Insurance pricing

The price of insurance is set before its cost is known. That inversion of the production cycle is what makes pricing an actuarial discipline: one has to estimate how many claims will occur, how large they will be, how much it will cost to handle them and sell the policies, and how much capital must be remunerated, all for a future period that may start six months from now and end two years from now.

A rate must meet three requirements that pull in different directions. It must be adequate to cover claims, expenses and the cost of capital; equitable, in the sense that each policyholder pays according to their expected risk; and competitive, because a technically perfect rate that the market will not buy is useless. The LISF (Insurance and Surety Institutions Law) captures the first two by requiring net risk premiums to be determined on technical bases and the technical note to justify their adequacy.

This page focuses on short-term insurance, mainly property and casualty and medical expense coverage. Premiums for life insurance with long-term contingencies are covered in Actuarial mathematics, and the credibility theory used by experience rating in Credibility.

The basic unit is exposure: the car-year in auto, the insured-year in medical expense, each thousand pesos of sum insured in fire. On it we measure frequency, the number of claims per unit of exposure, and severity, the average cost per claim. Their product is the pure premium, the expected claim cost per unit of exposure, which Mexican technical notes usually present as the risk premium.

Separating frequency and severity is not a whim. They respond to different drivers: auto frequency depends on vehicle use and territory; severity, on the price of spare parts, the exchange rate and hospital costs. They have different trends and different volatility: frequency is stable with a few thousand exposures, while severity can hinge on a handful of large claims. That is why large claims are usually capped at a limit and loaded separately.

The pure premium must be based on ultimate claims: experience data for recent years is incomplete, so it is first developed with the factors from the claims reserve. It is then brought to the cost level of the period in which the new rate will be in effect.

Loadings are added to the risk premium to arrive at the gross premium. Expenses are split according to their behavior: fixed per-policy expenses (issuance, systems, collection) do not depend on the size of the premium, while variable expenses do (agent commissions, production bonuses, expenses proportional to premium). The profit margin is also expressed as a proportion of premium. Because variable expenses and profit depend on the price itself, the rate formula divides by one minus those percentages instead of adding them.

The profit provision in the rate should not be confused with return on capital. A 5% margin on premium may be a lot or a little depending on the capital the product consumes and the investment income the reserves generate while claims are paid. In long-tailed lines such as liability, investment income allows lower underwriting margins; in volatile lines such as earthquake, the cost of capital calls for higher loadings.

In Mexico the premium the customer pays also includes items outside the technical rate: the derecho de póliza (a fixed policy issuance fee), the surcharge for installment payment, which pays for financing and cancellation risk when the premium is paid in installments, and 16% IVA (value added tax) on taxable lines, since life insurance is exempt. These items must be provided for in the technical note and in the contract documentation.

The pure premium method builds the rate from scratch: ultimate, trended pure premium plus expenses, divided by one minus variable expenses and profit. It is the natural choice for new products, for class rates where exposures are known and when the rate structure is overhauled.

The loss ratio method adjusts an existing rate: it compares the observed loss ratio with the permissible one and calculates the percentage change needed. It requires bringing historical premium to the level of the current rate (on-level premium), for example with the parallelogram method or by rerating policy by policy at current rates. It is the usual approach in periodic reviews because it does not need homogeneous exposures.

If the data are consistent, both methods give exactly the same price: the example on this page confirms it. When they differ, it is almost always because of an error in the on-level premium or in the definition of exposure.

Experience data describe the past. To price, they must be projected to the future period: from the average occurrence date of the experience period to the average occurrence date of the policies that will be written at the new rate. If the experience is calendar year 2024 (midpoint July 1, 2024) and the rate will apply to annual policies written during 2026, the future average occurrence date is January 1, 2027 and the trend period is 2.5 years.

Severity trend in Mexico is related to the INPC (National Consumer Price Index) published by INEGI, but it is rarely the same: medical inflation has persistently outpaced general inflation, and auto severity depends on the price of imported parts and the exchange rate. Frequency trend captures changes in road safety, policyholder behavior and coverage. It is estimated with exponential regressions on quarterly frequency and severity series, excluding catastrophe claims and adjusting for deductible changes.

Catastrophe risks are not priced from a few years of a company’s own experience. In Mexico, the earthquake and hurricane risk premium is calculated with the models and technical bases the CNSF sets out in Annexes 5.1.5-a and 5.1.6-a of the CUSF (Single Insurance and Surety Circular), and it is added to the risk premium for non-catastrophe coverages.

The base rate tells you what the average policyholder costs; relativities tell you how much more or less each profile costs. Estimating them one variable at a time (univariate analysis) biases the results when variables are correlated: if young drivers are concentrated in certain territories, the territory effect absorbs part of the age effect. Generalized linear models (GLMs) solve this by estimating all relativities at once.

The standard specification is multiplicative: log link, Poisson distribution for frequency with exposure as an offset, and gamma for severity; or a Tweedie with a power parameter between 1 and 2 to model the pure premium directly, since it has a mass at zero and a continuous tail. With a log link, each coefficient becomes a relativity: e^β is the factor that multiplies the base rate. Off-balancing is then applied so that the total portfolio premium under the new relativities matches the indicated premium.

Validation requires out-of-sample data: lift curves, the Gini index, double lift tests against the current rate and stability of the coefficients over time. Machine learning techniques such as boosted trees improve prediction, but the filed rate must be explainable and justifiable in the technical note, so they are usually used to discover interactions that are then built into the GLM. The loss functions behind these models are explained in Loss functions.

When experience is thin, the indicated change is weighted against a complement: pure trend applied to the current rate, the indication for a larger segment or market experience. Under limited fluctuation credibility, the classic full credibility standard for frequency is 1,082 claims (a 90% probability of being within 5% of the expected value), and partial credibility follows the square root rule. For experience rating of groups and fleets, the Bühlmann approach assigns Z = n/(n + k).

A rate that is adequate when filed can stop being so within months if inflation, business mix or competition change. Monitoring compares the observed ultimate loss ratio with the permissible one every quarter, isolates the effect of large and catastrophe claims and checks that commercial discounts are not eroding the risk premium. The LISF specifically prohibits discounts, dividends or rebates from reducing the net risk premium.

Article 200 of the LISF requires insurers to determine net risk premiums on technical bases that guarantee, with a high degree of certainty, that their obligations will be met, and requires that dividends, rebates and refunds not reduce the net risk premium. Article 201 provides that each product consists of a technical note, contract documentation and an opinion of consistency; the technical note must contain the actuarial premium procedures, the technical justification of their adequacy, the interest rate, the demographic and statistical bases and the loadings for acquisition, administration and profit, and it must be prepared and signed by an actuary holding a cédula profesional and current certification from the professional college or accreditation by the Commission.

The CUSF details the content in Provision 4.1.6: frequency, severity or loss ratio assumptions, consolidated statistical information with its source, risk premium and gross premium formulas, justification of risk-based loadings and discounts, and the option of an experimental rate when no reliable, homogeneous and sufficient information exists. The actuarial standards of practice for insurance premiums are in Annex 4.3.1-a. Adhesion products, as well as group and collective life and accident and health insurance, require prior registration; registration does not prejudge the truth of the technical note’s assumptions, and the contract documentation is entered in CONDUSEF’s Registro de Contratos de Adhesión (Register of Adhesion Contracts).

In market practice, AMIS (the Mexican insurers’ association) compiles industry statistics by line that serve as market experience when a company’s own is insufficient, and earthquake and hurricane catastrophe risks are priced with the CNSF’s system and technical bases. Certification of the actuary who signs technical notes is, by law, a product requirement.