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 $8 million vs.$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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