Deductible Factors Relative to a Base Deductible

Why Deductible Factors Need a Base

A deductible changes the portion of each covered loss paid by the insurer. To compare deductible options, an actuary can compare their expected insurer payments under the same severity assumptions. A deductible factor expresses that comparison as a ratio to a selected base deductible. The base is a reference point: changing it changes the reported factor scale, but it does not change the expected payment costs being compared.

The calculations below concern an ordinary per-loss deductible and insurer payment before any policy limit. They use the severity distribution or empirical claim data as given; they do not estimate deductible-dependent claim frequency. The factor is meaningful only when the expected payment at the base deductible is positive.

Definition: Let $X$ be a nonnegative ground-up loss and let $d$ be an ordinary per-loss deductible. The insurer payment per loss is $Y^L=(X-d)_+=\max(X-d,0)$. Write $C(d)=E[(X-d)_+]$ for the expected insurer payment per loss at deductible $d$. For a base deductible $d_b$ with $C(d_b)>0$, the deductible factor at $d$ relative to $d_b$ is $$ R(d;d_b)=\frac{C(d)}{C(d_b)} =\frac{E[(X-d)_+]}{E[(X-d_b)_+]}. $$

Thus $R(d_b;d_b)=1$. If $d$ is higher than $d_b$, then every loss produces an insurer payment no greater than it would at the base, so $R(d;d_b)\leq 1$. If $d$ is lower than the base, the factor can exceed 1. The factor is a relative expected payment, not a dollar cost: the expected payment per loss remains $C(d)$.

If the expected number of claims per exposure is the same at both deductible options, multiplying each per-loss expected payment by that common claim frequency gives expected aggregate payments per exposure. The frequency then cancels in the ratio. If frequency differs by deductible, a severity-only factor does not by itself represent the ratio of aggregate costs.

Calculating Expected Payments and Factors

For a discrete severity distribution, calculate the insurer payment at each possible loss, multiply each payment by its probability, and add. For an empirical sample of $n$ losses given equal weight, use the sample average of the payments. In either case, calculate the alternative and base expected payments separately before taking their ratio.

For a nonnegative severity distribution with survival function $S_X(x)=P(X>x)$, the same expected payment can be written as an area under the survival curve. This gives a useful calculation formula and explains why a higher deductible cannot increase the expected insurer payment.

Derivation: For any fixed nonnegative loss $X$, the excess above deductible $d$ is the length of the interval from $d$ to $X$, when $X>d$, and is zero otherwise. Therefore $$ (X-d)_+=\int_d^\infty \mathbf{1}_{\{X>x\}}\,dx. $$ Taking expectations and interchanging the expectation and integral for this nonnegative integrand gives $$ C(d)=E[(X-d)_+] =\int_d^\infty E[\mathbf{1}_{\{X>x\}}]\,dx =\int_d^\infty S_X(x)\,dx. $$ As $d$ increases, the integral is taken over a smaller interval, so $C(d)$ cannot increase. The formula applies to nonnegative losses and an ordinary deductible; a different payment function, such as one involving a limit, requires a different calculation.

Worked Example: A Discrete Severity Distribution

A hypothetical equipment policy has ground-up claim severity $X$ equal to $1,000, $3,000, $6,000, or $10,000, with respective probabilities 0.20, 0.30, 0.30, and 0.20. Calculate the factor for a $4,000 deductible relative to a $1,000 base deductible. Use the method of probability-weighted insurer payments.

Step 1: Calculate payments at the base deductible. With $d_b=\$1{,}000$, the payments at the four severities are $$ (1{,}000-1{,}000)_+=0,\quad (3{,}000-1{,}000)_+=2{,}000,\quad (6{,}000-1{,}000)_+=5{,}000,\quad (10{,}000-1{,}000)_+=9{,}000. $$ Weighting by probability gives $$ C(1{,}000)=0.20(0)+0.30(2{,}000)+0.30(5{,}000)+0.20(9{,}000) =0+600+1{,}500+1{,}800=\$3{,}900. $$

Step 2: Calculate payments at the alternative deductible. At $d=\$4{,}000$, the payments are $0, $0, $2,000, and $6,000. Hence $$ C(4{,}000)=0.20(0)+0.30(0)+0.30(2{,}000)+0.20(6{,}000) =0+0+600+1{,}200=\$1{,}800. $$

Step 3: Divide alternative by base. $$ R(4{,}000;1{,}000)=\frac{1{,}800}{3{,}900} =\frac{6}{13}=0.461538\ldots\approx 0.4615. $$ The expected insurer payment at the $4,000 deductible is 46.15% of the expected payment at the $1,000 base. The check is directionally sensible: the alternative deductible is higher, and its expected payment, $1,800, is lower than $3,900.

Method and mistake avoided. The method is to calculate both expected payments from the same severity distribution, then divide alternative by base. Reversing the division would produce $3{,}900/1{,}800=2.1667$, which is the factor for the $1,000 deductible relative to a $4,000 base—not the factor requested.

Model-Based and Empirical Calculations

A fitted severity model can make it possible to evaluate deductible options without summing over a finite list of loss amounts. For example, under an exponential severity model, the survival function has a particularly simple form. An empirical calculation instead uses the observed losses directly and does not require fitting a parametric distribution. Both approaches still rely on the same definition: compare expected insurer payments, not ground-up mean losses.

Worked Example: An Exponential Severity Model

For a hypothetical property portfolio, assume the ground-up severity follows an exponential distribution with mean $\theta=\$5{,}000$, so $S_X(x)=e^{-x/5{,}000}$ for $x\geq0$. Calculate the factor for a $3,000 deductible relative to a $1,000 base deductible. There is no policy limit.

Step 1: Obtain the expected payment formula. Using the survival-function result, $$ C(d)=\int_d^\infty e^{-x/\theta}\,dx =\theta e^{-d/\theta}. $$ The integral follows because an antiderivative of $e^{-x/\theta}$ is $-\theta e^{-x/\theta}$, and $e^{-x/\theta}$ tends to zero as $x$ tends to infinity.

Step 2: Evaluate the base expected payment. With $\theta=5{,}000$ and $d_b=1{,}000$, $$ C(1{,}000)=5{,}000e^{-1{,}000/5{,}000} =5{,}000e^{-0.2} \approx \$4{,}093.65. $$

Step 3: Evaluate the alternative expected payment. $$ C(3{,}000)=5{,}000e^{-3{,}000/5{,}000} =5{,}000e^{-0.6} \approx \$2{,}744.06. $$

Step 4: Calculate and check the factor. $$ R(3{,}000;1{,}000) =\frac{5{,}000e^{-0.6}}{5{,}000e^{-0.2}} =e^{-0.4} \approx 0.6703. $$ The common scale parameter cancels in the ratio, and the result is below 1 because the alternative deductible is higher. The expected payments above are rounded to cents; the factor is rounded to four decimal places.

Method and mistake avoided. The method is to integrate the exponential survival function, or use its resulting expected-payment formula, at each deductible. Using $E[X]=\$5{,}000$ as the expected payment at either deductible would ignore the portion of each loss retained by the insured.

An empirical sample can be used in the same way as a discrete distribution. If each observed claim is given equal weight, $C(d)$ is the average of $(X_j-d)_+$ over the observations. If observations have different weights, such as claim weights chosen to represent a target mix, use those weights consistently at both deductible levels and normalize them to sum to 1.

Worked Example: Empirical Costs for Two Classes

A hypothetical portfolio has two claim classes. Assume 60% of claims are in class A and 40% are in class B, and use these proportions at both deductible levels. Class A has severities $1,000 and $5,000 with equal probability. Class B has severities $3,000 and $8,000 with equal probability. Find the portfolio factor for a $2,000 deductible relative to a $500 base deductible. The class proportions refer to claim mix, not exposure mix.

Step 1: Calculate class A costs. At the $500 base, class A’s payments are $500 and $4,500, so $$ C_A(500)=0.50(500)+0.50(4{,}500)=250+2{,}250=\$2{,}500. $$ At $2,000, its payments are $0 and $3,000, giving $$ C_A(2{,}000)=0.50(0)+0.50(3{,}000)=\$1{,}500. $$

Step 2: Calculate class B costs. At $500, its payments are $2,500 and $7,500: $$ C_B(500)=0.50(2{,}500)+0.50(7{,}500)=1{,}250+3{,}750=\$5{,}000. $$ At $2,000, its payments are $1,000 and $6,000: $$ C_B(2{,}000)=0.50(1{,}000)+0.50(6{,}000)=500+3{,}000=\$3{,}500. $$

Step 3: Weight expected payments by claim mix. $$ C(500)=0.60(2{,}500)+0.40(5{,}000)=1{,}500+2{,}000=\$3{,}500, $$ $$ C(2{,}000)=0.60(1{,}500)+0.40(3{,}500)=900+1{,}400=\$2{,}300. $$

Step 4: Divide the mixed costs. $$ R(2{,}000;500)=\frac{2{,}300}{3{,}500} =0.657142\ldots\approx 0.6571. $$ This is also a base-cost-weighted average of the class factors. Class A’s factor is $1{,}500/2{,}500=0.6000$, and class B’s is $3{,}500/5{,}000=0.7000$. Their weights in base expected payment are $1{,}500/3{,}500$ and $2{,}000/3{,}500$, respectively, so $$ \frac{1{,}500}{3{,}500}(0.6000)+\frac{2{,}000}{3{,}500}(0.7000) =\frac{900+1{,}400}{3{,}500} =0.6571. $$

Method and mistake avoided. The method is to mix the expected payments first and then form their ratio, or equivalently to weight class factors by each class’s contribution to base expected payment. A simple claim-mix-weighted average of class factors, $0.60(0.6000)+0.40(0.7000)=0.6400$, is incorrect because the classes have different base expected payments per claim.

Interpreting the Base and Applying the Factor

The denominator determines the scale of every factor in a schedule. For the same pair of deductible choices, reversing which deductible is the base gives the reciprocal ratio, provided both expected payments are positive: $$ R(d_1;d_2)=\frac{C(d_1)}{C(d_2)} =\frac{1}{R(d_2;d_1)}. $$ This is a change of reference, not a change in severity or in the underlying expected payment at either deductible. A factor should therefore always be reported with its base deductible stated.

A base deductible with zero expected insurer payment cannot serve as the denominator: the ratio is undefined. For instance, if no loss in the severity distribution exceeds a proposed base deductible, then $C(d_b)=0$, and no finite factor relative to that base can be calculated. A factor calculated from one assumed severity distribution also need not apply to a different class or portfolio mix; those may have different expected payment ratios.

Key takeaway: First calculate $C(d)$, the expected insurer payment at each deductible, using a consistent severity model, sample, and mix. Then calculate the alternative-to-base ratio. A deductible factor is a normalized comparison of costs; it does not replace the underlying expected payments or automatically account for a change in claim frequency.

Common Mistakes

Reversing the ratio. For a factor at an alternative deductible relative to a base, divide the alternative expected payment by the base expected payment.
Using ground-up severity instead of insurer payment. Apply the deductible to each loss before taking its expectation.
Averaging class factors with the wrong weights. Aggregate expected payments first, or use base expected payment contributions as the weights.
Leaving the base unstated. A factor such as 0.70 has no complete interpretation without its reference deductible.
Ignoring the assumptions behind the ratio. The severity basis and mix must be consistent; a severity-only comparison does not capture a change in expected claim frequency.

Check Your Understanding

For each problem, calculate the factor relative to the stated base deductible. Round factors to four decimal places.

A hypothetical severity distribution has losses of $2,000 and $6,000 with probabilities 0.40 and 0.60. Find the factor for a $3,000 deductible relative to a $1,000 base. (Answer: 0.5294)
Assume hypothetical claim severities follow an exponential distribution with mean $4,000. Find the factor for a $2,500 deductible relative to a $500 base. (Answer: 0.6065)
An empirical sample has three equally weighted losses: $1,000, $4,000, and $9,000. Find the factor for a $5,000 deductible relative to a $2,000 base. (Answer: 0.4444)
Two claim classes have base expected payments of $2,000 and $5,000 per claim, and alternative expected payments of $1,000 and $3,000 per claim. If 30% of claims are class A and 70% class B at both options, find the portfolio factor for the alternative relative to the base. (Answer: 0.5854)
A hypothetical severity distribution has losses of $1,000 and $5,000 with equal probability. Find the factor for a $1,000 deductible relative to a $3,000 base deductible. Explain why the factor exceeds 1. (Answer: 2.0000; the alternative deductible is lower, so its expected payment is twice the base expected payment.)

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