Moments of Loss and Payment Random Variables

Expected payment alone does not set the capital requirement. The CFO asks: "What is the standard deviation?" If it is $8millionvs.$2 million, the reserve changes dramatically, even with the same expected cost.

For the payment per loss $Y^L=(X-d{)}_{+}$, the variance requires the second moment:

$$\text{Var}(Y^L)=E[(Y^L{)}^2]-(E[Y^L]{)}^2$$

Compute $E[(Y^L{)}^2]$ using LOTUS:

$$E[((X-d{)}_{+}{)}^2]={\int }_d^{\infty }(x-d{)}^2{f}_X(x)dx$$

HIGH-FREQUENCY: For the exponential with mean $\theta$, the memoryless property gives a clean shortcut:

$$E[((X-d{)}_{+}{)}^2]=S(d)\cdot 2{\theta }^2=e^{-d/\theta }\cdot 2{\theta }^2$$

Given $X>d$, the excess $X-d$ has the same distribution as $X$ itself. Its second moment is $E[X^2]=2{\theta }^2$. You multiply by the probability of exceeding $d$.

$$\text{Var}(Y^L)=2{\theta }^2{e}^{-d/\theta }-(\theta e^{-d/\theta }{)}^2={\theta }^2{e}^{-d/\theta }(2-e^{-d/\theta })$$

For $Y^L=\min ((X-d{)}_{+},u)$, the second moment splits into three regions:

Common mistakes

• Forgetting the cap contribution to the second moment. With $d=500$ and $u=2000$, the second moment has two parts. Omitting $u^2\cdot S(d+u)=1,500,000$ gives $E[(Y^L{)}^2]=666,667$. Then $\text{Var}=666,667-1,562,500<0$. Negative variance signals the error. Trap: 666,667 as the second moment.
• Using $c$ instead of $c^2$ for coinsurance variance. With $c=0.80$, variance multiplies by 0.64, not 0.80. Using 0.80 gives $483,333$ instead of 386,667. Trap: 483,333.
• Computing per-payment variance when asked for per-loss. Per loss includes zero payments from losses below the deductible. For an exponential, $\text{Var}(Y^P)={\theta }^2=640,000$, but $\text{Var}(Y^L)=577,416$. Trap: 640,000.

Bottom line

• Variance of payment: $\text{Var}(Y^L)=E[(Y^L{)}^2]-(E[Y^L]{)}^2$; compute both moments separately.
• Second moment with cap: variable-region integral plus $u^2\cdot S(d+u)$ for the capped region.
• Coinsurance enters squared: $\text{Var}(cY)=c^2\text{Var}(Y)$, so 80% coinsurance multiplies variance by 0.64.
• Exponential shortcut: $E[((X-d{)}_{+}{)}^2]=S(d)\cdot 2{\theta }^2$ via the memoryless property.

Exam shortcut

Split the second-moment computation into three regions: (1) below deductible = 0, (2) variable region = integrate $(x-d{)}^{2}f(x)$, (3) above cap = $u^2\cdot S(d+u)$. Write the three regions before integrating. "Coinsurance squares into variance." $c=0.80$ for the mean becomes $c^2=0.64$ for the variance. "Variance = second moment minus mean squared." Write this identity at the top of every variance problem.

The full lesson (about 3,454 words, 23 min read) adds 5 worked examples, all 8 common mistakes, a self-check, free in the app.

Learning objectives

• 2f

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