32  Insurance Related Concepts

32.1 Insurance with Deductible

The amount paid on an insurance with loss random variable \(X\) and ordinary deductible \(d\) is written as \((X-d)_+\), where \[(X-d)_+ = \begin{cases}X - d & \text{if } X > d \\ 0 & \text{otherwise} \end{cases}\]

Suppose the amount paid on an insurance with loss random variable \(X\) and franchise deductible \(d\) is \(Y\), then \[Y = \begin{cases}X &\text{if } X > d \\ 0 &\text{otherwise } \end{cases}\]

Example: The amount of a single loss \(X\) for an insurance policy is exponential, with density function \[f(x) = .002e^{-.002x}\] for \(x \ge 0\). The expected value of a single loss is \[\text E[X] = \frac{1}{.002} = 500\]Suppose the insurance in has an (ordinary) deductible of \(\$100\) for each loss. Find the expected value of a single claim.

Solution: The expected amount of a single claim is \[\begin{align}\text E[(X-100)_+] &= \int_0^\infty (x-100)_+ f(x)dx \\ &= \int_{100}^\infty (x-100)(0.002e^{-0.002}) \\ &= -e^{-0.002x}(x+400)\Big|_{100}^\infty = 500e^{-.2}\end{align}\]

32.2 Insurance with Policy Limit

The amount paid on the insurance with loss random variable \(X\) and policy limit (cap) \(c\) is written as \((X\wedge c)\), where \[X\wedge c = \begin{cases}X & \text{if } X \le c \\ c & \text{otherwise} \end{cases}\]

Example: The amount of a single loss \(X\) for an insurance policy is exponential, with density function \(f(x) = .002e^{-.002x}\). Suppose the insurance in has a deductible of \(\$100\) per claim and the restriction that the largest amount paid on any claim will be \(\$700\). Find the expected value of a single claim.

Solution: Let the random variable \(Y\) be the amount paid by the insurer for a single claim. Then \[Y = \begin{cases}0 & \text{if } X < 100 \\ X - 100 & \text{if } 100 \le X \le 700 \\ 700 & \text{if } 700 < X \end{cases}\] The expected amount of a single claim is \[\begin{align}\text E[Y] &= \int_{100}^{800} (x-100)(0.002e^{-0.002}) + \int_{800}^{\infty} 700(0.002e^{-0.002}) \\ &= (x+400)(-e^{-0.002x})\Big|_{100}^{800} + 700(-e^{-0.002x})\Big|_{800}^{\infty} \\ &= 500e^{-0.2} - 500e^{-1.6}\end{align}\]

Proposition: * For an ordinary deductible, the expected cost per loss is \[\text E[X] − \text E[X\wedge d]\] * For a franchise deductible, the expected cost per loss is \[\text E[X] − \text E[X\wedge d]+d[1-F(d)]\] ## Loss Elimination Ratio and the Effect of Inflation for Ordinary Deductibles ### Loss Elimination Ratio The loss elimination ratio is the ratio of the decrease in the expected payment with an ordinary deductible to the expected payment without the deductible.

With the deductible, the expected payment is \(\text E[X] − \text E[X\wedge d]\). Therefore, the loss elimination ratio is \[\frac{\text E[X\wedge d]}{\text E[X]}\]

32.3 Effect of Inflation

Inflation increases costs, but when there is a deductible, the effect of inflation is magnified. First, some events that formerly produced losses below the deductible will now lead to payments. Second, the relative effect of inflation is magnified because the deductible is subtracted after inflation.

For example, suppose that an event formerly produced a loss of 600. With a 500 deductible, the payment is 100. Inflation at 10% will increase the loss to 660 and the payment to 160, a 60% increase in the cost to the insurer.

After inflation, losses are given by the random variable \((1 + r)X\). Then, with an ordinary deductible of \(d\), the amount paid per loss is: \[Y = \begin{cases}0 &\text{if }X < \frac{d}{1+r} \\ (1+r)X-d &\text{if }X \ge \frac{d}{1+r}\end{cases}\]

32.4 Coinsurance

For coinsurance, the insurance company pays a proportion, \(\alpha\), of the loss and the policyholder pays the remaining fraction. If coinsurance is the only modification, this changes the loss variable \(X\) to the payment variable, \(Y=\alpha X\). When all four items covered in this chapter are present (ordinary deductible, limit, coinsurance, and inflation), we create the following per-loss random variable: \[Y = \begin{cases}0 &\text{if }X < \frac{d}{1+r} \\ \alpha[(1+r)X-d] &\text{if }\frac{d}{1+r} \le X \le \frac{c}{1+r} \\ \alpha(c-d) &X \ge \frac{c}{1+r} \end{cases}\]