Overview

Insurance is a means of protection from financial loss. It is a form of risk management, primarily used to hedge against the risk of a contingent or uncertain loss

An entity which provides insurance is known as an insurer, insurance company, insurance carrier or underwriter. A person or entity who buys insurance is known as an insured or as a policyholder. The insurance transaction involves the insured assuming a guaranteed and known relatively small loss in the form of payment to the insurer in exchange for the insurer's promise to compensate the insured in the event of a covered loss. The loss may or may not be financial, but it must be reducible to financial terms, and usually involves something in which the insured has an insurable interest established by ownership, possession, or pre-existing relationship.

The insured receives a contract, called the insurance policy, which details the conditions and circumstances under which the insurer will compensate the insured. The amount of money charged by the insurer to the Policyholder for the coverage set forth in the insurance policy is called the premium. If the insured experiences a loss which is potentially covered by the insurance policy, the insured submits a claim to the insurer for processing by a claims adjuster. The insurer may hedge its own risk by taking out reinsurance, whereby another insurance company agrees to carry some of the risk, especially if the primary insurer deems the risk too large for it to carry.

Characteristics of Insurable Losses

Risk which can be insured by private companies typically shares seven common characteristics:

Characteristic	Description
Large number of similar exposure units	Since insurance operates through pooling resources, the majority of insurance policies are provided for individual members of large classes, allowing insurers to benefit from the law of large numbers in which predicted losses are similar to the actual losses. Exceptions include Lloyd's of London, which is famous for ensuring the life or health of actors, sports figures, and other famous individuals. However, all exposures will have particular differences, which may lead to different premium rates.
Definite loss	The loss takes place at a known time, in a known place, and from a known cause. The classic example is death of an insured person on a life insurance policy. Fire, automobile accidents, and worker injuries may all easily meet this criterion. Other types of losses may only be definite in theory. Occupational disease, for instance, may involve prolonged exposure to injurious conditions where no specific time, place, or cause is identifiable. Ideally, the time, place, and cause of a loss should be clear enough that a reasonable person, with sufficient information, could objectively verify all three elements.
Accidental loss	The event that constitutes the trigger of a claim should be fortuitous, or at least outside the control of the beneficiary of the insurance. The loss should be pure, in the sense that it results from an event for which there is only the opportunity for cost. Events that contain speculative elements such as ordinary business risks or even purchasing a lottery ticket are generally not considered insurable.
Affordable premium	If the likelihood of an insured event is so high, or the cost of the event so large, that the resulting premium is large relative to the amount of protection offered, then it is not likely that the insurance will be purchased, even if on offer.
Limited risk of catastrophically large losses	Insurable losses are ideally independent and non-catastrophic, meaning that the losses do not happen all at once and individual losses are not severe enough to bankrupt the insurer; insurers may prefer to limit their exposure to a loss from a single event to some small portion of their capital base. Capital constrains insurers' ability to sell earthquake insurance as well as wind insurance in hurricane zones. In the United States, flood risk is insured by the federal government. In commercial fire insurance, it is possible to find single properties whose total exposed value is well in excess of any individual insurer's capital constraint. Such properties are generally shared among several insurers, or are insured by a single insurer who syndicates the risk into the reinsurance market.

Coverage Modifications

Deductibles

In an insurance policy, the deductible is the amount of expenses that must be paid out of pocket before an insurer will pay any expenses. In general usage, the term deductible may be used to describe one of several types of clauses that are used by insurance companies as a threshold for policy payments.

Deductibles are typically used to deter the large number of claims that a consumer can be reasonably expected to bear the cost of. By restricting its coverage to events that are significant enough to incur large costs, the insurance firm expects to pay out slightly smaller amounts much less frequently, incurring much higher savings. As a result, insurance premiums are typically cheaper when they involve higher deductibles.

When a policy contains a deductible, the ultimate claim amount will depend on the type of deductible in question. There are generally two types of deductibles: an ordinary deductible and a franchise deductible. In what follows, we let [math]d[/math] denote the deductible and let [math]X[/math] the loss incurred by the insured.

Ordinary Deductible

When the deductible is ordinary, the claim amount is:

[[math]] \begin{equation*} (X-d)^+ = \begin{cases} X-d & \text{if}\,\, X \geq d \\ 0 & \text{otherwise} \end{cases} \end{equation*} [[/math]]

Franchise Deductible

A franchise deductible is similar to an ordinary deductible except that the total loss to the insured can be claimed when that loss exceeds the deductible. More precisely, the claim amount is:

[[math]] \begin{align*} X & \quad \text{if}\,\, X\geq d \\ 0 & \quad \text{otherwise} \end{align*} [[/math]]

Policy Limits

Policy limits simply set a limit on the amount of the claim. More precisely, if [math]l[/math] is the policy limit then the claim amount is:

[[math]] \begin{equation*} X \wedge l = \begin{cases} l & \text{if}\,\, X \gt l \\ X & \text{if}\,\, X \leq l \end{cases} \end{equation*} [[/math]]

Coinsurance

This usually means that the loss to the insurer is a fraction of what it would be without coinsurance. The fraction is usually called the coinsurance factor and is expressed in %. For instance, if a policy has a deductible [math]d[/math], a limit [math]l[/math] and a coinsurance factor [math]f[/math], then the claim amount is:

[[math]] \begin{equation*} f \cdot \left[(X-d)^+ -(X-l)^+ \right]. \end{equation*} [[/math]]

Expected Loss and Mean Excess Loss

In probability theory, the expected value of a random variable, intuitively, is the long-run average value of repetitions of the experiment it represents. For example, within an insurance context, the expected loss can be thought of as the average loss incurred by an insurer on a very large portfolio of policies sharing a common loss distribution (similar risk profile). Less roughly, the law of large numbers states that the arithmetic mean of the values almost surely converges to the expected value as the number of repetitions approaches infinity.

The expected value does not exist for random variables having some distributions with large "tails", such as the Cauchy distribution. For random variables such as these, the long-tails of the distribution prevent the sum/integral from converging. That being said, most loss models encountered in insurance implicitly assume finite expected losses.

The expected value is also known as the expectation, mathematical expectation, EV, average, mean value, mean, or first moment.

Expected Values

We give a brief mathematical review of expected values and provide some special formulas that apply for loss variables.

Univariate discrete random variable

Let [math]X[/math] be a discrete random variable taking values [math]x_1,x_2,\ldots[/math] with probabilities [math]p_1,p_2,\ldots[/math] respectively. Then the expected value of this random variable is the infinite sum

[[math]] \operatorname{E}[X] = \sum_{i=1}^\infty x_i\, p_i,[[/math]]

provided that this series converges absolutely (that is, the sum must remain finite if we were to replace all [math]x_i[/math]s with their absolute values). If this series does not converge absolutely, we say that the expected value of [math]X[/math] does not exist.

Univariate continuous random variable

If the probability distribution of [math]X[/math] admits a probability density function [math]f(x)[/math], then the expected value can be computed as

[[math]] \operatorname{E}[X] = \int_{-\infty}^\infty x f(x)\, \mathrm{d}x . [[/math]]

Properties

• The expected value of a constant is equal to the constant itself; i.e., if [math]c[/math] is a constant, then [math]\operatorname{E}[c]=c[/math].

• If [math]X[/math] and [math]Y[/math] are random variables such that [math]X \le Y[/math] almost surely, then [math]\operatorname{E}[X] \le \operatorname{E}[Y][/math].

• The expected value operator (or expectation operator) [math]\operatorname{E}[\cdot][/math] is linear in the sense that

[[math]]\begin{align*} \operatorname{E}[X + c] &= \operatorname{E}[X] + c \\ \operatorname{E}[X + Y] &= \operatorname{E}[X] + \operatorname{E}[Y] \\ \operatorname{E}[aX] &= a \operatorname{E}[X] \end{align*}[[/math]]

Combining the results from previous three equations, we can see that

[[math]]\operatorname{E}[a X + b Y + c] = a \operatorname{E}[X] + b \operatorname{E}[Y] + c\,[[/math]]

for any two random variables [math]X[/math] and [math]Y[/math] and any real numbers [math]a[/math],[math]b[/math] and [math]c[/math].

Layer Cake Representation

When a continuous random variable [math]X[/math] takes only non-negative values, we can use the following formula for computing its expectation (even when the expectation is infinite):

[[math]] \operatorname{E}[X]=\int_0^\infty \operatorname{P}(X \ge x)\; \mathrm{d}x[[/math]]

Similarly, when a random variable takes only values in {0, 1, 2, 3, ...} we can use the following formula for computing its expectation:

[[math]] \operatorname{E}[X]=\sum\limits_{i=1}^\infty \operatorname{P}(X\geq i).[[/math]]

Residual Life Distribution

Suppose [math]X[/math] is a non-negative random variable which can be thought of as representing the lifetime for some entity of interest. A family of residual life distributions can be constructed by considering the conditional distribution of [math]X[/math] given that [math]X[/math] is beyond some level [math]d[/math],i.e., the distribution of lifetime given that death (failure) hasn't yet occurred at time [math]d[/math]:

[[math]] \begin{align} R_d(t) &= \operatorname{P}(X \leq d + t \mid X \gt d) \\ &= \frac{1 - S(t+d)}{S(d)} \end{align} [[/math]]

with [math]S(t)[/math] denoting the survival function for [math]X[/math] representing the probability that [math]X[/math] is greater than [math]t[/math] (the lifetime of [math]X[/math] is greater than [math]t[/math]).

Residual life distributions are relevant for insurance policies with deductibles. Since a claim is made when the loss to the insured is beyond the deductible, the loss to the insurer given that a claim was made is precisely the residual life distribution [math]R_d(t)[/math].

Mean Excess Loss Function

If [math]X[/math] represents loss to the insured with an insurance policy with a deductible [math]d[/math], then the expected loss to the insurer given that a claim was made is the mean excess loss function evaluated at [math]d[/math]:

[[math]] m(d) = \operatorname{E}[X-d \mid X \gt d] = \int_{0}^{\infty}\frac{S(t + d)}{S(d)} \,dt \,. [[/math]]

This function is also called the mean residual life function when [math]X[/math] is a general non-negative random variable. When the distribution of [math]X[/math] has a density say [math]f(x)[/math], then the mean excess loss function equals

[[math]] m(d) = \frac{\int_{d}^{\infty} (x-d) f(x) \, dx}{S(d)} \,. [[/math]]

Effects of Coverage Modifications and Inflation

How is the expected loss on a policy affected by coverage modifications and inflation?

Loss Elimination Ratio

Loss elimination ratios are useful to quantify the impact of coverage modifications on the insurer. Let [math]X[/math] represent the loss to the insured and let [math]L[/math] represent loss to the insurer. If [math]M[/math] represents some coverage modification, then we let [math]L_M[/math] represent the loss to the insurer if [math]M[/math] is in effect. For any modification [math]M[/math], the loss elimination ratio equals

[[math]] \begin{equation}\operatorname{LER}(M) = \frac{\operatorname{E}[L_{M}] - \operatorname{E}[L]}{\operatorname{E}[L]}. \end{equation} [[/math]]

Deductible

If [math]M[/math] denotes the modification corresponding to adding an ordinary deductible [math]d[/math] to the policy, then the expected loss to the insurer can equal any of the following expressions:

[[math]] \operatorname{E}[(L-d)^+] = \operatorname{E}[L] - \operatorname{E}[L \wedge d] = S(d) \cdot m(d) [[/math]]

The loss elimination ratio equals

[[math]] \operatorname{LER}(M) = \frac{\operatorname{E}[L \wedge d]}{\operatorname{E}[L]}\,. [[/math]]

If [math]M[/math] denotes the modification corresponding to adding a franchise deductible [math]d[/math] to the policy, then the expected loss to the insurer equals any of the following expressions:

[[math]] \operatorname{E}[L \cdot \operatorname{1}_{L \gt d}] = \int_{0}^{\infty}S(d +t) \,dt = S(d) \cdot (m(d) + d). [[/math]]

The loss elimination ratio equals any of the following expressions ([math]f(x)[/math] is the density function for the loss):

[[math]] \frac{\operatorname{E}[L \cdot \operatorname{1}_{L \leq u}]}{\operatorname{E}[L]} = \frac{\int_{0}^{u}S(t)\, dt}{\int_{0}^{ \infty}S(t)\,dt} = \frac{\int_{0}^{u}x f(x) dx}{\int_{0}^{\infty}x f(x) dx}. [[/math]]

Limit

If [math]M[/math] denotes the modification corresponding to adding a limit [math]u[/math] to the policy, then the expected loss equals any of the following expressions:

[[math]] \operatorname{E}[L \wedge u] = \int_{0}^{u}S(t)\, dt + S(u)\cdot u = \int_{0}^{u}x f(x) \, dx + S(u)\cdot u. [[/math]]

The loss elimination ratio equals any of the following expressions:

[[math]] \frac{\operatorname{E}[L] - \operatorname{E}[L \wedge u]}{\operatorname{E}[L]} = \frac{\operatorname{E}[(L-u)^+] }{\operatorname{E}[L]} = \frac{S(u) \cdot m(u)}{\operatorname{E}[L]}. [[/math]]

Coinsurance

If [math]M[/math] denotes the modification corresponding to adding a coinsurance factor [math]\alpha[/math] to the policy, then

[[math]] \operatorname{E}[L_M] = \alpha \cdot \operatorname{E}[L],\ \ \operatorname{LER}(M) =\alpha. [[/math]]

Inflation

The loss to the insured may be subject to inflation from one period to another. For instance, if the loss [math]X[/math] to the insured is inflated by [math]r[/math], how does it affect the loss variable for the insurer?

If the policy has an ordinary deductible [math]d[/math], then the loss to the insurer equals

[[math]] L = (1+r)\left[X-d/(1+r)\right]^+ [[/math]]

and the expected loss equals

[[math]] \operatorname{E}[L] = (1+r) \cdot \left(\operatorname{E}[X] - \operatorname{E}[X \wedge \frac{d}{1+r} ] \right). [[/math]]

If the policy has a limit [math]u[/math], then the loss to the insurer equals

[[math]] L = (1+r)\left[X\wedge\frac{u}{1+r}\right] [[/math]]

and the expected loss equals

[[math]] \operatorname{E}[L] = (1+r) \cdot \operatorname{E}\left[X \wedge \frac{u}{1+r} \right]. [[/math]]

Finally, suppose the policy has a regular deductible [math]d[/math], a limit [math]u[/math] (greater than [math]d[/math]) and a coinsurance factor [math]\alpha[/math], then the loss to the insurer equals

[[math]] L = \alpha (1 + r)\left[ X \wedge \frac{u}{1+r} - X \wedge \frac{d}{1+r} \right]. [[/math]]

References

• Wikipedia contributors. "Insurance". Wikipedia. Wikipedia. Retrieved 7 June 2019.
• Wikipedia contributors. "Deductible". Wikipedia. Wikipedia. Retrieved 7 June 2019.
• Wikipedia contributors. "Co-insurance". Wikipedia. Wikipedia. Retrieved 7 June 2019.
• Wikipedia contributors. "Expected Value". Wikipedia. Wikipedia. Retrieved 7 June 2019.