Building a Deductible Factor Schedule

Why Build a Deductible Factor Schedule?

A deductible factor schedule translates expected insurer payments at several deductible amounts into a common set of relative values. The schedule can then support rating calculations that use a chosen deductible as the reference. Its meaning depends on two choices that must be explicit: the payment basis being measured and the deductible selected as the base.

This tutorial concerns ordinary per-claim deductibles and expected insurer payments without an upper policy limit. A schedule built on that basis is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus. The factors describe expected loss payments; they are not, by themselves, premium factors incorporating expenses or profit.

Definition: Let $X$ be a nonnegative ground-up loss and let $Y_d^L=(X-d)_+$ be the insurer's payment under an ordinary per-claim deductible $d$, where $(z)_+=\max(z,0)$. Write $L(d)=E[Y_d^L]$ for the expected payment per ground-up claim. For a selected base deductible $b$ with $L(b)>0$, the relative deductible factor at $d$ is $$ F(d\mid b)=\frac{L(d)}{L(b)}. $$ Thus $F(b\mid b)=1$. The factor is relative to the stated base, not an absolute measure of claim cost.

For a fixed ground-up severity distribution, increasing the deductible cannot increase the insurer's payment on any individual loss. Therefore, expected payments and factors should be nonincreasing as the deductible rises. If claim frequency is common across deductible options, multiplying every expected payment per claim by the same expected claim count does not change the relative factors. If frequencies differ by option or class, calculate the relevant aggregate expected payments first.

Why Expected Payments Have a Survival-Integral Form

A useful calculation for constructing a schedule is the survival-integral expression for an ordinary deductible payment. It follows by viewing a payment as the accumulated amount of loss above the deductible. For a fixed loss value $x$, the amount $(x-d)_+$ equals the length of the interval from $d$ to $x$ when $x>d$, and is zero otherwise. Equivalently, it is the integral of the indicator that the loss exceeds each threshold above $d$.

(X-d)_+=\int_d^\infty \mathbf{1}_{\{X>t\}}\,dt \quad\Longrightarrow\quad L(d)=E[(X-d)_+]=\int_d^\infty P(X>t)\,dt =\int_d^\infty S_X(t)\,dt.

The expectation and integral can be interchanged because the integrand is nonnegative. The result applies to nonnegative losses with a finite expected payment; it does not assume a particular continuous severity distribution. For an empirical distribution, the same expected payment can be calculated directly by averaging the deductible payments for the observed losses.

For an exponential severity distribution with mean $\theta>0$, $S_X(t)=e^{-t/\theta}$ for $t\geq 0$. Integrating gives

L(d)=\int_d^\infty e^{-t/\theta}\,dt =\theta e^{-d/\theta}.

This formula is for an unlimited ordinary-deductible payment. A policy limit, a different deductible contract, or a different claim-payment basis changes the payment calculation and must not be silently folded into this schedule.

A Reliable Construction Process

Fix the basis.
Specify the loss population, payment function, deductible options, and whether values are per claim or aggregate. Confirm that the compared options use compatible terms.

Calculate expected payments.
Use the specified severity model or average the payments in an empirical distribution. If the schedule is for a portfolio, weight classes by their expected claim counts before forming ratios.

Normalize to the base.
Divide every expected payment by the expected payment at the selected base deductible. Keep the base value at exactly 1 before display rounding.

Check the schedule.
Verify the base value, the direction of the factors, and the ratio relationships among deductible levels. Investigate any violation rather than smoothing it away without justification.

Worked Schedules

Worked Example: Exponential Severity and a Middle Base

A hypothetical product has ground-up claim severity $X$ that is exponentially distributed with mean $\theta=\$8{,}000$. Consider deductibles $d=\$0,\$4{,}000,\$8{,}000,\$12{,}000$, and select $\$4{,}000$ as the base. Calculate the expected payment schedule and relative factors. Payments are undiscounted; no interest or mortality assumption is involved.

Method and assumptions. For this continuous exponential model, $S_X(x)=e^{-x/8000}$ for $x\geq0$, and the contract is an ordinary per-claim deductible with no upper limit. Therefore $L(d)=8000e^{-d/8000}$, with all dollar amounts in dollars.

At $d=\$0$, $L(0)=8000e^0=\$8{,}000.00$. At $d=\$4{,}000$, $L(4000)=8000e^{-0.5}=\$4{,}852.25$. At $d=\$8{,}000$, $L(8000)=8000e^{-1}=\$2{,}943.04$. At $d=\$12{,}000$, $L(12000)=8000e^{-1.5}=\$1{,}785.04$. The exponential values are evaluated before rounding; payments are shown to the nearest cent.

Divide each payment by the base payment, $L(4000)=8000e^{-0.5}$. The factors are $F(0\mid4000)=e^{0.5}=1.6487$, $F(4000\mid4000)=1.0000$, $F(8000\mid4000)=e^{-0.5}=0.6065$, and $F(12000\mid4000)=e^{-1}=0.3679$, rounded to four decimals.

Check and common mistake avoided. The base factor is 1 and the factors decrease as the deductible rises. A common error is to interpret the larger payment at the zero deductible as a factor below 1. Since the selected base is $\$4{,}000$, the zero-deductible payment is larger than the base payment, so its relative factor must exceed 1.

Worked Example: An Empirical Loss Distribution

A hypothetical portfolio has five equally weighted ground-up claims of $\$2{,}000,\$6{,}000,\$10{,}000,\$18{,}000,$ and $\$30{,}000$. Treat these observations as the full empirical severity distribution, with probability $1/5$ on each loss. Construct factors for deductibles of $\$0,\$5{,}000,\$10{,}000,$ and $\$20{,}000$, using $\$5{,}000$ as the base. There is no upper limit, and payments are undiscounted.

Method. For each loss, calculate $(X-d)_+$, then average the five payments. At deductible $\$0$, the total payment is $2{,}000+6{,}000+10{,}000+18{,}000+30{,}000=\$66{,}000$, so $L(0)=\$66{,}000/5=\$13{,}200.00$.

At deductible $\$5{,}000$, the five payments are $\$0,\$1{,}000,\$5{,}000,\$13{,}000,\$25{,}000$. Their total is $\$44{,}000$, so $L(5000)=\$8{,}800.00$. At deductible $\$10{,}000$, payments are $\$0,\$0,\$0,\$8{,}000,\$20{,}000$, giving $L(10000)=\$28{,}000/5=\$5{,}600.00$. At deductible $\$20{,}000$, only the $\$30{,}000$ loss produces payment, $\$10{,}000$, so $L(20000)=\$2{,}000.00$.

Normalize to $L(5000)=\$8{,}800$. The factors, rounded to four decimals, are $F(0\mid5000)=13{,}200/8{,}800=1.5000$, $F(5000\mid5000)=1.0000$, $F(10000\mid5000)=5{,}600/8{,}800=0.6364$, and $F(20000\mid5000)=2{,}000/8{,}800=0.2273$.

Check and common mistake avoided. Each payment and factor decreases with the deductible. Reversing the ratio at $\$10{,}000$ would give $8{,}800/5{,}600=1.5714$; that is a base-to-option ratio, not the defined factor. Under this convention the correct option-to-base factor is $5{,}600/8{,}800=0.6364$.

Worked Example: Weighting Classes Before Normalizing

A hypothetical portfolio has two claim classes. Class A has an exponential ground-up severity with mean $\theta_A=\$5{,}000$ and an expected 100 claims per year. Class B has an exponential severity with mean $\theta_B=\$15{,}000$ and an expected 50 claims per year. Calculate the portfolio factors for deductibles $\$0,\$5{,}000,\$10{,}000$, using $\$5{,}000$ as the base. Assume these frequencies apply across the deductible options, and use undiscounted payments with no upper limits.

Method. Calculate each class's expected payment per claim using $L(d)=\theta e^{-d/\theta}$, multiply by its expected claim count, and add the class totals. At $\$0$, the annual expected payment is $100(5000)+50(15000)=\$1{,}250{,}000.00$.

At $\$5{,}000$, Class A pays $5000e^{-1}=\$1{,}839.40$ per claim, or $\$183{,}939.72$ for 100 expected claims. Class B pays $15000e^{-1/3}=\$10{,}747.97$ per claim, or $\$537{,}398.48$ for 50 expected claims. The portfolio total is $\$721{,}338.20$.

At $\$10{,}000$, Class A pays $5000e^{-2}=\$676.68$ per claim, or $\$67{,}667.64$. Class B pays $15000e^{-2/3}=\$7{,}701.26$ per claim, or $\$385{,}062.84$. The total is $\$452{,}730.48$. Class-level amounts are rounded to cents for display; ratios below use the unrounded exponential calculations.

The factors relative to the $\$5{,}000$ portfolio payment are $1{,}250{,}000/721{,}338.20=1.7329$ at $\$0$, $1.0000$ at $\$5{,}000$, and $452{,}730.48/721{,}338.20=0.6276$ at $\$10{,}000$, rounded to four decimals.

Check and common mistake avoided. The portfolio factors are ordered correctly and the base is 1. Do not take an unweighted average of Class A and Class B factors: their expected claim counts and base payments differ. The correct portfolio factor is the ratio of aggregate expected payments, with each class weighted by its expected claims.

Consistency Checks and Interpretation

A schedule should be treated as a connected set of ratios, not merely a list of numbers. If the same payment basis is used at every deductible and all expected payments are positive, relative factors satisfy

F(d_1\mid b)=F(d_1\mid d_2)\,F(d_2\mid b), \qquad F(d_1\mid d_2)=\frac{L(d_1)}{L(d_2)}.

This follows by cancellation: $[L(d_1)/L(d_2)][L(d_2)/L(b)]=L(d_1)/L(b)$. It provides a useful check when a schedule is restated relative to a different base. Factors are also nonincreasing as the deductible increases because $(X-d)_+$ is nonincreasing in $d$ for every fixed loss $X$. Taking expectations preserves that ordering, and division by the same positive base payment preserves it again.

A practical validation should confirm all of the following:

The chosen base deductible has factor exactly 1 before rounding.
For a fixed ordinary-deductible payment basis, higher deductibles do not have higher expected payments or factors.
Every factor uses the same numerator and denominator convention: option payment divided by base payment.
Portfolio factors use aggregate expected payments, not an unweighted average of class factors.
The underlying terms are comparable. A schedule that includes a policy limit or a different deductible contract requires the corresponding payment calculation.

A severity-based factor also does not automatically capture changes in claim frequency, claim reporting, expenses, or other rating components. If a deductible changes the expected number of claims included in the data or the rating basis, address that effect separately rather than assuming the severity ratio covers it. Finally, if $L(b)=0$, the factor is undefined; select a base with positive expected payment or report expected payments directly.

Key takeaway: Calculate expected payments on a consistent basis, form each factor as option payment divided by the selected base payment, and then test the schedule's base value, ordering, and ratio consistency. A factor schedule is only as sound as the payment basis and weights used to construct it.

Check Your Understanding

For each problem, state the relevant base and use the option-to-base convention defined above.

A hypothetical portfolio has expected payments of $\$900,\$600,\$300$ at deductibles $\$0,\$2{,}000,\$5{,}000$, respectively. If $\$2{,}000$ is the base, give all three factors. (Answer: $1.5000,\ 1.0000,\ 0.5000$, in deductible order.)
An empirical severity distribution assigns equal probability to losses of $\$1{,}000,\$4{,}000,\$9{,}000,\$16{,}000$. With no upper limit, calculate expected payments at deductibles $\$0,\$5{,}000,\$10{,}000$, then factors relative to $\$5{,}000$. (Answer: Expected payments are $\$7{,}500,\ \$3{,}750,\ \$1{,}500$; factors are $2.0000,\ 1.0000,\ 0.4000$.)
Class A has 10 expected claims and expected payments per claim of $\$100,\$60,\$30$ at deductibles $\$0,\$5{,}000,\$10{,}000$. Class B has 5 expected claims and respective payments of $\$200,\$100,\$50$. Calculate portfolio factors relative to $\$5{,}000$. (Answer: Aggregate payments are $\$2{,}000,\ \$1{,}100,\ \$550$; factors are $1.8182,\ 1.0000,\ 0.5000$.)
A proposed schedule reports expected payments of $\$12{,}000,\$8{,}000,\$9{,}000$ at deductibles $\$0,\$5{,}000,\$10{,}000$. Normalize to $\$5{,}000$ and assess whether the schedule is internally consistent for the same ordinary-deductible basis. (Answer: Factors are $1.5000,\ 1.0000,\ 1.1250$; the increase from $\$5{,}000$ to $\$10{,}000$ violates the required nonincreasing order.)
Expected payments at deductibles $\$2{,}000,\$6{,}000,\$10{,}000$ are $\$12{,}000,\$8{,}000,\$5{,}000$. Give factors relative to $\$6{,}000$, and check the $\$2{,}000$-to-$\$10{,}000$ ratio by multiplying the adjacent relative factors. (Answer: Factors are $1.5000,\ 1.0000,\ 0.6250$; the direct ratio is $12{,}000/5{,}000=2.4000$, and $1.5000(8{,}000/5{,}000)=2.4000$.)

What You'll Learn