Loss Quantiles and Confidence Levels

Why a Loss Quantile Has Two Probabilities

A loss quantile identifies a threshold on the loss scale. Its confidence level describes the probability accumulated at or below that threshold. The probability remaining above the threshold is the upper-tail probability. These two probabilities are complements, but they refer to different sides of the same threshold. Confusing them can lead to selecting a loss threshold that is far too low.

For example, a 95% confidence level means that 95% of the modeled loss distribution is at or below the specified quantile, under the chosen quantile definition. The corresponding upper-tail probability is 5%. It does not mean that the probability of exceeding the quantile is 95%.

Definition: Let $X$ be a loss with cumulative distribution function $F_X(x)=P(X\le x)$. For a confidence level $0<\alpha<1$, the loss quantile is $$ q_\alpha(X)=\inf\{x:F_X(x)\ge \alpha\}. $$ Thus, $q_\alpha(X)$ is the smallest loss threshold at which the cumulative probability reaches or exceeds $\alpha$. The corresponding upper-tail probability level is $1-\alpha$.

The infimum definition applies to continuous and discrete loss distributions. For a continuous distribution that increases smoothly near the quantile, the definition usually gives $F_X(q_\alpha)=\alpha$. For a discrete distribution, the CDF can jump over $\alpha$, so it is possible that $F_X(q_\alpha)>\alpha$. The definition deliberately uses “at least” $\alpha$, rather than requiring equality.

A loss quantile is read from the CDF by moving up to probability $\alpha$, then across to the loss scale. On a loss-tail diagram, the same threshold is marked on the horizontal loss axis, with the region to its right representing losses above the threshold. The tail probability associated with strict exceedance is $P(X>q_\alpha)=1-F_X(q_\alpha)$. If there is probability at the threshold itself, the probability of a loss at least as large as the threshold is instead $P(X\ge q_\alpha)$, which can be greater.

\begin{array}{c} \text{CDF view: } \quad F_X(q_\alpha)\ge \alpha \\[4pt] \text{Loss axis: } \quad \underbrace{\hspace{1.2cm}}_{\text{probability at or below }q_\alpha} \quad \big|_{q_\alpha} \quad \underbrace{\hspace{1.2cm}}_{\text{probability above }q_\alpha} \\[4pt] \text{Target tail level: } \quad 1-\alpha \end{array}

The line at $q_\alpha$ separates the two regions. For continuous losses, the probability exactly on that line is zero, so “above” and “at least” have the same probability. For discrete losses, an outcome can occur exactly at the threshold. In that case, the CDF includes the probability at the threshold, while the strict upper tail does not.

Reading the Confidence Level and Tail Level

The CDF $F_X(x)$ answers the question, “What is the probability that the loss is no greater than $x$?” Its complement $1-F_X(x)$ answers, “What is the probability that the loss is greater than $x$?” At a loss quantile, the confidence level $\alpha$ is the target cumulative probability; $1-\alpha$ is the corresponding nominal probability left in the upper tail.

Confidence level $\alpha$	Upper-tail level $1-\alpha$	Interpretation
0.90	0.10	90% of losses are at or below the quantile; the target upper tail is 10%
0.95	0.05	95% of losses are at or below the quantile; the target upper tail is 5%
0.99	0.01	99% of losses are at or below the quantile; the target upper tail is 1%

For a continuous distribution with no probability concentrated at the threshold, the probability strictly above $q_\alpha$ is exactly $1-\alpha$. For a discrete distribution, the generalized-inverse definition ensures $F_X(q_\alpha)\ge\alpha$, but the CDF may have jumped past $\alpha$. Then the actual strict exceedance probability can be less than $1-\alpha$. The confidence level still identifies the quantile correctly; the difference comes from probability concentrated at one or more loss values.

Worked Quantiles

Worked Example: A Quantile from a Continuous Loss Model

A hypothetical equipment insurer models one contract’s annual loss $X$ as uniform on the interval from $0 to $20,000. Find the 97.5% loss quantile and the associated upper-tail probability.

Method. Use the uniform CDF to solve $F_X(q_\alpha)=\alpha$. The common mistake to avoid is using 0.025 as the cumulative probability, even though it is the upper-tail probability.

For $0\le x\le 20{,}000$, the uniform model has CDF $F_X(x)=x/20{,}000$. Here, $\alpha=0.975$, so

\begin{aligned} F_X(q_{0.975})&=0.975,\\ \frac{q_{0.975}}{20{,}000}&=0.975,\\ q_{0.975}&=0.975(20{,}000)=19{,}500. \end{aligned}

The quantile is $19,500. The upper-tail probability is $1-\alpha=1-0.975=0.025$, or 2.5%. Checking directly, $F_X(19{,}500)=19{,}500/20{,}000=0.975$, and $P(X>19{,}500)=500/20{,}000=0.025$. Because the uniform distribution is continuous, $P(X\ge19{,}500)$ is also 0.025.

Worked Example: A Quantile at a Discrete Probability Mass

A hypothetical small portfolio has annual loss $X$ with the following distribution: loss $0 with probability 0.55, loss $12,000 with probability 0.25, loss $40,000 with probability 0.15, and loss $100,000 with probability 0.05. Find the 80% quantile, then calculate the probabilities of a loss strictly greater than and at least as great as that quantile.

Method. Accumulate the probabilities from the smallest loss upward and select the first loss at which the CDF is at least 0.80. The common mistake to avoid is assuming the CDF must equal 0.80 exactly.

At loss zero, the cumulative probability is 0.55. At loss $12,000, it is $0.55+0.25=0.80$. Thus, the first loss reaching the target is $12,000, so $q_{0.80}(X)=\$12{,}000$.

\begin{aligned} F_X(0)&=0.55,\\ F_X(12{,}000)&=0.55+0.25=0.80,\\ P(X>12{,}000)&=0.15+0.05=0.20,\\ P(X\ge12{,}000)&=0.25+0.15+0.05=0.45. \end{aligned}

The strict upper-tail probability is 0.20, which equals $1-0.80$ in this example. The inclusive probability is 0.45 because it also counts the 0.25 probability of a loss exactly equal to $12,000. The check $F_X(12{,}000)=0.80$ confirms the quantile; the mass at the threshold explains the difference between strict and inclusive exceedance.

Worked Example: Quantiles from Equally Likely Loss Scenarios

A hypothetical risk team assigns equal probability to ten annual loss scenarios, measured in thousands of dollars. The losses, after sorting from smallest to largest, are 4, 7, 9, 9, 13, 16, 22, 22, 31, and 48. Find the 70% and 90% quantiles, and interpret the strict upper-tail probabilities.

Method. Each scenario has probability $1/10=0.10$, so the empirical CDF increases by 0.10 at each ordered observation, counting repeated losses with their full probability. The common mistake to avoid is interpolating between scenario losses when the stated model is a discrete distribution.

At the 70% level, the first seven ordered scenarios account for cumulative probability $7/10=0.70$. The seventh loss is 22, so $q_{0.70}=\$22{,}000$. At the 90% level, the first nine scenarios account for $9/10=0.90$. The ninth loss is 31, so $q_{0.90}=\$31{,}000$.

\begin{aligned} F_X(22{,}000)&=\frac{8}{10}=0.80, & P(X>22{,}000)&=\frac{2}{10}=0.20,\\ F_X(31{,}000)&=\frac{9}{10}=0.90, & P(X>31{,}000)&=\frac{1}{10}=0.10. \end{aligned}

At $22,000, eight scenarios are at or below the threshold because $22,000 occurs twice; therefore the CDF is 0.80, not 0.70. That value is still at least 0.70, and no smaller loss reaches 0.70. The strict tail at $22,000 is 0.20, whereas the nominal level $1-0.70$ is 0.30. At $31,000, the CDF is 0.90 and the strict tail is 0.10. Both results agree with the generalized-inverse definition and illustrate how ties can make the CDF jump over a target level.

Why the Quantile Formula Works

The definition is a direct way to turn a cumulative probability target into a loss threshold. To see why it captures the intended threshold, consider the set of losses $x$ for which $F_X(x)\ge\alpha$. As $x$ increases, $F_X(x)$ cannot decrease. The set therefore begins at the first loss level where the cumulative probability has reached the target. Taking the infimum selects that boundary, rather than a larger threshold that would include unnecessary loss values.

Set the target cumulative probability.
Choose the confidence level $\alpha$, such as 0.95. This is the probability target on the CDF scale.

Locate where the CDF reaches the target.
Find the smallest loss $x$ for which $F_X(x)\ge\alpha$. For a discrete model, check the CDF at each possible loss.

Read the upper tail separately.
The nominal tail level is $1-\alpha$. The actual strict exceedance probability at the selected threshold is $1-F_X(q_\alpha)$, which can be smaller for a discrete distribution.

This reasoning also explains why a quantile should be understood as a threshold, not as a loss outcome that must occur with probability $\alpha$. The probability $\alpha$ is accumulated over all losses at or below the threshold. A single threshold may itself have zero probability in a continuous model or positive probability in a discrete model.

Common Interpretation Traps

Putting $1-\alpha$ on the CDF scale. For a 99% confidence level, the target CDF probability is 0.99, not 0.01. The 0.01 is the nominal upper-tail probability.
Equating “at or below” with “below.” The CDF uses $P(X\le x)$. At a discrete threshold, its value includes any probability mass at that loss.
Assuming the tail probability must equal $1-\alpha$. That equality holds when the CDF at the quantile is exactly $\alpha$, as is typical for a continuous model. A discrete jump may make $F_X(q_\alpha)>\alpha$, reducing $P(X>q_\alpha)$.
Ignoring the direction of the tail. Loss quantiles use the lower cumulative probability to locate a threshold, then the right-hand side of the loss axis to interpret unusually large losses.

Key takeaway: A confidence level $\alpha$ is a cumulative probability, not an upper-tail probability. The loss quantile is the smallest threshold whose CDF is at least $\alpha$; $1-\alpha$ is the nominal probability assigned to the upper tail, with exact exceedance probabilities depending on whether there is mass at the threshold.

Check Your Understanding

For each problem, identify the cumulative target before calculating or interpreting the upper tail.

A hypothetical loss is uniform from $0 to $50,000. Find its 92% quantile and the probability of a loss strictly above it. (Answer: $46,000; upper-tail probability 0.0800.)
A hypothetical loss is $0 with probability 0.40, $5,000 with probability 0.35, and $20,000 with probability 0.25. Find the 75% quantile and the probabilities of a loss strictly greater than, and at least as great as, the quantile. (Answer: $5,000; strictly greater 0.2500; at least as great 0.6000.)
Eight equally likely hypothetical loss scenarios, in thousands of dollars, are 3, 5, 5, 8, 11, 14, 14, and 30. Find the 75% quantile and its strict upper-tail probability. (Answer: $14,000; upper-tail probability 0.1250.)
A hypothetical discrete loss model has $F_X(\$18{,}000)=0.94$, and the CDF next jumps to 0.97 at $20,000. Find the 95% quantile and the strict probability of exceeding it. (Answer: $20,000; strict exceedance probability 0.0300.)
For a 99% confidence level, state the target cumulative probability and the nominal upper-tail probability. (Answer: cumulative probability 0.9900; nominal upper-tail probability 0.0100.)

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