Why Risk Mix Matters
A portfolio can contain risks drawn from populations with different average expected losses. Even if each population has little variation in its own hypothetical means, pooling them can produce a substantially larger overall variance: the pooled population now varies both within each group and between groups. This distinction matters when interpreting or estimating the Bühlmann variance components. A larger pooled between-risk component may reflect a different mix of populations, not more heterogeneity among risks within each population.
The quantities below refer to hypothetical means on the same basis, such as expected annual losses per exposure unit. Assume a risk is selected at random from the pooled population, and let \(G\) identify its group. Group \(g\) has selection weight \(w_g=P(G=g)\), with \(w_g\geq 0\) and \(\sum_g w_g=1\). Let \(M\) denote the risk’s hypothetical mean, such as its conditional expected loss. Assume the relevant first and second moments are finite.
These are variances of conditional expected losses, not variances of individual observed losses. For an observed loss \(X\), the process variance describes variation in \(X\) around its conditional mean; the variance of hypothetical means describes variation in those conditional means across risks. The distinction is essential when a group mix changes.
Separating Within-Group and Between-Group Variation
The pooled variance has two parts. The first averages the hypothetical-mean variances within the groups. The second measures how far each group’s average hypothetical mean lies from the pooled average.
The first term, \(\sum_g w_gV_g\), is the weighted within-group variance. The second term is the between-group variance. It is nonnegative, and it is zero exactly when every group with positive weight has the same average hypothetical mean as the pooled mean. Thus, for a fixed set of groups, weights, and within-group variances, the pooled variance is at least the weighted within-group variance.
This does not mean that every possible mixture has a larger pooled variance than every constituent group. The weighted within-group variance itself depends on the weights, and a group with a high within-group variance may receive a small weight. The precise comparison is between pooled variance and the weighted within-group variance for that same mix.
Why the Decomposition Holds
The key is to compare each risk’s hypothetical mean with the pooled mean by adding and subtracting its group average. Conditional on group \(G\), the deviation \(M-\mu\) is the sum of the within-group deviation \(M-\mu_G\) and the group-average deviation \(\mu_G-\mu\).
Square both sides and take expectations. The cross-product has expectation zero because, within each group, the average deviation from that group’s mean is zero:
The remaining squared terms are the within-group and between-group contributions. In particular, \(E[(M-\mu_G)^2]=\sum_g w_gV_g\), while \(E[(\mu_G-\mu)^2]=\sum_g w_g(\mu_g-\mu)^2\). Therefore,
and \(E[(M-\mu)^2]=\operatorname{Var}(M)\), proving the result. No normality assumption is needed: the derivation requires only the group means, variances, and finite second moments.
Worked Example: Pooling Two Populations With Different Averages
A hypothetical insurer combines two risk populations. Group A is 75% of the risks and has an average annual hypothetical loss of $300, with within-group hypothetical-mean variance \(1{,}600\) dollars squared. Group B is 25% of the risks and has an average of $500, with within-group variance \(3{,}600\) dollars squared. Find the pooled mean and pooled hypothetical-mean variance.
Model and assumptions. A risk is drawn from the two-group mixture with \(P(G=A)=0.75\) and \(P(G=B)=0.25\). Conditional on group, \(M\) has the stated mean and variance; no particular parametric distribution is assumed. Losses are on a common annual basis, with no discounting or mortality assumption involved.
Step 1: Find the pooled mean.
Step 2: Find the weighted within-group variance.
Step 3: Find the between-group variance.
Step 4: Add the components.
The pooled variance is therefore \(9{,}600\) dollars squared, and the pooled standard deviation is \(\sqrt{9{,}600}\approx \$97.98\), rounded to the nearest cent. As a check, \(E[M^2]=2{,}100+0.75(300^2)+0.25(500^2)=132{,}100\); subtracting \(\mu^2=350^2=122{,}500\) also gives \(9{,}600\). The method avoids treating the weighted within-group variance alone as the pooled variance.
The Portfolio Mix Can Change the Result
Group means and within-group variances do not, by themselves, determine a single pooled variance. The weights matter. A change in the portfolio mix changes both the pooled average and the contributions of each group. Even if every group’s underlying characteristics stay fixed, the pooled hypothetical-mean variance can move.
Worked Example: Comparing Two Portfolio Mixes
Two hypothetical risk populations have fixed average annual hypothetical losses of $200 and $400. Their within-group hypothetical-mean variances are \(900\) and \(2{,}500\) dollars squared, respectively. Compare the pooled variances when the first population’s weight is 50% and 90%.
Model and assumptions. In each portfolio, \(M\) is drawn from the two populations with the stated weights. Conditional on population, the means and variances remain as given; no parametric distribution is assumed. The loss basis is annual, and no discounting or mortality assumption is needed.
At equal weights: the pooled mean is \(0.5(200)+0.5(400)=300\). The weighted within-group variance is \(0.5(900)+0.5(2{,}500)=450+1{,}250=1{,}700\). The between-group variance is \(0.5(200-300)^2+0.5(400-300)^2=5{,}000+5{,}000=10{,}000\). Thus the pooled variance is \(1{,}700+10{,}000=11{,}700\) dollars squared.
When the first population has 90% weight: the pooled mean is \(0.9(200)+0.1(400)=220\). The weighted within-group variance is \(0.9(900)+0.1(2{,}500)=810+250=1{,}060\). The between-group variance is \(0.9(200-220)^2+0.1(400-220)^2=0.9(400)+0.1(32{,}400)=360+3{,}240=3{,}600\). The pooled variance is \(1{,}060+3{,}600=4{,}660\) dollars squared.
The second mix has a lower pooled variance because it places much less weight on the higher-average population, reducing the between-group contribution. This is not a change in either population’s within-group variance. The check is that both calculations use the same fixed group means and within-group variances; only the weights and resulting pooled mean differ.
Calculation With Unequal Group Sizes
In a finite portfolio, group weights are often calculated as the proportion of risks in each group. Use weights that represent the population being analyzed, and make sure the hypothetical means and variances are measured on comparable units. For example, an annual expected loss per exposure unit should not be mixed with an unadjusted total loss over several years.
Worked Example: A Weighted Portfolio With Unequal Groups
A hypothetical portfolio has 60% of its risks in Group A and 40% in Group B. Group A has an average annual hypothetical loss of $1,200 and within-group variance \(10{,}000\) dollars squared. Group B has an average of $900 and within-group variance \(6{,}400\) dollars squared. Calculate the pooled hypothetical-mean variance and identify the effect of pooling.
Model and assumptions. The mixture weights are 0.60 and 0.40. Within each group, \(M\) has the specified mean and variance, with no assumed parametric form. All amounts refer to the same annual loss basis; interest and mortality are not applicable.
Step 1: Pooled mean.
Step 2: Weighted within-group variance.
Step 3: Between-group variance.
Therefore, the pooled hypothetical-mean variance is \(8{,}560+21{,}600=30{,}160\) dollars squared. A common incorrect calculation reports only \(8{,}560\), the weighted within-group component, and calls it the pooled variance. That omits the variation due to the $300 difference in group averages. To check, the weighted second moment is \(8{,}560+0.60(1{,}200^2)+0.40(900^2)=1{,}196{,}560\); subtracting \(1{,}080^2=1{,}166{,}400\) gives \(30{,}160\) dollars squared.
Interpretation and Exam Traps
A pooled variance is a property of both the risk populations and their mixture weights. If the groups have different average expected losses, the between-group term is positive. It can be material even when each group is internally quite homogeneous. Consequently, comparing variance components across portfolios without checking whether they use the same classification and mix can lead to misleading conclusions.
- Do not omit the between-group term. A weighted average of group variances is not generally the pooled variance.
- Use the pooled mean in the deviations. The between-group contribution is measured around \(\mu=\sum_gw_g\mu_g\), not around one selected group mean.
- Apply the correct weights. Equal weights are appropriate only when the groups have equal representation in the population being modeled.
- Keep units consistent. Variances are in squared loss units, and the means must be on the same basis for their differences to be meaningful.
- Keep process variation separate. If the quantity of interest is an observed loss \(X\), then its variance also includes variation of observations around their conditional means. The mixture decomposition for hypothetical means alone does not include that process component.
For observed losses, the law of total variance makes the distinction explicit. If \(M=E[X\mid\text{risk characteristics}]\), then \(\operatorname{Var}(X)=E[\operatorname{Var}(X\mid\text{risk characteristics})]+\operatorname{Var}(M)\). The pooled hypothetical-mean variance is the second term; it should not be mistaken for the total variance of observed losses.
Check Your Understanding
For each mixture, use the stated risk weights and hypothetical-mean variances. No parametric distribution is assumed; all means and variances use a common loss basis.
- Two groups have weights 0.40 and 0.60, average hypothetical losses of $100 and $250, and within-group variances of 400 and 900 dollars squared. Find the pooled mean and pooled variance. (Answer: pooled mean $190; pooled variance \(6{,}100\) dollars squared.)
- A portfolio is 80% Group A and 20% Group B. Their average hypothetical losses are $500 and $800, and their within-group variances are \(2{,}500\) and \(10{,}000\) dollars squared. Find the weighted within-group, between-group, and pooled variances. (Answer: \(4{,}000\), \(14{,}400\), and \(18{,}400\) dollars squared, respectively.)
- Three groups have weights 0.20, 0.50, and 0.30; average hypothetical losses of $40, $60, and $100; and within-group variances of 4, 9, and 16 dollars squared. Calculate the pooled mean and pooled variance. (Answer: pooled mean $68; pooled variance \(506.1\) dollars squared.)
- Two groups have averages of $120 and $220 and within-group variances of 100 and 400 dollars squared. Compare pooled variances when the lower-average group has weight 0.75 versus 0.25. (Answer: \(2{,}050\) versus \(2{,}200\) dollars squared, respectively.)
- Two groups have weights 0.70 and 0.30, average hypothetical losses of $200 and $300, within-group hypothetical-mean variances of \(4{,}000\) and \(9{,}000\) dollars squared, and process variances of \(2{,}000\) and \(3{,}000\) dollars squared. Find the pooled hypothetical-mean variance and the variance of an observed loss, assuming the stated process variances are conditional variances of that loss. (Answer: pooled hypothetical-mean variance \(7{,}600\); pooled process variance \(2{,}300\); observed-loss variance \(9{,}900\) dollars squared.)