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CAS Exam 5 · Premium On-Leveling Methods · Tutorial 129 of 989

Direction and Size of On-Level Factors

Compare the target and embedded average rate levels to calculate, interpret, and apply an on-level factor correctly.

Intermediate 9 min read

What You'll Learn

  • Define an on-level factor as the target rate level divided by the rate level embedded in historical premium
  • Determine whether on-leveling increases or decreases premium before calculating
  • Convert a factor into a percentage adjustment and check the direction
  • Calculate a weighted average rate level using appropriate exposure weights
  • Combine period-specific adjustments to find an aggregate on-level factor
  • Recognize when a simple average of factors gives the wrong result

Why Direction Matters

On-leveling adjusts historical premium to reflect a specified target rate level. The essential comparison is between the target rate level and the average rate level already embedded in the historical premium. If the target is higher, on-leveling increases premium; if it is lower, on-leveling decreases it.

The factor’s direction is easy to reverse when working quickly. A reliable check is to ask what the adjustment should do before calculating: would the target level make the same exposures produce more or less premium than the historical average level? The factor should agree with that answer.

Definition: An on-level factor for historical premium is the ratio of the target rate level to the average rate level embedded in that premium. If \(L_T\) is the target rate level and \(\bar L_H\) is the historical average rate level, then \(F=L_T/\bar L_H\), and on-leveled premium is \(P_{\text{on}}=P_HF\). Both rate levels must be expressed on the same index basis.

Rate levels are indices, not dollar amounts. They might be expressed relative to a common base level of 1.00, 100, or another positive number. The ratio is unchanged if both levels are rescaled by the same constant: for example, \(1.10/1.00=110/100\). What matters is that the numerator and denominator use a consistent basis.

This is a deterministic adjustment for rate level. It does not itself adjust for changes in exposure, business mix, coverage, loss costs, or the time value of money. The interpretation assumes those other factors are held constant or handled separately. No interest rate, mortality assumption, or loss distribution is used in the examples below.

Formula, Direction, and Interpretation

The formula follows by comparing the premium for the same underlying exposures at two rate levels. If \(B\) denotes the premium those exposures would produce at the common base level, historical premium is \(P_H=B\bar L_H\). At the target level, it would be \(P_T=BL_T\). Dividing cancels the common base premium:

$$ \frac{P_T}{P_H} = \frac{BL_T}{B\bar L_H} = \frac{L_T}{\bar L_H} = F. $$

Thus, multiplying historical premium by \(F\) converts its rate-level component from \(\bar L_H\) to \(L_T\). This justification requires the comparison to use the same underlying exposure basis; the factor is not a correction for a change in the quantity or mix of exposures.

ComparisonFactorPremium adjustment
\(L_T>\bar L_H\)\(F>1\)Increase
\(L_T=\bar L_H\)\(F=1\)No change
\(L_T<\bar L_H\)\(F<1\)Decrease

The percentage change in premium is \(F-1\), expressed as a percentage. A factor of 1.06 means an increase of 6%, not 106%; a factor of 0.94 means a decrease of 6%, not a 94% decrease. For a factor below 1, the decrease is \(1-F\).

Worked Example: On-Leveling to a Lower Target

A hypothetical insurer has $2,240,000 of historical earned premium. The average rate level embedded in that premium is 1.12, and the target level is 1.05. Find the on-level factor, on-leveled premium, and percentage change. Name the method and state the direction before calculating. No interest rate, mortality assumption, or loss distribution is used; this is a deterministic rate-level adjustment.

Method: direct level ratio. First compare the levels: \(1.05<1.12\). The factor must therefore be below 1, and the premium must decrease.

Calculate the factor:

$$ F=\frac{1.05}{1.12}=0.9375. $$

Apply it to historical premium:

$$ P_{\text{on}} = \$2{,}240{,}000(0.9375) = \$2{,}100{,}000. $$

The percentage change is \(0.9375-1=-0.0625\), so premium decreases by 6.25%. As a check, the difference is \(\$2{,}100{,}000-\$2{,}240{,}000=-\$140{,}000\), which is \(6.25\%\) of the original premium. The factor is below 1 and the adjusted amount is lower, as required. This method avoids confusing the target-to-historical ratio with its reciprocal.

Finding an Embedded Average Rate Level

Historical premium can reflect more than one rate level. In that case, use the appropriate weights to find the average level embedded in the premium before comparing it with the target. For a fixed mix of exposures, a weighted average can be written as:

$$ \bar L_H=\sum_j w_j L_j, \qquad \sum_j w_j=1, \qquad F=\frac{L_T}{\sum_j w_jL_j}. $$

Here, \(L_j\) is a rate level for group \(j\), and \(w_j\) is that group’s share of the relevant base-level premium for the exposure mix being analyzed. The weight reflects how much of the underlying exposure or base premium belongs to the group; it is not automatically the group’s share of premium already written at its current rate level. The weighting basis must match the rate-level calculation.

Worked Example: Weighted Average Level Across Two Groups

A hypothetical portfolio has two fixed exposure groups. Group A represents 25% of the portfolio’s base-level premium and has a rate level of 0.96. Group B represents 75% and has a rate level of 1.08. Historical premium is $840,000, and the target level is 1.1025. Calculate the weighted historical level, factor, and on-leveled premium. No interest rate, mortality assumption, or loss distribution is used.

Method: fixed-mix weighted level, followed by the level ratio. The stated shares are the appropriate weights, and they sum to \(0.25+0.75=1.00\).

Calculate each group’s contribution to the average level:

$$ 0.25(0.96)=0.24, \qquad 0.75(1.08)=0.81. $$

Add the contributions:

$$ \bar L_H=0.24+0.81=1.05. $$

The target is higher than the embedded average, so the adjustment should increase premium. The factor is:

$$ F=\frac{1.1025}{1.05}=1.05. $$

Therefore:

$$ P_{\text{on}} = \$840{,}000(1.05) = \$882{,}000. $$

The result is a 5% increase, or \(\$42{,}000\). As a check, the factor is above 1 and the adjusted premium exceeds the historical premium. The method avoids taking an unweighted average of group levels, which would ignore that Group B represents three times the base-level premium share of Group A.

Combining Period-Specific Adjustments

When historical premium is divided into periods with different embedded rate levels, calculate each period’s factor against the target and apply it to that period’s premium. If \(P_j\) is the observed premium for period \(j\), the adjusted total and the factor for the combined premium are:

$$ P_{\text{on,total}} = \sum_j P_j\frac{L_T}{L_j}, \qquad F_{\text{aggregate}} = \frac{\sum_j P_j(L_T/L_j)}{\sum_j P_j}. $$

The aggregate factor is a premium-weighted average of the period-specific factors, with observed historical premiums as the weights. It is not generally the simple arithmetic average of those factors. A period with more historical premium has more influence on the aggregate adjustment.

Worked Example: Combining Two Historical Periods

A hypothetical insurer is combining premium from two historical periods to a target rate level of 1.00. Period 1 has $500,000 of historical premium at an average level of 0.80; Period 2 has $750,000 at an average level of 1.25. Find each period’s factor, the adjusted total, and the aggregate factor. No interest rate, mortality assumption, or loss distribution is used.

Method: period-by-period on-leveling, then aggregation. The target exceeds Period 1’s level, so its factor should increase premium. The target is below Period 2’s level, so its factor should decrease that premium.

Calculate and apply the Period 1 factor:

$$ F_1=\frac{1.00}{0.80}=1.25, \qquad P_{1,\text{on}}=\$500{,}000(1.25)=\$625{,}000. $$

Calculate and apply the Period 2 factor:

$$ F_2=\frac{1.00}{1.25}=0.80, \qquad P_{2,\text{on}}=\$750{,}000(0.80)=\$600{,}000. $$

Add the adjusted premiums and the original premiums:

$$ P_{\text{on,total}} = \$625{,}000+\$600{,}000 = \$1{,}225{,}000, \qquad P_{\text{historical,total}} = \$500{,}000+\$750{,}000 = \$1{,}250{,}000. $$

The aggregate factor is:

$$ F_{\text{aggregate}} = \frac{\$1{,}225{,}000}{\$1{,}250{,}000} = 0.98. $$

The combined premium decreases by 2%. As a check, the period-specific changes are \(+\$125{,}000\) and \(-\$150{,}000\), for a net change of \(-\$25{,}000\). The method avoids taking the simple average \((1.25+0.80)/2=1.025\), which would incorrectly imply a 2.5% increase and would give equal weight to periods with different amounts of historical premium.

Common Mistakes and Practical Checks

The most important exam check is directional: compare the target with the embedded average before dividing. For the first example, reversing the ratio gives \(1.12/1.05=1.0667\) (rounded), which would increase the premium even though the target rate level is lower. That is the wrong calculation. The correct factor is \(1.05/1.12=0.9375\), which decreases premium as required.

  • Reversing the ratio. For historical premium moved to a target level, use target divided by embedded historical level. The reciprocal applies to the reverse movement, from target back to the historical level.
  • Reading a factor as a percentage change. A factor of 0.92 means a 8% decrease, not a 92% decrease. Subtract 1 to interpret the signed percentage change.
  • Using the latest rate level instead of the embedded average. Historical premium may reflect several levels. The denominator must represent the average level embedded in the premium being adjusted.
  • Averaging factors without considering premium weights. When combining periods, use the adjusted premiums divided by total historical premium, or equivalently use a premium-weighted average of the factors.
  • Confusing rate-level changes with other changes. On-leveling does not by itself adjust for a changed exposure mix or a changed amount of exposure. Those effects require separate treatment.

A final reasonableness check should confirm three things: the target and historical levels use the same index basis; the factor’s direction agrees with their comparison; and the on-leveled premium changes in the direction implied by the factor. These checks catch many arithmetic and interpretation errors before a result is reported.

Key takeaway: To on-level historical premium, divide the target rate level by the average level embedded in that premium, then multiply by the historical premium. A factor above 1 increases premium; a factor below 1 decreases it. For multiple periods, adjust each period and aggregate the resulting premiums.

Check Your Understanding

For each problem, identify the direction before calculating. Amounts are hypothetical, and no interest rate, mortality assumption, or loss distribution is needed.

  1. A historical premium of $540,000 has an embedded average rate level of 1.08. The target is 0.972. Find the factor, adjusted premium, and percentage change. (Answer: \(F=0.9000\); adjusted premium = $486,000; decrease of 10%.)
  2. A portfolio’s base-level premium shares are 30% at a rate level of 0.95 and 70% at 1.10. Historical premium is $1,250,000, and the target is 1.1183. Find the weighted historical level and on-leveled premium. (Answer: weighted level = 1.055; \(F=1.0600\); adjusted premium = $1,325,000, an increase of 6%.)
  3. Historical premium is $300,000 at an average level of 0.75 and $700,000 at an average level of 1.25. Both periods are to be adjusted to level 1.00. Find the adjusted total and aggregate factor. (Answer: adjusted total = $960,000; aggregate factor = 0.9600, a decrease of 4%.)
  4. Historical premium is $2,400,000 at level 1.20, and the target is 1.15. Calculate the factor and adjusted premium to the nearest cent. (Answer: \(F=1.15/1.20=0.958333\ldots\); adjusted premium = $2,300,000.00; decrease of approximately 4.1667%.)
  5. Two periods have historical premiums of $400,000 and $600,000, with embedded levels of 0.80 and 1.25, respectively. The target is 1.00. Find both period factors and the aggregate factor, and explain why their simple average is not the aggregate factor. (Answer: factors are 1.25 and 0.80; adjusted premiums are $500,000 and $480,000; aggregate factor = $980,000/$1,000,000 = 0.9800. Their simple average, 1.025, gives equal weight to periods with unequal historical premiums.)

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