Purpose and Sequence
An integrated adjustment puts historical premium and losses on the bases needed for a ratemaking analysis: premium at a specified rate level, and losses at a specified prospective date. The two adjustments answer different questions. On-leveling changes premium to reflect rate-level changes; trending projects loss experience from its historical date to a future date. Applying a correct factor to the wrong quantity, weight basis, or time interval can still produce an incorrect result.
This tutorial is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus. The calculations below assume that the historical premium and losses represent a consistent coverage, population, and exposure basis. They do not allow for changes in mix or coverage unless stated.
On-Leveling With Earned-Premium Shares
Suppose historical earned premium \(P_H\) was written at several rate levels. Let \(L_j\) be the relative rate level for portion \(j\), and let \(w_j\) be that portion’s share of historical earned premium, so \(w_j=P_j/P_H\) and \(\sum_j w_j=1\). If the target rate level is \(L_T\), then the portion of premium earned at level \(L_j\) is multiplied by \(L_T/L_j\).
The reciprocal in this formula matters: each premium portion is adjusted from its own historical level to the target. Averaging the historical levels using earned-premium shares and dividing the target by that average is generally not equivalent. An alternative formula is available when the weights are exposure shares, not premium shares. If \(q_j\) is the share of exposures at level \(L_j\), then the average historical premium per exposure is proportional to \(\sum_j q_jL_j\), and the factor is \(L_T/\sum_jq_jL_j\). Do not substitute one set of weights into the other formula.
Worked Example: On-Leveling Earned Premium at Multiple Rate Levels
A hypothetical insurer earned $680,000 of premium. Of that premium, 40% was earned at rate level 1.02 and 60% at rate level 1.09. Find the premium at target rate level 1.13.
Method and basis. The stated percentages are earned-premium shares, so use \(L_T\sum_jw_j/L_j\).
Step 1: Calculate the factor.
Step 2: Adjust the premium. Multiplying before rounding gives \( \$680{,}000(1.0651556035)=\$724{,}305.8104\), or $724,305.81 to the nearest cent.
Check and common mistake. The result is about 6.52% above historical premium, which is plausible because the target level is above both historical levels. A tempting but incorrect calculation is to average the levels using the premium shares, giving \(0.40(1.02)+0.60(1.09)=1.062\), and then divide \(1.13/1.062\). Those are not the right operations for earned-premium shares. The correct factor is 1.065155604, not approximately 1.064030.
Why the On-Level Formula Holds
The formula follows by adjusting each premium portion separately. For a portion \(P_j\) earned at level \(L_j\), the amount at target level \(L_T\) is \(P_jL_T/L_j\): the ratio removes the old level and applies the target level. Adding those adjusted portions gives the target-level premium. Since \(P_j=P_Hw_j\), factoring out \(P_HL_T\) gives the formula above.
The distinction from exposure shares can also be seen directly. If \(E_j\) exposures are rated at level \(L_j\), then premium is proportional to \(E_jL_j\), assuming the same base premium per exposure. The historical average level per exposure is therefore exposure-weighted, \(\sum_jq_jL_j\), where \(q_j=E_j/\sum_kE_k\). That is why the exposure-share formula uses an average of levels, while the earned-premium-share formula uses an average of their reciprocals.
Worked Example: Keeping Exposure Shares and Premium Shares Separate
A hypothetical portfolio has 60 exposures at level 1.00 and 40 exposures at level 1.20. The base premium at level 1.00 is $5,000 per exposure. Find historical premium and the premium at target level 1.10. Assume the exposures and base premium are otherwise comparable.
Method. The supplied counts are exposure information. Calculate the premiums by rate level, then check using the exposure-share formula.
Step 1: Historical premium. The first group contributes \(60(\$5{,}000)=\$300{,}000\). The second group has premium per exposure \(1.20(\$5{,}000)=\$6{,}000\), so it contributes \(40(\$6{,}000)=\$240{,}000\). Total historical premium is \(\$300{,}000+\$240{,}000=\$540{,}000\).
Step 2: Target-level premium. At level 1.10, each of the 100 exposures has premium \(1.10(\$5{,}000)=\$5{,}500\). Thus target-level premium is \(100(\$5{,}500)=\mathbf{\$550{,}000}\).
Step 3: Check with exposure shares. The exposure shares are \(q_1=0.60\) and \(q_2=0.40\). The historical average level per exposure is \(0.60(1.00)+0.40(1.20)=1.08\). The factor is \(1.10/1.08=1.018518519\), and \(\$540{,}000(1.018518519)=\$550{,}000\), subject only to displayed rounding.
Common mistake. Treating the exposure percentages as if they were earned-premium shares would use the wrong formula. The actual historical premium shares here are \(300/540=0.5556\) and \(240/540=0.4444\), not 0.60 and 0.40.
Trending Losses to a Prospective Date
A trend factor depends on the time from the loss observation date to the prospective date. Under a constant annual effective trend rate \(r\), compounded over \(t\) years, the factor is \((1+r)^t\). If the interval is \(m\) months, then \(t=m/12\). For losses summarized by an accident-period midpoint, use the interval from that midpoint to the prospective-period midpoint, unless a different date convention is specified.
The exponent is a year fraction, not a count of months. For example, 18 months is \(18/12=1.5\) years, so the factor at an annual effective rate of 6% is \(1.06^{1.5}\), not \(1.06^{18}\) and not \(1+0.06(1.5)\). The latter would be simple interest, not annual effective compounding.
Worked Example: Trending Loss Cohorts From Their Own Midpoints
A hypothetical book has $240,000 of losses with a midpoint of July 1, 2024, and $160,000 with a midpoint of January 1, 2025. Project both amounts to January 1, 2026, using a constant 5% annual effective loss trend rate.
Method. Trend each cohort from its own midpoint; then add the projected amounts.
Step 1: Find each interval. July 1, 2024, to January 1, 2026, is 18 months or 1.5 years. January 1, 2025, to January 1, 2026, is 12 months or 1 year.
Step 2: Calculate each factor. The first factor is \(1.05^{1.5}=1.075929830\) (displayed to nine decimal places). The second factor is \(1.05^1=1.05\).
Step 3: Project each loss amount.
Adding the unrounded amounts gives \(\$426{,}223.1593\), or $426,223.16 to the nearest cent.
Check and common mistake. The earlier cohort receives the larger factor because it is farther from the prospective midpoint. Applying the 18-month factor to both cohorts would overstate the second cohort; applying the one-year factor to both would understate the first.
Putting On-Leveling and Trending Together
In a complete adjustment, first state what each amount represents and the dates and levels to which it must be moved. On-level premium to the target rate level using the supplied premium or exposure weights. Trend losses from their historical loss date to the prospective loss date. Keep the adjustments on their appropriate data components: a rate-level factor does not trend losses, and a loss trend factor does not adjust premium for rate changes.
Record the target rate level for premium and the prospective midpoint for losses. Confirm the historical amounts and periods refer to a consistent coverage and experience basis.
For premium shares, use \(L_T\sum_jw_j/L_j\). For exposure shares, use \(L_T/\sum_jq_jL_j\). Confirm the relevant shares sum to 1.
Multiply historical earned premium by its on-level factor. Multiply each historical loss amount by the trend factor for its own date interval.
Verify that the trend exponent is in years and that neither adjustment has been applied twice. Retain precision during calculations; round final dollar amounts to cents.
Worked Example: Integrated On-Level and Trend Adjustment
A hypothetical insurer reports $1,200,000 of historical earned premium. Its earned-premium shares are 30% at level 1.00, 45% at level 1.08, and 25% at level 1.12. The target rate level is 1.15. Historical losses are $960,000 with midpoint July 1, 2024; the prospective loss midpoint is January 1, 2026. Use a 6% annual effective loss trend rate.
Method. On-level the premium using earned-premium shares, and trend the losses for 18 months. Do not apply the premium factor to losses.
Step 1: Calculate the on-level factor.
Step 2: On-level premium. Multiplying the historical earned premium by this factor gives \(\$1{,}200{,}000(1.0808630952)=\$1{,}297{,}035.7142\), or $1,297,035.71 to the nearest cent.
Step 3: Calculate the loss interval and factor. The period from July 1, 2024, to January 1, 2026, is 18 months, or 1.5 years. The trend factor is \(1.06^{1.5}=1.0913367953\).
Step 4: Trend losses. The projected losses are \(\$960{,}000(1.0913367953)=\$1{,}047{,}683.3235\), or $1,047,683.32 to the nearest cent.
Adjusted data and check. The resulting pair is $1,297,035.71 of on-level premium and $1,047,683.32 of losses trended to January 1, 2026. The premium increased because the target level exceeds each historical level; the losses increased because the trend rate is positive and the prospective date is later. The common mistake avoided is using earned-premium shares as exposure shares or applying the on-level factor to losses.
Checks That Prevent Misapplication
A complete answer should make the factor’s base visible, not merely report a number. Before accepting a calculation, check the following:
- Weights: Are the stated percentages shares of premium or shares of exposures? Use the matching formula, and confirm the shares total 100%.
- Rate levels: Is each \(L_j\) the historical level for the corresponding premium portion, and is \(L_T\) the intended target?
- Dates: Does the trend interval run from the loss midpoint to the prospective midpoint? Do not substitute policy dates or premium earning dates without a stated reason.
- Time units: Convert months to years before using an annual effective rate. Use compound growth over fractional years.
- Components: On-level premium and trend losses as separate operations. Do not apply a premium adjustment to loss data or trend premium unless the problem specifically calls for it.
- Precision: Keep unrounded factors in intermediate steps and round final dollar amounts only at the end.
These checks are useful because the arithmetic in a combined adjustment is usually straightforward; the main risk is attaching a correct-looking factor to the wrong weights, component, or date. Stating the target bases and the weight definition makes that risk easier to detect.
Check Your Understanding
For each problem, identify the relevant weight basis or date interval before calculating. Report final dollar amounts to the nearest cent where applicable.
- A hypothetical book has $420,000 of earned premium: 50% at level 1.00 and 50% at level 1.08. Find the on-level factor and premium at target level 1.08. (Answer: factor 1.0400; $436,800.00)
- Trend $300,000 of losses for 18 months at a 4% annual effective trend rate. (Answer: factor \(1.04^{1.5}=1.0605961\); $318,178.82)
- A portfolio has 25% of its exposures at level 0.90 and 75% at level 1.10. Find the on-level factor to target level 1.05. (Answer: factor 1.0000)
- A hypothetical insurer earned $600,000 of premium, with 20% at level 1.00 and 80% at level 1.10. Its target is 1.10. Historical losses are $500,000 at midpoint January 1, 2025; the prospective midpoint is January 1, 2026. Use a 5% annual effective loss trend rate. Find on-level premium and trended losses. (Answer: premium $612,000.00; losses $525,000.00)