Deriving the Log-Additive Form of a Rating Plan

Why Write a Rating Plan in Log Form?

A multiplicative rating plan combines a base rate with factors for the characteristics of a risk. The product is natural for calculating the rate, but it can be useful to express the same plan as a sum. Taking logarithms provides that equivalent additive form: the logarithm of the rate is the logarithm of the base rate plus the logarithm of each factor. The transformation does not change the indicated rate; it changes how the calculation is represented.

This relationship is an algebraic property of positive products. It does not require a particular distribution for insurance losses or an assumption that factors are additive on the dollar-rate scale. The critical condition is positivity: the base rate, each relativitiy, and therefore the resulting rate must be greater than zero. If any input is zero or negative, its real logarithm is not defined, so this log form cannot be used as written.

Definition: For a rating plan with positive base rate $B$ and positive relativities $r_1,\ldots,r_k$, the indicated rate is $R=B\prod_{j=1}^{k}r_j$. Taking natural logarithms gives the equivalent log-additive form $\ln R=\ln B+\sum_{j=1}^{k}\ln r_j$. Applying the exponential function recovers the original rate.

From a Product to a Sum

The base rate $B$ is the rate before the listed relativities are applied. Each $r_j$ is a multiplicative factor for a rating characteristic, and $k$ is the number of factors applied to the risk. A relativitiy greater than 1 increases the rate relative to the rate before that factor is applied; one between 0 and 1 decreases it. The plan combines those effects by multiplication, not by adding the relativities to the base rate.

The standard logarithm rules state that the logarithm of a product of positive values equals the sum of their logarithms. Applying that rule to $R=B r_1 r_2 \cdots r_k$ gives the log-additive expression. The natural logarithm, $\ln$, is used here; any one logarithm base could be used consistently, but the inverse function must use that same base.

\begin{aligned} R &= B r_1 r_2 \cdots r_k,\\ \ln R &= \ln(B r_1 r_2 \cdots r_k)\\ &= \ln B+\ln r_1+\ln r_2+\cdots+\ln r_k\\ &= \ln B+\sum_{j=1}^{k}\ln r_j. \end{aligned}

The additive equation is on the log-rate scale. It does not say that the dollar rate equals the base rate plus the relativities, nor that the raw relativities can be added. To return to the rate scale, exponentiate both sides. Since the exponential function reverses the natural logarithm, $\exp(\ln R)=R$.

R=\exp\left(\ln B+\sum_{j=1}^{k}\ln r_j\right) =\exp(\ln B)\prod_{j=1}^{k}\exp(\ln r_j) =B\prod_{j=1}^{k}r_j.

This identity is a useful check on the transformation: the log-additive calculation and the original multiplicative calculation must give the same rate, apart from any rounding used in the intermediate calculations. It also explains why the operation is reversible. The logarithm changes multiplication into addition, and exponentiation changes that sum back into the original product.

Write the product.
Identify the positive base rate and every relativitiy applied to the risk.

Take logarithms.
Use the product rule to write the log of the rate as the sum of the log of the base rate and the logs of the factors.

Evaluate or rearrange.
Calculate the sum, or isolate an unknown log factor if the other terms are known.

Return to the rate scale.
Apply the exponential function to the complete log-rate value and check against the product where practical.

Worked Calculations

Worked Example: Converting Three Relativities to a Log Sum

A hypothetical insurer uses a base rate of $600 for a coverage unit. For one risk, the applicable relativities are 1.20, 0.85, and 1.10. Calculate the indicated rate using both the product and log-additive forms. Use natural logarithms and round the final rate to the nearest cent.

Method and condition. All four inputs are positive, so their natural logarithms are defined. First calculate the product directly:

$600(1.20)=720.00$, then $720.00(0.85)=612.00$, and $612.00(1.10)=673.20$. The indicated rate from the product form is therefore $673.20.

For the log-additive form, the individual log values, shown to nine decimal places for the calculation, are $\ln(600)=6.396929655$, $\ln(1.20)=0.182321557$, $\ln(0.85)=-0.162518929$, and $\ln(1.10)=0.095310180$. Add them:

$6.396929655+0.182321557-0.162518929+0.095310180=6.512042463$. Thus $\ln R=6.512042463$, and $R=\exp(6.512042463)\approx673.20$. Rounded to the nearest cent, the log form also gives $673.20.

Check and trap avoided. The direct product and the exponentiated log sum agree. A common error is to add the relativities themselves to $600; that would give $600+1.20+0.85+1.10=$603.15$, which mixes dollars and dimensionless factors and is not the rating formula.

Worked Example: A Decrease and Two Increases

A hypothetical policy has a base rate of $840. Its rating plan applies relativities of 0.75, 1.40, and 1.08. Find the indicated rate in product form and verify it through the log-additive form. Round the final rate to the nearest cent.

Product method. Apply each factor in turn: $840(0.75)=630.00$; $630.00(1.40)=882.00$; and $882.00(1.08)=952.56$. The product-form indicated rate is $952.56.

Log-additive method. The log values, to nine decimal places, are $\ln(840)=6.733401892$, $\ln(0.75)=-0.287682072$, $\ln(1.40)=0.336472237$, and $\ln(1.08)=0.076961041$. Therefore,

$\ln R=6.733401892-0.287682072+0.336472237+0.076961041=6.859153098$. Exponentiating gives $R=\exp(6.859153098)\approx952.56$, or $952.56 to the nearest cent.

Check and trap avoided. The two calculations agree. In the log sum, the factor 0.75 contributes a negative term because its logarithm is negative, while the factors above 1 contribute positive terms. The factor 0.75 is still applied by multiplication in the original plan; it is not subtracted as a dollar amount.

Worked Example: Recovering a Missing Relativity

A hypothetical rating plan has a base rate of $500 and three positive factors. Two known relativities are 1.25 and 0.96. The indicated rate for a risk is $720. Find the missing relativitiy, first using the log-additive equation and then check it in the product form. Round the final factor to four decimal places.

Set up the log equation. Let $r$ denote the missing factor. The plan gives $720=500(1.25)(0.96)r$, so the log-additive equation is $\ln(720)=\ln(500)+\ln(1.25)+\ln(0.96)+\ln(r)$. Isolate the unknown term:

$\ln(r)=\ln(720)-\ln(500)-\ln(1.25)-\ln(0.96)$. The needed log values are $\ln(720)=6.579251212$, $\ln(500)=6.214608098$, $\ln(1.25)=0.223143551$, and $\ln(0.96)=-0.040821995$. Hence $\ln(r)=6.579251212-6.214608098-0.223143551-(-0.040821995)=0.182321558$.

Exponentiate to obtain $r=\exp(0.182321558)\approx1.2000$. Check in the product form: $500(1.25)=625$, $625(0.96)=600$, and $600(1.2000)=720$. The missing relativitiy is 1.2000, as required.

Method and trap avoided. This method isolates the missing factor on the log scale, then exponentiates to recover it. The known factor 0.96 has a negative logarithm; subtracting that log therefore adds 0.040821995. Dropping the negative sign would produce an incorrect missing factor.

Conditions, Interpretation, and Calculation Checks

The log-additive form is an exact restatement of a positive multiplicative rating formula. It is useful for expressing the combined contribution of several factors as a sum of log terms and for rearranging a rating equation. It is not a different pricing plan, and it does not make the original dollar rate additive. After adding the terms on the log scale, the result must be exponentiated to obtain a rate.

The formula requires $B>0$ and $r_j>0$ for every factor. A zero factor would make the product rate zero, but $\ln(0)$ is undefined; a negative factor is not a valid positive relativitiy in this setting and likewise has no real natural logarithm. If the inputs are positive but the product is extremely large or small, the log form can also be convenient for calculation because it delays multiplying all factors together. Any rounding of log values should be limited to intermediate steps, with rounding applied to the requested final result.

The log base must be used consistently. For example, if common logarithms are used instead of natural logarithms, the inverse operation is raising 10 to the resulting sum, not applying the natural exponential function. Using $\ln$ throughout avoids that mismatch: sum natural logarithms, then use $\exp$.

Key takeaway: For a positive multiplicative rating plan, $\ln R=\ln B+\sum_j\ln r_j$. The terms add only after taking logarithms; exponentiating their sum recovers the original indicated rate. Check the result against the product form when possible.

Common Mistakes

Adding factors on the rate scale. The product $B r_1 r_2$ is not $B+r_1+r_2$. Relativities are multipliers, not dollar amounts.
Taking the log of each value but forgetting the base rate. The complete sum includes $\ln B$ as well as every $\ln r_j$.
Stopping at the log rate. A value for $\ln R$ is not the indicated rate. Apply $\exp$ to return to the rate scale.
Using inconsistent inverse operations. If the sum uses natural logarithms, use $\exp$; if it uses common logarithms, use 10 raised to that sum.
Ignoring the sign of a factor's logarithm. A positive factor below 1 has a negative logarithm. Preserve that sign when adding or rearranging terms.

Check Your Understanding

For each problem, write or use the log-additive form as requested. Rates are hypothetical. Round dollar rates to the nearest cent and factors to four decimal places.

A plan has a $400 base rate and relativities 1.15 and 0.90. Write the log-additive expression and calculate the indicated rate. (Answer: $\ln R=\ln(400)+\ln(1.15)+\ln(0.90)$; $414.00)
A hypothetical policy uses a $1,250 base rate and factors 0.80, 1.12, and 1.05. Calculate $\ln R$ using the given log values $\ln(1,250)=7.130898830$, $\ln(0.80)=-0.223143551$, $\ln(1.12)=0.113328685$, and $\ln(1.05)=0.048790164$, then find the rate. (Answer: $\ln R=7.069874128$; $1,176.00)
A $700 base rate is multiplied by 1.30, 0.92, and 1.05. Calculate the rate in product form and state the corresponding log-additive expression. (Answer: $\ln R=\ln(700)+\ln(1.30)+\ln(0.92)+\ln(1.05)$; $879.06)
A plan's base rate is $600. One known factor is 0.90, and the indicated rate after applying that factor and one unknown positive factor is $810. Use the log-additive equation to find the unknown factor and check it by multiplication. (Answer: $r=1.5000$; $600(0.90)(1.5000)=$810.00$)
A candidate calculates $\ln R=\ln(900)+\ln(1.25)+\ln(0.80)=\ln(900)$, then reports $6.8024 as the rate. Identify the error and give the correct indicated rate. (Answer: The log sum is $\ln(900)$, but that is not the rate; exponentiate it to obtain $900.00)

What You'll Learn