Variance, Standard Deviation, and Coefficient of Variation

Two portfolios each expect $10M in annual claims. One has wild year-to-year swings, the other is stable. Variance tells the actuary whether to hold $2M or $20M in surplus.

$$\text{Var}(X)=E[(X-\mu {)}^2]=E[X^2]-(E[X]{)}^2$$

HIGH-FREQUENCY: The shortcut is used in the vast majority of variance calculations on Exam P. Compute $E[X]$ and $E[X^2]$ separately, then subtract.

$$\text{Var}(aX+b)=a^2\text{Var}(X)$$

The constant $b$ disappears. The coefficient $a$ enters squared.

Variance is always non-negative, equaling zero only for constants.

$${\sigma }_X=\sqrt{\text{Var}(X)}$$

Same units as $X$. Under scaling:

$$\text{SD}(aX+b)=|a|\cdot \text{SD}(X)$$

$$\text{CV}(X)=\frac{{\sigma }_X}{{\mu }_X}=\frac{\text{SD}(X)}{E[X]}$$

Meaningful only when $E[X]>0$. The CV is dimensionless and allows comparing variability across distributions with different means.

HIGH-FREQUENCY: Under positive scaling, $\text{CV}(aX)=\text{CV}(X)$. This makes CV the natural measure for comparing loss distributions at different coverage levels.

Common mistakes

• Computing $\text{Var}(aX+b)=a^2\text{Var}(X)+b$. The constant $b$ does not contribute to variance. $\text{Var}(3X+100)=9\text{Var}(X)$, not $9\text{Var}(X)+100$. Trap: adding the constant.
• Using $E[X^2]=(E[X]{)}^2$. For this example: $E[X^2]=9.9$ but $(E[X]{)}^2=8.41$. These are equal only if $\text{Var}=0$. Trap: zero variance.
• Poisson mean vs. variance. Poisson($\lambda =4$): $\text{Var}=4$, not 16. The variance is $\lambda$, not ${\lambda }^2$. Trap: 16.

Bottom line

• Variance shortcut: $\text{Var}(X)=E[X^2]-(E[X]{)}^2$; always compute each piece separately.
• Scaling rule: $\text{Var}(aX+b)=a^2\text{Var}(X)$; the constant drops out, the coefficient squares.
• Standard deviation is $\sqrt{\text{Var}}$ and carries the same units as $X$.
• CV = $\sigma /\mu$: dimensionless and invariant to positive scaling.

Exam shortcut

When two distributions have the same mean and the question asks to compare risk, compute CVs. The CV normalizes away the mean and isolates relative volatility. "Second minus squared." $E[X^2]-(E[X]{)}^2$. Write this at the top of every variance problem. "CV of 1 means exponential." Among standard Exam P distributions with non-negative support, the exponential is characterized by CV = 1.

The full lesson (about 2,560 words, 17 min read) adds 3 worked examples, all 7 common mistakes, a self-check, free in the app.

Learning objectives

• 2d

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