This theory post will cover how to derive the Expected Cost Per Loss and the Expected Cost Per Payment using a franchise deductible.

For clarity, we’ll use a subscript of $F$ if we want to talk about a franchise deductible and a subscript of $O$ if we want to work with an ordinary deductible.

Recall the per-loss variable $Y^L$ for losses with a franchise deductible of $d$:

$$

Y_F^L = \left\{

\begin{array}{l l}

0 & X \leq d \\

X & X > d

\end{array}

\right. $$

In words, this describes the payout structure to be 0 if the loss is less than or equal to $d$ and the full amount of the loss if it is greater than $d$.

Now what happens when we throw in a maximum covered loss $u$, annual rate of inflation $r$, and a coinsurance factor $\alpha$? (Review Chapter 5 if any of these terms don’t ring a bell.) We now have:

$$

Y_F^L = \left\{

\begin{array}{l l}

0, & X < \frac{d}{1+r}\\
\alpha(1+r)X & \frac{d}{1+r} \leq X < \frac{u}{1+r} \\
\alpha u & X \geq \frac{u}{1+r}
\end{array}
\right.$$
In the text, we give the per-loss variable for an ordinary deductible:
$$
Y_O^L = \left\{
\begin{array}{l l}
0, & X < \frac{d}{1+r}\\
\alpha[(1+r)X - d]& \frac{d}{1+r} \leq X < \frac{u}{1+r} \\
\alpha(u-d) & X \geq \frac{u}{1+r}
\end{array}
\right.$$
Note the similarities. In the first interval, when losses are below the inflation-adjusted deductible of $\frac{d}{1+r}$, $Y_O^L$ is identical to $Y_F^L$. When losses are above the deductible, $Y_F^L=Y_O^L+\alpha d$. We can mathematically express this observation as follows:
$$Y_F^L = Y_O^L + \alpha d \cdot \mathbb 1 \{X \geq \frac{d}{1+r}\}$$
Here, the $``\mathbb 1 \{X \geq \frac{d}{1+r}\}''$ is what we call an indicator function, evaluating to 0 when the condition inside the curly braces is false, and evaluating to 1 when the condition is true. (Skip over to Chapter 14 to get the scoop on indicator functions.) The take-away point is that the expectation of this indicator function is:
$$E(\mathbb 1 \{X \geq \frac{d}{1+r}\}) = Pr\left(X \geq \frac{d}{1+r}\right) = S_X\left(\frac{d}{1+r}\right)$$
Now, let's try to compute the expected cost per-loss for a franchise deductible:
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