# Tag Archives: Deductible

## Other Coverage Modifications

Coinsurance $\alpha$ is the fraction of losses covered by the policy.  For example, $\alpha = 0.8$ means if a loss is incurred, 80% will be paid by the insurance company.  A claims limit $u$ is the maximum amount that will be paid.  The order in which coinsurance, claims limits, and deductibles is applied to a loss is important and will be specified by the problem.  The expected payment per loss when all three are present in a policy is given by

$E\left[Y\right] = \alpha \left[E\left[X\wedge u\right] - E\left[X \wedge d\right]\right]$

where $Y$ is the payment variable and $X$ is the original loss variable.  The second moment is given by

$E\left[Y^2\right] = \alpha^2\left(E\left[(X\wedge u)^2\right] - E\left[(X \wedge d)^2\right]-2d\left(E\left[X \wedge u\right]-E\left[X \wedge d\right]\right)\right)$

The second moment can be used to find the variance of payment per loss.  If inflation $r$ is present, multiply the second moment by $(1+r)^2$ and divide $u$ and $d$ by $(1+r)$.   For payment per payments, divide the expected values by $P(X>d)$ or $1-F(d)$.

## The Loss Elimination Ratio

If you impose a deductible $d$ on an insurance policy that you’ve written, what fraction of expected losses do you eliminate from your expected liability?  This is measured by the Loss Elimination Ratio $LER(d)$.

$\displaystyle LER(d) = \frac{E\left[X \wedge d\right]}{E\left[X\right]}$

Definitions:

1. Ordinary deductible $d$— The payment made by the writer of the policy is the loss $X$ minus the deductible $d$.  If the loss is less than $d$, then nothing is paid.
2. Franchise deductible $d_f$—  The payment made by the writer of the policy is the complete amount of the loss $X$ if $X$ is greater than $d_f$.
A common type of question considers what happens to LER if an inflation rate $r$ increases the amount of all losses, but the deductible remains unadjusted.  Let $X$ be the loss variable.  Then $Y=(1+r)X$ is the inflation adjusted loss variable.  If losses $Y$ are subject to deductible $d$, then
$\begin{array}{rll} \displaystyle LER_Y(d) &=& \frac{E\left[(1+r)X\wedge d\right]}{E\left[(1+r)X\right]} \\ \\ \displaystyle &=&\frac{(1+r)E\left[X\wedge \frac{d}{1+r}\right]}{(1+r)E\left[X\right]} \\ \\ &=& \frac{E\left[X \wedge \frac{d}{1+r}\right]}{E\left[X\right]}\end{array}$
Memorize:
$\displaystyle E\left[X \wedge d\right] = \int_0^d{x f(x) dx} + d\left(1-F(x)\right)$

## Expected Values for Insurance

Before I begin, please note: I hated this chapter.  If there are any errors please let me know asap!

A deductible $d$ is an amount that is subtracted from an insurance claim.  If you have a $500 deductible on your car insurance, your insurance company will only pay damages incurred beyond$500.  We are interested in the following random variables: $(X - d)_+$ and $(X\wedge d)$.

Definitions:

1. Payment per Loss: $(X-d)_+ = \left\{ \begin{array}{ll} X-d &\mbox{ if } X>d \\ 0 &\mbox{ otherwise} \end{array} \right.$
2. Limited Payment per Loss:  $(X\wedge d) = \left\{ \begin{array}{ll} d &\mbox{ if } X>d \\ X &\mbox{ if } 0
Expected Values:
1. $\begin{array}{rll} E[(X-d)_+] &=& \displaystyle \int_{d}^{\infty}{(x-d)f(x)dx} \\ \\ &=& \displaystyle \int_{d}^{\infty}{S(x)dx} \end{array}$

2. $\begin{array}{rll} E[(X\wedge d)] &=& \displaystyle \int_{0}^{d}{xf(x)dx +dS(x)} \\ \\ &=& \displaystyle \int_{0}^{d}{S(x)dx} \end{array}$
We may also be interested in the payment per loss, given payment is incurred (payment per payment) $X-d|X>d$.
By definition:
$E[X-d|X>d] = \displaystyle \frac{E[(X-d)_+]}{P(X>d)}$
Since actuaries like to make things more complicated than they really are, we have special names for this expected value.  It is denoted by $e_X(d)$ and is called mean excess loss in P&C insurance and $\displaystyle {\mathop{e}\limits^{\circ}}_d$ is called mean residual life in life insurance.  Weishaus simplifies the notation by using the P&C notation without the random variable subscript.  I’ll use the same.
Memorize!
1. For an exponential distribution,
$e(d) = \theta$
2. For a Pareto distribution,
$e(d) = \displaystyle \frac{\theta +d}{\alpha - 1}$
3. For a single parameter Pareto distribution,
$e(d) = \displaystyle \frac{d}{\alpha - 1}$
Useful Relationships:
1. $\begin{array}{rll} E[X] &=& E[X\wedge d] + E[(X-d)_+] \\ &=& E[X\wedge d] + e(d)[1-F(d)] \end{array}$
Actuary Speak (important for problem comprehension):
1. The random variable $(X-d)_+$ is said to be shifted by $d$ and censored.
2. $e(d)$ is called mean excess loss or mean residual life.
3. The random variable $X\wedge d$ can be called limited expected value, payment per loss with claims limit, and amount not paid due to deductible.  $d$ can be called a claims limit or deductible depending on how it is used in the problem.
4. If data is given for $X$ with observed values and number of observations or probabilities, the data is called the empirical distribution.  Sometimes empirical distributions may be given for a problem, but you are still asked to assume an parametric distribution for $X$.