Why the Unit of Weighting Matters
An exposure curve expresses limited expected loss as a fraction of ground-up expected loss. When loss information is organized by policy, the calculation is not just a matter of capping losses at a chosen limit. It also requires deciding what portfolio quantity the curve should represent and using weights that correspond to that quantity at every limit.
A pooled set of claims describes the severity distribution of the observed claims. A policy-level calculation can instead describe an average across policies, or the aggregate loss experience of a portfolio of policies. These are different estimands: policies with many or large claims can have a different influence on a claim-sample curve than on an equally policy-weighted curve. The data unit and the intended portfolio measure should therefore be settled before calculating the curve.
The examples below use hypothetical data. There is no interest or mortality assumption: neither is relevant to this calculation. Unless a problem explicitly supplies a parametric severity model, the empirical calculations make no parametric distribution assumption. Losses are assumed to be ground-up and measured on the same basis at every limit.
Policy-Level Quantities and the Portfolio Curve
Suppose policy \(i\) has exposure weight \(w_i\), and let \(\hat m_i(u)\) be its estimated limited loss per unit of that exposure at limit \(u\). Let \(\hat m_i(\infty)\) denote its ground-up loss per unit of exposure. For an aggregate portfolio exposure curve, combine the loss amounts first, using the policy exposure weights, and then take the ratio:
The denominator is the weighted ground-up loss on the same exposure basis as the numerator. In particular, it does not change when the limit changes. If the policy values are already total losses rather than loss rates per unit of exposure, use the corresponding policy weights consistently with those totals; do not multiply by exposure a second time.
A distinct quantity is the exposure-weighted average of policy-specific exposure curves:
This gives each unit of policy exposure the same influence on its normalized policy curve, regardless of that policy's ground-up loss rate. The aggregate curve instead gives more influence, in its effective weighting, to policies with more ground-up loss. Neither quantity is universally preferable; they answer different questions. For an aggregate portfolio loss ratio, the ratio of weighted losses is the relevant construction. If the target is the average policy curve, normalize within each policy and then average those curves.
Worked Example: Aggregate Curve From Policy Loss Rates
A hypothetical portfolio has three policy groups. Their exposure weights and estimated loss rates are shown below. The limited rates are per unit of exposure. Calculate the aggregate curve at limits \(100\) and \(200\), and compare the curve at \(100\) with the weighted average of the policy-specific curves. No parametric severity distribution is assumed.
| Policy group | Weight \(w_i\) | \(\hat m_i(100)\) | \(\hat m_i(200)\) | \(\hat m_i(\infty)\) |
|---|---|---|---|---|
| A | 1 | 60 | 90 | 100 |
| B | 2 | 75 | 115 | 125 |
| C | 1 | 40 | 70 | 80 |
Step 1: Calculate weighted ground-up loss. The denominator is the same at both limits: \(1(100)+2(125)+1(80)=100+250+80=430\).
Step 2: Calculate weighted limited loss at 100. The numerator is \(1(60)+2(75)+1(40)=60+150+40=250\). Thus, \(\hat G_{\mathrm{agg}}(100)=250/430=0.5814\), to four decimal places.
Step 3: Calculate weighted limited loss at 200. The numerator is \(1(90)+2(115)+1(70)=90+230+70=390\). Thus, \(\hat G_{\mathrm{agg}}(200)=390/430=0.9070\), to four decimal places.
Step 4: Compare with an average of policy curves at 100. Each policy curve is \(60/100=0.60\), \(75/125=0.60\), and \(40/80=0.50\). Their exposure-weighted average is \(\{1(0.60)+2(0.60)+1(0.50)\}/4=2.30/4=0.5750\). This differs from the aggregate curve, \(0.5814\), because the aggregate calculation weights loss rates before taking the ratio.
Check and method. The aggregate curve increases from \(0.5814\) to \(0.9070\), and both values lie between 0 and 1. The method is to apply the same policy weights to limited and ground-up loss rates, then divide the weighted totals. The common mistake is to replace that ratio with the average of policy-specific ratios when the target is an aggregate portfolio curve.
Why the Aggregate Formula Holds
The formula follows directly from aggregating limited and ground-up loss on the same exposure basis. For each policy group, \(w_i\hat m_i(u)\) is its weighted limited loss, and \(w_i\hat m_i(\infty)\) is its weighted ground-up loss. Summing across groups gives the portfolio's total limited and ground-up loss estimates. Dividing the first sum by the second therefore estimates the fraction of portfolio ground-up loss retained at limit \(u\).
Equivalently, if each policy has a curve \(\hat G_i(u)=\hat m_i(u)/\hat m_i(\infty)\), then \(\hat m_i(u)=\hat G_i(u)\hat m_i(\infty)\). Substituting this into the aggregate formula gives:
Thus, the aggregate curve is a weighted average of policy curves with weights proportional to \(w_i\hat m_i(\infty)\), not generally a simple policy-exposure-weighted average. Policies with higher ground-up loss per exposure contribute more ground-up loss to the aggregate portfolio, so their curves appropriately contribute more to this particular ratio.
Policy Weighting Versus Claim-Sample Weighting
For observed claim severities \(x_1,\ldots,x_N\), a common empirical claim-sample curve is \(\sum_j\min(x_j,u)/\sum_jx_j\). Each observed claim enters the sums once. This estimate describes the limited-to-ground-up ratio for the observed claim sample; policies that generate more observed claims contribute more observations. By contrast, averaging policy-specific curves gives a policy-based target, with the policy weights chosen to represent the intended policy population.
A claim-sample curve is not the same as an equal average of claim-level fractions \(\min(x_j,u)/x_j\). The ratio of sums can be written as a weighted average of those fractions, with weights proportional to claim size \(x_j\). State which calculation is being used; “claim-weighted” should not obscure whether the weights are claim counts or ground-up loss amounts.
Worked Example: A Policy Average Is Not a Claim-Sample Curve
Two hypothetical policies each have policy weight 1. Policy A has one ground-up claim of \(100\). Policy B has three ground-up claims, each \(20\). At limit \(50\), calculate the equal-policy average of the two policy curves and the pooled claim-sample curve. No parametric distribution is assumed; all losses are ground-up. No interest or mortality assumption applies.
Step 1: Calculate Policy A's curve. Its ground-up total is \(100\), and its limited total is \(\min(100,50)=50\). Its curve is \(50/100=0.50\).
Step 2: Calculate Policy B's curve. Its ground-up total is \(20+20+20=60\). Since each claim is below the limit, its limited total is \(20+20+20=60\). Its curve is \(60/60=1.00\).
Step 3: Average by policy. With equal policy weights, the average is \(\{1(0.50)+1(1.00)\}/2=0.75\).
Step 4: Pool claims. The pooled ground-up total is \(100+20+20+20=160\). The pooled limited total is \(50+20+20+20=110\). The claim-sample curve is \(110/160=0.6875\).
Check and method. The policy average is \(0.75\), while the claim-sample ratio is \(0.6875\). Policy A supplies half the policy weight but one of the four claims; policy B supplies the other half of policy weight but three claims. The difference is therefore expected, not an arithmetic error. The common mistake is to use the pooled claim result while describing it as an equally policy-weighted estimate.
Computing a Curve at Several Limits
When policy-level data are claim lists rather than pre-calculated loss rates, cap each ground-up claim at the target limit, sum within policy, and then apply the policy weights. Reuse the same ground-up denominator at every limit. If policies have different exposure units, first express policy loss on the stated exposure basis or use weights that consistently convert it to the desired portfolio basis.
Identify whether the requested result is an aggregate portfolio curve, a policy-average curve, or a claim-sample curve. Confirm that losses are ground-up and consistently defined.
For each claim, use \(\min(x,u)\). Sum those amounts within the applicable policy or group, without applying a policy limit to a claim amount already capped on a different basis.
For an aggregate curve, weight the limited and ground-up policy loss amounts or rates consistently. Use the weighted ground-up total as the denominator at every limit.
Each curve value must be between 0 and 1, and it must be nondecreasing as the limit rises. Investigate violations before interpreting the curve.
Worked Example: Applying Weights to Raw Policy Claims
Two hypothetical policy groups have frequency weights of 2 and 1, respectively. Group A has observed ground-up claims of \(40\) and \(160\); Group B has one claim of \(80\). Treat the frequency weights as multipliers for the corresponding policy-group experience. Calculate the aggregate empirical curve at limits \(50\) and \(100\). No parametric distribution is assumed; interest and mortality are not relevant.
Step 1: Fix the denominator. Group A's ground-up total is \(40+160=200\), and Group B's is \(80\). Applying the weights gives \(2(200)+1(80)=480\).
Step 2: Calculate the limit-50 numerator. Group A's capped claims are \(\min(40,50)=40\) and \(\min(160,50)=50\), totaling \(90\). Group B's capped claim is \(\min(80,50)=50\). The weighted limited total is \(2(90)+1(50)=180+50=230\). The curve is \(230/480=0.4792\), to four decimal places.
Step 3: Calculate the limit-100 numerator. Group A's capped claims are \(40\) and \(100\), totaling \(140\). Group B's capped claim is \(80\). The weighted limited total is \(2(140)+1(80)=280+80=360\). The curve is \(360/480=0.7500\), to four decimal places.
Check and method. The curve rises from \(0.4792\) to \(0.7500\), remains below 1, and uses the same ground-up denominator, \(480\), at both limits. The common mistake is to weight the limited amounts but leave the policy weights out of the ground-up denominator. That would compare differently weighted quantities and would not estimate the stated aggregate curve.
Common Mistakes and Practical Checks
- Changing the estimand without noticing. A ratio of weighted loss totals, an average of policy curves, and a pooled claim-sample curve answer different questions. Name the target before choosing the formula.
- Using inconsistent weights. If policy exposure weights apply to the limited loss amounts, apply them to the ground-up amounts as well. Otherwise the numerator and denominator refer to different portfolios.
- Changing the denominator by limit. The ground-up total is the reference amount for the curve and stays fixed as the limit changes.
- Confusing weights with claim counts. A frequency weight, an exposure weight, and an equal-policy weight need not mean the same thing. Follow the problem's definition and state the interpretation.
- Ignoring basic curve checks. A limited loss cannot exceed its ground-up loss, and raising the limit cannot reduce a capped claim amount. A curve outside \([0,1]\) or one that decreases as the limit rises signals an inconsistent calculation or data basis.
Check Your Understanding
For each problem, identify the target quantity and keep its weighting basis consistent.
- Two policy groups have weights 1 and 3. At limit \(100\), their limited loss rates are \(50\) and \(80\), and their ground-up rates are \(100\) and \(160\). Calculate the aggregate curve. (Answer: \(0.5000\))
- Two equally weighted policies have policy-specific curves of \(0.40\) and \(0.80\). Calculate their equal-policy average. (Answer: \(0.6000\))
- A claim sample has ground-up amounts \(30\), \(70\), and \(150\). Calculate the pooled claim-sample curve at limit \(60\). (Answer: \(150/250=0.6000\))
- Two groups have exposure weights 2 and 1. At limit \(100\), their limited loss rates are \(40\) and \(60\); their ground-up rates are \(80\) and \(120\). Calculate the aggregate curve. (Answer: \(140/280=0.5000\))
- Using the same two groups as in question 4, suppose their limited loss rates at limit \(200\) are \(70\) and \(110\), while the ground-up rates remain \(80\) and \(120\). Calculate the aggregate curve at \(200\), and state whether the pair of results passes the monotonicity check. (Answer: \(250/280=0.8929\); yes, it rises from \(0.5000\))