LESSON 6.3 · ABOUT 20 MINUTES

Payment moments & risk

Calculate the average and variability of what the insurer actually pays.

By the end of this lesson

Calculate payment mean, variance, and standard deviation.
Compare loss and payment moments, including coefficient of variation.

Transform first

For discrete losses, list each loss x, its probability, and its payment g(x). Then E[Y]=Σg(x)p(x) and E[Y²]=Σg(x)²p(x). Distinct losses may produce the same payment.

Continuous losses

Integrate g(x)f(x) for the first moment and g(x)²f(x) for the second. Split the integral at the deductible and cap thresholds. Alternatively, for nonnegative Y, E[Y]=∫₀∞P(Y>y)dy.

Summarize spread

Var(Y)=E[Y²]−E[Y]² and SD(Y)=√Var(Y). For positive mean, CV(Y)=SD(Y)/E[Y]. Deductibles and caps are nonlinear, so you cannot generally subtract a deductible from the mean or leave the loss variance unchanged.

E[Y]=Σg(x)p(x); Var(Y)=E[Y²]−E[Y]²; CV=SD(Y)/E[Y]

Worked example

Losses 0,100,300 have probabilities 0.5,0.3,0.2. Deductible=100; no binding cap. Find payment moments.

Payments are 0,0,200. E[Y]=40; E[Y²]=8000; Var(Y)=8000−1600=6400; SD(Y)=80; CV=2. Payment occurs with probability 0.2, so mean per positive payment=200.