The Severity Survival Function

Why Focus on the Probability Above a Threshold?

In severity modeling, a question such as “What is the probability that one claim exceeds $5,000?” is about the portion of the loss distribution above a specified amount. This probability can be important when assessing the likelihood of unusually large claims, comparing thresholds, or describing the tail of a hypothetical portfolio’s claim severity. The survival function gives this exceedance probability directly.

The threshold wording matters. “Exceeds $5,000” means strictly greater than $5,000; a claim of exactly $5,000 does not count. This distinction is especially important when severity can take particular values with positive probability. In that setting, a probability table may assign a substantial probability to a claim exactly at the threshold.

Definition: For a claim severity random variable $X$, the survival function is $$ S(x)=P(X>x). $$ It gives the probability that one individual claim is strictly greater than the amount $x$. The threshold $x$ and the severity $X$ must be measured in the same units.

The survival function does not count claims; it describes the probability for a single claim severity. For nonnegative severity, a negative threshold is exceeded with probability 1. At a nonnegative threshold, the probability depends on the severity model. If the severity has a largest possible value $b$, then $S(b)=0$, because no claim can be strictly greater than $b$.

Properties That Help Check a Tail Probability

The survival function must take values between 0 and 1. It must also be nonincreasing: raising the threshold cannot make it more likely that a claim exceeds that threshold. These facts follow directly from the exceedance events.

0\leq S(x)\leq 1

x_1<x_2 \quad\Longrightarrow\quad \{X>x_2\}\subseteq\{X>x_1\} \quad\Longrightarrow\quad S(x_2)\leq S(x_1).

The set inclusion is the justification: every value greater than the higher threshold $x_2$ is also greater than the lower threshold $x_1$, but a value between the two thresholds exceeds only the lower one. This provides a useful calculation check. For example, a reported probability of exceeding $8,000 should not be larger than the probability of exceeding $6,000.

The strict inequality in the definition also determines how to handle equality at a threshold. The event $X\geq a$ can be partitioned into the disjoint events $X=a$ and $X>a$. Therefore,

P(X\geq a)=P(X=a)+S(a).

This is a direct event decomposition, not an approximation. If there is no probability at exactly $a$, then $P(X=a)=0$ and the two threshold probabilities agree. If there is positive probability at $a$, they differ by exactly that amount.

Calculating the Survival Function from a Discrete Model

For a discrete severity model, identify every possible claim amount strictly greater than the threshold and add their probabilities. Do not include the probability at the threshold itself. The probabilities in a complete discrete model sum to 1, which is a helpful check that the table has been read correctly.

Worked Example: A Threshold That Is a Possible Claim Amount

A hypothetical equipment policy has individual claim severity $X$, in dollars, with the following distribution. Calculate the probability that a claim exceeds $1,000, the probability it exceeds $500, and the probability it is at least $500.

Claim amount $x$	Probability $P(X=x)$
$0	0.10
$500	0.25
$1,000	0.40
$2,500	0.25

Method. For each requested threshold, select the table entries strictly above it and add their probabilities. First check the table: $0.10+0.25+0.40+0.25=1.00$.

Exceeds $1,000. The only listed amount strictly greater than $1,000 is $2,500. Hence $$ S(1000)=P(X>1000)=0.25. $$

Exceeds $500. The amounts strictly greater than $500 are $1,000 and $2,500. Thus $$ S(500)=P(X>500)=0.40+0.25=0.65. $$

At least $500. This event includes the $500 claim as well as the amounts above it: $$ P(X\geq 500)=0.25+0.40+0.25=0.90. $$ As a check, $P(X\geq 500)=P(X=500)+S(500)=0.25+0.65=0.90$.

Common mistake avoided. Including the $500 entry when calculating $S(500)$ would give 0.90. That is $P(X\geq 500)$, not the defined strict exceedance probability $P(X>500)$.

Calculating Tail Probabilities from a Model

When a severity model is specified by a formula, substitute the threshold into its survival function, while checking that the threshold lies in the model’s stated range. Two useful examples are a uniform severity on a bounded interval and an exponential severity on the nonnegative amounts.

Worked Example: A Uniform Claim Amount

For a hypothetical coverage, suppose claim severity is uniformly distributed over the interval from $0 to $12,000, including the endpoints. Find the probability that a claim exceeds $7,500, and state the survival probability at $12,000.

Model and method. Here $X\sim\operatorname{Uniform}(0,12000)$. For a threshold $x$ between 0 and 12,000, the values exceeding $x$ occupy the interval from $x$ to 12,000. Under a uniform model, probability is proportional to interval length, so $$ S(x)=\frac{12000-x}{12000},\qquad 0\leq x\leq 12000. $$

At $x=7{,}500$, the length of the amount range above the threshold is $12{,}000-7{,}500=4{,}500$. Therefore, $$ S(7500)=\frac{4500}{12000}=0.3750. $$ The probability that the claim exceeds $7,500 is 0.3750.

At the upper endpoint, no claim amount is greater than $12,000, so $S(12000)=0$. This agrees with the formula, which gives $(12000-12000)/12000=0$. The probabilities decrease as the threshold increases, as required.

Common mistake avoided. Using $7{,}500/12{,}000=0.6250$ would give the fraction of the interval up to $7,500, not the fraction above it. The exceedance event uses the remaining interval length, $4,500.

Worked Example: Exponential Severity at Two Thresholds

A hypothetical repair-cost severity $X$, measured in dollars, follows an exponential model with rate $\lambda=1/4{,}000$ per dollar. Its mean severity is $1/\lambda=\$4{,}000$, and the model’s survival function is $S(x)=e^{-\lambda x}$ for $x\geq0$. Find the probability that a repair cost exceeds $6,000, and the probability that it is greater than $6,000 but no greater than $10,000.

Method. Evaluate the survival function at each threshold. The event $6000<X\leq10000$ is the portion exceeding $6,000 after removing the portion exceeding $10,000.

For the first threshold, $$ S(6000)=e^{-(1/4000)(6000)}=e^{-1.5}=0.2231. $$ Thus, the probability of a repair cost greater than $6,000 is 0.2231.

At the second threshold, $$ S(10000)=e^{-(1/4000)(10000)}=e^{-2.5}=0.0821. $$ Since $\{X>10000\}$ is contained within $\{X>6000\}$, removing the first event from the second leaves $6000<X\leq10000$. Its probability is $$ P(6000<X\leq10000)=S(6000)-S(10000) =0.223130\ldots-0.082085\ldots=0.141045\ldots. $$ Rounded to four decimal places, this probability is 0.1410. As a check, it is positive and smaller than $S(6000)=0.2231$.

Common mistake avoided. The rate is per dollar, so the exponent uses dollar thresholds with $\lambda=1/4{,}000$. Replacing the threshold by 6 or 10 without also expressing the rate per thousand dollars would give an inconsistent exponent.

A Reliable Threshold Calculation

A short sequence of checks prevents most errors. First identify the event in words; then write it as a strict or non-strict inequality. Next use the specified severity model to identify the amounts or range that satisfy the event. Finally check the answer against the threshold ordering and the model’s support.

Translate the wording.
“Exceeds $x$” means $X>x$, while “at least $x$” means $X\geq x$. Equality can matter for discrete severities.

Use the relevant model information.
For a probability table, add probabilities at values strictly above the threshold. For a specified model, evaluate its survival function within its valid range.

Check the result.
A probability must be between 0 and 1. If the threshold is raised, the exceedance probability cannot increase; beyond a finite upper bound, it is zero.

A survival probability is a compact description of how much probability lies above a chosen amount. It is not a claim amount, a percentage of the threshold, or a count of claims. Keeping those interpretations separate is useful when moving from individual severity to more involved actuarial models.

Key takeaway: The severity survival function is $S(x)=P(X>x)$. Evaluate it using amounts strictly above $x$, respect the units and support of the severity model, and account separately for any probability at exactly the threshold.

Check Your Understanding

For each hypothetical severity model, calculate the requested probability. Use strict exceedance exactly as stated.

A claim severity takes values $0, $400, $900, and $1,800 with respective probabilities 0.20, 0.30, 0.35, and 0.15. Find $S(900)$ and $P(X>400)$. (Answer: $S(900)=0.1500$; $P(X>400)=0.5000$)
Claim severity is uniform on the interval from $0 to $20,000. Find the probability that one claim exceeds $12,500. (Answer: 0.3750)
Claim severity follows an exponential model with mean $2,500. Find the probability that one claim exceeds $5,000. (Answer: $e^{-2}=0.1353$)
A discrete severity model has $S(1000)=0.72$ and $S(2500)=0.31$. Find the probability that severity is greater than $1,000 but no greater than $2,500. (Answer: 0.4100)
An exponential severity model has $S(2000)=0.70$. Find $S(8000)$. (Answer: $0.70^4=0.2401$)

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