Quantiles of Modified Loss Payments

Why Payment Quantiles Need Care

A loss percentile does not always become an ordinary percentile of the insurer’s payment in an intuitive way. A deductible can make many different losses produce a zero payment, while a cap can make a range of large losses produce the same maximum payment. Those flat portions of the payment function create atoms—positive probability at particular payment amounts—and can leave gaps in the payment distribution.

The key is to define the payment transformation precisely and use the generalized quantile, rather than trying to solve for a payment percentile as if its distribution were continuous. This approach applies to continuous and discrete loss distributions and handles percentile levels that land at the boundaries of atoms.

Definition: For a random variable $Z$ with cumulative distribution function $F_Z$, its $p$-quantile, for $0<p<1$, is the generalized inverse $$ Q_Z(p)=\inf\{z:F_Z(z)\geq p\}. $$ This convention selects the smallest value at which the CDF reaches or exceeds $p$. If $Z$ has an atom, the $p$-quantile can be at that atom even though $F_Z(z)=p$ has no solution.

Payment Transformation and Quantile Rule

Let $X$ be a nonnegative loss. Consider an ordinary deductible $d$, a cap $L$ on the amount paid before coinsurance, and an insurer’s proportional share $c$, where $d\geq0$, $L>0$, and $0<c\leq1$. The payment is

Y=T(X)=c\min\bigl((X-d)_+,L\bigr), \qquad (z)_+=\max(z,0).

The cap is reached at loss $d+L$, and the maximum payment is $cL$. This definition makes the order explicit: apply the deductible, cap the remaining amount at $L$, and then apply coinsurance. If a contract instead specifies a cap on the final insurer payment, its payment function must be written using that contract’s convention.

Theorem: If $T$ is nondecreasing and continuous, then for $0<p<1$, $$ Q_Y(p)=T\bigl(Q_X(p)\bigr). $$ For the deductible, pre-coinsurance cap, and coinsurance above, $T$ is continuous and nondecreasing, so the rule applies even when $X$ is discrete or has atoms.

The rule follows from the generalized-inverse definition. Let $x_p=Q_X(p)$. By definition, $F_X(x_p)\geq p$, so $P(X\leq x_p)\geq p$. Since $T$ is nondecreasing, every $X\leq x_p$ gives $T(X)\leq T(x_p)$. Therefore $F_Y(T(x_p))\geq p$, and $Q_Y(p)\leq T(x_p)$. For any $y<T(x_p)$, monotonicity ensures that $T(X)\leq y$ can occur only for losses below $x_p$; their cumulative probability is less than $p$. Hence $F_Y(y)<p$, which proves $Q_Y(p)=T(x_p)$. Continuity ensures the threshold value $T(x_p)$ is attained.

Find the loss quantile.
Use the generalized inverse $Q_X(p)$ for the stated loss distribution.

Apply the contract in its stated order.
For this contract, calculate $c\min((Q_X(p)-d)_+,L)$.

Check the payment region.
Determine whether the quantile is at zero, in the paying layer, or at the cap. This check also reveals whether the requested percentile falls in an atom or across a gap.

The payment CDF makes the atoms visible. At zero, $Y=0$ exactly when $X\leq d$, so $P(Y=0)=F_X(d)$. For $0\leq y<cL$, the event $Y\leq y$ is equivalent to $X\leq d+y/c$, giving $F_Y(y)=F_X(d+y/c)$. At the maximum payment, all larger losses are included, and

P(Y=cL)=P(X\geq d+L)=1-F_X\bigl((d+L)-\bigr).

Here $F_X(a-)$ denotes the left limit of the CDF at $a$. The left limit matters when $X$ has probability exactly at the cap threshold: that loss receives the capped payment too. If the loss distribution is continuous, the expression simplifies to $P(X\geq d+L)=1-F_X(d+L)$.

Worked Examples

Worked Example: An Exponential Loss With a Deductible and Cap

A hypothetical policy covers a loss $X$ that is exponentially distributed with mean $\theta=\$400$, so $F_X(x)=1-e^{-x/400}$ for $x\geq0$. The insurer pays 75% of the amount above a $\$100$ deductible, subject to a $\$500$ cap before coinsurance. Find the 70th percentile of the insurer payment and the probability that the payment is capped. No interest or mortality adjustment applies.

Method. Find the loss quantile, then apply the payment transformation. The cap is reached at loss $d+L=\$100+\$500=\$600$, and the maximum payment is $cL=0.75(\$500)=\$375$.

For an exponential loss, $F_X(x)=p$ gives $x=-400\ln(1-p)$. Thus

Q_X(0.70)=-400\ln(0.30)=\$481.5891216\ldots.

This loss is above the deductible and below the cap threshold, since $\$100<\$481.5891216<\$600$. Therefore the payment quantile is

Q_Y(0.70)=0.75(481.5891216-100) =0.75(381.5891216) =\$286.1918412\ldots.

Rounded to the nearest cent, the 70th payment percentile is $\$286.19$. The cap-payment probability is

P(Y=375)=P(X\geq600)=e^{-600/400}=e^{-1.5}=0.2231

to four decimal places. As a check, the zero-payment probability is $P(X\leq100)=1-e^{-0.25}=0.2212$; the 70th percentile is above this zero atom and below the cap atom’s starting percentile, consistent with an interior payment.

Common mistake avoided: A rough or inaccurate logarithm can shift a reported payment percentile. Keeping the loss quantile unrounded until after applying coinsurance gives $\$286.19$, not a value obtained by rounding the loss prematurely.

Worked Example: A Uniform Loss and a Pre-Coinsurance Cap

A hypothetical loss is uniformly distributed on $[0,\$1{,}000]$. A contract has a $\$100$ deductible, pays 50% of the amount above the deductible, and caps that amount at $\$400$ before coinsurance. Find the 95th percentile of payment and the probability of reaching the cap. No interest or mortality adjustment applies.

Method. The loss quantile for a uniform distribution on this interval is $Q_X(p)=1000p$. At $p=0.95$,

Q_X(0.95)=1000(0.95)=\$950.

The cap threshold is the deductible plus the pre-coinsurance cap: $100+400=\$500$. Since $\$950\geq\$500$, this percentile receives the maximum payment:

Q_Y(0.95)=0.50\min(950-100,400) =0.50\min(850,400) =\$200.00.

The cap-payment probability is

P(Y=200)=P(X\geq500)=\frac{1000-500}{1000}=0.5000.

Thus the 95th payment percentile is $\$200.00$, and the cap-payment probability is 0.5000. The result is consistent with the cap atom: half the loss distribution lies at or above the $\$500$ threshold, so half of all payments are the maximum $\$200$.

Common mistake avoided: Treating $\$400$ as a cap on the final payment would give a different threshold and is not what the contract states. The cap here applies before coinsurance; the maximum payment is $0.50(\$400)=\$200$, reached when $X\geq\$500$.

Worked Example: A Gap in the Payment Distribution

A hypothetical loss has a mixture distribution: with probability $0.4$, $X$ is uniform on $[\$0,\$100]$; with probability $0.6$, $X$ is uniform on $[\$300,\$500]$. A contract has a $\$50$ deductible, a $\$150$ pre-coinsurance cap, and 50% coinsurance. Find the median payment and describe the relevant gap in the payment distribution. No interest or mortality adjustment applies.

Method. The payment is $Y=0.5\min((X-50)_+,150)$. In the first mixture component, losses from $\$0$ through $\$50$ produce zero payment. Losses above $\$50$ and through $\$100$ produce payments from $\$0$ through $\$25$. In the second component, every loss is at least $\$300$, above the cap threshold $50+150=\$200$, so every such loss produces the maximum payment, $0.5(150)=\$75$.

The first component has total probability $0.4$, and the second has total probability $0.6$. Consequently the payment CDF has accumulated probability $0.4$ by $\$25$, then remains at $0.4$ for payments between $\$25$ and $\$75$. At $\$75$, it jumps to 1 because the entire second component pays exactly $\$75$. In particular,

F_Y(y)<0.5 \quad\text{for }y<75, \qquad F_Y(75)=1.

By the generalized-inverse definition, $Q_Y(0.50)=\$75.00$. The median lies at the cap atom, not in the flat region of the CDF. The probability of the top payment is $0.6$, and the probability of zero payment is $0.4(50/100)=0.2$.

Common mistake avoided: A flat CDF segment does not mean every payment in that segment is a possible percentile value. The generalized inverse selects the first payment at which the CDF reaches the requested percentile; here that is $\$75$.

Reading Atoms and Flat Regions Correctly

A deductible and a cap create endpoint atoms even when the underlying loss has a smooth density. Their sizes depend on the loss probabilities below the deductible and at or above the cap threshold. Other features of the loss distribution can create additional gaps or atoms in the payment distribution. The transformation rule remains useful in all of these cases, because it works with generalized quantiles and does not require the payment CDF to be strictly increasing.

Payment region	Loss condition	Quantile interpretation
Zero payment	$X\leq d$	A percentile at or below $F_X(d)$ has payment quantile zero.
Uncapped positive payment	$d<X<d+L$	Transform the loss quantile using $c(Q_X(p)-d)$, provided the quantile is in this region.
Maximum payment	$X\geq d+L$	A percentile falling in the cap atom has payment quantile $cL$.
Gap in possible payments	A range of loss values has no probability	The payment CDF can be flat; use the first payment value where the CDF reaches $p$.

The transformation formula can be used directly, but checking the region is a valuable exam discipline. It identifies when the deductible or cap dominates the calculation and helps catch an incorrectly ordered coverage modification. The same logic applies to other continuous, nondecreasing payment functions, including a proportional payment without a deductible or a capped layer with different endpoints.

Common Exam Traps

Solving $F_Y(y)=p$ as though the CDF were continuous. At an atom, the CDF jumps over $p$; use $Q_Y(p)=\inf\{y:F_Y(y)\geq p\}$.
Ignoring the deductible atom. All losses $X\leq d$ pay zero, so the zero-payment probability is $F_X(d)$, not just $P(X<d)$.
Misreading the cap basis. A cap before coinsurance $L$ gives maximum payment $cL$ and is reached at $X=d+L$. A final-payment cap is a different contract specification.
Dropping a boundary atom. Losses equal to $d+L$ reach the cap. For a distribution with an atom there, use $P(X\geq d+L)=1-F_X((d+L)-)$.
Rounding too early. Carry the loss quantile through the payment transformation before rounding the final money amount.

Key takeaway: Define the payment function first, find the generalized loss quantile, and then apply the nondecreasing payment transformation. Check the zero-payment and capped-payment atoms explicitly, and use the generalized-inverse convention whenever the payment CDF jumps or is flat.

Check Your Understanding

For each problem, use the stated cap convention and the generalized-inverse definition. All scenarios are hypothetical; no interest or mortality adjustment applies.

A loss is exponential with mean $\$300$. The insurer pays 80% above a $\$60$ deductible, with no cap. Find the 80th percentile of payment. (Answer: $\$338.27$)
A loss is uniform on $[0,\$600]$. The insurer pays 50% above a $\$100$ deductible, subject to a $\$200$ cap before coinsurance. Find the 60th percentile of payment. (Answer: $\$100.00$)
A loss is exponential with mean $\$250$. The insurer pays 40% above a $\$50$ deductible, with a $\$150$ cap before coinsurance. Find the 95th percentile of payment and the probability of the maximum payment. (Answer: $\$60.00$; 0.4493)
A loss has probability $0.4$ of being uniform on $[\$0,\$100]$ and probability $0.6$ of being uniform on $[\$300,\$500]$. The insurer pays 50% above a $\$50$ deductible, with a $\$150$ cap before coinsurance. Find the 30th percentile of payment. (Answer: $\$12.50$)
A loss is exponential with mean $\$500$. The insurer pays 60% above a $\$100$ deductible, with a $\$300$ cap before coinsurance. Find the 40th percentile of payment and the probability of reaching the cap. (Answer: $\$93.25$; 0.4493)

Payment region	Loss condition	Quantile interpretation
Zero payment	\(X\leq d\)	A percentile at or below \(F_X(d)\) has payment quantile zero.
Uncapped positive payment	\(d<X<d+L\)	Transform the loss quantile using \(c(Q_X(p)-d)\), provided the quantile is in this region.
Maximum payment	\(X\geq d+L\)	A percentile falling in the cap atom has payment quantile \(cL\).
Gap in possible payments	A range of loss values has no probability	The payment CDF can be flat; use the first payment value where the CDF reaches \(p\).

What You'll Learn