Quantiles Of Capped And Deductible Payments

Why Payment Percentiles Need Care

A percentile describes a point in the distribution of the amount actually paid, not necessarily a percentile of the underlying loss. A deductible can put a positive probability at zero payment. A loss model with point masses can also create jumps at positive payment amounts, and a policy limit puts a probability mass at the maximum payment. These features matter because a percentile can land exactly on a jump.

The method is to write the payment as a function of the loss, derive its cumulative distribution function (cdf), and use the quantile definition. In particular, do not solve a continuous equation across a jump without checking the cdf immediately before and at that payment.

Definition: For a payment random variable $Y^L$ and percentile level $p$, where $0<p<1$, the $p$-quantile is $q_p=\inf\{y:F_{Y^L}(y)\ge p\}$, where $F_{Y^L}(y)=P(Y^L\le y)$. The $p$th percentile is $100p\%$. This lower-quantile convention selects the first value at which the cdf reaches or exceeds $p$, including when the cdf jumps over $p$.

Payment With a Deductible and a Cap

Assume a nonnegative loss $X$, a per-loss deductible $d\ge0$, and a maximum insurer payment $u>0$. In this tutorial, $u$ is a cap on the payment after the deductible. The payment per loss is

Y^L=\min\bigl((X-d)_+,u\bigr),\qquad (X-d)_+=\max(X-d,0).

Thus the insurer pays zero for a loss at or below $d$, pays the excess over $d$ for losses between $d$ and $d+u$, and pays $u$ for losses at or above $d+u$. This model assumes no coinsurance and no inflation adjustment. If a contract instead defines its limit as a cap on the loss amount before applying the deductible, its payment function must be specified separately; the threshold need not be $d+u$.

Theorem: For $0\le y<u$, the payment cdf is $F_{Y^L}(y)=F_X(d+y)$. At $y<0$, it is zero, and at $y\ge u$, it is one. In particular, $F_{Y^L}(0)=F_X(d)$, while the probability of a capped payment is $P(Y^L=u)=P(X\ge d+u)=1-F_X((d+u)-)$. The left-limit notation accounts for a possible atom at $X=d+u$.

The formula for $0\le y<u$ follows from an event equivalence. A payment is at most $y$ precisely when the loss is at most $d+y$: losses at or below $d$ pay zero; losses between $d$ and $d+u$ pay $X-d$; and losses that reach the cap pay $u>y$, so they are excluded. Therefore

\{Y^L\le y\}=\{X\le d+y\},\qquad 0\le y<u.

Taking probabilities gives the cdf formula. At $y=u$, every possible payment is at most $u$, so the cdf is one. The distinction between $F_X(d+u)$ and $F_X((d+u)-)$ matters when the loss distribution has an atom at the cap threshold: a loss exactly equal to $d+u$ already produces payment $u$.

A Reliable Quantile Procedure

For a mixed or discrete loss model, write the cdf including its jumps; do not assume it has a density everywhere. Use the following procedure for a requested percentile.

Map loss values to payments.
Mark the regions $X\le d$, $d<X<d+u$, and $X\ge d+u$. Identify any loss atoms and their corresponding payments.

Build the payment cdf.
For $0\le y<u$, use $F_{Y^L}(y)=F_X(d+y)$. Include all mixture components and atoms, and state the cdf at the cap.

Locate the first crossing.
Find the smallest $y$ with $F_{Y^L}(y)\ge p$. At a suspected jump, compare the cdf just below the payment with the cdf at the payment.

Check the result.
Confirm that the cdf at the answer is at least $p$, and that all smaller payment values have cdf below $p$. Check that the answer lies between zero and $u$.

An atom in $X$ at a point $x$ strictly between $d$ and $d+u$ becomes an atom in payment at $x-d$. The deductible also creates an atom at zero whenever $P(X\le d)>0$, and the cap creates an atom at $u$ whenever $P(X\ge d+u)>0$. These masses can make $F_{Y^L}$ jump over a requested percentile, so equality with $p$ is not required.

Worked Example: A Discrete Loss Model With Three Payment Outcomes

A hypothetical one-period policy has deductible $d=\$200$ and payment cap $u=\$300$. There is no discounting or mortality model; all amounts are one-period payments. The loss distribution is $P(X=\$0)=0.10$, $P(X=\$100)=0.20$, $P(X=\$250)=0.25$, $P(X=\$500)=0.25$, and $P(X=\$900)=0.20$. Find the 40th and 60th percentiles of $Y^L$.

Method. Apply the payment function to each possible loss. The threshold for the cap is $d+u=\$500$.

For losses $\$0,\$100,$ and $\$250$, the payments are $\$0,\$0,$ and $\$50$, respectively. For $X=\$500$, the payment is $\min(\$500-\$200,\$300)=\$300$. For $X=\$900$, the payment is also capped at $\$300$. Therefore the payment distribution is

P(Y^L=0)=0.10+0.20=0.30,\quad P(Y^L=50)=0.25,\quad P(Y^L=300)=0.25+0.20=0.45.

The cdf is $0.30$ at payment $\$0$, $0.55$ at payment $\$50$, and $1.00$ at payment $\$300$. Since $0.30<0.40\le0.55$, the first payment at which the cdf reaches $0.40$ is $\$50$. Since $0.55<0.60\le1.00$, the first payment at which it reaches $0.60$ is $\$300$.

Answers: The 40th percentile is $\$50$, and the 60th percentile is $\$300$. The cdf at each answer is at least its requested percentile, and the cdf at every smaller attainable payment is below that percentile. The common mistake avoided here is averaging payment values or interpolating through a discrete jump.

Worked Example: A Mixture Atom and the Jump It Creates

In a hypothetical one-period contract, $d=\$200$, $u=\$500$, and there is no discounting or mortality assumption. With probability $0.25$, the loss is exactly $\$300$. With probability $0.75$, the loss is uniform on $[\$0,\$1{,}000]$. Find the 45th and 70th percentiles of the payment per loss.

The atom at loss $\$300$ produces payment $\$100$. In the uniform component, payment is zero when $X\le\$200$, is $X-\$200$ when $\$200<X<\$700$, and is capped at $\$500$ when $X\ge\$700$. For $0\le y<\$500$, the uniform branch contributes $0.75(200+y)/1000=0.15+0.00075y$ to the payment cdf, with $y$ measured in dollars. The loss atom contributes an additional $0.25$ when $y\ge\$100$. Thus

F_{Y^L}(y)=0.15+0.00075y+0.25\,\mathbf{1}_{\{y\ge100\}}, \qquad 0\le y<500.

Just below $\$100$, the cdf approaches $0.15+0.00075(100)=0.225$. At $\$100$, it jumps to $0.225+0.25=0.475$. Since $0.225<0.45\le0.475$, the 45th percentile is $\$100$.

For the 70th percentile, the cdf after the jump must reach $0.70$. Solve $0.15+0.00075y+0.25=0.70$, giving $0.00075y=0.30$ and $y=400$. At $\$400$, the cdf is $0.15+0.30+0.25=0.70$; below that point but above $\$100$, it is less than $0.70$. The 70th percentile is therefore $\$400$.

Answers: 45th percentile: $\$100$; 70th percentile: $\$400$. A tempting but incorrect calculation for the 45th percentile is to ignore the atom and solve $0.15+0.00075y=0.45$, which gives $\$400$. That equation omits the atom's probability and does not describe the cdf after the jump; at $\$400$, the actual cdf is $0.70$, not $0.45$. Always check both sides of a jump.

Worked Example: An Exponential Loss With a Capped Upper Percentile

A hypothetical one-period loss $X$ is exponential with mean $\$400$, so its rate is $\lambda=1/400$ per dollar and $F_X(x)=1-e^{-x/400}$ for $x\ge0$. The policy has deductible $d=\$100$ and payment cap $u=\$300$. There is no discounting or mortality model. Find the 50th and 80th percentiles of $Y^L$, reporting money to the nearest cent.

At zero payment, $F_{Y^L}(0)=F_X(100)=1-e^{-0.25}=0.221199\ldots$, so the median is not zero. For $0\le y<300$,

F_{Y^L}(y)=1-e^{-(100+y)/400}.

For the 50th percentile, solve $1-e^{-(100+y)/400}=0.50$. Then $e^{-(100+y)/400}=0.50$, so $(100+y)/400=-\ln(0.50)=0.693147\ldots$. Hence $y=400(0.693147\ldots)-100=177.2588\ldots$, which rounds to $\$177.26$.

The cap threshold is $d+u=\$400$. Just below payment $\$300$, the cdf approaches $F_X(400)=1-e^{-1}=0.6321206\ldots$. At payment $\$300$, it is one, because all losses $X\ge\$400$ pay the cap. Since $0.6321206\ldots<0.80\le1$, the 80th percentile is the cap, $\$300.00$.

Answers: The 50th percentile is $\$177.26$, and the 80th percentile is $\$300.00$. The checks are $F_{Y^L}(177.26)$ approximately $0.5000$, and the capped payment cdf is one. The trap avoided is extending the uncapped inverse beyond $u$; a percentile beyond the cdf just below the cap must be the cap itself.

Common Checks Before Reporting a Percentile

A percentile answer should be supported by a cdf check, especially if the payment distribution has atoms. For a candidate $q$, verify $F_{Y^L}(q)\ge p$. If $q$ is an atom, also verify that the cdf immediately below $q$ is less than $p$; otherwise an earlier payment may already be the quantile. For a continuous section, solve the cdf equation only within the section's valid range.

At zero: $F_{Y^L}(0)=P(X\le d)$. A requested percentile at or below this mass has quantile zero.
At an interior atom: include the full probability mass at the payment value. A percentile can be reached at that atom even if no continuous solution equals $p$.
At the cap: compare $p$ with the cdf immediately below $u$. If that cdf is below $p$, the quantile is $u$.
At a threshold: use the contract's precise payment function. With the model here, a loss exactly equal to $d$ pays zero, and a loss exactly equal to $d+u$ pays $u$.

The central exam skill is not merely solving an inverse-cdf equation. It is determining which part of the payment distribution contains the requested percentile, including whether it lies at an atom or at the cap. This method is aligned with the SOA Exam P (Probability) syllabus.

Check Your Understanding

For each hypothetical one-period policy, use the stated payment function $Y^L=\min((X-d)_+,u)$. There is no discounting or mortality model. Check any jump before solving a continuous cdf equation.

$X$ takes values $\$100,\$200,\$400$ with probabilities $0.20,0.30,0.50$, respectively. The deductible is $d=\$150$ and the payment cap is $u=\$200$. Find the 40th percentile. (Answer: $\$50$)
$X$ is uniform on $[\$0,\$600]$, with $d=\$100$ and $u=\$250$. Find the 50th percentile. (Answer: $\$200$)
With probability $0.20$, $X=\$500$; otherwise $X$ is uniform on $[\$0,\$1{,}000]$. For $d=\$200$ and $u=\$400$, find the 25th percentile. (Answer: $\$112.50$)
$X$ is exponential with mean $\$200$, so its rate is $1/200$ per dollar. If $d=\$100$ and $u=\$300$, find the 80th percentile to the nearest cent. (Answer: $\$221.89$)
$X=\$200$ with probability $0.40$ and $X=\$500$ with probability $0.60$. With $d=\$200$ and $u=\$200$, find the median under the lower-quantile definition. (Answer: $\$200$)

What You'll Learn