Measuring Portfolio-Mix Effects on Premium and Capital

Why Portfolio Mix Matters

Two portfolios can have different claim frequencies, severities and dependence on shared events. Combining them changes more than the expected loss: it can also change the variability of the aggregate and the capital associated with a chosen risk level. Consequently, a change in portfolio mix can alter an indicated premium even when the overall exposure base is unchanged.

The key is to be precise about what is being mixed. This tutorial treats each segment's aggregate loss as a random loss rate for a common reference exposure base. The portfolio loss rate is a weighted blend of those rates. If instead the problem gives disjoint independent groups of policies and asks for their total claims, use the sum of the groups' aggregate losses; the variance then scales with the actual group sizes. These are different models and their variance formulas are not interchangeable.

Definition: Let $L_A$ and $L_B$ be aggregate losses for two segments, each normalized to the same reference exposure base. For segment A weight $w$, where $0\leq w\leq1$, define the mixed aggregate loss rate by $L(w)=wL_A+(1-w)L_B$. Its mean is $E[L(w)]=wE[L_A]+(1-w)E[L_B]$. Its variance is $w^2\operatorname{Var}(L_A)+(1-w)^2\operatorname{Var}(L_B)+2w(1-w)\operatorname{Cov}(L_A,L_B)$.

Here $w$ is a share of the exposure base, not a share of the variance or standard deviation. The covariance term represents dependence between the segments' aggregate loss rates. It is positive when high losses in one segment tend to accompany high losses in the other, negative when they tend to move in opposite directions, and zero when the aggregate losses are uncorrelated. Independence implies zero covariance, but zero covariance alone does not generally imply independence.

From Segment Models to Portfolio Risk

For a compound Poisson aggregate $S=\sum_{j=1}^{N}X_j$, assume $N\sim\operatorname{Poisson}(\lambda)$, the severities $X_j$ are identically distributed with finite second moment, and $N$ is independent of the severities. Then $E[S]=\lambda E[X]$ and $\operatorname{Var}(S)=\lambda E[X^2]$. These moments summarize each segment's aggregate risk for the calculations below.

A common capital measure is the excess of a loss quantile over the mean. Under a normal approximation with mean $\mu$, standard deviation $\sigma$, and confidence level $q$, the capital is $K_q=z_q\sigma$, where $z_q$ is the standard normal quantile. The corresponding approximate loss quantile is $\mu+K_q$. This approximation is most defensible when the aggregate is sufficiently diversified; it can be poor for small or strongly tail-driven portfolios. It is not an exact formula for the quantile of an arbitrary aggregate distribution.

Premium indication: In a one-period model, suppose $P$ is gross premium, $e$ is the variable expense proportion of premium, $F$ is fixed expense, and $rK_q$ is a risk charge proportional to capital. If the premium is set to cover expected loss, fixed expense and the risk charge, then $(1-e)P=E[L]+F+rK_q$, so $P=(E[L]+F+rK_q)/(1-e)$. This setup assumes the stated expense and capital-charge rules; a different pricing basis requires a different equation.

All dollar examples below are hypothetical, use an annual aggregate-loss horizon, and make no present-value adjustment (effective annual interest rate $i=0$ for this one-period calculation). Mortality assumptions do not apply to these property-loss models. Unless stated otherwise, the normal approximation is used only for the capital illustration, not to define the underlying claim model.

Worked Example: Combining Different Compound Poisson Segments

A hypothetical insurer normalizes two segments to the same reference exposure base. In segment A, the annual claim count is Poisson with $\lambda_A=25$, and every claim is exactly $4,000. In segment B, the annual count is Poisson with $\lambda_B=40$, and every claim is exactly $4,000. Assume the two segment loss rates are independent. Find the expected loss rate and variance for a 50/50 mix, then estimate 95% capital and the indicated premium. Use $z_{0.95}=1.645$, $e=0.10$, $F=\$2{,}000$, and $r=0.05$.

Method and segment moments. For a Poisson frequency with fixed severity $c$, $E[S]=\lambda c$ and $\operatorname{Var}(S)=\lambda c^2$. Thus for A, $E[L_A]=25(4{,}000)=\$100{,}000$, and $\operatorname{Var}(L_A)=25(4{,}000^2)=\$400{,}000{,}000$. For B, $E[L_B]=40(4{,}000)=\$160{,}000$, and $\operatorname{Var}(L_B)=40(4{,}000^2)=\$640{,}000{,}000$.

Mix mean and variance. Here $w=0.5$ and the covariance is zero. The mean is $0.5(100{,}000)+0.5(160{,}000)=\$130{,}000$. The variance is $0.5^2(400{,}000{,}000)+0.5^2(640{,}000{,}000)=\$260{,}000{,}000$. Therefore $\sigma=\sqrt{260{,}000{,}000}=\$16{,}124.5155\ldots$.

Capital and premium. The 95% capital is $K_{0.95}=1.645(16{,}124.5155\ldots)=\$26{,}524.828\ldots$, and the approximate 95% loss quantile is $130{,}000+26{,}524.828\ldots=\$156{,}524.828\ldots$. The premium is

P=\frac{130{,}000+2{,}000+0.05(26{,}524.828\ldots)}{0.90} =\$148{,}140.27.

The checks are that the mixed mean lies between the segment means and the variance calculation uses squared weights. The common mistake avoided here is adding the segment variances as if this were a sum of two unweighted aggregate losses; the stated model is a weighted blend of normalized loss rates.

Why Covariance Changes Capital

The variance formula follows from expanding the centered weighted sum. Let $\mu_A=E[L_A]$ and $\mu_B=E[L_B]$. Then

\begin{aligned} \operatorname{Var}(L(w)) &=E\!\left[\left(w(L_A-\mu_A)+(1-w)(L_B-\mu_B)\right)^2\right]\\ &=w^2\operatorname{Var}(L_A)+(1-w)^2\operatorname{Var}(L_B)\\ &\quad+2w(1-w)E[(L_A-\mu_A)(L_B-\mu_B)]\\ &=w^2\operatorname{Var}(L_A)+(1-w)^2\operatorname{Var}(L_B) +2w(1-w)\operatorname{Cov}(L_A,L_B). \end{aligned}

The cross-product is exactly the covariance, so dependence affects the variance even though it does not change the weighted mean. A shared catastrophe, for example, may generate positive covariance. Treating such segments as independent would understate the portfolio variance and, under the normal approximation, its capital.

Worked Example: Measuring the Effect of a Shared Shock

Use the two segment means and variances from the previous example, but suppose a hypothetical shared event produces $\operatorname{Cov}(L_A,L_B)=\$200{,}000{,}000$. For the 50/50 mix, calculate 95% capital and compare it with the independent-segment result. Use $z_{0.95}=1.645$.

Method and dependence check. The covariance is feasible: its absolute value must not exceed $\sqrt{\operatorname{Var}(L_A)\operatorname{Var}(L_B)}=\sqrt{400{,}000{,}000(640{,}000{,}000)}$, approximately $\$506{,}000{,}000$. Thus the specified covariance does not violate the covariance bound.

The mean remains $0.5(100{,}000)+0.5(160{,}000)=\$130{,}000$. The variance is now

0.25(400{,}000{,}000)+0.25(640{,}000{,}000) +2(0.5)(0.5)(200{,}000{,}000) =\$360{,}000{,}000.

The standard deviation is $\sqrt{360{,}000{,}000}=\$18{,}973.6650\ldots$. Hence $K_{0.95}=1.645(18{,}973.6650\ldots)=\$31{,}211.68$, and the approximate loss quantile is $\$130{,}000+\$31{,}211.68=\$161{,}211.68$, rounding only at the end to cents.

With independence, capital was $\$26{,}524.83$. The shared-shock covariance therefore increases estimated capital by $\$4{,}686.85$, while leaving expected loss unchanged. The check is that positive covariance raises the variance relative to the independent case.

Mix Decisions Affect Premium Through More Than Expected Loss

A pricing comparison should report the change in expected loss and the change in capital separately. The premium formula transmits both: expected loss changes the pure-loss component, while capital changes the risk charge. Expenses and the premium denominator can make the gross premium change differ from either change considered alone.

Worked Example: Comparing Two Mixes Under Dependence

A hypothetical pair of normalized segment loss rates has means $\mu_A=\$100{,}000$ and $\mu_B=\$160{,}000$, variances $\$400{,}000{,}000$ and $\$900{,}000{,}000$, and covariance $\$150{,}000{,}000$. Compare a 50/50 mix with a 75/25 mix, where the first percentage is segment A. Use a 95% normal approximation with $z_{0.95}=1.645$, $e=0.10$, $F=\$2{,}000$, and $r=0.05$.

Method and 50/50 mix. Its mean is $0.5(100{,}000)+0.5(160{,}000)=\$130{,}000$. Its variance, in millions of dollars squared, is $0.25(400)+0.25(900)+2(0.5)(0.5)(150)=400$, or $\$400{,}000{,}000$. Thus $\sigma=\$20{,}000$, $K_{0.95}=1.645(20{,}000)=\$32{,}900$, and

P_{50/50}=\frac{130{,}000+2{,}000+0.05(32{,}900)}{0.90} =\frac{133{,}645}{0.90} =\$148{,}494.44.

75/25 mix. Its mean is $0.75(100{,}000)+0.25(160{,}000)=\$115{,}000$. Its variance, again in millions of dollars squared, is $0.75^2(400)+0.25^2(900)+2(0.75)(0.25)(150)=225+56.25+56.25=337.5$. Therefore $\sigma=\sqrt{337{,}500{,}000}=\$18{,}371.1730709\ldots$, and $K_{0.95}=1.645(18{,}371.1730709\ldots)=\$30{,}220.5797016\ldots$. Carrying the unrounded capital into the premium calculation gives

P_{75/25} =\frac{115{,}000+2{,}000+0.05(30{,}220.5797016\ldots)}{0.90} =\$131{,}678.92.

The 75/25 mix has expected loss lower by $\$15{,}000$, capital lower by $\$2{,}679.42$, and indicated premium lower by $\$16{,}815.52$ when the unrounded premiums are differenced and the result is rounded to cents. The premium difference reflects both the mean and the capital change.

Common mistake and correction. It is wrong to average the segment standard deviations and then use that as the mixed standard deviation. For the 75/25 mix, that incorrect calculation gives $0.75(20{,}000)+0.25(30{,}000)=\$22{,}500$, implying capital of $\$37{,}012.50$. The correct variance calculation, including the covariance, gives a standard deviation of $\$18{,}371.17$ and capital of $\$30{,}220.58$. Always mix variances with squared weights and include the covariance term before taking the square root.

Practical Checks for Mix Analysis

A sound calculation starts by identifying whether the model is a weighted blend of normalized loss rates or a sum of actual segment totals. State the segment means, variances and covariance on a consistent exposure and time basis. Then calculate the mixed mean and variance, obtain the capital measure specified in the problem, and apply the pricing equation. This sequence prevents a unit mismatch from entering both capital and premium.

Check the weights: exposure shares should be between zero and one and sum to one for a two-segment blend.
Check the mean: a weighted mean must lie between the segment means.
Check the variance: it cannot be negative. The covariance must also satisfy $|\operatorname{Cov}(L_A,L_B)|\leq\sqrt{\operatorname{Var}(L_A)\operatorname{Var}(L_B)}$.
Check the interpretation: a normal-approximation capital is not a distribution-free quantile. Use it only when the stated approximation is appropriate.
Check rounding: retain unrounded intermediate values in the premium formula and round the final reported amounts to cents.

Portfolio-mix analysis is useful because it separates expected cost from aggregate risk. A segment with a higher expected loss can still contribute differently to capital depending on its variance and dependence with the rest of the portfolio. Conversely, a reduction in expected loss does not by itself establish that the mix is safer. This framework is aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.

Check Your Understanding

Use the stated model in each problem, and retain unrounded values until the final reported answer.

A segment has annual Poisson claim frequency with $\lambda=12$ and fixed claim severity of $2,500. Find its annual aggregate mean and variance. (Answer: mean $30,000; variance $75,000,000.)
Two independent normalized segment loss rates have means $80,000 and $140,000, and variances $100,000,000 and $400,000,000. Find the mean, variance and 95% normal-approximation capital of a 40/60 mix, using $z_{0.95}=1.645$. (Answer: mean $116,000; variance $160,000,000; capital $20,807.79.)
For a 50/50 mix, segment variances are $100,000,000 and $400,000,000, with covariance $100,000,000. Find the mixed variance and 95% capital using $z_{0.95}=1.645$. (Answer: variance $175,000,000; capital $21,761.30.)
A portfolio has expected loss $90,000 and capital $20,000. Its fixed expense is $1,500, variable expense ratio is 8%, and capital charge rate is 4%. Find the indicated gross premium. (Answer: $100,326.09.)
For segment A weight $w$, suppose the variance in millions of dollars squared is $V(w)=100w^2+400(1-w)^2+100w(1-w)$. Find the weight that minimizes variance over $0\leq w\leq1$. (Answer: $w=0.875$, or an 87.5% share of segment A.)

What You'll Learn