Why Portfolio Mix Matters
A deductible reduces the amount paid on each claim, but the resulting factor for a portfolio depends on more than the deductible itself. It also depends on which kinds of risks make up the portfolio. Two segments can have the same number of exposures but very different expected ground-up losses per exposure; the segment with more expected losses should then have more influence on the portfolio factor.
This distinction matters in ratemaking because a portfolio factor is intended to describe the ratio of total expected payments after the deductible to total expected ground-up losses. It is not generally the simple average of segment factors. The calculation below is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.
The portfolio calculation assumes that each segment’s exposure basis, observation period, and coverage definition are consistent. Let \(e_j\) be the number of exposure units in segment \(j\), \(\lambda_j\) its expected claim count per exposure unit, \(\mu_j=E[X_j]\) its expected ground-up severity, and \(m_j=E[(X_j-d)_+]\) its expected payment per claim after the deductible. Then the expected ground-up loss per exposure unit is \(\lambda_j\mu_j\), and expected payment per exposure unit is \(\lambda_jm_j\).
When segment \(j\) has a positive expected ground-up loss, its segment factor is \(F_j=m_j/\mu_j\). Substituting \(m_j=F_j\mu_j\) into the portfolio expression gives an equivalent weighted-average formula:
Thus, the relevant weight is segment \(j\)’s expected ground-up loss, not its exposure count by itself. Exposure count is sufficient as a weight only when expected ground-up loss per exposure is the same across the segments being combined.
Why the Ground-Up Loss Weights Are Correct
The weighting rule follows directly from the definition of the portfolio factor. For one segment, expected ground-up loss across \(e_j\) exposures is \(e_j\lambda_j\mu_j\). Its expected payment is \(e_j\lambda_jm_j\). Since \(m_j=F_j\mu_j\), that expected payment can also be written as \(e_j\lambda_j\mu_jF_j\), or \(w_jF_j\). Adding the expected payments and ground-up losses across segments gives the formula.
This is a ratio of portfolio totals, not an average of the segment ratios with arbitrary weights. The derivation uses linearity of expectation: expected amounts across segments add. It does not require losses in different segments to be independent. In applications where expected aggregate losses are already available, those expected losses can be used directly as weights without separately estimating frequency and severity.
Calculating from Segment Expected Values
If a problem supplies expected severity and expected payment per claim, first multiply each by the segment’s expected claim frequency and exposure count. This puts both numerator and denominator on the same aggregate basis. The expected payment should already reflect the stated deductible and any other coverage terms included in the problem.
Worked Example: Combining Two Segments Correctly
A hypothetical portfolio has two segments. Segment A has 800 policy-years, expected claim frequency 0.25 per policy-year, mean ground-up severity of $2,000, and expected payment per claim after the deductible of $1,200. Segment B has 400 policy-years, frequency 0.10, mean ground-up severity of $5,000, and expected payment per claim of $3,500. Find the portfolio deductible factor.
Method and assumptions. Use the ratio of aggregate expected payments to aggregate expected ground-up losses. The expected severities and payments are supplied segment-level estimates; no parametric severity distribution is assumed. There is no interest or mortality assumption in this expected-loss calculation.
Segment A ground-up losses. Expected ground-up loss per policy-year is \(0.25(2{,}000)=\$500\). Across 800 policy-years, it is \(800(500)=\$400{,}000\).
Segment A payments. Expected payment per policy-year is \(0.25(1{,}200)=\$300\). Across 800 policy-years, it is \(800(300)=\$240{,}000\).
Segment B ground-up losses. Expected ground-up loss per policy-year is \(0.10(5{,}000)=\$500\). Across 400 policy-years, it is \(400(500)=\$200{,}000\).
Segment B payments. Expected payment per policy-year is \(0.10(3{,}500)=\$350\). Across 400 policy-years, it is \(400(350)=\$140{,}000\).
Portfolio calculation. Total expected ground-up loss is \(\$400{,}000+\$200{,}000=\$600{,}000\). Total expected payment is \(\$240{,}000+\$140{,}000=\$380{,}000\). Therefore, \(F_{\text{portfolio}}=380{,}000/600{,}000=0.633333\ldots\), or 0.6333 to four decimal places.
Check and common mistake avoided. The segment factors are \(1{,}200/2{,}000=0.60\) and \(3{,}500/5{,}000=0.70\). Their ground-up loss weights are $400,000 and $200,000, so the weighted result is \((400{,}000(0.60)+200{,}000(0.70))/600{,}000=0.6333\). A simple average, \((0.60+0.70)/2=0.65\), is wrong here because the two segments do not contribute equal ground-up expected losses.
A Change in Mix Can Change the Factor
If the deductible and each segment’s risk characteristics stay fixed, changing the proportion of exposures in the segments can still change the portfolio factor. The following example computes payments from explicitly stated severity distributions, so the segment factors can be checked rather than taken as unexplained inputs.
Worked Example: Reweighting a Portfolio with a $2,000 Deductible
A hypothetical book has a per-claim deductible of $2,000 and two segments. In A, claim frequency is 0.20 per exposure-year. Conditional on a claim, severity \(X_A\) is $500 with probability \(2/3\) and $14,000 with probability \(1/3\). In B, frequency is 0.10, and every claim has severity \(X_B=\$5{,}000\). Initially, there are 1,000 exposure-years in A and 500 in B. Find the portfolio factor, then recalculate it if the mix changes to 500 exposure-years in A and 1,000 in B.
Model and assumptions. The stated discrete distributions describe claim severity conditional on a claim; claim frequencies are expected counts per exposure-year. Payment per claim is \((X-2{,}000)_+\). No interest or mortality assumption applies. Severity parameters are the probabilities and amounts stated above.
Segment A severity and payment. Its expected ground-up severity is \((2/3)(500)+(1/3)(14{,}000)=\$5{,}000\). The payment is $0 for a $500 loss and $12,000 for a $14,000 loss, so expected payment per claim is \((2/3)(0)+(1/3)(12{,}000)=\$4{,}000\). Its factor is \(4{,}000/5{,}000=0.80\).
Segment B severity and payment. Expected ground-up severity is $5,000. Payment on every claim is \(5{,}000-2{,}000=\$3{,}000\), so its factor is \(3{,}000/5{,}000=0.60\).
Initial mix. A has expected ground-up loss per exposure-year \(0.20(5{,}000)=\$1{,}000\) and expected payment of \(0.20(4{,}000)=\$800\). Across 1,000 exposure-years, these totals are $1,000,000 and $800,000. B has $500 ground-up loss and $300 payment per exposure-year; across 500 exposure-years, the totals are $250,000 and $150,000. Therefore, the portfolio factor is \((800{,}000+150{,}000)/(1{,}000{,}000+250{,}000)=950{,}000/1{,}250{,}000=\mathbf{0.76}\).
Changed mix. With 500 exposure-years in A, expected ground-up losses and payments are $500,000 and $400,000. With 1,000 in B, they are $500,000 and $300,000. The new factor is \((400{,}000+300{,}000)/(500{,}000+500{,}000)=\mathbf{0.70}\).
Check and common mistake avoided. The changed mix has equal ground-up loss weights, so its factor is \((0.80+0.60)/2=0.70\). In the initial mix, simply averaging by exposure count gives \((1{,}000(0.80)+500(0.60))/1{,}500=0.7333\), which is not the portfolio factor of 0.76. Segment A has twice the ground-up loss per exposure-year as B, so exposure counts alone understate A’s weight.
Using Supplied Segment Factors and Loss Weights
Sometimes a problem supplies segment factors and expected ground-up losses directly. In that case, use each segment’s expected ground-up loss as its weight. If it supplies only the number of exposures and segment factors, that information is not generally enough: the expected ground-up loss per exposure, or an equivalent frequency and severity measure, is also needed.
Worked Example: Three Segments with Different Loss Costs
A hypothetical insurer has three segments. Each segment’s expected ground-up loss and expected payment per exposure-year are estimated from its own claim experience under the same deductible and coverage basis. No parametric severity distribution is assumed; the expected values below are supplied estimates. Segment A has 200 exposure-years, ground-up loss of $400 per exposure-year, and factor 0.55. Segment B has 100 exposure-years, ground-up loss of $1,200 per exposure-year, and factor 0.80. Segment C has 300 exposure-years, ground-up loss of $600 per exposure-year, and factor 0.70. Calculate the portfolio factor.
Method. Find each segment’s total expected ground-up loss, then multiply that amount by its factor to obtain expected payment.
Segment A. Ground-up total is \(200(400)=\$80{,}000\). Payment total is \(80{,}000(0.55)=\$44{,}000\).
Segment B. Ground-up total is \(100(1{,}200)=\$120{,}000\). Payment total is \(120{,}000(0.80)=\$96{,}000\).
Segment C. Ground-up total is \(300(600)=\$180{,}000\). Payment total is \(180{,}000(0.70)=\$126{,}000\).
Portfolio calculation. Total ground-up loss is \(80{,}000+120{,}000+180{,}000=\$380{,}000\). Total payment is \(44{,}000+96{,}000+126{,}000=\$266{,}000\). Thus the portfolio factor is \(266{,}000/380{,}000=\mathbf{0.7000}\) to four decimal places.
Check and common mistake avoided. A simple average of the three factors is \((0.55+0.80+0.70)/3=0.6833\), and an exposure-weighted average is \((200(0.55)+100(0.80)+300(0.70))/600=0.6667\). Neither uses the correct loss weights. Weighting by $80,000, $120,000, and $180,000 gives \((80{,}000(0.55)+120{,}000(0.80)+180{,}000(0.70))/380{,}000=0.7000\), matching the ratio of totals.
Practical Checks and Common Traps
A reliable calculation keeps the numerator and denominator on a common basis. If the denominator uses expected ground-up loss per exposure-year, the numerator must use expected payment per exposure-year. If claim-level expectations are supplied instead, multiply both by the expected number of claims. Do not multiply severity by exposure count alone when claim frequency differs.
Confirm the exposure unit, time period, deductible, and coverage terms are consistent across segments.
Use exposure count times expected ground-up loss per exposure for the denominator, and exposure count times expected payment per exposure for the numerator.
Add expected payments across segments, add expected ground-up losses across segments, then divide the two sums.
Confirm that any weighted-average calculation uses each segment’s expected ground-up loss, not automatically its exposure count.
A frequent exam trap is averaging ratios before considering the segment denominators. Segment factors describe proportions; they do not say how much ground-up loss each segment contributes. Another trap is mixing a claim-level expected payment with an exposure-level ground-up loss. Convert both quantities to aggregate expected amounts before forming the portfolio ratio.
Check Your Understanding
For each hypothetical portfolio, calculate the requested factor using consistent expected-loss weights.
- Segment A has 100 exposure-years, ground-up loss of $600 per exposure-year, and factor 0.75. Segment B has 50 exposure-years, ground-up loss of $200 per exposure-year, and factor 0.50. Find the portfolio factor. (Answer: 0.7143)
- Two segments have total ground-up losses of $120,000 and $80,000, with factors 0.65 and 0.90. Find the portfolio factor and compare it with the simple average. (Answer: portfolio factor 0.7500; simple average 0.7750)
- A segment has 250 exposure-years, frequency 0.20 per exposure-year, mean severity $4,000, and mean payment after the deductible of $2,800. Find its ground-up expected loss, expected payment, and deductible factor. (Answer: $200,000 ground-up loss; $140,000 payment; factor 0.7000)
- A book has 500 exposure-years in A and 500 in B. Expected ground-up loss per exposure-year is $800 in A and $400 in B; segment factors are 0.60 and 0.90. Find the portfolio factor. (Answer: 0.7000)
- Segment A contributes $240,000 of ground-up expected loss and has factor 0.70. Segment B contributes $240,000 and has an unknown factor. If the portfolio factor is 0.75, find B’s factor. (Answer: 0.8000)