Expected Loss for Discrete Severity

Why Expected Loss Matters

A discrete severity model assigns probabilities to a set of possible loss amounts. Its expected loss summarizes the model with one dollar amount: the probability-weighted average loss for an individual claim. It is not generally a loss that must occur, nor does it predict the exact amount of any particular claim. Rather, over many claims generated under the same model, the average loss per claim tends toward this value.

The distinction between a per-claim average and a total for a portfolio is important. Expected loss for one severity variable describes one individual loss. It does not, by itself, account for how many claims occur.

Definition: Suppose a discrete loss $X$ can take the distinct values $x_1,x_2,\ldots$, with point probabilities $p_i=P(X=x_i)$. Its expected loss, or mean severity, is $$ E[X]=\sum_i x_i p_i, $$ provided the sum is finite. The probabilities must satisfy $p_i\geq 0$ and $\sum_i p_i=1$. For a finite support, the expected loss is the sum of each possible loss amount multiplied by its point probability.

Loss amounts and the expected loss must use the same units. If the amounts are given in dollars, the result is in dollars; if the amounts are in thousands of dollars, the result is in thousands of dollars. A zero-loss outcome contributes zero to the weighted sum, but its probability still belongs in the probability total.

Why the Weighted-Sum Formula Holds

For a finite discrete model, let $I_i$ equal 1 when $X=x_i$, and 0 otherwise. Exactly one of these indicators equals 1 on any outcome, so the loss can be written as $X=\sum_i x_i I_i$. By linearity of expectation, $E[X]=\sum_i x_iE[I_i]$. Since $I_i$ is an indicator of the event $X=x_i$, its expectation is $P(X=x_i)=p_i$. Therefore,

E[X]=\sum_i x_i p_i.

This argument does not require independence: the indicators for the possible outcomes are mutually exclusive, but linearity of expectation applies whether or not random variables are independent. In the discrete severity setting, the formula is simply a weighted average, with the probabilities as the weights.

Identify the possible losses.
Read the support values carefully and keep their measurement units visible.

Check the point probabilities.
Confirm they are nonnegative and sum to 1. If a probability is missing, determine it before taking the weighted sum.

Multiply and add.
For each possible value, multiply the loss amount by its point probability, then sum the products.

Interpret and check.
State the answer per individual claim and check that it lies between the smallest and largest possible losses.

Calculating Expected Loss from a Probability Table

The table should list point probabilities, meaning probabilities of the individual outcomes $X=x_i$. Do not substitute cumulative probabilities $F_X(x_i)=P(X\leq x_i)$ as if they were point probabilities; CDF values combine probability at the current value with probability at all smaller values.

Worked Example: A Discrete Equipment Claim

A hypothetical equipment policy has an individual claim severity $X$ with the following distribution. Calculate the expected loss per claim and interpret the result.

Loss $x_i$ (dollars)	Point probability $P(X=x_i)$
0	0.20
500	0.50
1,500	0.20
4,000	0.10

Method: probability-weighted sum. First check the probabilities:

$$ 0.20+0.50+0.20+0.10=1.00. $$

The table is a valid probability distribution. Multiply each amount by its point probability:

\begin{aligned} E[X] &=(0)(0.20)+(500)(0.50)+(1{,}500)(0.20)+(4{,}000)(0.10)\\ &=0+250+300+400\\ &=\$950. \end{aligned}

The expected loss is $950 per individual claim. Under this model, the long-run average severity across many claims is $950 per claim; it does not mean that a particular claim will be exactly $950. As a check, $950 is between the smallest possible loss, $0, and the largest, $4,000.

Common mistake avoided: Dropping the zero-loss outcome from the probability check because it contributes zero to the sum. Its weighted contribution is indeed $(0)(0.20)=0$, but its probability remains part of the model and the total probability must still equal 1.

When a Probability Is Missing

A table may provide all but one of the point probabilities. Since the probabilities of all possible outcomes must sum to 1, the missing probability is 1 minus the sum of those provided. Calculate it before finding the mean. This also gives an immediate check that the entries describe a complete distribution.

Worked Example: Completing a Discrete Loss Distribution

A hypothetical cargo policy has possible individual losses of $200, $800, $2,000, and $5,000. The stated point probabilities for the first three losses are 0.15, 0.35, and 0.40. The remaining probability is assigned to the $5,000 loss. Find the expected loss per claim.

Method: complete the probability table, then take the weighted sum. The probabilities already given total

$$ 0.15+0.35+0.40=0.90. $$

Therefore, the probability at $5,000 is $1-0.90=0.10$. Check that the completed probabilities sum to 1:

$$ 0.15+0.35+0.40+0.10=1.00. $$

Now multiply each loss by its point probability and add:

\begin{aligned} E[X] &=(200)(0.15)+(800)(0.35)+(2{,}000)(0.40)+(5{,}000)(0.10)\\ &=30+280+800+500\\ &=\$1{,}610. \end{aligned}

The expected loss is $1,610 per claim. It lies between $200 and $5,000, as expected for a weighted average of those possible losses.

Common mistake avoided: Using 0.90 as the probability of the $5,000 outcome. The 0.90 is the total probability already assigned to the other three losses; the remaining probability is $1-0.90=0.10$.

Finding Point Probabilities from a CDF

Sometimes the information is given as CDF values rather than as a probability table. For a discrete severity model, the point probability at a value is the jump in the CDF there. If the support values are ordered $x_1<x_2<\cdots<x_n$, then the probability at $x_1$ is $F_X(x_1)$, and at each later support value $x_i$ it is $F_X(x_i)-F_X(x_{i-1})$. The final CDF value should be 1 when the listed support includes every possible outcome.

Worked Example: Calculating the Mean from CDF Values

A hypothetical repair claim severity $X$ has support $0, $1,000, $3,000, and $7,000. Its CDF is given at these values by $F_X(0)=0.10$, $F_X(1{,}000)=0.45$, $F_X(3{,}000)=0.85$, and $F_X(7{,}000)=1.00$. Calculate the expected loss.

Method: recover the point probabilities from the CDF jumps. At $0, the point probability is the CDF value itself. At later values, subtract the preceding CDF value:

\begin{aligned} P(X=0)&=0.10,\\ P(X=1{,}000)&=0.45-0.10=0.35,\\ P(X=3{,}000)&=0.85-0.45=0.40,\\ P(X=7{,}000)&=1.00-0.85=0.15. \end{aligned}

Check the recovered probabilities:

$$ 0.10+0.35+0.40+0.15=1.00. $$

Now calculate the weighted sum:

\begin{aligned} E[X] &=(0)(0.10)+(1{,}000)(0.35)+(3{,}000)(0.40)+(7{,}000)(0.15)\\ &=0+350+1{,}200+1{,}050\\ &=\$2{,}600. \end{aligned}

The expected loss is $2,600 per claim. The answer is between $0 and $7,000, the smallest and largest possible losses.

Common mistake avoided: Multiplying each loss by the CDF value at that loss. For example, 0.45 is $P(X\leq 1{,}000)$, not $P(X=1{,}000)$. The probability of exactly $1,000 is the jump $0.45-0.10=0.35$.

Interpretation, Checks, and Common Traps

Expected loss is a model-based long-run average. Imagine many independent claims generated from the same severity distribution and calculate the arithmetic average loss across them. As the number of claims grows, that average tends toward $E[X]$, provided the expected loss is finite. This interpretation explains why the mean is useful for summarizing a severity model, but it does not imply that every claim, or even most claims, will be close to the mean.

For a finite set of possible losses, the expected loss must be between the smallest and largest support values. This is because it is a weighted average with nonnegative weights that sum to 1. A result outside that range usually signals an arithmetic error, an invalid probability table, or a unit mismatch. If the support is countably infinite, the same weighted-sum definition applies when the sum is finite; for nonnegative losses the sum may instead be infinite.

Using cumulative probabilities as weights: Convert CDF values to point probabilities by taking successive differences.
Forgetting to normalize or complete the table: Check that all point probabilities sum to 1 before calculating the mean.
Mixing units: Keep dollars, thousands of dollars, or another stated unit consistent through every product and the final answer.
Calling the mean a guaranteed payment: Expected loss is an average across outcomes, not a promise about an individual claim.
Changing the quantity being averaged: The result is per individual claim under the specified severity model, not an expected total for a group of claims.

Key takeaway: For a discrete severity model, multiply each possible loss by its point probability and add the products. First verify the probability weights, preserve the loss units, and interpret the result as a long-run average per individual claim.

Check Your Understanding

For each problem, calculate the expected loss per individual claim. Check the probabilities and keep the stated units in your calculation.

A hypothetical appliance claim has losses of $0, $250, and $1,000 with probabilities 0.30, 0.50, and 0.20, respectively. What is the expected loss? (Answer: $325 per claim)
A hypothetical bicycle policy has possible losses of $400, $1,200, and $3,600 with probabilities 0.25, 0.50, and 0.25. Calculate the mean severity and state whether it is between the smallest and largest possible losses. (Answer: $1,600 per claim)
A hypothetical service contract has losses of $0, $1,000, $2,500, and $5,000. The first three point probabilities are 0.08, 0.42, and 0.30; all remaining probability is at $5,000. Complete the distribution and calculate the expected loss. (Answer: The remaining probability is 0.20; expected loss is $2,170 per claim)
A hypothetical claim severity has support $0, $2,000, and $6,000, with CDF values $F_X(0)=0.12$, $F_X(2{,}000)=0.62$, and $F_X(6{,}000)=1.00$. Recover the point probabilities and calculate the mean. (Answer: Point probabilities are 0.12, 0.50, and 0.38; expected loss is $3,280 per claim)
A hypothetical tool-replacement policy has losses of $750, $1,750, and $4,250 with probabilities 0.40, 0.45, and 0.15. A reviewer incorrectly reports the mean as $1,750 because that outcome has the largest probability. Calculate the correct expected loss and explain the error. (Answer: $1,725 per claim; the most probable loss is not necessarily the probability-weighted mean)

What You'll Learn