Deductibles And ILF Loss Bases

Why the Loss Basis Matters

An increased limits factor compares expected losses at different limits. With a deductible, the expected amount paid by the insurer is not the same as the expected ground-up loss capped at a limit. The deductible changes which part of each claim is insured, so it changes both the limited expected loss and the ILF. A calculation is meaningful only after the loss basis and the meaning of “limit” have been specified.

For this tutorial, assume $X$ is a nonnegative ground-up loss for one claim, $d$ is a per-claim deductible, and there are no coinsurance provisions, expenses, or aggregate limits. The deductible is applied separately to each claim. We will distinguish a policy limit stated as the maximum insurer payment from a policy limit stated as the maximum total loss covered, inclusive of the deductible. These conventions can give different payments for the same stated number.

Definition: The ground-up loss is $X$, before applying policy terms. With deductible $d$ and maximum insurer payment $u$, the insurer’s payment per loss is $Y^L=\min((X-d)_+,u)$, where $(z)_+=\max(z,0)$. Here $u$ is a limit in excess of the deductible: the total loss at which the insurer’s payment reaches its cap is $d+u$. If instead the policy limit $L$ is inclusive of the deductible, the payment is $Y^L=\min((X-d)_+,L-d)$, for $L\ge d$.

The ground-up loss limited at an inclusive policy limit $L$ is $\min(X,L)$. That is not the insurer’s payment under a deductible policy. The latter is $\min((X-d)_+,L-d)$: no payment is made below the deductible, and the insurer’s maximum payment is $L-d$. When $L<d$, there can be no positive payment under this convention.

Quantity	Definition	What it represents
Ground-up limited loss	$\min(X,L)$	Loss after an inclusive cap, before subtracting a deductible
Payment, limit above deductible	$\min((X-d)_+,u)$	Insurer payment capped at $u$; maximum covered total loss is $d+u$
Payment, inclusive limit	$\min((X-d)_+,L-d)$	Insurer payment when the total covered loss is capped at $L$

Expected Payment and the Survival Function

For a deductible $d$ and an insurer-payment limit $u$, the limited expected payment is $E[Y^L]$. Define the survival function by $S_X(x)=P(X>x)$. For nonnegative losses, the expected payment can be written as an integral over the layer in which the insurer pays:

E[Y^L] = \int_0^u P(X>d+t)\,dt = \int_d^{d+u} S_X(x)\,dx.

The lower endpoint is the deductible attachment point. The upper endpoint is $d+u$, because the insurer pays at most $u$ above the deductible. If the policy limit $L$ is inclusive of the deductible, substitute $u=L-d$; then the upper endpoint is $L$.

Why the Formula Holds

Set $W=(X-d)_+$, the amount of loss above the deductible. For any nonnegative value of $W$, the amount capped at $u$ can be represented as the length of the interval from zero up to $\min(W,u)$. Equivalently, it is the integral of an indicator:

\min(W,u)=\int_0^u \mathbf{1}_{\{W>t\}}\,dt.

Taking expectations and interchanging the expectation and integral gives an integral of the probability that $W>t$. Since $W>t$ holds exactly when $X>d+t$, this produces the first expression. Changing variables to $x=d+t$ produces the survival-function expression. The argument applies to nonnegative severities; it also works when the severity distribution has atoms, since changing a probability at a single integration endpoint does not change the integral.

Worked Example: Discrete Losses With an Inclusive Policy Limit

A hypothetical insurer models one claim with the following ground-up severity distribution. Each limit below is the maximum total covered loss, inclusive of a deductible of $500. Calculate the expected insurer payment at each limit and the ILF for the higher limit relative to the lower limit. Also calculate the ground-up limited expected losses to show why the loss basis matters.

Ground-up loss $X$	Probability
$0	0.20
$500	0.20
$1,000	0.25
$2,000	0.20
$4,000	0.15

Method. At an inclusive limit $L$, payment is $\min((X-500)_+,L-500)$. For $L=$1,500$, the insurer’s payment for losses $0, 500, 1,000, 2,000,$ and $4,000$ is respectively $0, 0, 500, 1,000,$ and $1,000$. The expected payment is:

E[Y^L\mid L=1{,}500] =(0.20)(0)+(0.20)(0)+(0.25)(500)+(0.20)(1{,}000)+(0.15)(1{,}000) =475.

For $L=$3,000$, the insurer’s cap is $L-d=$2,500$. The payments at the same five loss values are $0, 0, 500, 1,500,$ and $2,500$. Thus:

E[Y^L\mid L=3{,}000] =(0.20)(0)+(0.20)(0)+(0.25)(500)+(0.20)(1{,}500)+(0.15)(2{,}500) =800.

The deductible-adjusted ILF is $800/475=1.6842105$, or 1.6842 to four decimal places. As a check, increasing the limit has increased expected payment from $475 to $800.

For comparison, the ground-up limited losses at limits $1,500 and $3,000 have expected values $(0.20)(500)+(0.25)(1,000)+(0.20)(1,500)+(0.15)(1,500)=$875$ and $(0.20)(500)+(0.25)(1,000)+(0.20)(2,000)+(0.15)(3,000)=$1,200$. Their ratio is $1,200/875=1.3714$ to four decimal places. This is a different ILF: it does not represent insurer payments after the $500 deductible. The common mistake avoided here is using $\min(X,L)$ as though it were the payment under the deductible contract.

Worked Example: Exponential Severity With a Limit Above the Deductible

Suppose a hypothetical ground-up claim severity has an exponential distribution with mean $1,000. Equivalently, its rate parameter is $1/1{,}000$ per dollar, and $S_X(x)=e^{-x/1{,}000}$ for $x\ge0$. A $500 per-claim deductible applies. Calculate expected payments at insurer-payment limits of $500 and $1,500 above the deductible, then calculate the ILF.

Method. The limits are stated above the deductible, so the upper integration endpoints are $d+u=$1,000$ and $d+u=$2,000$. Using the survival-function formula:

E[Y^L\mid u] =\int_{500}^{500+u}e^{-x/1{,}000}\,dx =1{,}000\left(e^{-0.5}-e^{-(500+u)/1{,}000}\right).

For $u=$500$, the upper endpoint is $1,000. Using $e^{-0.5}=0.6065307$ and $e^{-1}=0.3678794$:

E[Y^L\mid u=500] =1{,}000(0.6065307-0.3678794) =$238.65.

For $u=$1,500$, the upper endpoint is $2,000. Using $e^{-2}=0.1353353$:

E[Y^L\mid u=1{,}500] =1{,}000(0.6065307-0.1353353) =$471.20.

The ILF is $471.1954/238.6512=1.9744$ to four decimal places. The higher expected payment is consistent with the higher cap. The common mistake avoided is treating a $1,500 limit above the deductible as an inclusive $1,500 policy limit; the corresponding total-loss endpoint is $500+$1,500=$2,000.

Calculating an ILF on a Consistent Basis

An ILF is a ratio of limited expected losses at two limits. With a fixed deductible and insurer-payment limits measured above that deductible, define the deductible-adjusted ILF relative to base limit $u_0$ by:

\operatorname{ILF}_d(u;u_0) = \frac{E[\min((X-d)_+,u)]} {E[\min((X-d)_+,u_0)]}.

The denominator must be positive. The same severity distribution, deductible, payment basis, and limit convention must be used in numerator and denominator. If limits are inclusive of the deductible, use $u=L-d$ for each inclusive limit $L$. Using an inclusive limit in one part of the ratio and a limit above the deductible in the other produces a ratio that compares different contracts.

If the same expected claim frequency applies at both limits, the frequency cancels in the ratio. For example, if the expected number of claims is $E[N]$ and claim severities have the same distribution at both limits, expected annual payment is $E[N]E[Y^L]$. The annual-loss ratio then equals the per-claim expected-payment ratio. This cancellation does not justify changing the deductible or severity basis between the two calculations.

Worked Example: Annual Losses and Two Inclusive Limits

A hypothetical portfolio has a Poisson claim count $N$ with mean 3 per year. Claim severities are independent of the count and identically distributed: $X=$500$ with probability 0.40, $X=$1,500$ with probability 0.35, and $X=$3,500$ with probability 0.25. Each claim has a $500 deductible. Compare annual expected payments at policy limits of $2,500 and $4,500, each inclusive of the deductible, and calculate the ILF.

Method. For the $2,500 inclusive limit, the insurer’s payment cap is $2,500-$500=$2,000. The three claim payments are $0$, $1,000$, and $2,000$, so expected payment per claim is:

E[Y^L_{2{,}500}] =(0.40)(0)+(0.35)(1{,}000)+(0.25)(2{,}000) =$850.

For the $4,500 inclusive limit, the payment cap is $4,000. The claim payments are $0$, $1,000$, and $3,000$, giving:

E[Y^L_{4{,}500}] =(0.40)(0)+(0.35)(1{,}000)+(0.25)(3{,}000) =$1{,}100.

Since $E[N]=3$, the annual expected payments are $3($850)=$2,550$ and $3($1,100)=$3,300$. The annual ILF is $3,300/2,550=1.2941$ to four decimal places, equal to the per-claim ratio $1,100/850$. The check is that both annual expected payments use the same expected claim count. The common mistake avoided is applying the deductible once to annual aggregate losses rather than separately to each claim.

Common Mistakes and Practical Checks

Deductible-adjusted ILFs are most often miscalculated because the policy terms are not translated into a payment formula before the arithmetic begins. Use the following checks:

Check where the limit applies. A $2,000 limit above a $500 deductible allows insurer payment up to $2,000 and reaches that cap at a $2,500 ground-up loss. A $2,000 inclusive limit allows payment of at most $1,500.
Do not subtract the deductible from ground-up limited loss without checking the payment floor. The correct inclusive-limit payment is $\min((X-d)_+,L-d)$, not simply $\min(X,L)-d$, which can be negative when $X<d$.
Keep the basis consistent in the ILF ratio. A ground-up limited expected loss in the numerator and a deductible-adjusted expected payment in the denominator do not form a deductible-adjusted ILF.
Check the direction of the result. With the same deductible and severity distribution, increasing a per-claim limit cannot decrease the payment on any claim. Therefore expected payment should not decrease.
State the unit of the limit. Record explicitly whether the limit is total covered loss or insurer payment above the deductible. Do not infer the convention from a number alone.

A correctly specified deductible-based ILF describes the change in expected insurer payments under the stated terms. It is not interchangeable with a ground-up severity ILF. Keeping those bases distinct is essential in calculations aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus and in practical comparisons of policy limits.

Check Your Understanding

For each problem, identify whether the stated limit is inclusive of the deductible or above it before calculating.

A hypothetical severity distribution is $X=$1,000$ with probability 0.50, $X=$3,000$ with probability 0.30, and $X=$5,000$ with probability 0.20. With a $1,000 deductible and a $4,000 inclusive limit, calculate expected payment per claim. (Answer: $1,200)
Use the same severity distribution and deductible as in Question 1. Calculate expected payment at $3,000 and $5,000 inclusive limits and the ILF for the higher limit over the lower limit. (Answer: $1,000 and $1,400; ILF = 1.4000)
Ground-up severity is exponential with mean $2,000. With a $400 deductible and an insurer-payment limit of $600 above the deductible, calculate expected payment per claim to the nearest cent. (Answer: $424.40)
A hypothetical claim has severity $1,000, $2,000, or $5,000 with probabilities 0.20, 0.50, and 0.30, respectively. The deductible is $1,000. Calculate expected payment at a $4,000 inclusive policy limit. (Answer: $1,400)
Severity is $0, $1,000, or $3,000 with probabilities 0.20, 0.50, and 0.30. With a $500 deductible, calculate the deductible-adjusted ILF from a $2,000 inclusive limit to a $4,000 inclusive limit. Then calculate the corresponding ground-up limited expected losses and their ratio. (Answer: Deductible-adjusted expected payments are $700 and $1,000, so ILF = 1.4286; ground-up limited expected losses are $1,100 and $1,400, so ratio = 1.2727)

Quantity	Definition	What it represents
Ground-up limited loss	\(\min(X,L)\)	Loss after an inclusive cap, before subtracting a deductible
Payment, limit above deductible	\(\min((X-d)_+,u)\)	Insurer payment capped at \(u\); maximum covered total loss is \(d+u\)
Payment, inclusive limit	\(\min((X-d)_+,L-d)\)	Insurer payment when the total covered loss is capped at \(L\)

What You'll Learn