A Diagnostic Checklist for Combined Adjustments

Why Combined Adjustments Need a Diagnostic Checklist

A combined ratemaking calculation can use familiar adjustments and still produce an unreliable result. The problem is often not the arithmetic itself: a loss trend may start from the wrong date, a premium amount may use a different earning basis from the losses, or a factor may be applied in the wrong direction. A correct-looking total is not evidence that the underlying periods and bases agree.

A reliable review starts by identifying what each amount represents, then checks the endpoint and direction of each adjustment. Only after those checks should the factors be multiplied and the indication calculated. The approach below is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Definition: An on-level factor $O$ converts premium earned at a historical rate level to the selected current rate level. When $O$ is defined as current rate level divided by historical rate level, the premium is multiplied by $O$. A loss-trend factor $T$ converts losses from their historical measurement date to a prospective measurement date. With a constant annual effective trend rate $r$ over $t$ years, $T=(1+r)^t$. Each factor is valid only for the basis and endpoints for which it was developed.

For an adjusted loss-ratio calculation, let $L$ be historical losses and $P$ be earned premium on a compatible experience basis. If the losses are trended to the prospective period and the premium is on-leveled to the selected current rate level, then

\text{Adjusted loss ratio} = \frac{L T}{P O}.

This form assumes that $L$ and $P$ describe compatible experience, that the loss-trend factor moves losses to the intended endpoint, and that $O$ converts the premium to the intended rate level. It does not mean that every data set needs both factors: a factor must not be applied again if the input is already adjusted to that basis.

A Sequence of Checks Before Calculating

Treat each adjustment as a statement about a particular amount, time period, and basis. The checklist below is designed to reveal inconsistencies before they are hidden inside a product of factors.

Label every input.
Record whether losses are historical or already trended, whether premium is written or earned, and what period and rate level each amount represents.

Match the experience bases.
Confirm that losses and premium relate to compatible coverage and experience periods. If a written-premium amount must be converted to earned premium, use the stated earning assumption before on-leveling.

Set the trend endpoints.
Identify the date represented by the historical losses and the prospective date required. Convert the interval to years using the stated time convention; do not substitute a policy, calendar, or evaluation date unless it is the intended endpoint.

Check for previous adjustments.
Establish whether a trend, rate-level change, or earning conversion is already reflected in an input. Apply only the remaining adjustment needed to reach the target basis.

Verify factor direction.
Read the numerator and denominator in the factor definition. A factor defined as current level divided by historical level multiplies historical premium; its reciprocal is used only if the desired conversion is the reverse.

Recalculate and label the result.
Show adjusted losses and adjusted premium separately, then calculate the ratio or other requested quantity. State the final basis so another reviewer can see what the result represents.

Why the Adjusted Loss-Ratio Formula Has This Form

The formula follows from putting the numerator and denominator on their intended bases. Suppose historical losses $L$ are expected to become $L T$ at the prospective loss-cost level. Suppose historical earned premium $P$, when repriced at the selected current rate level, becomes $P O$. The ratio of those two adjusted quantities is therefore

\frac{\text{prospective-level losses}}{\text{current-level premium}} = \frac{L T}{P O}.

The directions are not interchangeable. Trend changes the loss amount in the numerator; on-leveling changes the premium amount in the denominator. This is also why a factor-direction error can have a large effect: dividing premium by $O$ instead of multiplying it changes the denominator in the opposite direction.

The formula is not a license to adjust every amount mechanically. If losses are already at the prospective level, $T$ is not applied again. If premium is already at the selected rate level, $O$ is not applied again. If the premium is written rather than earned, it cannot be treated as earned merely because an on-level factor is available.

Worked Diagnostics

Worked Example: Wrong Trend Endpoint and Reversed On-Level Factor

A hypothetical book has historical losses of $800,000 with a loss midpoint of January 1, 2024, and earned premium of $1,000,000 at the same historical experience basis. The loss trend is 5% per year. Losses are required at January 1, 2026. The on-level factor from the historical premium level to the selected current level is 1.10. A draft calculation uses three years of trend and divides the premium by 1.10. Diagnose and correct it.

Method. Check the dates first, then apply the factor in the direction stated. January 1, 2024 to January 1, 2026 is two years, not three:

$$ T=(1.05)^2=1.1025. $$

The correctly trended losses are

L T = \$800{,}000(1.1025) = \$882{,}000.

The stated on-level factor converts historical premium to current-level premium, so multiply:

P O = \$1{,}000{,}000(1.10) = \$1{,}100{,}000.

The adjusted loss ratio is

\frac{\$882{,}000}{\$1{,}100{,}000} = 0.801818\ldots \approx 80.18\%.

Check. Losses increased by 10.25% over two years, and premium increased by 10%. A ratio near 80%, slightly higher than the historical ratio of 80%, is consistent with those relative changes.

Common mistake avoided. The draft's incorrect three-year trend is $(1.05)^3=1.157625$, and dividing premium by 1.10 gives about $909,090.91. Its resulting ratio would be $\$800{,}000(1.157625)/\$909{,}090.91\approx101.87\%$. Both the endpoint and factor direction are wrong; the corrected ratio is 80.18%.

Worked Example: Written Premium Used as Earned Premium

A hypothetical portfolio has $2,500,000 of written premium. The stated earning pattern says that 80% of this amount is earned in the experience interval being analyzed. Losses of $1,600,000 relate to that same coverage interval. The losses are measured at a midpoint one year before the prospective midpoint, and the annual loss trend is 3%. The factor to bring the earned premium from its historical level to the selected current level is 1.05. Find the adjusted loss ratio and identify the earning-basis trap.

Method. First convert written premium to the earned amount for this interval:

P_{\text{earned}} = \$2{,}500{,}000(0.80) = \$2{,}000{,}000.

The losses need one year of trend, so

T=(1.03)^1=1.03, \qquad L T = \$1{,}600{,}000(1.03) = \$1{,}648{,}000.

On-level the earned premium, not the original written premium:

P_{\text{earned}} O = \$2{,}000{,}000(1.05) = \$2{,}100{,}000.

Thus,

\text{Adjusted loss ratio} = \frac{\$1{,}648{,}000}{\$2{,}100{,}000} = 0.7847619\ldots \approx 78.48\%.

Check. The denominator is the $2,000,000 earned amount increased by 5% to $2,100,000. The earning conversion has been applied once, before on-leveling.

Common mistake avoided. Treating all $2,500,000 of written premium as earned gives an incorrect current-level denominator of $2,625,000 and a ratio of about 62.78%. The opposite error is to apply the 80% earning factor again after starting with the $2,000,000 earned amount; that would also misstate the basis.

Worked Example: Detecting a Trend Already Included in the Losses

A hypothetical analysis reports historical losses of $500,000 at July 1, 2023. A data-preparation step has already trended those losses at 3% per year to the prospective midpoint of July 1, 2025 and reports the adjusted amount as $530,450. Earned premium is $700,000 at the historical rate level, and the on-level factor to the selected current level is 1.04. A draft worksheet applies the two-year trend factor to the reported $530,450 again. Find the correct adjusted loss ratio and explain the duplicate adjustment.

Method. Verify the stated adjustment:

(1.03)^2=1.0609, \qquad \$500{,}000(1.0609)=\$530{,}450.

The reported $530,450 is already at the prospective midpoint, so the remaining trend factor is 1. The on-leveled premium is

\$700{,}000(1.04)=\$728{,}000.

The correct ratio is therefore

\frac{\$530{,}450}{\$728{,}000} = 0.7286401\ldots \approx 72.86\%.

Check. The numerator remains $530,450 because its trend adjustment is complete; only the historical-level premium is on-leveled in this calculation.

Common mistake avoided. Multiplying by 1.0609 again would produce losses of $\$530{,}450(1.0609)=\$562{,}754.405$, and a ratio of approximately 77.30%. That result trends the same two-year interval twice.

What the Checklist Can—and Cannot—Validate

A sound calculation is internally consistent, but consistency alone does not establish that every assumption is appropriate. For example, the checklist can verify that a trend factor uses the stated start and end dates; it does not determine whether the selected trend rate is the best estimate. Likewise, it can expose a written-versus-earned mismatch, but it cannot replace an appropriate earning pattern or an appropriate on-level factor.

Keep a short audit trail alongside the arithmetic. For each input, record its period, basis, and adjustment status. For each factor, record what it converts from and to. If a factor combines multiple rate changes, confirm that it applies to the premium being adjusted rather than to a different policy or exposure mix. A weighted factor may be appropriate for a premium amount made up of different rate levels, but its weights must correspond to that amount's actual composition.

A useful final check is to compare the direction and relative size of the changes. With positive loss trend and an on-level increase in premium, the numerator and denominator both rise, so the ratio depends on which rises more. A result that moves sharply in the opposite direction from what the factors imply should trigger a review of the dates, factor definitions, and bases before any further interpretation.

Key takeaway: Do not begin by multiplying every available factor. First establish the input's period and basis, identify the required endpoint, check which adjustments are already included, and confirm each factor's direction. Then adjust losses and premium separately and calculate the ratio.

Check Your Understanding

For each problem, check periods and bases before calculating. Round percentages to two decimal places where needed.

A hypothetical segment has losses of $360,000 and earned premium of $500,000 at a historical rate level. The losses need two years of trend at 4%, and the premium on-level factor is 1.08. Calculate the adjusted loss ratio. (Answer: $389,376 of trended losses; $540,000 of on-level premium; 72.11%.)
Losses of $720,000 are measured at January 1, 2023 and must be trended to January 1, 2025 at 4% annually. Historical earned premium is $900,000, and its on-level factor is 1.05. Calculate the adjusted loss ratio. (Answer: $778,752 of trended losses; $945,000 of on-level premium; 82.41%.)
A hypothetical portfolio reports $1,800,000 of written premium, of which 75% is earned in the matching experience interval. Matching losses are $1,080,000 and require one year of trend at 5%. The on-level factor for earned premium is 1.04. Calculate the adjusted loss ratio. (Answer: earned premium $1,350,000; trended losses $1,134,000; on-level premium $1,404,000; 80.77%.)
A worksheet reports losses of $420,000 that have already been trended to the prospective midpoint. Earned premium is $600,000 at a historical rate level, and the factor to current level is 1.10. What loss ratio should be used if no other adjustment is required? (Answer: $660,000 of on-level premium; $420,000 of losses; 63.64%. Do not apply another trend factor.)
A hypothetical segment has $300,000 of losses at October 1, 2022, to be trended to April 1, 2024 at 4% annually. Earned premium is $400,000 and its on-level factor is 1.05. Use the actual 1.5-year interval and calculate the adjusted loss ratio. (Answer: trend factor $1.04^{1.5}\approx1.060596$; trended losses about $318,179; on-level premium $420,000; 75.76%.)

What You'll Learn