Mathematics and insurance have developed along parallel paths during the past 350 years. It is difficult to identify an economic activity more closely tied to mathematics than insurance. Since the genesis of probability ideas in the mid-17th century, there have been times when mathematical developments were ahead of insurance practice. At other times, commercial necessity required improvisations that did not rest on solid mathematical foundations. In general the science and the application moved together.

Reserves and Premiums: Long-Term Coverages

Two related valuation problems are to establish a price, or premium, and to estimate the liability created by a contract. The basic tools for solving these problems are expected values and compound interest combined with an economic concept, the equivalence principle. The equivalence principle requires, at the time the coverage is activated, that the expected present value of premiums equals to expected present value of benefits. Following the issuance of the coverage, the principle can be extended to define the liability of the insurer as the expected present value of future benefits less the expected present value of future premiums.

Long-term insurance contracts have the possibility of extending for many years. The time of benefit payment, and for some contracts the amount of payments, often depend on the length of the survival time of the insured. Specifically, let T denote the random variable time until death. One example of a long-term coverage is life insurance with a single payment of benefits at time T Another example is a life annuity with many payments of benefits paid during survival, up to time T The life insurance model would apply to financing the replacement of equipment from light bulbs to generators. The mathematics of annuities would apply to funding equipment maintenance costs.

To illustrate, consider a life insurance policy paying a benefit b at death to be funded by a premium π, paid at a continuous annual rate until death. For the time until death random variable, let s(t) = Pr (T > t) be the survival function. Then the equivalence principle determines the premium π by the equation

$$ - b{\int}_0^{\infty}\;{e}^{-\delta t}{\mathrm{s}}^{\prime }(t) dt=\pi {\int}_0^{\infty}\;{e}^{-\delta t}\mathrm{s}(t) dt. $$
(1)

Here, −s′(t) is the probability density function of time until death and δ is the continuous interest rate, also called the force of interest. It will be assumed constant for simplicity. It is defined through the relation 1 + i = e δ, where i is the annual effective rate of interest. The premium rate π is known as a ‘benefit premium’; it is computed assuming that firms are risk neutral and that there are no transactions costs. In commercial practice, the benefit premium π will be increased to a contract premium G ,  G > π. The contract premium will contain provisions for expenses, profits and risk. The equivalence principle can be extended to include these elements.

The liability of the insurer, denoted by s V given survival to s ,  s ≥ 0, would be given by the equivalence principle as

$$ {}_s V+\pi {\int}_s^{\infty}\;{e}^{-\delta \left( t- s\right)}\mathrm{s}\left( t- s| s< T\right) dt=- b{\int}_s^{\infty}\;{e}^{-\delta \left( t- s\right)}{\mathrm{s}}^{\prime}\left( t- s| s< T\right) dt. $$
(2)

In words, the liability is the expected present value, also called the actuarial present value, of future benefits less the actuarial present value of future premiums. In this equation, the conditional survivorship function is s(t − s| s < T) = s(t)/s(s).

In Eq. (1) for the premium rate and Eq. (2) for the reserves, making benefits and premiums a function of time survived, b(t) and π(t), creates no conflict with the equivalence principle. There are practical reasons for requiring s V ≥ 0. This prevents a voluntarily withdrawing insured from leaving the insurer with a negative liability, a non-collectable asset.

As another special case, we now consider a life annuity with benefits at an annual rate b starting at retirement time r, funded by a continuously paid premium rate π paid during survival to r. Such a contact would be a building block of a pension plan. The equivalence principle yields

$$ \pi {\int}_0^r\;{e}^{-\delta t}\mathrm{s}(t) dt= b\;{\int}_r^{\infty}\;{e}^{-\delta t}\mathrm{s}(t) dt, $$
(3)

allowing us to compute the premium rate π based on survivorship and interest information.

To illustrate how other contracts can be accommodated, we consider the life annuity case that also includes a so-called ‘return of premiums’ feature. With this feature, there is an additional benefit consisting of the accumulated premiums (with interest) that are paid at death before time r. The benefit side of formula (3) is increased by

$$ -\pi {\int}_0^r\left({\int}_0^t{e}^{\delta x} dx\right){e}^{-\delta t}{\mathrm{s}}^{\prime }(t) dt=-\pi \left(\mathrm{s}(r){\int}_0^r{e}^{-\delta t} dt+{\int}_0^r{e}^{-\delta t}\mathrm{s}(t) dt\right), $$
(4)

where the right-hand side is from an integration by parts. With this additional benefit, from formula (3) we have

$$ \pi \mathrm{s}(r){\int}_0^r{e}^{-\delta t} dt= b{\int}_r^{\infty }{e}^{-\delta t}\mathrm{s}(t) dt, $$

a result that might have been derived by general reasoning from the equivalence principle.

We return to the life annuity premium displayed in formula (3). The equivalence principle yields a reserve liability at time s,  0 ≤ s, of

$$ \begin{array}{ll}\hfill & sV+\pi {\int}_s^r{e}^{-\delta \left( t- s\right)}\mathrm{s}\left( t- s| s< T\right) dt\\ {}& = b{\int}_s^{\infty }{e}^{-\delta \left( t- s\right)}\mathrm{s}\left( t- s| s< T\right) dt 0< s< r\operatorname{}\operatorname{}{\operatorname{}}_s V\hfill \\ {}& = b{\int}_s^{\infty }{e}^{-\delta \left( t- s\right)}\mathrm{s}\left( t- s| s< T\right) dt r< s.\hfill \end{array} $$

The key role played by the survival function and the assumed interest rate in these typical formulas is clear.

Reserves and Premiums: Short-Term Coverages

Short-term coverages include most individual property/casualty, health and group insurance policies. They are characterized by the reduced role of present values. In addition, the benefit amount is typically a random variable. Its value will depend in health insurance on the services provided, and in property insurance on the extent of the property damage. Premiums and reserves will continue to be determined by the equivalence principle. In the time period between the occurrence of a loss event and its settlement, available information about the loss event will determine reserve amounts.

The expected value of benefit payments for short-term coverages is given by

$$ \pi =\mathrm{E}\left(\sum_{i=1}^N{X}_i\right)=\mathrm{E}\left(\mathrm{E}\left(\sum_{i=1}^N{X}_i\right)| N= n\right)=\mu \mathrm{E}(N), $$

where N denotes the random number of losses during the insurance period, X i is the loss amount arising from loss i and E X i = μ.

If the distribution of N is Poisson and N and the loss amounts are independent, then

$$ S={X}_1+\dots +{X}_N $$
(5)

has a compound Poisson distribution. Clearly many distributions, such as the binomial or negative binomial, could be used for the distribution of N.

The reserve liability for short-term coverages uses information about loss events and the loss reserve is

$$ \mathrm{E}\left({X}_1+\dots +{X}_N| N= n\right)= n\mathrm{E}(X)= n\mu $$

and n is the number of losses incurred.

Risk theory is the study of the distribution of total losses and the management of their inconvenient consequences. The earliest contributions to risk theory build on the model for long-term coverages. We start with loss variables

$$ {L}_j={b}_j{e}^{-\delta {T}_i}-{\pi}_j{\int}_0^{T_i}{e}^{-\delta s} ds, $$

and study the distribution of S = L 1 + … + L n . Here, T i is a random variable representing the future lifetime of an individual. This study is known as individual risk theory because the variable S is based on n individual loss variables. If the loss variables are assumed to be mutually independent, then

$$ Z=\frac{S-\mathrm{E}(S)}{\sqrt{\mathrm{Var}(S)}} $$

will have, as a result of an extension of the central limit theorem, an approximate normal distribution, with mean zero and variance one. In contrast, the direct study of the distribution of S as in formula (5) is called collective risk theory. Approximating the distribution of S has been an active topic in actuarial research since early in the 20th century.

Experience Adjustment: Long-Term Coverages

Valuation of long-term coverages requires assumptions about the realizations about interest rates and mortality in the distant future. In this dynamic world it is almost certain that the results expected by an insurance system will not be obtained. For many contracts, it has become customary for insurers to make assumptions that many financial analysts would view as conservative for pricing at contract initiation. As better than anticipated experience is realized, excess funds are realized that can be directed to the insured in a mutual insurance organization or to owners of the insurance company. For the insured, these are additional (non-contractual) benefits; depending on the regulatory environment, these additional benefits come in the form of dividends or bonuses.

Reconciling anticipated to actual experience is done periodically, not just at the conclusion of the contract. Because of this periodic reconciliation, recursion relationships are important tools for measuring and adjusting for deviations from expected results.

Specifically, let s − 1 F be the fund, possibly the insurance reserve, at the end of policy year s-l. Define s P to be the premium paid at the beginning of policy year s ,  E( s B) the expected benefits paid at the end of policy year s and s F the expected fund at the end of policy year s. We simplify and assume that E( s B) = b q s . A basic recursive relationship is

$$ \left({}_{s-1} F{+}_s P\right)\left(1+ i\right)-{bq}_s{=}_s F {p}_s, $$
(5a)

where i is the expected annual interest rate and p s = 1 − q s . Formula (5) can be written as

$$ \left({}_{s-1} F{+}_s P\right)\left(1+ i\right)-{q}_s\left( b{-}_s F\right){=}_s F. $$
(6)

If the actual experience yields i ′ and q s ', then formula (6) can be written as

$$ \left({}_{s-1} F{+}_s P\right)\left(1+{i}^{\prime}\right)-{q}_s\hbox{'}\left( b{-}_s F\right){=}_s F+ D, $$
(7)

where D is a deviation of actual from expected results. If D > 0, the amount might be paid to the insured in a mutual insurance organization or to owners of the insurance company.

Subtracting formula (6) from (7), yields

$$ D=\left({q}_s-{q}_s\hbox{'}\right)\left( b{-}_s F\right)+\left({i}^{\prime }- i\right)\left({}_{s-1} F{+}_s P\right). $$
(8)

The first term on the right-hand side of formula (8) is called the mortality contribution and the second term the interest contribution. Formulae used in practice also contain a term for the difference between expense loading and actual expenses.

To study life annuities, the general formula (6) can be modified during the benefit payment period to yield

$$ {}_{s-1} F\left(1+ i\right)-{p}_S b={p}_{S S} F. $$
(9)

Replacing the expected parameters with experience parameters, we have

$$ {}_{s-1} F\left(1+{i}^{\prime}\right)-{p}_s^{\prime } b={p}_s^{\prime}\left({}_s F+ D\right). $$
(10)

Subtracting formula (9) from formula (10) yields

$$ {}_{s-1} F\left({i}^{\prime }- i\right)+\left({p}_s-{p}_s^{\prime}\right)\left( b{+}_s F\right)={p}_s^{\prime } D. $$

If D > 0, this expression could be the basis of a dividend to surviving annuitants.

These recursion relationships are also the basis for flexible coverages where premiums and benefits can be changed by the insured within contractual limits.

Experience Adjustment: Short-Term Coverages

In the first decade of the 20th century industrial accidents were a leading cause of death, a source of much litigation and a major social concern. The advent of workers’ compensation insurance replaced litigation with a system based on defined benefits. Employers, in most cases, were required by statute to provide workers’ compensation benefits. Because of great variation in the hazards faced in different industries and the lack of loss statistics, initial premiums were set by judgement. The goal was to develop a self-correcting rate estimation process that would also provide incentives to employers to improve industrial safety.

The solution came from the formula

$$ \left( New rate\right)= Z(n)\times \left( observed\ average\ losses\right)+\left[1- Z(n)\right]\times \left( Initial\ Rate\right), $$

where the credibility factor is Z(n), n a measure of exposure and 0 ≤ Z(n) ≤ 1. To provide intuition, consider the case where Z(n) = 1, known as the ‘full credibility’ case. Here, the employer’s next period premium would consist entirely of observed average losses from the prior period. If the employer had introduced practices to improve industrial safety then this would be reflected in a lower premium. In contrast, consider the case where Z(n) = 0. Here, the premium would consist of an initial rate that presumably would reflect industry results but not the employer’s actual experience. The case Z(n) = 0 is the standard for individual coverages. Many employers would fall in the intermediate case, 0 < Z(n) < 1, known as ‘partial credibility’. Premiums for employers in this category would reflect their own industrial safety records as well as benefit from the pooling of risks within an industry.

For the credibility factor, one typically requires Z ′(n) > 0 and Z ″(n) < 0. Thus, other things equal, employers with larger exposure (n) enjoy larger credibility but the rate of increase decreases with exposure. A typical credibility function is of the form Z(n) = n/(n + k), k > 0. The establishment of k with a satisfactory intellectual foundation has come from Bayesian statistics after its introduction into practice. The credibility idea for experience adjustments is now used in many short-term coverages.

Another type of insurance plan available for groups is known as ‘stop-loss’ or ‘excess of loss’ coverage. Large group insurance plans, usually based on employee groups, have distinctly different risk characteristics from individual policies. The sponsor, usually a large organization, is typically willing and able to absorb some variation in benefit payments. Only large and unexpected payments are financially inconvenient to the sponsor. The insurance company is paid to adjudicate and pay benefit claims, and to absorb large and inconvenient benefit payments. Typically, the sponsor maintains an internal account of losses known as an ‘experience account’. This account records premiums as income and losses and expenses as expenditures.

We let X be the losses in an experience period and d be the stop loss amount (or d for ‘deductible’). The experience account is charged for losses up to d. If X > d, then X-d is not charged to the experience account of the sponsor. A risk premium for this experience adjustment is charged on the basis of

$$ {\int}_d^{\infty}\left( x- d\right) f(x) dx. $$

Model Calibration: Experience Studies

The models introduced suggest that extensive work must be done in estimating survival functions in implementing long-term insurance models. For short-term models, the distribution of the number of losses N, per policy period, and the distribution of X, the loss amount, must be estimated. These efforts are in most applications special cases of statistical estimation.

These estimation projects are generally observational studies. The data come from insurance experience and the subjects have purchased insurance or gained insurance as an employee benefit. The use of general population statistics for insurance purposes has hazards because of potential biases. To illustrate, when studying annuitant mortality, it is well-known that mortality is substantially lower than the general population mortality. This is a selection bias issue; seldom do those in substandard health purchase a life annuity.

Rapid increases in the cost of health services and jury awards in some areas have increased the need to estimate time trends for the distribution of X, loss costs. Because of longer settlement time in some coverages, this estimation has become a major project in loss reserve determination. The rate of increase in health care costs in recent years has been such that estimates of the distribution of X, benefit amount random variable, using information from previous years would result in a distribution significantly to the left of the distribution for the current year. The rate of growth of health care costs is the most important single pricing decision for health insurance.

Model Calibration: Classification

The distributions that enter insurance models are all conditional distributions. Clearly, the distribution of X, loss amount, depends on the time, location and other facts surrounding the insurance loss incident. The distribution of T, time until death, in life insurance depends on a set of classification variables. The purpose of observing these classification variables is to increase the likelihood that the assumed distribution of T will be approximately realized.

The selection of these classification variables may be constrained by law and expense. For example, a determination of the degree of aggression of an applicant for automobile insurance might have a significant impact on the distribution of N, but the expense of collecting the information might be greater than its value in reducing variability.

Model calibration: financial economics

The critical role played by the force of interest δ in premiums and reserves for long-term coverage is clear. The use of an assumed force of interest for an extended period of time will lead, according to common experience, to serious deviations between actual and expected results. Options for moderating these deviations are numerous.

• A statistical model for the force of interest, estimated from past data, could be constructed and the equivalence principle extended to take expectations over the joint distribution of δ T and T. The joint distribution might also be used to fix an interest rate risk loading into premiums to minimize the inconvenient consequence of variations in δ.

• Arrange the timing of investment cash flows to approximately match the expected cash flows from the insurance operations.

• Pass variations in interest earnings directly to the policy owner as indicated in the section on Experience adjustment, long-term coverages. The insured’s account F would absorb variation in investment earnings.

• Use a program of financial derivative contracts to stabilize, for a price, variations in investment income.

Financial economics has not only enriched insurance mathematics by providing risk management tools for the investment risk in conventional insurance contracts, it has also created the possibility of absorbing many traditional insurance risks into special securities traded in worldwide investment markets. The idea is to use the capital in investment markets, and not just the capital held by insurance companies, to manage risk.

The idea of special securities with contractual payments that approximately match payments from an insurance system has already been developed for several coverages –for example, catastrophe bonds with modified payments following a catastrophe, fitting the definition of the security. A second example is a survivorship bond with regular coupon payments proportional to the number of survivors in a defined group. Such bonds could spread the risk if mortality improvement exceeds the capacity of the sponsor of a pension system.

The market for such special securities is determined, in part, from ideas in financial economics. Portfolio theory would predict that investors would seek securities that have cash flows that are not positively correlated with the regular business cycle. Tying security payments to natural disasters, such as earthquakes and hurricanes, might achieve the sought for independence.

See Also