Deriving the Rate Change Formula

Why the Rate Change Formula Works

A rate indication asks how the current rate level should change to support projected costs while meeting a target financial result. The percentage change is not obtained by subtracting a loss ratio from a target loss ratio. It is obtained by comparing the premium needed at the target loss ratio with the premium at the current rate level. That comparison produces a rate-level factor; subtracting one from the factor expresses the change as a percentage.

The calculation only has meaning when the loss and premium amounts are comparable. Projected losses must reflect the exposure period and coverage being priced, and the on-level premium must represent those same exposures at a consistent current rate level. The permissible loss ratio must also use the same definition of losses as the numerator. These matching-basis principles are essential to interpreting the result.

Definition: Let $L^{\text{proj}}$ be projected losses on the selected exposure and coverage basis, $P^{\text{on}}$ be the premium for that same basis restated to the current rate level, and $r_p$ be the permissible loss ratio. The permissible loss ratio is the share of premium available for the losses included in $L^{\text{proj}}$, after allowing for the other provisions included in the rate. The indicated rate-level factor is $$ F=\frac{L^{\text{proj}}}{r_p P^{\text{on}}}. $$ The indicated percentage rate change is $$ c=F-1=\frac{L^{\text{proj}}}{r_p P^{\text{on}}}-1. $$

If $c>0$, the indicated change is an increase; if $c<0$, it is a decrease; and if $c=0$, the current rate level is indicated. For example, $F=1.08$ means the target rate level is 1.08 times the current rate level, or an indicated increase of 8%. It does not say that every policyholder’s dollar premium will increase by exactly 8%.

Deriving the Formula

The formula follows directly from the definition of a loss ratio. At the indicated rate level, the premium required on the chosen basis must be sufficient for projected losses to equal the permissible proportion of premium. Denote this required premium by $P^*$. Then the target condition is $L^{\text{proj}}/P^*=r_p$. Solving for $P^*$, and then comparing it with current on-level premium, gives the result.

Set the target condition.
At the indicated premium, projected losses are the permissible share of that premium: $L^{\text{proj}}/P^*=r_p$.

Solve for required premium.
Rearranging gives $P^*=L^{\text{proj}}/r_p$. This is the premium needed on the selected basis to meet the target ratio.

Compare target with current level.
The factor relating the target rate level to the current one is $F=P^*/P^{\text{on}}$. Substitution gives $F=L^{\text{proj}}/(r_pP^{\text{on}})$.

Express the factor as a change.
A factor of 1.00 means no change, so the percentage change is $c=F-1$. Multiply $c$ by 100 to state it as a percentage.

The same formula can be understood using the projected loss ratio at the current rate level, $r_c=L^{\text{proj}}/P^{\text{on}}$. Substitution gives $F=r_c/r_p$. Thus the factor is the current-basis projected loss ratio divided by the permissible loss ratio. A projected ratio above the permissible ratio leads to an increase; a projected ratio below it leads to a decrease.

F=\frac{L^{\text{proj}}/P^{\text{on}}}{r_p} =\frac{r_c}{r_p}, \qquad c=\frac{r_c}{r_p}-1

Worked Calculations

In each calculation, first identify what the loss amount includes, what premium basis is being compared, and how the permissible ratio is defined. The formula is not a substitute for matching those inputs.

Worked Example: Indicating a Rate Decrease

A hypothetical personal property program has projected losses of $1,260,000 for a set of exposures. The on-level premium for those same exposures is $2,000,000. The permissible loss ratio is 70%, with the numerator defined to include exactly the losses in the projected amount. Find the indicated percentage rate change.

Method. Apply the loss ratio rate indication formula, comparing the premium required at the permissible ratio with current on-level premium.

First calculate the current-basis projected loss ratio:

r_c=\frac{\$1{,}260{,}000}{\$2{,}000{,}000}=0.6300=63.00\%.

The premium required at the 70% permissible loss ratio is:

P^*=\frac{\$1{,}260{,}000}{0.70} =\$1{,}800{,}000.

Therefore, the indicated factor and percentage change are:

F=\frac{\$1{,}800{,}000}{\$2{,}000{,}000}=0.9000, \qquad c=0.9000-1=-0.1000.

The indicated change is a 10.00% decrease. Check: at the indicated premium of $1,800,000, projected losses are $1,260,000, and $1,260,000/$1,800,000=0.70, the permissible ratio. The calculation avoids treating the difference between 63% and 70% as the rate change.

Worked Example: Keeping Loss Adjustment Expense in the Correct Basis

A hypothetical liability portfolio has projected losses of $780,000 and projected loss adjustment expenses of $60,000. Its on-level premium is $1,400,000. The permissible ratio is 70% of premium for losses and loss adjustment expenses combined. Find the indicated percentage rate change.

Method. Match the numerator to the permissible-ratio definition by combining losses and loss adjustment expenses before applying the formula.

The amount governed by the 70% ratio is:

L^{\text{proj}}=\$780{,}000+\$60{,}000=\$840{,}000.

The required premium is:

P^*=\frac{\$840{,}000}{0.70}=\$1{,}200{,}000.

The factor and change are:

F=\frac{\$1{,}200{,}000}{\$1{,}400{,}000} =\frac{6}{7} =0.8571429, $$ $$ c=0.8571429-1=-0.1428571.

Rounded to two decimal places as a percentage, the indication is a 14.29% decrease. Check: $0.70\times \$1{,}200{,}000=\$840{,}000$, exactly the combined projected amount. The calculation avoids omitting the expense from the numerator or adding it again after the permissible ratio has already been defined to include it.

Worked Example: Trending Losses Before Indicating a Change

A hypothetical program has historical losses of $900,000. For this calculation, those losses are projected forward by two years at an annual trend of 8%. The matching on-level premium is $1,600,000, and the permissible loss ratio is 68%. Find the indicated percentage change. Assume the stated premium and loss basis otherwise match.

Method. Project the losses to the target period first, then compare the projected amount with the premium required at the permissible ratio.

The two-year trend factor is:

$$ (1+0.08)^2=1.08^2=1.1664. $$

The projected losses are:

L^{\text{proj}}=\$900{,}000(1.1664)=\$1{,}049{,}760.

The premium required at a 68% permissible ratio is:

P^*=\frac{\$1{,}049{,}760}{0.68} =\$1{,}543{,}764.7059.

Compare this with the $1,600,000 on-level premium:

F=\frac{\$1{,}543{,}764.7059}{\$1{,}600{,}000} =0.9648529, $$ $$ c=0.9648529-1=-0.0351471.

Rounded to two decimal places as a percentage, the indicated change is a 3.51% decrease. Check: the indicated premium is lower than the current on-level premium, and $0.68\times \$1{,}543{,}764.7059$ is approximately $1,049,760. The calculation avoids applying the formula to historical losses before projecting them to the target period.

Rate Level Is Not Dollar Premium

A rate level describes the scale of the rates used to price a defined portfolio or exposure basis. A dollar premium is the amount charged for a particular policy or exposure. An indicated rate factor compares two rate levels; it does not by itself calculate each insured’s renewal premium. A policy’s premium can also depend on its exposure amount, classification, limits, deductibles, rating factors, and any applicable policy-level changes.

For example, suppose a rate indication gives $F=1.06$. This says the indicated overall rate level is 6% above the current rate level on the basis used in the analysis. If a hypothetical policy’s current premium were $500 and every component of its price were subject to exactly that same factor, its resulting premium would be $1.06\times \$500=\$530$. But a portfolio indication alone does not establish that each policy has the same factors or receives the same dollar change.

Likewise, a dollar difference between two premiums is not a percentage rate change until the relevant base is stated. A $30 difference on a $500 premium is 6%; a $30 difference on a $1,000 premium is 3%. The indicated factor is the comparison of rate levels on a consistent basis, not a standalone dollar adjustment.

Assumptions and Common Traps

The formula is appropriate when the required premium can be represented by the projected loss amount divided by a permissible loss ratio, and when $P^{\text{on}}$ is a valid current-level comparison for the same basis. It does not repair mismatched data. For example, if projected losses include loss adjustment expenses but the permissible ratio applies only to losses, the numerator and ratio do not agree. Either use a ratio defined for the combined amount or separate the components and handle each consistently.

A further condition is that the permissible loss ratio correctly reflects the provisions excluded from the loss numerator. If expense, profit, or contingency provisions are already reflected in the permissible ratio, do not add those amounts again to the required premium. Conversely, if the selected ratio does not account for a component that must be funded, the simple calculation is incomplete until that component is incorporated consistently.

Common exam trap: Subtracting the projected loss ratio from the permissible loss ratio gives a difference in ratios, not the required percentage rate change. If the current-basis ratio is 63% and the permissible ratio is 70%, the incorrect calculation $63\%-70\%=-7\%$ is not the rate indication. The correct factor is $0.63/0.70=0.90$, so the correct indicated change is $0.90-1=-10\%$. The ratio of ratios is required because a rate change scales premium, and therefore scales the loss ratio inversely.

Before accepting an answer, check its direction and its implied target. If the projected loss ratio exceeds the permissible ratio, the indicated factor must exceed one. If it is below the permissible ratio, the factor must be below one. Finally, multiplying the target premium by the permissible ratio should recover the projected losses used in the calculation. These checks catch sign errors, mismatched bases, and confusion between the rate level and the dollar premium.

Key takeaway: Calculate the premium required at the permissible loss ratio, divide it by on-level premium to obtain the indicated rate-level factor, and subtract one to obtain the percentage change. Matching the loss, premium, and ratio bases is as important as the arithmetic.

Check Your Understanding

For each problem, use consistent loss and premium bases and state the indicated change.

A hypothetical book has projected losses of $525,000, on-level premium of $750,000, and a permissible loss ratio of 70%. What is the indicated percentage rate change? (Answer: 0.00%)
A hypothetical equipment portfolio has projected losses of $960,000, on-level premium of $1,200,000, and a permissible loss ratio of 64%. Find the indicated percentage change. (Answer: 25.00% increase)
A hypothetical book has projected losses of $640,000, on-level premium of $1,000,000, and a permissible loss ratio of 80%. Find the indicated percentage change. (Answer: 20.00% decrease)
A hypothetical program has losses of $500,000 and loss adjustment expenses of $40,000, with a combined permissible ratio of 60% and on-level premium of $900,000. Find the indicated change. (Answer: 0.00%)
Historical losses for a hypothetical portfolio are $400,000 and are projected forward one year at 10%. On-level premium is $600,000 and the permissible loss ratio is 75%. Calculate the indicated change after projecting losses. (Answer: 2.22% decrease)

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