Technical reserves

An insurer collects today and pays later. Between collecting the premium and making the last payment on a claim, days may pass in auto insurance, or decades in a life annuity or a liability claim. Technical reserves are the estimate, in present value, of everything still to be paid on contracts already written. If they are understated, reported profit and solvency are fictitious; if they are overstated, capital is tied up and pricing is distorted.

The Mexican framework, in force since 2015 under the Ley de Instituciones de Seguros y de Fianzas (LISF, Insurance and Surety Institutions Law) and the Circular Única de Seguros y Fianzas (CUSF, Single Insurance and Surety Circular), adopted the economic approach of Solvency II: the reserve is the best estimate of future cash flows discounted with risk free curves plus a risk margin calculated on a cost of capital basis. Each company develops its own estimation methods, but they must be registered with the CNSF (National Insurance and Surety Commission) and comply with actuarial standards of practice.

This page focuses on short term reserves and on estimating incurred claims, which is where actuarial judgment carries the most weight. Life mathematical reserves are covered in Actuarial mathematics, and a quick chain-ladder calculation is available in the Calculators.

Article 216 of the LISF lists the reserves that insurance companies must establish. Each responds to a different obligation or risk, and the distinction matters because they are calculated, invested and released differently. The first two, the unearned premium reserve and the reserve for outstanding obligations, are valued with company specific methods registered with the CNSF; the rest follow rules set by the Commission itself.

It helps to picture a timeline for each policy. Before the claim occurs, the obligation sits in the unearned premium reserve. Once it occurs, it moves to outstanding obligations, even if the insurer does not know about it yet: that is IBNR. Catastrophe and contingency reserves do not relate to specific policies but to possible deviations in aggregate claims experience.

In its traditional accounting form, the unearned premium reserve for an annual policy is simply the unearned premium: if an annual premium was collected and four months have elapsed, eight twelfths are reserved. That approach assumes the premium is adequate and that risk is spread evenly over the term. Under the LISF, the reserve is the best estimate of future claims and expenses for the remaining term, plus the risk margin. If the rate is inadequate, the best estimate exceeds the unearned premium and the loss is recognized immediately, which is precisely the point.

For short term insurance, the best estimate is usually obtained by applying to the unearned premium an expected ultimate loss ratio, derived from development triangles by origin year, plus administrative and adjustment expenses. For long term insurance (whole life, endowments, multiyear medical expense) cash flows are projected policy by policy using mortality, morbidity, lapse and expense assumptions, and discounted with the risk free curve published by the CNSF. The CUSF includes in this reserve premiums written in advance whose coverage has not yet started.

Seasonality and the distribution of risk over the term matter: a rain fed crop policy or a hurricane policy does not accumulate risk evenly throughout the year. Prorating the premium linearly in those cases releases reserve before the risk has passed.

Once a claim has occurred, the obligation has three layers. The first is the case reserve the adjuster sets for each known claim. The second is the deficiency in those case reserves: reported claims tend to grow (or shrink) over their life because of new information, litigation or inflation; in the English language literature this is called IBNER. The third is claims that have already occurred but the insurer does not yet know about, pure IBNR. In Mexico, the LISF groups the last two under siniestros ocurridos y no reportados (SONR, incurred but not reported claims), together with allocated loss adjustment expenses.

Loss adjustment expenses are split into allocated (fees of lawyers, experts and external adjusters attributable to a claim) and unallocated (the cost of the internal claims department). The former are estimated with the same triangles as the claims or with their own triangles; the latter, with expense ratios to paid and outstanding claims.

The basic input is the development triangle: rows by origin year (accident, underwriting or report year), columns by development age and cells with cumulative paid or incurred amounts. The consistency of the triangle matters more than the method: changes in case reserving policy, in payment speed or in the treatment of salvage break the patterns and make any technique fail.

The chain-ladder assumes that each origin year will develop in the future with the same average factors observed in the past. Age to age factors are calculated as weighted ratios of consecutive columns, multiplied together to obtain the cumulative development factors to ultimate (CDF) and applied to the latest diagonal. Its great advantage is that it relies solely on the company's own experience and requires no a priori loss assumption.

Its weakness shows up in the most recent years: with a CDF of 1.6 or 3, any random deviation in the first diagonal is amplified. A large claim reported early inflates the projected ultimate; a year with slow reporting depresses it. It is also sensitive to the tail: if the triangle does not reach full development, a tail factor is needed, fitted with curves (exponential, inverse power) or with market information.

In 1972 Ronald Bornhuetter and Ronald Ferguson proposed estimating IBNR as the unreported portion of an a priori expected loss. The unreported proportion, 1 - 1/CDF, comes from the development pattern; the level, from premium multiplied by an expected loss ratio (ELR) taken from the business plan, the technical note or mature years. As a result, IBNR does not depend on what has been reported to date, and an early large claim is not multiplied.

BF is equivalent to a credibility weighted average of the chain-ladder and the expected loss ratio method, with weight Z = 1/CDF on the chain-ladder: the more mature the year, the more credibility given to experience. The Benktander method iterates BF using the BF ultimate itself as the new a priori and lands halfway between the two. Its weak point is the ELR: if it is optimistic, BF under-reserves persistently.

The Cape Cod method (or Stanard-Bühlmann) removes the arbitrariness of the ELR by estimating it from the data itself: it divides reported claims for all years by used premium, that is, premium weighted by the reported proportion. That common ELR is then plugged into the BF formula. It works well when rates have been stable or when premiums are first adjusted to current rate and trend levels.

A point estimate of the reserve says nothing about how wrong it could be. In 1993 Thomas Mack showed that the chain-ladder is the estimator of a distribution free model in which the conditional expectation of C_i,j+1 is f_j·C_i,j and its variance is proportional to C_i,j. With that model he derived closed form expressions for the mean squared error of prediction of IBNR, which combines process error (future randomness) and estimation error (uncertainty in the factors).

The bootstrap on the over-dispersed Poisson model, popularized by England and Verrall, resamples the Pearson residuals of the incremental triangle, rebuilds thousands of pseudo observed triangles, re-estimates the factors and simulates future payments. The result is a full distribution of IBNR, from which percentiles are taken for the risk margin, economic capital or the IFRS 17 risk adjustment. In both cases, the estimated variability is only as good as the stability of the triangle: it does not capture regime changes.

Article 218 of the LISF provides that technical reserves equal the sum of the best estimate and a risk margin, calculated separately. The best estimate is the probability weighted mean of future cash flows, discounted with market risk free curves, on a gross basis and taking into account all policyholder options. The risk margin is what a third party would require, on top of the best estimate, to take over the obligations.

The margin is calculated as the cost of holding, over the life of the obligations, the own funds backing their solvency capital requirement (RCS). The CUSF sets a 10% net cost of capital rate, the same for all companies, and describes how to allocate the RCS by line of business. The projection of the future RCS is usually approximated with a pattern proportional to the remaining best estimate, which gives long tailed lines proportionally larger margins.

Articles 216 to 219 of the LISF define the reserves, their purpose and their valuation principles, and require the actuarial methods for the unearned premium reserve and the reserve for outstanding obligations to be registered with the CNSF. Title 5 of the CUSF develops the regime: Chapter 5.1 governs the unearned premium reserve, 5.2 outstanding obligations, 5.3 the statutory method the Commission assigns when a company's method does not adequately reflect its obligations, 5.4 the risk margin, 5.5 the registration of methods and 5.6 catastrophe reserves. Triangles are classified by origin and development year under Annex 5.3.2, and the actuarial standards of practice for reserving are in Annex 4.3.1-b.

Earthquake reserves and hurricane and other hydrometeorological risk reserves are cumulative: they increase every month with the earned portion of retained risk premiums, calculated with the technical models in Annexes 5.1.5-a and 5.1.6-a, plus investment income at the 28 day Cetes rate, and they have a ceiling linked to the retained probable maximum loss. The agricultural and livestock reserve increases with 35% of the earned retained gross premium; the credit and surety (caución) reserves, with 75% of the annual excess of earned retained risk premium over retained claims. Drawing on them requires authorization from the Commission, except in automatic cases.

Reserve adequacy is the subject of an annual independent actuarial opinion, subject to the standard in Annex 23.3.1 of the CUSF, and each company's actuarial function must express a view on methodology and assumptions. International groups that report to their parent under IFRS 17 calculate, in parallel, a risk adjustment for non-financial risk using an approach that does not necessarily match the regulatory risk margin.