Quantiles for Discrete and Continuous Losses

Why Loss Quantiles Matter

A loss quantile describes a threshold in the loss distribution: at a selected probability level, how large a loss is needed for the cumulative probability to reach that level? For example, a risk analyst may need a threshold that is exceeded by only a small fraction of modeled outcomes. Calculating it correctly requires attention to the entire cumulative distribution, especially when the loss can take only selected values or has a probability mass at a particular amount.

This tutorial treats a loss $X$ as a nonnegative random variable measured in stated monetary units. A quantile is determined by a probability level and the cumulative distribution function, not by an average or by a visual interpolation between possible losses.

Definition: Let $F_X(x)=P(X\le x)$ be the cumulative distribution function of loss $X$. For a probability level $p$ with $0<p<1$, the lower $p$-quantile is $$ q_p(X)=\inf\{x:F_X(x)\ge p\}. $$ It is the smallest loss amount at which the cumulative probability is at least $p$. In risk measurement, this loss quantile is also commonly called the $p$-level Value-at-Risk, or $\operatorname{VaR}_p(X)$.

The word “smallest” matters. A cumulative distribution function is nondecreasing, but it need not increase smoothly. It can jump at losses with positive probability, and it can remain flat between possible values. The lower-quantile definition works for both discrete and continuous losses. Unless stated otherwise, this tutorial uses that convention.

Finding Quantiles From a Discrete Distribution

For a discrete loss, list the possible loss amounts in increasing order and add their probabilities cumulatively. The quantile is the first listed loss at which the cumulative probability reaches or exceeds $p$. Equality counts: if the cumulative probability is exactly $p$, that loss is the quantile.

q_p(X)=\min\{x_j:F_X(x_j)\ge p\}

Here $x_1<x_2<\cdots$ are the possible loss amounts with positive probability. The minimum exists for the usual discrete distributions whose cumulative probability eventually reaches 1. The rule does not mean to interpolate between adjacent losses: the loss variable cannot take an intermediate value unless that value is in its support.

Worked Example: A Discrete Portfolio Loss Table

A hypothetical portfolio has one-year loss $X$, measured in thousands of dollars. Its possible losses and probabilities are $0$ with probability $0.50$, $10$ with probability $0.20$, $25$ with probability $0.20$, and $60$ with probability $0.10$. Find $q_{0.7500}(X)$, $q_{0.9000}(X)$, and $q_{0.9100}(X)$. State the method and check the results.

Method. Order the losses and calculate the cumulative probabilities. First, the probabilities sum to $0.50+0.20+0.20+0.10=1.00$, so the table is a valid probability distribution.

Loss $x$ (thousands of dollars)	$P(X=x)$	$F_X(x)=P(X\le x)$
0	0.50	0.50
10	0.20	0.70
25	0.20	0.90
60	0.10	1.00

For $p=0.7500$, the cumulative probability at 10 is $0.70<0.7500$, while at 25 it is $0.90\ge0.7500$. Therefore $q_{0.7500}(X)=25$ thousand dollars.

For $p=0.9000$, the cumulative probability at 25 is exactly $0.9000$, so $q_{0.9000}(X)=25$ thousand dollars. For $p=0.9100$, the cumulative probability at 25 is too small, while at 60 it is $1.00$, so $q_{0.9100}(X)=60$ thousand dollars.

Check. At 25, $F_X(25)=0.90$, which reaches both 0.7500 and 0.9000; at the immediately preceding possible loss, 10, the cumulative probability is only 0.70. At 60, $F_X(60)=1.00$, and the preceding cumulative probability is 0.90, below 0.9100. Thus all three answers meet the definition and are the smallest qualifying losses.

Common mistake avoided. Interpolating between 10 and 25 for the 0.7500 quantile would produce $10+(0.75-0.70)(25-10)/(0.90-0.70)=13.75$ thousand dollars. That is not a possible loss in this model, and $F_X(13.75)=0.70$, which is below 0.7500. The quantile is 25, not 13.75.

Finding Quantiles From a Continuous Distribution

For a continuous loss with a strictly increasing cumulative distribution function over the relevant range, the quantile is found by solving $F_X(x)=p$. Strict increase ensures there is only one solution. If the cumulative distribution is continuous but has flat portions, use the lower-quantile definition: a value satisfying $F_X(x)=p$ may not be unique, and the quantile is the smallest qualifying value.

Result: If $F_X$ is continuous and strictly increasing at the probability level $p$, then $q_p(X)$ is the unique solution to $$ F_X(q_p(X))=p. $$ After solving, check that the result is in the support of $X$ and that substituting it into $F_X$ returns $p$.

Worked Example: Quantile of an Exponential Loss

Suppose a hypothetical one-year loss $X$, measured in thousands of dollars, has an exponential distribution with scale parameter $\theta=20$ thousand dollars. Thus $X\ge0$, its density is $f_X(x)=\frac{1}{20}e^{-x/20}$ for $x\ge0$, and its cumulative distribution is $F_X(x)=1-e^{-x/20}$ for $x\ge0$. Find the 0.9500-quantile in dollars, rounded to the nearest cent.

Method. This continuous exponential cumulative distribution is strictly increasing for $x\ge0$, so solve $F_X(x)=0.9500$:

1-e^{-x/20}=0.9500

Rearrange in steps:

e^{-x/20}=1-0.9500=0.0500

-\frac{x}{20}=\ln(0.0500)

x=-20\ln(0.0500)=20(2.9957322736)=59.91464547

The result is in thousands of dollars, so the loss amount is $59.91464547\times \$1{,}000=\$59{,}914.64547$. Rounding only at the final step to the nearest cent gives $q_{0.9500}(X)=\$59{,}914.65$.

Check. At the unrounded value in thousands, $e^{-x/20}=0.0500$, so $F_X(x)=1-0.0500=0.9500$. The result is nonnegative and therefore lies in the model’s support.

Common mistake avoided. The scale parameter is 20 thousand dollars, not 20 dollars. The calculation produces $x$ in thousands because the model is specified in those units; converting to dollars is a separate final step.

Why the Quantile Rule Works

The definition follows directly from the cumulative distribution function’s role. $F_X(x)$ is the probability accumulated at or below $x$. As $x$ increases, that probability cannot decrease. A quantile therefore marks the first point where the accumulated probability meets the target $p$.

Start below the threshold.
For any $x$ below $q_p(X)$, the definition of the infimum and the nondecreasing nature of $F_X$ mean that $F_X(x)<p$; otherwise an earlier point would already qualify.

Reach the threshold.
At the lower quantile, the cumulative probability has reached or passed $p$: $F_X(q_p(X))\ge p$, for the ordinary distributions considered here.

Apply the rule to a discrete loss.
The cumulative probability stays constant between possible losses and jumps at a loss with positive probability. The first support point whose cumulative probability is at least $p$ is consequently the smallest qualifying amount.

Apply the rule to a strictly increasing continuous loss.
There is no jump to carry $F_X$ past $p$; continuity gives a point where the cumulative probability equals $p$. Strict increase makes that solution unique, so solving $F_X(x)=p$ yields the quantile.

This reasoning also explains why different quantile conventions can matter at jumps. A continuous distribution with a strictly increasing cumulative distribution has an unambiguous inverse at the selected probability. A discrete distribution generally has an interval of probability levels that all map to the same loss amount. The lower-quantile convention specifies which value to report consistently.

Quantiles When the Distribution Has a Jump

A jump in $F_X$ occurs when $X$ has positive probability at a particular loss amount. If $X$ has probability $m$ at $x_0$, then $F_X(x_0)-F_X(x_0-)=m$, where $F_X(x_0-)$ is the cumulative probability immediately to the left of $x_0$. Every probability level strictly above $F_X(x_0-)$ and at or below $F_X(x_0)$ has lower quantile $x_0$.

F_X(x_0-)<p\le F_X(x_0) \quad\Longrightarrow\quad q_p(X)=x_0

The condition includes the upper endpoint because the cumulative probability at $x_0$ includes the entire mass at $x_0$. It excludes the lower endpoint: if $p=F_X(x_0-)$, an earlier loss may already be the smallest value at which the cumulative probability reaches $p$.

Worked Example: A Loss With a Point Mass at Zero

A hypothetical policy portfolio has a 0.3000 probability of no loss. With probability 0.7000, its positive loss $X$, in thousands of dollars, is uniformly distributed between 0 and 100. Find $q_{0.2000}(X)$, $q_{0.3000}(X)$, and $q_{0.6500}(X)$. Treat the zero-loss probability as a point mass, and round monetary answers to the nearest cent.

Method. The model has a probability mass of 0.3000 at zero, plus a continuous component. For $0\le x\le100$, the conditional probability that the positive loss is at most $x$ is $x/100$. Therefore, including the probability that there is no loss,

F_X(x)=0.3000+0.7000\left(\frac{x}{100}\right), \qquad 0\le x\le100.

For $x<0$, $F_X(x)=0$, and at zero $F_X(0)=0.3000$. Thus $q_{0.2000}(X)=0$, since the cumulative probability at zero already exceeds 0.2000. Likewise, $q_{0.3000}(X)=0$, since $F_X(0)=0.3000$ exactly.

For $p=0.6500$, solve the cumulative distribution equation on the continuous part:

0.3000+0.7000\left(\frac{x}{100}\right)=0.6500

Subtracting 0.3000 gives $0.7000(x/100)=0.3500$. Dividing by 0.7000 gives $x/100=0.5000$, so $x=50$ thousand dollars. Converting units, $q_{0.6500}(X)=\$50{,}000.00$.

Check. The CDF at zero is 0.3000, so both lower quantiles are zero and no negative loss can qualify. At 50 thousand dollars, $F_X(50)=0.3000+0.7000(50/100)=0.6500$, as required. All reported monetary amounts are rounded to the nearest cent.

Common mistake avoided. The 0.7000 probability belongs to the positive-loss component; it is not the CDF at zero. Ignoring the 0.3000 mass at zero and solving $x/100=0.6500$ would give 65 thousand dollars, which is incorrect. The continuous component’s cumulative probability must be weighted by its probability of occurring and added to the zero-loss mass.

Common Quantile Errors

Quantile calculations are short, but they are easy to misread when probabilities are near a jump or when amounts use different units. Use these checks before reporting a result:

Do not interpolate between discrete outcomes. The lower quantile must be a loss amount at which the cumulative probability reaches $p$, and for a discrete model it must be a support point.
Use “at least,” not “strictly greater than,” in the definition. If $F_X(x)=p$, that value qualifies. In a discrete table, equality can determine the quantile.
Account for point masses. A probability of no loss, a deductible structure, or another feature of the model may produce an atom. The jump can carry the cumulative probability across many possible values of $p$.
Keep the units attached to the calculation. A parameter or quantile expressed in thousands of dollars must be converted before giving a dollar answer.
Check both sides when possible. Verify that the cumulative probability at the proposed quantile is at least $p$, and that earlier support values have cumulative probability below $p$.

Key takeaway: A loss quantile is the smallest loss amount whose cumulative probability is at least the selected probability level. Use cumulative sums for discrete losses, invert the CDF for a strictly increasing continuous loss, and apply the same lower-quantile definition when a jump is present.

Check Your Understanding

For each problem, use the lower-quantile definition and check the relevant cumulative probability or distribution formula.

A hypothetical equipment-loss model has losses of 0, 5, 20, and 50 thousand dollars with probabilities 0.40, 0.35, 0.20, and 0.05, respectively. Calculate $q_{0.7000}(X)$ and $q_{0.9500}(X)$. (Answer: 5 thousand dollars and 20 thousand dollars, respectively.)
A hypothetical annual loss is uniform on the interval from 0 to 80 thousand dollars, with no probability mass at either endpoint. Find the 0.8750-quantile and report it in dollars. (Answer: \$70,000.00.)
A hypothetical loss in thousands of dollars has an exponential distribution with scale parameter 12. Find its 0.9000-quantile in dollars, rounded to the nearest cent. (Answer: \$27,631.02.)
A hypothetical loss has probability 0.4000 at zero; conditional on a positive loss, it is uniform from 0 to 60 thousand dollars. Find $q_{0.4000}(X)$ and $q_{0.7000}(X)$, in dollars. (Answer: \$0.00 and \$30,000.00, respectively.)
A discrete hypothetical loss has support 2, 8, 15, and 40 thousand dollars with probabilities 0.15, 0.45, 0.25, and 0.15, respectively. Find $q_{0.6000}(X)$ and $q_{0.6001}(X)$. Explain why the two answers differ despite the nearby probability levels. (Answer: 8 thousand dollars and 15 thousand dollars; the cumulative probability jumps from 0.15 to 0.60 at 8, then does not reach 0.6001 until 15.)

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