Deductibles and Policy Limits: Insurance Applications of Probability

Ordinary Deductibles

An ordinary (or per-loss) deductible d means the insurer pays (X - d) when the loss X exceeds d, and nothing when X <= d. The payment random variable is Y = max(X - d, 0), commonly written (X - d)+. This is the most fundamental insurance modification tested on Exam P.

The expected insurer payment can be computed in several equivalent ways. The direct approach is E[Y] = integral from d to infinity of (x - d) * f(x) dx. An alternative formula uses the limited expected value: E[Y] = E[X] - E[min(X, d)], where E[min(X, d)] = integral from 0 to d of x * f(x) dx + d * (1 - F(d)). The survival function approach gives E[Y] = integral from d to infinity of S(x) dx, where S(x) = 1 - F(x). Choose whichever form makes the integration easiest for the given distribution.

For the exponential distribution with mean theta, the memoryless property yields the clean result E[(X - d)+] = theta * e^(-d/theta). This follows because P(X > d) = e^(-d/theta) and E[X - d | X > d] = theta (by the memoryless property), so E[(X - d)+] = P(X > d) * E[X - d | X > d] = theta * e^(-d/theta). This formula is one of the most frequently tested results on Exam P.

The Limited Expected Value

The limited expected value E[min(X, d)] represents the expected amount of loss retained by the policyholder (up to the deductible). It can be computed as E[min(X, d)] = integral from 0 to d of S(x) dx, where S(x) = 1 - F(x). This formula uses the survival function method and avoids splitting the integral at d.

For common distributions on Exam P:

• Exponential(theta): E[min(X, d)] = theta * (1 - e^(-d/theta))
• Uniform(0, b): E[min(X, d)] = d - d^2/(2b) for d <= b
• Pareto(alpha, theta): E[min(X, d)] = theta/(alpha-1) * [1 - (theta/(theta+d))^(alpha-1)] for alpha > 1

Notice that E[min(X, d)] + E[(X - d)+] = E[X] always holds. This decomposition splits the total expected loss into the part borne by the policyholder and the part borne by the insurer. On Exam P, if you can compute one piece, you can find the other by subtraction.

Policy Limits

A policy limit u caps the insurer's payment at u. Without a deductible, the payment is min(X, u), and E[min(X, u)] = integral from 0 to u of S(x) dx. This is the same limited expected value formula, now applied to the insurer's perspective rather than the policyholder's.

With both a deductible d and a maximum payment u, the insurer's payment per loss is min(max(X - d, 0), u). This can be rewritten as min(X, d + u) - min(X, d), which expresses the payment as a difference of two limited expected values. The expected payment is therefore E[min(X, d + u)] - E[min(X, d)].

Example: If X ~ Exponential(1000), d = 200, and the maximum payment is u = 5000, then the insurer pays min(X - 200, 5000) when X > 200 and 0 otherwise. The expected payment is E[min(X, 5200)] - E[min(X, 200)] = 1000(1 - e^(-5.2)) - 1000(1 - e^(-0.2)) = 1000(e^(-0.2) - e^(-5.2)) = 1000(0.8187 - 0.0055) = 813.2.

Per-Payment vs. Per-Loss Variables

Exam P distinguishes between two types of random variables when a deductible is present. The per-loss variable Y^L = max(X - d, 0) assigns the value 0 to losses that fall below the deductible. The per-payment variable Y^P = X - d, defined only when X > d, conditions on a payment actually being made.

The relationship is E[Y^L] = P(X > d) * E[Y^P]. Since E[Y^P] = E[X - d | X > d], you can compute one from the other using the probability of exceeding the deductible.

For the exponential distribution with mean theta, E[Y^P] = E[X - d | X > d] = theta (by the memoryless property, the excess over any threshold has the same distribution as the original). So E[Y^L] = e^(-d/theta) * theta, confirming the earlier formula.

For non-memoryless distributions, E[Y^P] = E[(X - d)+] / P(X > d) = E[(X - d)+] / S(d). This per-payment expected value is the mean excess function e(d), which is an important concept in loss modeling.

Loss Elimination Ratio

The loss elimination ratio (LER) measures the proportion of expected losses eliminated by a deductible: LER = E[min(X, d)] / E[X]. It represents the fraction of total expected losses borne by the policyholder. Equivalently, 1 - LER = E[(X - d)+] / E[X] is the fraction retained by the insurer.

For the exponential distribution: LER = (1 - e^(-d/theta)) / 1 = 1 - e^(-d/theta). So a deductible of d = theta * ln(2) eliminates exactly half of the expected losses.

On Exam P, you may be asked to find the deductible that eliminates a given fraction of losses. This requires solving E[min(X, d)] / E[X] = target for d. For the exponential, this is straightforward. For other distributions, it may require solving a more complex equation.

Coinsurance and Inflation

A coinsurance factor alpha (0 < alpha < 1) means the insurer pays alpha * max(X - d, 0). The expected payment is simply alpha * E[(X - d)+]. This linear scaling is easy to handle on Exam P.

Loss inflation increases all losses by a factor (1 + r). If the original loss is X, the inflated loss is (1 + r) * X. With a deductible d, the payment on the inflated loss is max((1 + r)*X - d, 0) = (1 + r) * max(X - d/(1 + r), 0). The deductible effectively decreases to d/(1 + r) when measured in terms of the original loss distribution. The expected payment becomes (1 + r) * E[(X - d/(1+r))+].

For the exponential with mean theta: E[inflated payment] = (1 + r) * theta * e^(-d/((1+r)*theta)). Compared to the original E[(X - d)+] = theta * e^(-d/theta), the inflated expected payment grows faster than the inflation factor (1 + r) because the effective deductible has decreased. This phenomenon is called "leveraging" and is important in insurance pricing.

Common Pitfalls

The most common mistake is confusing per-loss and per-payment variables. The per-loss variable includes the zeros (losses below the deductible), while the per-payment variable conditions on a payment being made. Their expected values differ by the factor P(X > d).

Another error is using the wrong integration limits. When computing E[(X - d)+], integrate from d to the upper limit of the support (or infinity), not from 0. When computing E[min(X, d)], integrate from 0 to d using the survival function, or split the integral at d.

Students sometimes forget that the limited expected value formula E[min(X, d)] = integral from 0 to d of S(x) dx requires X >= 0. If the loss distribution can take negative values (unusual but possible in some contexts), the formula needs modification.

Study Tips

For Exam P, memorize the survival function formulas for E[min(X, d)] and E[(X - d)+] for the exponential, Pareto, and uniform distributions. Practice converting between per-loss and per-payment perspectives. Know the loss elimination ratio formula and how to solve for the deductible. Work through problems that combine deductibles with policy limits, using the difference of limited expected values. These insurance application problems appear in three to five questions per exam and are considered moderate to hard difficulty. Building speed with the formulas is essential, as the calculations can be lengthy if you do not use the most efficient approach.