Why Mixtures Need Their Own Exposure Curve
A portfolio may contain groups of risks with markedly different claim severities. For example, a hypothetical insurer might cover both small commercial service firms and large industrial contractors. Even if each group has a well-defined severity curve, the portfolio’s curve depends on how much each group contributes to ground-up expected loss. The number of risks or claims in a group, by itself, does not determine its weight in the aggregate curve.
This matters for increased limits indications: a change in the portfolio’s segment composition can change the aggregate limited-loss ratio and therefore the indicated factor between two limits. A curve or factor calculated for one mix should not automatically be applied to a different mix.
The \(n_j\) values can represent expected claim counts over the rating basis being studied. If a model is instead stated per unit of exposure, use each segment’s expected claim count per unit of that exposure. In either case, the weight is based on expected ground-up loss—not merely on the share of policies, risks, or claims.
This definition assumes that the segment severities and expected claim counts describe the portfolio on a consistent basis, and that the total ground-up expected loss is finite and positive. Limited and ground-up losses must use the same severity basis. The formula concerns expected losses; it does not require independent claims, since expectations add without an independence assumption.
Why Ground-Up Expected-Loss Weights Work
The aggregate curve follows directly from adding limited expected losses across segments. Within segment \(j\), the expected limited loss from \(n_j\) expected claims is \(n_jE[\min(X_j,u)]\). Summing these contributions gives the portfolio’s limited expected loss. The denominator is the sum of ground-up expected losses.
This rearrangement shows why claim-count weights alone generally fail: each segment’s limited-to-ground-up ratio must be weighted by the segment’s share of the portfolio’s ground-up expected loss. It also shows that the mixture curve is a weighted average of segment curves at each limit. Because the weights are nonnegative and sum to one, the mixture curve lies between the smallest and largest segment curves at that limit.
An increased limits factor from base limit \(b\) to limit \(u\), for a fixed portfolio mix, is the ratio of aggregate limited expected losses:
This is not generally the weighted average of the segment increased limits factors. Each segment factor has its own denominator \(H_j(b)\), whereas the aggregate factor has the single denominator \(\sum_jw_jH_j(b)\). The ratio must be calculated after aggregating the curves.
Worked Example: Combining Two Discrete Severity Segments
A hypothetical portfolio has three expected claims in segment A for every one expected claim in segment B. Segment A severity is 100 or 300 dollars, each with probability 0.5. Segment B severity is 200 or 600 dollars, each with probability 0.5. Calculate the aggregate exposure curve at limits 100, 200, and 400 dollars, then calculate the factor from 100 to 200 dollars. Use the segment-curve weighting method.
Step 1: Calculate segment means. For segment A, \(E[X_A]=0.5(100)+0.5(300)=50+150=200\) dollars. For segment B, \(E[X_B]=0.5(200)+0.5(600)=100+300=400\) dollars. The portfolio’s ground-up expected loss is \(3(200)+1(400)=1{,}000\) dollars.
Step 2: Calculate ground-up expected-loss weights. Segment A contributes \(3(200)=600\) dollars, and segment B contributes \(1(400)=400\) dollars. Therefore \(w_A=600/1{,}000=0.6\) and \(w_B=400/1{,}000=0.4\).
Step 3: Calculate the segment curves at each limit. At \(u=100\), segment A’s limited severities are 100 and 100, so its limited mean is \(0.5(100)+0.5(100)=100\), and \(H_A(100)=100/200=0.5\). Segment B’s limited severities are 100 and 100, so its limited mean is 100 and \(H_B(100)=100/400=0.25\).
At \(u=200\), segment A’s limited severities are 100 and 200, giving a mean of \(0.5(100)+0.5(200)=150\), so \(H_A(200)=150/200=0.75\). Segment B’s limited severities are 200 and 200, giving a mean of 200, so \(H_B(200)=200/400=0.5\).
At \(u=400\), segment A’s limited severities are 100 and 300, with mean 200, so \(H_A(400)=200/200=1\). Segment B’s limited severities are 200 and 400, with mean \(0.5(200)+0.5(400)=300\), so \(H_B(400)=300/400=0.75\).
Step 4: Aggregate the segment curves. At 100 dollars, \(H_{\mathrm{mix}}(100)=0.6(0.5)+0.4(0.25)=0.30+0.10=0.40\). At 200 dollars, \(H_{\mathrm{mix}}(200)=0.6(0.75)+0.4(0.5)=0.45+0.20=0.65\). At 400 dollars, \(H_{\mathrm{mix}}(400)=0.6(1)+0.4(0.75)=0.60+0.30=0.90\).
Step 5: Calculate and check the indicated factor. The factor from 100 to 200 dollars is \(0.65/0.40=1.625\). Directly, expected limited loss at 100 dollars is \(3(100)+1(100)=400\) dollars; at 200 dollars it is \(3(150)+1(200)=650\) dollars. Their ratio is \(650/400=1.625\), confirming the curve calculation. Method: weight the segment curves by ground-up expected-loss share, then take their ratio. Common mistake avoided: weighting the two segment curves by the claim-count shares of 3/4 and 1/4.
Claim-Count Weights Are Not Loss Weights
If segment means differ, a segment with relatively few claims can still contribute a large share of ground-up expected loss. The following example isolates that issue using deterministic severities. A deterministic severity is a valid severity distribution concentrated at one value.
Worked Example: Finding and Correcting a Claim-Count Weight Error
A hypothetical book has 75 expected claims in segment A and 25 in segment B. Each A claim has severity 100 dollars, and each B claim has severity 500 dollars. Calculate the aggregate curve at a 200-dollar limit and the factor from 100 to 200 dollars. Then compare the correct curve with the result from using claim-count weights.
Step 1: Calculate ground-up expected loss. Segment A contributes \(75(100)=7{,}500\) dollars and segment B contributes \(25(500)=12{,}500\) dollars. Total ground-up expected loss is \(7{,}500+12{,}500=20{,}000\) dollars. Thus the correct weights are \(w_A=7{,}500/20{,}000=0.375\) and \(w_B=12{,}500/20{,}000=0.625\).
Step 2: Calculate the segment curves. At a 100-dollar limit, each segment’s claim is limited to 100 dollars. Therefore \(H_A(100)=100/100=1\) and \(H_B(100)=100/500=0.2\). At a 200-dollar limit, segment A remains limited to 100 dollars, while segment B is limited to 200 dollars. Thus \(H_A(200)=1\) and \(H_B(200)=200/500=0.4\).
Step 3: Aggregate with the correct weights. At 100 dollars, \(H_{\mathrm{mix}}(100)=0.375(1)+0.625(0.2)=0.375+0.125=0.5\). At 200 dollars, \(H_{\mathrm{mix}}(200)=0.375(1)+0.625(0.4)=0.375+0.25=0.625\). The indicated factor is \(0.625/0.5=1.25\).
Step 4: Demonstrate the wrong calculation. Using claim-count weights gives the incorrect 200-dollar curve \(0.75(1)+0.25(0.4)=0.75+0.10=0.85\). The incorrect 100-dollar curve would be \(0.75(1)+0.25(0.2)=0.75+0.05=0.80\), and their ratio would be \(0.85/0.80=1.0625\), rather than 1.25.
Step 5: Check directly. Limited expected loss at 100 dollars is \(75(100)+25(100)=10{,}000\) dollars, or \(10{,}000/20{,}000=0.5\) of ground-up loss. At 200 dollars it is \(75(100)+25(200)=12{,}500\) dollars, or \(12{,}500/20{,}000=0.625\). The direct ratio \(12{,}500/10{,}000=1.25\) agrees. Method: convert claim counts to ground-up expected-loss shares before combining curves. Common mistake avoided: treating the claim-count share as the severity exposure-curve weight.
Composition Changes Can Change the Indication
Segment curves may remain unchanged while the aggregate curve moves because the weights change. This is a mix effect: the portfolio contains different proportions of expected ground-up loss from each segment. Recalculate the aggregate curve using the new weights before deriving its factor.
Worked Example: Reweighting After a Portfolio Mix Shift
A hypothetical insurer initially expects 80 claims from segment A and 20 from segment B. A claims severity is always 100 dollars; B claims severity is always 400 dollars. A revised portfolio is expected to have 60 A claims and 40 B claims. Calculate the factor from a 100-dollar base limit to a 200-dollar limit under both mixes, and quantify the change.
Step 1: Establish the segment curves. At a 100-dollar limit, both segments have limited severity 100 dollars. Therefore \(H_A(100)=100/100=1\) and \(H_B(100)=100/400=0.25\). At a 200-dollar limit, A remains limited to 100 dollars and B is limited to 200 dollars, giving \(H_A(200)=1\) and \(H_B(200)=200/400=0.5\).
Step 2: Calculate the original mix. Expected ground-up loss is \(80(100)+20(400)=8{,}000+8{,}000=16{,}000\) dollars. The weights are \(w_A=8{,}000/16{,}000=0.5\) and \(w_B=0.5\). Thus \(H_{\mathrm{mix}}(100)=0.5(1)+0.5(0.25)=0.625\), and \(H_{\mathrm{mix}}(200)=0.5(1)+0.5(0.5)=0.75\). The original indicated factor is \(0.75/0.625=1.2\).
Step 3: Calculate the revised mix. Revised expected ground-up loss is \(60(100)+40(400)=6{,}000+16{,}000=22{,}000\) dollars. The revised weights are \(w_A=6{,}000/22{,}000=3/11\) and \(w_B=16{,}000/22{,}000=8/11\). The revised base curve is \((3/11)(1)+(8/11)(0.25)=3/11+2/11=5/11\). The revised 200-dollar curve is \((3/11)(1)+(8/11)(0.5)=3/11+4/11=7/11\).
Step 4: Calculate the revised factor and check. The revised factor is \((7/11)/(5/11)=7/5=1.4\). Directly, revised limited expected loss at 100 dollars is \(60(100)+40(100)=10{,}000\) dollars; at 200 dollars it is \(60(100)+40(200)=14{,}000\) dollars. Their ratio is \(14{,}000/10{,}000=1.4\). The factor increases from 1.2 to 1.4, a change of 0.2. Method: recompute expected-loss weights for the revised portfolio before taking the ratio. Common mistake avoided: carrying forward the old factor merely because the segment severity distributions did not change.
Practical Checks and Common Traps
A mixture calculation is easier to audit when the numerator and denominator are kept visible. For each segment, calculate expected ground-up loss and expected limited loss at each relevant limit. Sum those amounts across segments, then divide. The curve-weighting method and this direct method should agree.
- Do not confuse the kinds of weight. A policy share or claim-count share is not automatically a ground-up expected-loss share. The conversion uses the segment mean severity.
- Do not average segment factors directly. First combine segment curves at the base limit and at the higher limit; then divide the two aggregate values.
- Revisit weights when composition changes. An aggregate curve describes a particular mix. Changed expected claim counts, segment shares, or mean severities can change its weights.
- Use consistent bases. Segment limited losses and ground-up losses must refer to the same severity definitions and portfolio basis. If those bases differ, the resulting ratio is not a consistent exposure curve.
For a fixed mix, each segment curve and the aggregate curve are nondecreasing in the limit and do not exceed 1. The aggregate curve is also bounded at each limit by the lowest and highest segment-curve values among segments with positive weight. These checks can reveal arithmetic or weighting errors, but they do not establish that the assumed severity distributions or segment mix are suitable for a real indication.
For ratemaking, the practical implication is direct: a factor is tied to the loss distribution and mix that generated it. When the portfolio mix changes, recompute the aggregate curve or otherwise justify why the old factor remains appropriate. The calculation is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.
Check Your Understanding
For each problem, calculate the requested aggregate quantity using the stated segment mix. Answers are rounded only where indicated.
- A hypothetical portfolio has two expected claims from segment A and one from segment B. A severity is always 100 dollars and B severity is always 300 dollars. Find the aggregate curve at limits 150 and 300 dollars, then the factor from 150 to 300 dollars. (Answer: \(H_{\mathrm{mix}}(150)=0.7\), \(H_{\mathrm{mix}}(300)=1\), and the factor is \(1.4286\).)
- A portfolio has four expected claims from segment A, each with severity 50 dollars, and one expected claim from segment B, with severity 250 dollars. Calculate the aggregate curve at a 100-dollar limit. (Answer: \(H_{\mathrm{mix}}(100)=2/3\), or \(0.6667\).)
- At a base limit, segment A’s curve is 0.6 and segment B’s is 0.8. At a higher limit, their curves are 0.9 and 0.95, respectively. If A contributes 25% and B contributes 75% of ground-up expected loss, calculate the aggregate curves and indicated factor. (Answer: base curve \(0.75\), higher-limit curve \(0.9375\), factor \(1.25\).)
- Two deterministic-severity segments have severities 100 and 400 dollars. The original expected claim counts are 80 and 20; revised counts are 60 and 40. Calculate the factor from 100 to 200 dollars for each mix and state the change. (Answer: original factor \(1.2\), revised factor \(1.4\), increase \(0.2\).)
- At a base limit, two segments have curves 0.5 and 0.9. At a higher limit, their curves are 0.8 and 0.95. Segment A has ground-up expected-loss weight \(w\), and segment B has weight \(1-w\). If \(w=0.4\), calculate the aggregate factor. (Answer: base curve \(0.4(0.5)+0.6(0.9)=0.74\); higher-limit curve \(0.4(0.8)+0.6(0.95)=0.89\); factor \(0.89/0.74=1.2027\).)