reinsurance"; the term "finite risk" here refers to property catastrophe covers for which the maximum loss amount is reduced relative to traditional covers, a distinction that is very soft because traditional covers already have a definite limit that can be paid.
    • Two characteristics are common to most finite risk covers: multiple year features, and loss sensitive features such as profit commissions and additional premium formulas — for example, a three-year contract cancelable after the first or second year only if premium to date exceeds the loss payments, with a profit commission returning 75% of premium if loss free for three years in exchange for a relatively high annual premium.
    • The ceding company can only consider this insurance if the reinsurer assumes significant insurance risk under the reinsured portions of the underlying insurance agreements and it is reasonably possible that the reinsurer may realize a significant loss from the transaction (risk transfer); the actuary may be called on to quantify the risk, evaluating timing risk as well as underwriting risk, with reference to the American Academy of Actuaries' "Risk Transfer Testing Practice Note".
    • Features like profit commissions may substantially reduce the "upside" of the contract from the reinsurer's perspective, and a finite cover can be compared with an equivalent traditional risk cover: a 2,500,000 annual premium (25% nominal rate on line) for a 10,000,000 occurrence limit with an 80% profit commission after a 10% margin and additional premium of 50% of (Loss + Margin − Annual Premium) gives an underwriting result of 700,000 loss free and (3,625,000) with one full loss.
    • The equivalent traditional cover has a 700,000 premium and a 4,325,000 loss in the one-full-loss scenario, a rate on line of 16%, with the same results; the question is whether the reinsurer would offer that cover on a traditional basis at a 16% rate on line, and if not, the pricing for the alternative cover is also inadequate.
    • The analysis becomes more complicated with reinstatement provisions, expenses and carryforward provisions, and more difficult with provisions in which additional premium and profit commission percents change each year or depend on whether the ceding company or reinsurer cancels the cover; the best approach is to estimate the possible outcomes for a one-year time horizon, assuming any penetration exhausts the full limit and assigning probabilities with a Poisson or other distribution, whose mean is related to the "payback period" and can be compared with catastrophe models or other pricing analysis.
    • A final consideration is the credit risk of the ceding company: the reinsurer depends on the contingent "additional premium" to minimize downside risk, but the ceding company may be unable to pay it after the loss that makes it necessary, so a careful review of the ceding company's annual statement is needed (reinsurance credit risk).

6 Calculating the Final Price

  • The final program must cover the reinsurer's expected losses, its expenses and the risk borne by the stockholder, with attention to the timing of payments because investment income contributes to profitability.
  • The reinsurer's expenses are not the same as those of the ceding company (reinsurers are not subject to premium tax) and fall into three types:
    • Expenses varying with premium — ceding commission paid to the reinsured, brokerage fees (where applicable) and federal excise tax (where applicable).
    • Fixed expenses — general overhead costs (salaries, real estate) and underwriting and claims audit expenses, which would not increase simply by taking a larger share of a given treaty and should be set independently of variable expenses such as ceding commissions; similarly, an excess treaty quoted with and without an annual aggregate deductible has lower expected loss net of the AAD but expenses that may not differ, so the expense structure must cover costs regardless of whether an AAD is selected.
    • Expenses varying with losses — the reinsurer's unallocated loss adjustment expenses (ULAE), which should vary with the type of reinsurance contract (a working layer excess treaty may require extensive work, a quota share contract a review of a loss bordereau but less claim file review) and can be estimated after discussion with the claims department.
  • Loading the losses for these expense categories, the final premium could be estimated as:

\text{Premium} = \frac{\text{Loss Cost} \cdot (1 + ULAE) + \text{Fixed Expense}}{1 - \text{Variable Expense \%}}

  • The "traditional" loading of 100/80, often applied on excess treaties, is an example of this formula in which all expenses are considered variable with premium and assumed to total 20%.
  • Consideration must also be given to the timing and risk elements of the contract: the cash flows on the treaty, including premium and loss payments and any adjustable features (swing plan premium), need to be estimated; the considerations for profit or risk load are more complex and beyond the scope of the note, with Feldblum describing various approaches to accounting for risk.

Bibliography

  • [1] Bear, Robert A. and Kenneth J. Nemlick, "Pricing the Impact of Adjustable Features and Loss Sharing Provisions of Reinsurance Treaties," 1990 PCAS Vol. LXXVII.
  • [2] Feldblum, Sholom, "Risk Loads for Insurers," 1990 PCAS Vol. LXXVII.
  • [3] Heckman, Philip E. and Glenn G. Meyers, "The Calculation of Aggregate Loss Distributions from Claim Severity and Claim Count Distributions," 1983 PCAS Vol. LXX.
  • [4] Ludwig, Stephen J., "An Exposure Rating Approach to Pricing Property Excess-of-Loss Reinsurance," 1991 PCAS Vol. LXXVIII.
  • [5] Mashitz, Isaac and Gary Patrik, "Credibility for Treaty Reinsurance Excess Pricing," 1990 CAS Discussion Paper Program.
  • [6] Panjer, Harry H. and Gordon E. Willmot, "Insurance Risk Models," Society of Actuaries, 1992.
  • [7] Robertson, John P., "The Computation of Aggregate Loss Distributions," 1992 PCAS Vol. LXXIX.
  • [8] Venter, Gary, "Transformed Beta and Gamma Distributions and Aggregate Losses," 1983 PCAS Vol. LXX.
  • Additional references in Revised Study Note
    • [9] Clark, David R., "Credibility for a Tower of Excess Layers," CAS Variance Journal, 2011 Volume 5 Issue 1.
    • [10] Mata, Ana J. and Mark A. Verheyen, "An Improved Method for Experience Rating Reinsurance Treaties using Exposure Rating Techniques," CAS Forum 2005 Spring.
    • [11] "Risk Transfer Testing Practice Note," American Academy of Actuaries Committee on Property and Liability Financial Reporting (COPLFR), January 2007.
    • [12] Wang, Shaun S., "Aggregation of Correlated Risk Portfolios: Models and Algorithms," 1998 PCAS Vol. LXXXV.

Supplement

  • Comparison of Primary and Reinsurance Development, from the RAA Historical Loss Development Study, 2012 Edition, reprinted with permission of the Reinsurance Association of America: four graphs of percentage of ultimate against maturity in years, primary against reinsurer, for case incurred losses in Automobile Liability, General Liability, Workers Compensation and Medical Malpractice.

Sources