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Concept

Variance

Variance measures how far a random outcome tends to sit from its mean. For actuaries, it is the first step from expected cost to volatility, capital, and aggregate risk.

What variance answers

Expected value tells you the center. Variance tells you how much outcomes move around that center. Two claim distributions can have the same mean and very different risk profiles if one has a heavier tail or more mass far from the mean.

Variance is measured in squared units, so it is usually interpreted through standard deviation. If claim amounts are measured in dollars, the standard deviation is back in dollars and is easier to compare with expected loss.

Core formulas

Variance is the expected squared distance from the mean. In exam problems, the computational form is often faster once you can find the first and second moments.

Definition
Var⁡(X)=E[(X−E[X])2]\operatorname{Var}(X)=E\left[(X-E[X])^2\right]
Computational form
Var⁡(X)=E[X2]−(E[X])2\operatorname{Var}(X)=E[X^2]-(E[X])^2
Standard deviation
σX=Var⁡(X)\sigma_X=\sqrt{\operatorname{Var}(X)}

Claim severity worked example

Suppose a claim amount is 0 with probability 0.82, 200 with probability 0.15, and 1,000 with probability 0.03. The expected claim is 60. The second moment is 36,000, so the variance is 32,400 and the standard deviation is 180.

The standard deviation is three times the mean. That is not a mistake: most policies have no claim, but the 1,000 claim outcome pulls the spread upward.

Second moment
E[X2]=02(0.82)+2002(0.15)+10002(0.03)=36000E[X^2]=0^2(0.82)+200^2(0.15)+1000^2(0.03)=36000
Variance and standard deviation
Var⁡(X)=36000−602=32400,σX=180\operatorname{Var}(X)=36000-60^2=32400,\quad \sigma_X=180

Scaling and aggregation

Adding a fixed expense shifts the mean but does not change the spread. Multiplying losses by a factor scales variance by the square of that factor. When independent risks are added, their variances add.

That is why diversification shows up through variance. A portfolio of independent policies can have a large total expected loss but a much smaller coefficient of variation than a single policy.

Linear transformation
Var⁡(aX+b)=a2Var⁡(X)\operatorname{Var}(aX+b)=a^2\operatorname{Var}(X)
Independent sum
Var⁡(X+Y)=Var⁡(X)+Var⁡(Y)\operatorname{Var}(X+Y)=\operatorname{Var}(X)+\operatorname{Var}(Y)

Conditional variance

When a portfolio is split into risk classes, total variance has two parts: average within-class variance and variance between class means. This is the spread version of total expectation.

The between-class term is easy to miss. If risk classes have very different expected losses, the portfolio variance can be high even if each class is stable internally.

Law of total variance
Var⁡(X)=E[Var⁡(X∣Y)]+Var⁡(E[X∣Y])\operatorname{Var}(X)=E[\operatorname{Var}(X\mid Y)]+\operatorname{Var}(E[X\mid Y])

Exam traps

Most variance errors are bookkeeping errors: forgetting to square the coefficient, subtracting the mean instead of the squared mean, or adding a constant to variance.

  • For a shifted random variable, the constant changes the mean but not the variance.
  • For a scaled random variable, the coefficient is squared in the variance.
  • For sums, do not add variances unless independence or zero covariance is justified.
  • For a discrete distribution, check that probabilities sum to 1 before computing moments.

Where variance connects next

Covariance and correlation explain how two random variables move together. Linear combinations use variance rules to price portfolios and approximate aggregate losses. The central limit theorem turns means and variances into normal approximations for sums.

Practice

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.

Variance Risk Spread Drill

Original variance checks for second moments, scaling, standard deviation, and class-mixture spread.

Exam P - 15 min
Source pattern: SOA Exam P variance and moment outcomes; original actuarial claim prompts.
  1. Question 1/Calculation

    Claim variance from a second moment

    A claim amount has mean 60 and second moment 36,000. Find the variance and standard deviation.

    Solution and grading points

    The variance is 36,000 minus 60 squared, which is 32,400. The standard deviation is the square root of 32,400, which is 180.

    • Uses the second moment minus the squared mean.
    • Computes the variance as 32,400.
    • Takes the square root to report standard deviation as 180.
  2. Question 2/Calculation

    Scaling a loss variable

    A loss variable has variance 400. A contract payment is three times the loss plus a fixed 50 expense. Find the payment variance.

    Solution and grading points

    The fixed expense does not change variance. Multiplying the loss by 3 multiplies variance by 9, so the payment variance is 9 times 400, or 3,600.

    • Ignores the fixed expense for variance.
    • Squares the scaling factor.
    • Reports the payment variance as 3,600.
  3. Question 3/Calculation

    Independent portfolio variance

    An insurer writes 100 independent policies. Each policy has expected loss 60 and variance 32,400. Find the expected total loss and total variance.

    Solution and grading points

    The expected total loss is 100 times 60, or 6,000. Since the policies are independent, the total variance is 100 times 32,400, or 3,240,000.

    • Adds expectations across policies.
    • Adds variances because independence is stated.
    • Does not multiply the variance by 100 squared.

References and official sources