These study notes are based on the chapter on reinsurance loss reserving in Foundations of Casualty Actuarial Science by G.S. Patrik. These notes describe some of the concerns that are specific to reinsurance, and provides details on the Stanard-Bühlmann (Cape Cod) approach, which is commonly used for long-tailed reinsurance lines.

Introduction

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Challenges with Reinsurance Reserving

Loss reserving for reinsurance is generally more difficult than it is for primary insurance due to the following seven problems.
  1. Report lags are longer, especially for casualty excess lines. Factors contributing to this include:
    • Longer reporting pipeline
    • Undervaluation of serious claims by the cedant (may be valued below the reinsurance reporting threshold)
    • Mass tort claims (e.g. asbestos) may have extreme delays in reporting
  2. There is a persistent upward development of most claim reserves. Factors contributing to this include:
    • Economic and social inflation
    • Tendency to underreserve ALAE
    • Early information may indicate that a claim will pierce the retention, but not the ultimate severity
  3. Reporting patterns differ greatly by reinsurance line, type of contract, by cedant, and possibly by intermediary.
    • Reinsurer experience is heterogenous, but most reserving methods rely on large volumes of homogeneous data.
    • Reinsurance contracts are unique
    • Claim frequency may be so low that there is extreme fluctuation in historical data
    • Generally, the reinsurer knows much less about the underlying exposures than the primary carrier
  4. Industry statistics are generally not useful, due to heterogeneity of the exposure and reporting differences by company.
    • Annual statements do not provide breakdowns that are useful for loss reserving, e.g. all excess casualty aggregated under one line.
    • The problem is compounded because non-experts may insist on evaluating reserves according to annual statement statistics.
    • Primary company loss development statistics are not directly applicable without significant adustments; the report lag increases with the attachment point.
  5. Reports sent to the reinsurer may lack important information.
    • Proportional covers require only summary claims information
    • Data may not be split by accident year, but may be reported by calendar year or UW year; this is not sufficient for reserving purposes.
    • For excess covers, obtaining additional information to evaluate each claim may require significant effort. (Important to have reinsurance claims staff even though the claims are handled by the cedant – reinsurer may be able to advise cedant on large claims.)
    • Exposure measures (e.g. premium) may not be cleanly split out by primary line of business; may be split based on distribution of subject premium by line, as estimated at the beginning of the contract.
    • Premiums and losses for a quarter may not be reported and paid until the following quarter, so there is an additional IBNR exposure in both premiums and losses. In particular, the latest year-end premiums may be incomplete.
  6. Reinsurers often have data coding and IT systems problem due to the size, complexity, and heterogeneity of the business.
  7. Adequate loss reserves are larger, relative to surplus, for a reinsurer. This is more of a management problem than a technical problem, e.g. need to overcome disbelief in the size of the reserves.

Estimation Procedures

A reinsurer’s loss reserves consist of six components:
  1. Case reserves reported by the ceding companies
  2. Additional case reserves (ACR) specified by the reinsurer’s claims department
  3. Actuarial estimate of future development on 1 and 2 (IBNER)
  4. Actuarial estimate of pure IBNR
  5. Discount for future investment income on the assets supporting the liabilities
  6. Risk load: keeps the reserve at a conservative level so as not to allow uncertain income to flow to profits too quickly. This may be built into the reserve estimate implicitly by employing conservative methods and assumptions, or it may be estimated separately.
For reinsurers specifically:

Partititoning data for Reserve Analysis

The first step in reinsurance loss reserving is to partition the data into exposure groups that are relatively homogeneous with respect to their loss development potential, and that are relatively consistent over time. Variables to consider when partitioning data include:
  1. Line of business: property, casualty
  2. Type of contract: treaty, facultative.
  3. Type of reinsurance cover: quota share, surplus share, excess per-risk, excess per-occurrence, aggregate excess, catastrophe
  4. Primary line of business
  5. Attachment point
  6. Contract terms (type of rating, claims made or occurrence, treatment of ALAE)
  7. Small or large cedant
  8. Intermediary
The first five are generally considered the most important, and depending on the categorization resulting from considering these five, the remaining three may have an important effect. General guidance on dividing categories includes:
  • Major categories should be refined into treaty and facultative
  • Significant excess exposure should be split by type of retention (per occurrence or aggregate)
  • Treaty casualty excess exposure should be split by attachment point range and primary line of business (auto liability, general liability, workers compensation, medical professional liability), since each of these lines has a distinctly different report lag.
  • For proportional business, consider that the exposure may not be a true proportional share of first-dollar primary layers, but a share of an excess of a higher layer.
  • In general, the categories should correspond closely to pricing categories.
Additional information to guide the decision includes:
  • Expertise from underwriters
  • Elementary data analysis (simple loss develpoment statistics)

In terms of selecting methodology, a key consideration is balancing accuracy against effort and cost. Generally, different approaches are used based on the expected reporting time (short-tailed, medium- tailed, or long-tailed) for the category.

Short-Tailed Lines

For short-tailed lines, elaborate loss development techniques are generally unnecessary. Examples of short-tailed categoires include:
  • Treaty property per-risk excess at lower layers
  • Treaty property proportional
  • Treaty property catastrophe
  • Facultative property, excluding construction risks
  • Fidelity proporitonal

For treaty property, caution should be exercised if there have been recent catastrophes that could increase report lag.

Approaches to estimating reserves for short-tailed lines include:
  • Property business is typically reserved by setting IBNR equal to some percentage of the latest year’s earned premium. This is appropriate for non-catastrophe (“filtered”) claims.
  • Recent catastrophes will cause IBNR to exceed the normal reserve, since claims from major catastophes may not be fully reported and finalized for years.
  • For new lines of business, reserve up to a selected loss ratio, provided the result is larger than already-reported claims.
  • If a proportional treaty is reported by calendar year, then they may need to be assigned to an approximate accident year based on the underwriting year and the general report lag for that type of exposure. However, this kind of adjustment is not needed if IBNR is based no a percentage of premium.

Medium-Tailed Lines

Lines are considered to be medium-tailed if claims are almost completely settled within five years, and the average report lag is between one and two years. Examples include:
  • Treaty property per-risk excess at high layers (especially if a time element coverage is involved)
  • Construction risks (discovery period may extend years beyond the contract period)
  • Surety (Consider gross losses and salvage separately. Gross losses are reported quickly, but salvage recoveries have a longer tail.)
  • Non-casualty aggregate excess (lags are generally longer than for the underlying exposure)
Approaches to estimating reserves for medium-tailed lines include:
  • Chain ladder method can be used, with or without ACR
    • Advantage: correlates future development with both an overall lag pattern and claims reported for each accident year.
    • Disadvantage: correlation between IBNR and reported claims is so great that for longer-tailed lines, random noise may produce extremely large tail estimates for immature years.
  • Paid loss development may be more stable than reported loss development, but this may increase the estimation error in immature accident years.
  • For surety salvage, ratios of salvage to gross loss in mature years may be helpful, since it tends to be fairly stable.

Long-Tailed lines

Exposures are considered long-tailed if the expected report lag is over two years, and claims are generally not settled for many years. For reinsurers, most of the loss reserve and almost all of the IBNR is associated with long-tailed exposures, so most estimation effort should be spent on these lines. Iteration on the first-pass categorization of exposures may be warranted based on the results of the analysis, and large contracts may be examined on a standalone basis. Examples of long-tailed lines include:
  • Asbestos, Pollution, and other Health Hazard (APH)
  • Treaty casualty per-risk excess (longest lags except for APH)
  • Treaty casualty proportional (some exposure may be medium-tailed)
  • Facultative casualty
  • Treaty casualty aggregate excess

Claims from commuted contracts should be excluded from the analysis, since development is artifically truncated.

APH has a much longer tail than any other exposure, with catastrophic significance, and should always be considered separately from other exposures. Reporting patterns typically involve nothing for many years, followed by a sudden gigantic total. As a result, their inclusion would distort claims development statistics. Actuarial consulting firms have developed specialized models for these exposures, and it is generally better to work with specialists.

Methods for estimating long-tailed liabilities include:
  • Chain ladder may be considered, but it will produce extremely variable results for immature years.
  • Bornhuetter-Ferguson may be used:
    • Advantage is that it correlates future development with an exposure measure, rather than reported losses
    • Disadvantages include its dependence on the selected loss ratio, and it does not reflect the losses to-date, unless these are taken into consideration when the loss ratio is selected. This can be exacerbated by the correlation between the loss ratio and the reinsurance profitability cycle, which is generally more extreme than primary insurance cycles.

Cape Cod (Stanard-Bühlmann) Method

The key idea behind the Cape Cod method that differentiates it from the Bornhuetter-Ferguson method is that the Cape Cod method uses a single expected ultimate loss ratio for all accident years. A key consideration is that this loss ratio needs to be calculated relative to rate-level adjusted risk pure premium:
  • Brokerage fees, reinsurance commissions, and internal expenses should be removed.
  • The adjustment removes the effect of rate level cycles. Such cycles are genrally stronger in reinsurance than in primary insurance.
  • A possible source of information to adjust rate levels is industry-wide data.
In these notes,
  • \(P_{a, k}\) denotes the adjusted risk pure premium for accident year \(k\), with \(P_a = \sum_k P_{a,k}\)
  • \(P_{e, k}\) denotes the earned risk pure premium for accident year \(k\)
  • \(L\) denotes the expected loss ratio
  • \(R_k\) denotes the cumulative reported reinsurance loss for accident year \(k\) as of the valuation date of the analysis, with \(R = \sum_k R_k\)
  • \(G(k)\) denotes the report lag for accident year \(k\), which is the percentage reported as of the valuation date of the analysis.
    • This is the inverse of the chain ladder age-to-ultimate factor. It can be interpreted as the probability that any particular claim dollar will be reported to the insurer by time \(t\).
    • The probabilistic interpretation allows us to calculate statistics such as the expected value of the reporting time, and compare report patterns.
    • This also allows for the calculation of values at intermediate points.
  • \(I_k\) denotes the IBNR estimate for accident year \(k\), with \(I = \sum_k I_k\)

In the Cape Cod method, the IBNR estimate is calculated as \[ I_k = P_{a,k} L (1 - G(k)) \] The above formula, with the implicit assumption of constant \(L\), completely determines the estimator for \(L\). Summing over all \(k\), \[ I = L \sum_k P_{a,k}(1 - G(k)) \] The overall ultimate loss ratio, by definition, is \[ L = \frac{R + I}{P_a} = \frac{R + L \sum_k P_{a,k}(1 - G(k))}{P_a} \] Solving for \(L\) gives \[ L = \frac{R}{\sum_k G(k) P_{a, k}} \] This estimator is also appealing on intuitive grounds: by applying the lag factor to premium, we are essentially spreading premium over the development years in proportion to the expected losses to emerge in each development year. The quantity \(G(k) P_{a,k}\) is called the used-up premium.

The method can be illustrated using the following table from Patrik’s paper:

patrik.5.year = fread("./patrik_5_year.csv")
datatable(patrik.5.year)

First, determine the expected ultimate loss ratio:

patrik.5.year[, used_up_premium := ARPP * report_lag]
reported.loss.5 = patrik.5.year[, sum(reported_loss)]
used.premium.5 = patrik.5.year[, sum(used_up_premium)]
LR.5 = reported.loss.5 / used.premium.5
print(paste0("The expected ultimate loss ratio is ", LR.5))
## [1] "The expected ultimate loss ratio is 0.865979381443299"

The IBNR for the Cape Cod and Chain Ladder methods can be calculated as follows:

patrik.5.year[, CC_IBNR := ARPP * LR.5 * (1 - report_lag)]
patrik.5.year[, CL_IBNR := reported_loss *(1 / report_lag - 1)]
patrik.5.year[, CC_loss_ratio := (reported_loss + CC_IBNR) / ERPP]
patrik.5.year[, CL_loss_ratio := (reported_loss + CL_IBNR) / ERPP]
datatable(patrik.5.year)

Note that the actual earned risk pure premium is used in calculating the ultimate loss ratio for each accident year – the adjusted premium is only used to project losses.

Incorporating Credibility

The method can be enhanced by using a credibility-weighted mixture of the IBNR estimates from the chain ladder and Cape Cod methods. Similar to the Benktander approach, Patrik recommends using a credibility weight of \[ Z = C G(k) \] where \(C\) is some constant with \(0 \leq C \leq 1\), with the chain ladder estimate receiving this credibility and the Cape Cod estimate being the complement. Note that this has the effect of hiving more weight to the Cape Cod estimate in the early years when the chain ladder method is less reliable. Using this approach with \(C = 0.5\), the results are as follows:

C = 0.5
patrik.5.year[, credibility_IBNR := C * report_lag * CL_IBNR + (1 - C * report_lag) * CC_IBNR]
patrik.5.year[, credibility_LR := (reported_loss + credibility_IBNR) / ERPP]
datatable(patrik.5.year[, .(AY, credibility_IBNR, credibility_LR, CC_loss_ratio, CL_loss_ratio)])
Other credibility approaches include:
  • Combine the IBNR estimates from reported and paid data, with weights proportional to the percentage reported / paid
    • Problem with reported claims data is that they include case reserves that are based on judgment of many people, which may vary over time
    • However, paid data has a longer reporting pattern, but may be more stable as long as the expected payment pattern isn’t changing over time
  • The Benktander approach: this is similar to the above approach, except that the Cape Cod reserve estimate is replaced with the Bornhuetter-Ferguson reserve estimate, which may be derived from loss ratio assumptions used in pricing, and \(C=1\).
  • As a general rule, use of multiple methods is recommended in order to see the range of estimates, and to test the sensitivity to the assumptions underlying various methodologies.

Testing Predictions Against Actual Emergence

Careful monitoring of predictions against actual experience is necessary for long-tailed lines in order to provide early warnings of problems. This includes:

In Patrik’s example, the report lag is assumed to follow a Gamma distribution with mean 5 years and standard deviation of 3 years, and the chain ladder method is used to determine IBNR, with the exception of AY2000. The table provided in the paper has 13 years from a 20-year range. The following code imports the loss values reported as of December 31, 2000 and June 30, 2001, and generates the rest of the table based on the Gamma assumption.

patrik.20.year = fread("./patrik_20_year.csv")
gamma.mean = 5
gamma.sd = 3
gamma.scale = gamma.sd^2 / gamma.mean
gamma.shape = gamma.mean / gamma.scale
patrik.20.year[, report_lag_2000 := pgamma(2000 - AY + 1, shape = gamma.shape, scale = gamma.scale)]
patrik.20.year[, report_lag_2001 := pgamma(2000 - AY + 1.5, shape = gamma.shape, scale = gamma.scale)]
patrik.20.year[AY != 2000, IBNR_2000 := reported_loss_12_2000 * (1 / report_lag_2000 - 1)]
patrik.20.year[AY == 2000, IBNR_2000 := 5182] # Unclear how this is calculated in the table -- likely an assumption based on expected loss ratio
patrik.20.year[, predicted_claims_2001 := reported_loss_12_2000 + IBNR_2000 / (1 - report_lag_2000) *(report_lag_2001 - report_lag_2000)]
patrik.20.year[, difference := predicted_claims_2001 - actual_claims_06_2001]
datatable(patrik.20.year)

Summarizing the table:

total.difference = patrik.20.year[, sum(difference)]
expected.emergence = patrik.20.year[, sum(predicted_claims_2001 - reported_loss_12_2000)]
percentage.difference = 100 * (total.difference / expected.emergence)
print(paste0("The predicted six-month emergence is off by ", total.difference))
## [1] "The predicted six-month emergence is off by -858.123220175018"
print(paste0("Expected emergence during this period is ", expected.emergence))
## [1] "Expected emergence during this period is 2222.87677982498"
print(paste0("The actual emergence is off by ", percentage.difference, " percent of the expected emergence"))
## [1] "The actual emergence is off by -38.6041740128566 percent of the expected emergence"
Note that it is important to compare the error to the expected emergence rather than the total reserve. Possible reasons for an underestimate could include: