Expected Value
Expected value is the probability-weighted center of a random variable. For actuaries, it is the first version of expected claim cost, expected payment, and model output.
What expected value answers
Expected value answers: if this risk process repeated many times under the same assumptions, what average value would we expect? It is not necessarily the most likely value, a guaranteed value, or even a value the random variable can take.
Actuarial work uses this idea constantly. A pure premium starts as an expected loss. A deductible problem asks for the expected insurer payment. A frequency-severity model uses expected value to turn random counts and random severities into an aggregate loss target.
Core formulas
For a discrete random variable, expected value is a probability-weighted average. For a continuous random variable, the sum becomes an integral against the density. For a transformed variable, average the transformed value, not the original value.
Linearity is the shortcut
Linearity of expectation is often the cleanest exam move. It works even when the random variables are dependent, as long as the expectations exist. Independence matters for many variance shortcuts, but not for adding expectations.
If an insurer writes 1,000 similar policies and the expected loss per policy is 70, the expected total loss is 70,000. That statement does not say the total loss will be close to 70,000 in one year. It says where the long-run average is centered.
Claim amount worked example
Suppose a claim amount X is 0 with probability 0.70, 100 with probability 0.20, and 500 with probability 0.10. Then E[X]=0(0.70)+100(0.20)+500(0.10)=70.
A candidate should read this as an expected claim amount of 70 per exposure under the stated model. It is not saying that 70 is a possible claim size. The support is 0, 100, and 500.
Deductible payment worked example
Now put a 100 ordinary deductible on the same claim amount. The insurer payment is Y=(X-100)_+. That makes the payment 0 when X is 0, 0 when X is 100, and 400 when X is 500. The expected insurer payment is 40.
This is where expected value becomes actuarial instead of abstract. The expected ground-up loss is 70, but the expected payment after the deductible is 40. Exam questions often test whether you define the payment variable before averaging.
Conditional expected value
The law of total expectation lets you average within groups and then average the group means. This is the right frame for mixtures, underwriting classes, claim-status splits, and experience rating.
If 70% of policies have expected claim 500 and 30% have expected claim 2,000, the portfolio mean is 0.70(500)+0.30(2,000)=950. You average the class means by class probability.
Exam traps
The biggest trap is replacing E[g(X)] with g(E[X]). That only works for linear functions. In general, squaring, exponentiating, capping, and applying deductibles must happen before you average.
- For a deductible or limit, define the payment variable first and then take its expectation.
- For mixtures, weight means by group probabilities. Do not average rates or parameters unless the question asks for that.
- For aggregate losses, separate frequency and severity assumptions before using any shortcut.
- For continuous variables, check the support and density before setting integration limits.
Where expected value connects next
Variance starts by asking how far the random variable tends to sit from this center. Conditional probability gives the information structure for conditional expectations. Compound loss models use expected value to move from claim counts and claim sizes to expected aggregate loss.
Original exam practice
3 questions built from syllabus outcomes and released-exam patterns. The prompts and answers are original, so they train the skill without copying official exam text.
Expected Value Claim Payment Drill
Original expected-value checks for claim amounts, deductible payments, and mixture class means.
- Question 1/Calculation
Ground-up expected claim
A claim amount X is 0 with probability 0.82, 200 with probability 0.15, and 1,000 with probability 0.03. Find E[X].
Solution and grading points
E[X] = 0(0.82) + 200(0.15) + 1,000(0.03) = 60. The expected claim amount is 60 per exposure under this model.
- Weights each claim amount by its probability.
- Includes the zero-claim probability without adding cost.
- Interprets 60 as an average, not a possible claim amount.
- Question 2/Calculation
Deductible payment variable
Use the same claim amount X. If the policy has a 200 ordinary deductible and no payment limit, find the expected insurer payment.
Solution and grading points
The payment is Y=(X-200)_+. The payments are 0, 0, and 800, so E[Y] = 800(0.03) = 24.
- Defines the insurer payment before averaging.
- Applies the deductible to the 1,000 claim only.
- Does not subtract 200 from the expected ground-up claim.
- Question 3/Calculation
Mixture class mean
A portfolio is 65 percent standard risks with mean claim 400 and 35 percent high risks with mean claim 1,200. Find the portfolio mean claim.
Solution and grading points
By total expectation, the portfolio mean is 0.65(400) + 0.35(1,200) = 680.
- Uses class probabilities as weights.
- Averages the class means, not individual distribution parameters.
- Recognizes this as a law-of-total-expectation calculation.