# 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 $d$ denote the deductible and let $X$ the loss incurred by the insured.

#### Ordinary Deductible

When the deductible is ordinary, the claim amount is:

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

### 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:

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

## Policy Limits

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

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

## 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 $d$, a limit $l$ and a coinsurance factor $f$, then the claim amount is:

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

## 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 $X$ represent the loss to the insured and let $L$ represent loss to the insurer. If $M$ represents some coverage modification, then we let $L_M$ represent the loss to the insurer if $M$ is in effect. For any modification $M$, the loss elimination ratio equals

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

### Deductible

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

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

The loss elimination ratio equals

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

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

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

The loss elimination ratio equals any of the following expressions ($f(x)$ is the density function for the loss):

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

### Limit

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

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

The loss elimination ratio equals any of the following expressions:

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

### Coinsurance

If $M$ denotes the modification corresponding to adding a coinsurance factor $\alpha$ to the policy, then

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

### Inflation

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

If the policy has an ordinary deductible $d$, then the loss to the insurer equals

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

and the expected loss equals

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

If the policy has a limit $u$, then the loss to the insurer equals

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

and the expected loss equals

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

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

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