Deriving the Overall Indication Relationship

Why the Overall Indication Is a Balance

An overall rate-level indication asks what change in the current rate level would make premium sufficient for the costs and other requirements included in the analysis. The calculation is a balance: premium at the indicated level must equal projected dollar costs plus any provisions that depend on premium. The distinction between those two kinds of amounts is central. A projected loss amount is a dollar requirement; a variable expense ratio is a proportion of the premium the indicated rates will produce.

The premium base must represent the premium for the business being evaluated at the current rate level, on the selected and consistent basis. Earlier material on the current-rate-level baseline and on-level premium explains why historical premium usually needs adjustment before serving as that base. Likewise, the projected cost amount should reflect the selected prospective cost measure and the same scope of business. This tutorial combines such inputs; it does not determine how to select or project them.

Definition: Let $B$ be the premium base at the current rate level for the selected business and period. Let $I$ be the indicated rate-level factor, so that $IB$ is the premium at the indicated level on that same base. Let $C$ be the total projected dollar costs included as fixed dollar requirements, and let $e$ and $q$ be, respectively, variable expense and profit or contingency provisions expressed as proportions of indicated premium. Under the assumptions below, the overall indication is the factor $I$ that balances indicated premium against these costs and provisions.

Here $C$ can include projected losses and loss adjustment expenses, as well as any other included dollar requirements that do not vary as a proportion of premium. If a cost is specified per exposure, first calculate the dollar amount for the selected exposures. The model assumes that the indicated premium is $IB$; that $e$ and $q$ apply to that indicated premium; and that $C$ is already projected to the prospective basis and is not itself a percentage of indicated premium. The proportions are assumed constant over the indication. If a provision is instead a fixed dollar amount, include it in $C$, not in $e$ or $q$.

Deriving the Indicated Factor

Write the balance before rearranging it. The required premium must fund the fixed dollar costs and the provisions calculated as proportions of that premium:

$$ IB = C + e(IB) + q(IB) $$

Since both percentage-based provisions use the same premium base, they combine to $(e+q)IB$. Moving that amount to the left gives the premium remaining after those provisions are allowed for. Provided $B>0$ and $1-e-q>0$, division yields the overall indication:

IB(1-e-q)=C \qquad\Longrightarrow\qquad I=\frac{C}{B(1-e-q)}

This is a direct derivation, not an extra assumption: the denominator is the current-rate-level premium base multiplied by the share of indicated premium available to cover $C$. The condition $1-e-q>0$ is necessary for a positive finite indication when $C>0$. If $e+q$ is at least 1, the modeled provisions consume all or more than the premium, so this equation cannot yield a positive finite factor for positive costs.

If there are also fixed dollar provisions, denote their total by $A$. Those amounts are added to the dollar costs, and the balance becomes $IB=C+A+(e+q)IB$. Thus:

I=\frac{C+A}{B(1-e-q)}

The formula is not appropriate without modification if a provision is not a constant percentage of indicated premium, if the rate change changes the exposure or coverage basis, or if the numerator includes amounts that have already been represented as premium-based provisions. In those cases, write the applicable premium-balance equation first rather than forcing the inputs into this form.

Worked Examples

Worked Example: Variable Expense as a Share of Indicated Premium

A hypothetical specialty insurer is evaluating a book with a current-rate-level premium base of $12,000,000. Projected losses and loss adjustment expenses are $8,600,000. Variable expenses are 23% of premium, and no separate profit, contingency, or fixed expense provision is included. Find the overall indicated factor and check the balance.

Method. Use the premium-balance equation with $B=\$12{,}000{,}000$, $C=\$8{,}600{,}000$, $e=0.23$, and $q=0$. The share of indicated premium available for projected losses and expenses is $1-0.23=0.77$.

The premium base after allowing for the variable expense proportion is:

B(1-e-q)=\$12{,}000{,}000(0.77)=\$9{,}240{,}000

Therefore, the factor is:

I=\frac{\$8{,}600{,}000}{\$9{,}240{,}000} =0.9307359\ldots

Rounded to four decimal places, $I=0.9307$. At this factor, indicated premium is $\$12{,}000{,}000(0.9307359\ldots)=\$11{,}168{,}831.17$. Variable expense is $0.23(\$11{,}168{,}831.17)=\$2{,}568{,}831.17$. Adding projected losses and LAE gives $\$8{,}600{,}000+\$2{,}568{,}831.17=\$11{,}168{,}831.17$, which reconciles to indicated premium. The calculation avoids treating a premium-based expense provision as a fixed dollar amount.

Worked Example: Adding a Fixed Dollar Provision

For a hypothetical commercial product, the current-rate-level premium base is $4,800,000. Projected losses and LAE are $3,120,000, and included fixed expenses are $240,000. Variable expenses are 18% of indicated premium, while the profit and contingency provision together are 4% of indicated premium. Find the indicated factor and verify its components.

Method. Combine the dollar requirements first: $C+A=\$3{,}120{,}000+\$240{,}000=\$3{,}360{,}000$. The total premium-based provision is $e+q=0.18+0.04=0.22$, leaving $1-0.22=0.78$ of premium to fund the dollar requirements.

B(1-e-q)=\$4{,}800{,}000(0.78)=\$3{,}744{,}000

Thus:

I=\frac{\$3{,}360{,}000}{\$3{,}744{,}000} =0.8974359\ldots

Rounded to four decimal places, the factor is $0.8974$. Using the unrounded factor, indicated premium is $\$4{,}800{,}000(0.8974359\ldots)=\$4{,}307{,}692.31$. The 18% variable expense is $\$775{,}384.62$, and the 4% profit and contingency provision is $\$172{,}307.69$. Total requirements are $\$3{,}120{,}000+\$240{,}000+\$775{,}384.62+\$172{,}307.69=\$4{,}307{,}692.31$, subject only to rounding cents. The fixed expense belongs in the dollar numerator; the two percentages reduce the premium available to fund it.

Worked Example: Reconciling All Included Requirements

A hypothetical insurer evaluates a selected segment with a current-rate-level premium base of $9,000,000. Its prospective loss estimate is $6,300,000, and projected LAE is $450,000. The indication also includes $300,000 of fixed dollar expenses, variable expenses equal to 20% of indicated premium, and a profit and contingency provision equal to 5% of indicated premium. Find the factor and test whether all requirements are funded.

Method. Add the projected loss and LAE amounts, then include fixed expenses: $C+A=\$6{,}300{,}000+\$450{,}000+\$300{,}000=\$7{,}050{,}000$. Premium-based provisions total $0.20+0.05=0.25$, so 75% of indicated premium remains for the dollar requirements.

B(1-e-q)=\$9{,}000{,}000(0.75)=\$6{,}750{,}000

The factor is:

I=\frac{\$7{,}050{,}000}{\$6{,}750{,}000} =1.044444\ldots

Rounded to four decimal places, $I=1.0444$. Using the unrounded factor, indicated premium is $\$9{,}000{,}000(1.044444\ldots)=\$9{,}400{,}000$. Variable expenses are $0.20(\$9{,}400{,}000)=\$1{,}880{,}000$, and the profit and contingency provision is $0.05(\$9{,}400{,}000)=\$470{,}000$. The check is $\$6{,}300{,}000+\$450{,}000+\$300{,}000+\$1{,}880{,}000+\$470{,}000=\$9{,}400{,}000$. The indicated premium exactly funds the stated components.

Common Trap: Using the Wrong Premium for a Variable Provision

A tempting shortcut is to multiply a variable expense ratio by the current premium base, add that amount to the projected costs, and divide by the base. For the first example, that calculation is:

\frac{\$8{,}600{,}000+0.23(\$12{,}000{,}000)} {\$12{,}000{,}000} = \frac{\$11{,}360{,}000}{\$12{,}000{,}000} =0.9467

The arithmetic is correct, but the method does not solve the stated model. It treats the variable expense as a fixed $2,760,000 requirement based on current-level premium. Under the stated assumption, variable expense is 23% of indicated premium, which is why it must be represented as $0.23IB$ in the balance equation. The correct factor is $0.9307$, as verified in the first example. The shortcut would apply only under a different model in which $2,760,000 were explicitly a fixed dollar cost.

A second useful check is to state what the factor multiplies. $I$ multiplies the current-rate-level premium base to give indicated premium; it is not itself a dollar amount. The model is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

A Practical Calculation Sequence

Confirm the base.
Identify $B$ as the selected premium base at the current rate level, for the same business and scope represented by the projected requirements.

Sort the requirements.
Sum projected dollar costs and fixed dollar provisions separately from provisions expressed as a percentage of premium.

Write the balance.
Set indicated premium $IB$ equal to all dollar requirements plus each premium-based proportion multiplied by $IB$.

Solve and check.
Rearrange for $I$, then calculate indicated premium and reconcile it to the individual components using the unrounded factor.

Key takeaway: The overall indication follows from a premium balance. Put fixed dollar requirements in the numerator and deduct premium-based provisions from the share of premium available to fund them: $I=(C+A)/[B(1-e-q)]$. The result is meaningful only when the premium base, projected costs, and provisions use a consistent scope and basis.

Check Your Understanding

For each hypothetical indication, write the premium balance before solving. Treat each stated expense or provision according to whether it is a dollar amount or a percentage of indicated premium.

A selected book has a current-rate-level premium base of $5,000,000 and projected dollar costs of $3,500,000. Variable expenses are 20% of indicated premium, with no other provisions. Find the factor. (Answer: 0.8750)
A segment has a $7,200,000 premium base, $4,500,000 of projected losses and LAE, and $300,000 of fixed expenses. Variable expenses are 15% and profit and contingency are 5% of indicated premium. Find the factor. (Answer: 0.8333)
A hypothetical product has a $10,000,000 premium base, $6,200,000 of projected costs, and $400,000 of fixed dollar requirements. Variable expenses are 22% and the profit provision is 3% of indicated premium. Find the factor. (Answer: 0.8800)
A premium base is $8,000,000, projected dollar costs are $5,200,000, and variable expenses are 25% of indicated premium. A calculation treats the expense as $0.25(\$8{,}000{,}000)$ and obtains $(\$5{,}200{,}000+\$2{,}000{,}000)/\$8{,}000{,}000=0.9000$. Find the factor under the stated model and identify why the other result is wrong. (Answer: 0.8667; the expense is a percentage of indicated premium, not a fixed $2,000,000 amount.)
Two selected experience components have current-rate-level premium bases of $3,000,000 and $5,000,000, with projected dollar costs of $2,400,000 and $3,500,000, respectively. Variable expenses are 20% and profit and contingency are 5% of indicated premium on the combined base. Find the overall factor by combining the dollar requirements and bases before applying the provisions. (Answer: 0.9833)

What You'll Learn