Disappearing Deductible Payment Functions

What Disappears Under a Disappearing Deductible?

A deductible usually reduces the insurer’s payment on each covered loss. A disappearing deductible instead sets a threshold: if the loss does not pass that threshold, the insurer pays nothing; once the loss passes it, the deductible disappears and the insurer pays the full loss. The jump in payment at the threshold is the key feature. It is not the same as subtracting a fixed amount from every sufficiently large loss.

The precise treatment of a loss equal to the threshold depends on the contract wording. For the calculations here, a loss must be strictly greater than the threshold for the deductible to disappear. A contract that says “equal to or greater than” uses a different boundary rule. This distinction matters when a loss distribution has a positive probability of a claim exactly at the threshold.

Definition: Let $X\ge 0$ be the amount of a covered loss and $d\ge 0$ the disappearing-deductible threshold. Under the strict-exceedance convention, the insurer payment per loss is $$ Y^L_{\mathrm{dis}} = \begin{cases} 0, & X\le d,\\ X, & X>d. \end{cases} $$ The insurer pays no amount at or below the threshold, and pays the full loss above it.

For comparison, under an ordinary deductible of amount $d$, the insurer pays the part of the loss above the deductible:

Y^L_{\mathrm{ord}}=(X-d)_+=\max(X-d,0).

Thus, for a loss above $d$, the disappearing-deductible payment exceeds the ordinary-deductible payment by exactly $d$. At or below $d$, both payments are zero under the strict-exceedance convention. Above the threshold, the insurer’s payment under the disappearing form jumps from zero to the full loss, while the ordinary form increases continuously from zero.

Loss amount	Disappearing deductible	Ordinary deductible
$X<d$	0	0
$X=d$, strict-exceedance convention	0	0
$X>d$	$X$	$X-d$

Calculate the Payment at Each Part of the Threshold

When applying a disappearing deductible, first identify the covered loss amount and the contract’s threshold rule. Compare the loss with the threshold before calculating the insurer’s payment. If it does not pass the threshold, the payment is zero. If it does pass, the payment is the full loss, not the excess over the threshold.

Worked Example: Three Losses Around a Threshold

A hypothetical equipment policy has a disappearing-deductible threshold of $d=\$5{,}000$. The contract requires the loss to exceed the threshold. Three covered losses are $\$4{,}200$, $\$5{,}000$, and $\$7{,}400$. Find the insurer payment for each loss under both a disappearing deductible and an ordinary deductible of $\$5{,}000$.

Method. Apply the threshold condition separately to each loss. There is no interest or mortality assumption, and no severity distribution is needed because the loss amounts are specified.

For $X=\$4{,}200$, the loss is below the threshold. The disappearing-deductible payment is $\$0$. The ordinary payment is $\max(\$4{,}200-\$5{,}000,0)=\$0$.

For $X=\$5{,}000$, the loss equals the threshold. Under the stated strict-exceedance wording, it does not pass the threshold. The disappearing-deductible payment is $\$0$. The ordinary payment is also $\max(\$5{,}000-\$5{,}000,0)=\$0$.

For $X=\$7{,}400$, the loss exceeds the threshold. The disappearing deductible disappears, so the insurer pays the full $\$7{,}400$. Under the ordinary deductible, the insurer pays only $\$7{,}400-\$5{,}000=\$2{,}400$.

Check. The disappearing-deductible payments are $\$0,\ \$0,\ \$7{,}400$; the ordinary-deductible payments are $\$0,\ \$0,\ \$2{,}400$. For the above-threshold loss, their difference is $\$7{,}400-\$2{,}400=\$5{,}000=d$, as expected. The common mistake is to pay only $\$2{,}400$ under the disappearing deductible; that incorrectly applies the ordinary-deductible formula.

The Boundary Rule and Discrete Loss Amounts

The words “exceeds,” “equals,” and “at least” determine whether a claim exactly at $d$ qualifies for full payment. Under the convention used here, $X=d$ produces no payment. If a contract instead says the deductible disappears when $X\ge d$, the payment at $X=d$ is the full amount $d$. The two rules give the same answer for losses strictly below or above $d$, but can produce different expected payments when claims can equal $d$.

This is not merely a technical distinction. A discrete severity model, or a data set with losses rounded to fixed increments, may assign positive probability to exactly the threshold. In that setting, the probability at the boundary must be included or excluded according to the actual wording. A continuous severity model assigns probability zero to one exact value, so changing only the boundary convention does not change its expected payment.

Worked Example: A Discrete Distribution With a Claim at the Threshold

A hypothetical policy has a disappearing-deductible threshold of $\$5{,}000$, with payment only when the loss is strictly greater than the threshold. The loss distribution is: $X=\$0$ with probability $0.20$, $X=\$4{,}000$ with probability $0.25$, $X=\$5{,}000$ with probability $0.15$, and $X=\$8{,}000$ with probability $0.40$. Calculate the expected insurer payment under the disappearing deductible and under an ordinary deductible of $\$5{,}000$.

Method. For each possible loss, calculate the payment and multiply it by its probability. The probabilities sum to $0.20+0.25+0.15+0.40=1.00$. No interest or mortality assumption is relevant.

Under the disappearing deductible, the payments at losses $\$0,\ \$4{,}000,\ \$5{,}000,\ \$8{,}000$ are respectively $\$0,\ \$0,\ \$0,\ \$8{,}000$. In particular, the probability $0.15$ at exactly $\$5{,}000$ produces no payment because the condition is strictly greater than the threshold.

The expected payment is $$ E[Y^L_{\mathrm{dis}}] =0.20(0)+0.25(0)+0.15(0)+0.40(\$8{,}000) =\$3{,}200. $$

Under the ordinary deductible, the corresponding payments are $\$0,\ \$0,\ \$0,\ \$3{,}000$. Therefore, $$ E[Y^L_{\mathrm{ord}}] =0.20(0)+0.25(0)+0.15(0)+0.40(\$3{,}000) =\$1{,}200. $$

Check. The expected payments differ by $\$3{,}200-\$1{,}200=\$2{,}000$, which is the threshold amount multiplied by the probability of exceeding it: $\$5{,}000(0.40)=\$2{,}000$. The common mistake is to include the $\$5{,}000$ outcome as a full payment despite the strict wording. If the contract instead specified payment for $X\ge d$, the expected disappearing-deductible payment would increase by $0.15(\$5{,}000)=\$750$, to $\$3{,}950$.

Expected Payment Under Continuous Severity

For a nonnegative continuous loss $X$ with survival function $S_X(x)=P(X>x)$, the disappearing-deductible payment is $X$ when $X>d$ and zero otherwise. Its expected value can be expressed using the survival function. This form separates the threshold contribution from the expected amount by which the loss exceeds the threshold.

Result: For a nonnegative continuous severity distribution with finite expected payment, $$ E[Y^L_{\mathrm{dis}}] =E[X\,\mathbf{1}_{\{X>d\}}] =dS_X(d)+\int_d^\infty S_X(x)\,dx. $$ For an ordinary deductible under the same assumptions, $$ E[Y^L_{\mathrm{ord}}] =E[(X-d)_+] =\int_d^\infty S_X(x)\,dx. $$

To justify the disappearing-deductible formula, on the event $X>d$ write $X=d+(X-d)$. Multiplying by the indicator of that event gives

X\mathbf{1}_{\{X>d\}} =d\mathbf{1}_{\{X>d\}}+(X-d)_+.

Taking expectations gives $E[X\mathbf{1}_{\{X>d\}}]=dP(X>d)+E[(X-d)_+]$. For a nonnegative loss, the positive-part expectation is the area under its excess survival function:

E[(X-d)_+] =\int_0^\infty P((X-d)_+>t)\,dt =\int_0^\infty P(X>d+t)\,dt =\int_d^\infty S_X(x)\,dx.

Substituting $P(X>d)=S_X(d)$ establishes the result. The derivation also shows why the expected disappearing-deductible payment exceeds the ordinary-deductible payment by $dS_X(d)$: for each claim that exceeds the threshold, the disappearing form pays an additional $d$.

Worked Example: Expected Payments Under Exponential Severity

For a hypothetical class of covered equipment losses, suppose $X$ has an exponential distribution with mean $\theta=\$5{,}000$, so $S_X(x)=e^{-x/5000}$ for $x\ge0$. The disappearing-deductible threshold is $d=\$5{,}000$, and the contract uses the strict-exceedance convention. Calculate expected insurer payments under the disappearing and ordinary deductible forms.

Method and assumptions. Use the survival-function formulas above. The loss is continuous, nonnegative, and exponentially distributed with scale (and mean) $\theta=\$5{,}000$. No interest or mortality assumption applies. Since the distribution is continuous, the probability of a loss exactly equal to $\$5{,}000$ is zero.

First, $S_X(d)=e^{-5000/5000}=e^{-1}$. The expected disappearing-deductible payment is $$ E[Y^L_{\mathrm{dis}}] =\$5{,}000e^{-1}+\int_{\$5{,}000}^{\infty}e^{-x/5000}\,dx. $$

The integral is $$ \int_{\$5{,}000}^{\infty}e^{-x/5000}\,dx =5{,}000e^{-1} =\$1{,}839.397\ldots. $$

Thus, $$ E[Y^L_{\mathrm{dis}}] =\$10{,}000e^{-1} =\$3{,}678.794\ldots =\$3{,}678.79. $$ The expected ordinary-deductible payment is just the integral: $$ E[Y^L_{\mathrm{ord}}] =\$5{,}000e^{-1} =\$1{,}839.397\ldots =\$1{,}839.40. $$

Check. The difference is $\$1{,}839.39$ using the rounded displayed amounts, and before rounding it is $\$5{,}000e^{-1}=\$1{,}839.397\ldots$, exactly $dP(X>d)$. Both amounts are rounded to the nearest cent only at the final step. The common mistake is to use the ordinary-deductible expected payment as the disappearing-deductible payment, omitting the threshold amount paid on every loss above $d$.

Practical Checks and Exam Traps

The central modeling decision is not an algebraic one: it is identifying which payment function matches the contract. Once the wording has been translated into a threshold rule, the calculation is direct. The following checks help prevent the most consequential errors.

Do not subtract the threshold after it disappears. For $X>d$, the disappearing-deductible payment is $X$, while $X-d$ is the ordinary-deductible payment.
Read the equality condition. A strict “exceeds” condition excludes $X=d$; an inclusive “equals or exceeds” condition includes it. State the convention before calculating.
Do not treat the function as continuous at the threshold. Under strict exceedance, the payment is zero at $d$ and approaches $d$ from above. That jump is intentional, not a calculation error.
Use the right probability event in expected values. The disappearing payment is tied to $P(X>d)$, not $P(X\ge d)$, under the convention used here.
Check the comparison with an ordinary deductible. For each loss above $d$, the disappearing payment is larger by $d$; for losses at or below $d$, both are zero.

In ratemaking, changing the deductible form changes the expected insurer payment and therefore the loss cost associated with the coverage. The difference can be especially material when many claims exceed the threshold, because each qualifying claim adds the threshold amount relative to an ordinary deductible. The contract boundary also matters when claim amounts have probability mass at that point. These payment-function distinctions are aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Check Your Understanding

For each problem, use the stated threshold convention and distinguish the disappearing-deductible payment from an ordinary-deductible payment where requested.

A hypothetical policy has a strict disappearing-deductible threshold of $\$2{,}500$. Find both payment amounts for losses of $\$1{,}800,\ \$2{,}500,$ and $\$3{,}900$, using an ordinary deductible of $\$2{,}500$. (Answer: Disappearing payments $\$0,\ \$0,\ \$3{,}900$; ordinary payments $\$0,\ \$0,\ \$1{,}400$.)
A contract says the disappearing deductible is waived when a loss is “equal to or greater than” $\$6{,}000$. What is the insurer payment for a loss of exactly $\$6{,}000$? How would the strict-exceedance convention change it? (Answer: Inclusive wording pays $\$6{,}000$; strict exceedance pays $\$0$.)
A discrete loss distribution has $P(X=\$0)=0.10$, $P(X=\$3{,}000)=0.20$, $P(X=\$5{,}000)=0.30$, and $P(X=\$9{,}000)=0.40$. With a strict disappearing-deductible threshold of $\$5{,}000$, calculate expected payments for the disappearing and ordinary forms. (Answer: Disappearing $\$3{,}600$; ordinary $\$1{,}600$.)
Suppose $X$ is exponential with mean $\theta=\$4{,}000$, and the strict disappearing-deductible threshold is $d=\$4{,}000$. Calculate both expected payments to the nearest cent. (Answer: Disappearing $\$8{,}000/e=\$2{,}943.04$; ordinary $\$4{,}000/e=\$1{,}471.52$.)
A hypothetical discrete severity model has losses $\$2{,}000,\ \$4{,}000,\ \$6{,}000,\ \$10{,}000$ with respective probabilities $0.10,\ 0.20,\ 0.40,\ 0.30$. For a strict threshold of $\$4{,}000$, calculate the expected disappearing-deductible and ordinary-deductible payments. (Answer: Disappearing $\$5{,}400$; ordinary $\$2{,}600$.)

Loss amount	Disappearing deductible	Ordinary deductible
\(X<d\)	0	0
\(X=d\), strict-exceedance convention	0	0
\(X>d\)	\(X\)	\(X-d\)

What You'll Learn