From Separate Indications to a Treaty Price
An excess-treaty indication is not just a selected loss estimate with a margin added. The final price must reconcile the expected recoveries, the selected risk load, premium-based expenses, and any expected reinstatement premium. These components interact: when a paid reinstatement premium is expected, it increases expected premium income, while brokerage and other proportional expenses may apply to that income as well.
The workshop below treats the treaty as hypothetical. It assumes a fixed annual premium, proportional expenses applied to both original and reinstatement premiums, and a reinstatement provision paid pro rata as to amount. Unless stated otherwise, there is no time proration, no fixed expense, and the stated loss indications already reflect the treaty terms and the selected rating basis.
This formula applies only when all included expenses are proportional to total premium, including reinstatement premium, and the risk load is a dollar amount required in addition to expected losses. A fixed expense must be handled separately. A free reinstatement has \(\rho=0\); a time-prorated reinstatement requires an expected time factor in addition to the amount reinstated.
Worked Example: Credibility-Weighted Layer Loss Estimate
Worked Example: Credibility-Weighted Layer Loss Estimate
A hypothetical insurer is pricing a property layer for a group of solar farms. Historical layer losses have been adjusted to the prospective year and portfolio: the historical estimate is $410,000 before adjustment, the loss trend factor is 1.08, and the portfolio adjustment factor is 1.05. The exposure-based estimate on the same prospective basis is $620,000. The selected credibility assigned to the adjusted historical estimate is 0.40. Calculate the selected expected annual layer loss.
Method. Put the historical estimate on the prospective basis first, then blend it with the exposure indication using the stated credibility. This avoids assigning credibility to a loss estimate that is not yet comparable with the exposure estimate.
Step 1: Adjust the historical estimate. Applying the trend factor gives \(410{,}000(1.08)=442{,}800\). Applying the portfolio factor gives \(442{,}800(1.05)=464{,}940\). Thus the adjusted historical indication is $464,940.
Step 2: Apply the credibility weights. The historical weight is 0.40 and the complementary exposure weight is \(1-0.40=0.60\). The selected estimate is $$ 0.40(464{,}940)+0.60(620{,}000) =185{,}976+372{,}000 =557{,}976. $$
Answer and check. The selected expected annual layer loss is $557,976. It lies between $464,940 and $620,000, as a convex weighted average with weights summing to 1 must. A common mistake is to blend the unadjusted $410,000 historical figure; that would mix unlike bases and produce \(0.40(410{,}000)+0.60(620{,}000)=536{,}000\), which is not the requested prospective estimate.
Reinstatement and Expected Premium Income
A reinstatement restores some or all of a treaty limit after it has been used. With a paid reinstatement, the cedent pays an additional premium for the restored capacity. For a single reinstatement paid at a rate of 100% pro rata as to amount, using one-half of the reinstated limit generates a reinstatement premium equal to one-half of the original premium. The expected factor \(\rho\) is based on the expected fraction of the limit reinstated, not simply the probability of a loss.
Worked Example: Expected Premium for a Paid Reinstatement
A hypothetical municipal water-liability treaty has one reinstatement paid at 100% pro rata as to amount. Based on a pricing review, the annual outcomes are: no limit reinstated with probability 0.7000; 20% of the limit reinstated with probability 0.2000; and 80% reinstated with probability 0.1000. The original premium is $480,000. Find the expected reinstatement premium and expected total premium. Assume there is no time proration.
Step 1: Calculate the expected fraction reinstated. $$ E[R]=0.7000(0)+0.2000(0.20)+0.1000(0.80) =0+0.0400+0.0800=0.1200. $$
Step 2: Convert the expected fraction into a premium. The reinstatement rate is 100%, so \(\rho=1.00(0.1200)=0.1200\). Expected reinstatement premium is \(480{,}000(0.1200)=\$57{,}600\).
Step 3: Add the original premium. Expected total premium is \(480{,}000+57{,}600=\$537{,}600\).
Answer and check. The expected reinstatement premium is $57,600, and expected total premium is $537,600. The reinstatement premium is 12% of the original premium, matching the expected reinstated fraction. A common mistake is to use the 30% probability of any reinstatement as the premium factor. That ignores whether 20% or 80% of the limit is restored; the correct factor is the expected amount fraction, 12%.
Why the Premium Formula Holds
The original premium is not the only expected premium income when reinstatements are paid. Under the stated assumptions, expected total premium is \(P_{\mathrm{orig}}+ \rho P_{\mathrm{orig}}=P_{\mathrm{orig}}(1+\rho)\). The proportion \(e\) of that total is used for proportional expenses and brokerage, leaving \((1-e)P_{\mathrm{orig}}(1+\rho)\) after those expenses.
To fund expected losses and the required risk load, that remaining amount must equal \(L_E+K\). Equating the two amounts gives $$ (1-e)P_{\mathrm{orig}}(1+\rho)=L_E+K, $$ and division by the positive factors \((1-e)(1+\rho)\) gives the indicated-premium formula. This derivation also explains why grossing up for expenses and then simply adding expected reinstatement premium can be wrong: the expense percentage applies to that additional premium too.
Worked Example: Complete Excess-Treaty Indication
Worked Example: Complete Excess-Treaty Indication
A hypothetical satellite-liability treaty has a $2,000,000 annual limit and one paid reinstatement, pro rata as to amount at 100%. The historical and exposure-based estimates, each on the prospective treaty basis, are $540,000 and $660,000. The selected credibility on the historical estimate is 0.40. A separate annual loss model has outcomes of $0 with probability 0.6000, $1,040,000 with probability 0.3000, and $3,000,000 with probability 0.1000. For pricing, use a risk load equal to 5% of the annual loss standard deviation. Acquisition and administration expenses are 8% of total premium, brokerage is 5%, and tax is 2%. Calculate the original premium and check the expected total premium.
Step 1: Select expected losses. The historical weight is 0.40 and the exposure weight is \(1-0.40=0.60\): $$ L_E=0.40(540{,}000)+0.60(660{,}000) =216{,}000+396{,}000 =\$612{,}000. $$
Step 2: Calculate the variability-based risk load. Express the annual losses in millions of dollars. The mean is \(0.6000(0)+0.3000(1.04)+0.1000(3)=0.612\) million. The second moment is \(0.6000(0^2)+0.3000(1.04^2)+0.1000(3^2)=0.32448+0.90000=1.22448\) million squared. Therefore, $$ \operatorname{Var}(X)=1.22448-(0.612)^2 =1.22448-0.374544 =0.849936 \text{ million}^2. $$ The standard deviation is \(\sqrt{0.849936}=0.921920\) million dollars, or approximately $921,920. The risk load is \(0.05(921{,}920)=\$46{,}096\). The 5% factor is a stated pricing assumption, not a universal standard.
Step 3: Determine the expected reinstatement factor. In the $3,000,000 outcome, $2,000,000 uses the initial limit and the remaining $1,000,000 uses one-half of the reinstated limit. The other outcomes do not use reinstated capacity. Thus $$ E[R]=0.6000(0)+0.3000(0)+0.1000\left(\frac{1{,}000{,}000}{2{,}000{,}000}\right)=0.0500. $$ At a 100% reinstatement rate, \(\rho=0.0500\).
Step 4: Combine proportional expenses. The total expense rate is \(e=0.08+0.05+0.02=0.15\). The amount required after expenses is \(L_E+K=612{,}000+46{,}096=\$658{,}096\). Therefore, $$ P_{\mathrm{orig}} =\frac{658{,}096}{(1.05)(0.85)} =\frac{658{,}096}{0.8925} =\$737{,}362.46. $$ The calculation is rounded to the nearest cent only at the final step.
Step 5: Reconcile expected premium and expenses. Expected reinstatement premium is \(0.05(737{,}362.46)=\$36{,}868.12\), so expected total premium is $774,230.58. Expenses are \(0.15(774{,}230.58)=\$116{,}134.59\). The amount remaining is \(774{,}230.58-116{,}134.59=\$658{,}095.99\), which agrees with the $658,096 target to the cent-level rounding shown.
Answer and check. The indicated original premium is $737,362.46, with expected reinstatement premium of $36,868.12. Expected total premium after expenses funds expected losses plus the risk load. A common mistake is to calculate \(658{,}096/0.85=\$774{,}230.59\) as the original premium and then add reinstatement premium on top. That treats the target as if reinstatement premium were not additional income; it also overlooks that proportional expenses apply to the reinstatement premium. The correct denominator includes both \(1+\rho\) and \(1-e\).
Pricing Checks and Limits of the Calculation
A useful final review is to trace each input to the amount it affects. The selected expected loss comes from the loss indications and credibility decision. The risk load is an explicit addition to expected loss. Reinstatement changes expected premium income under the stated paid terms. Proportional expenses reduce the amount retained from all premium income included in the calculation.
Check that historical and exposure indications reflect the same prospective period, layer, and portfolio basis before blending.
Apply the stated credibility weights and verify that they sum to 1.
State the load method and add the resulting dollar amount once.
Use expected limit utilization under the actual reinstatement terms, including any time proration when applicable.
Apply proportional expenses to the correct premium base, then check that expected premium after expenses equals expected losses plus the risk load.
The calculation is a structured indication, not a substitute for reviewing contract wording or judgmental assumptions. It does not automatically account for fixed expenses, multiple reinstatement layers, reinstatement timing, taxes assessed on a different base, or uncertainty in the selected indications. If those apply, the premium equation must be adapted rather than forcing them into a proportional expense percentage. The workflow is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.
Check Your Understanding
For each hypothetical treaty, calculate the requested indication and check that the relevant weights or premium components reconcile.
- A layer has a historical estimate of $400,000 before a 1.10 trend factor and a 1.05 portfolio factor. Its exposure estimate is $700,000, and credibility on the adjusted historical estimate is 0.25. Find the selected expected loss. (Answer: $640,500)
- A single paid reinstatement is 100% pro rata as to amount. The probability-weighted fractions reinstated are 0 with probability 0.7000, 0.20 with probability 0.2000, and 0.80 with probability 0.1000. If original premium is $500,000, find expected reinstatement premium. (Answer: $60,000)
- A treaty has expected losses of $460,000, a dollar risk load of $40,000, an expected reinstatement factor of 0.1000, and proportional expenses of 20% on all premium. Find the original premium. (Answer: $568,181.82)
- An annual treaty loss is $0 with probability 0.5000, $1,000,000 with probability 0.4000, and $3,000,000 with probability 0.1000. The initial limit is $2,000,000 and one paid reinstatement is available. Find expected annual treaty loss and expected fraction of limit reinstated. (Answer: Expected loss $700,000; expected reinstated fraction 0.0500)
- A selected expected loss is $480,000, the risk load is $40,000, the expected reinstatement factor is 0.0800, and proportional expenses are 20%. Find the original premium to the nearest cent. (Answer: $601,851.85)