From Projected Costs to an Overall Indication
An overall rate indication puts projected claim costs and expense, profit, and contingency provisions on a consistent basis. The central calculation is a balance: the premium must fund costs stated in dollars and also fund provisions stated as percentages of that premium. Because the percentage provisions depend on the premium being calculated, they generally belong in a denominator, not as an amount simply added to losses.
For this calculation, use amounts for the same prospective period and the same exposure base. Assume projected losses already reflect the selected loss and loss adjustment expense basis, and that the current premium used for comparison has been put on the corresponding basis. This tutorial focuses on combining the supplied components, not on how to project losses or select each provision.
A fixed expense may be supplied as a total dollar amount or as a dollar amount per exposure. In the latter case, multiply the amount per exposure by the applicable number of exposures before using it in the indication. A provision stated as a percentage of premium is different: its dollar amount cannot be known until the required premium is found.
The Overall Indication Formula
Let \(P_{\mathrm{req}}\) denote required premium and \(P_{\mathrm{current}}\) denote the current premium for the same prospective exposure base. Under the stated provision model, the premium balance is
The premium pays for projected losses and fixed costs, \(L+F\), as well as premium-based provisions equal to \(rP_{\mathrm{req}}\). Rearranging gives the overall indication formula, provided that \(r<1\):
The indicated rate change, expressed as a proportion of current premium, is
A positive result indicates an increase relative to the current premium; a negative result indicates a decrease. When the comparison is on a common exposure base, the ratio of required to current premium is also the overall rate level factor. If there are no premium-based provisions, \(r=0\), and required premium is simply \(L+F\). If \(r\) is at least 1, this model does not produce a finite positive required premium for positive costs; the assumptions or provisions must be reviewed.
Why the Premium-Based Provisions Go in the Denominator
The denominator follows directly from the premium balance. The dollar amount of a premium-based provision is not fixed in advance: for example, a 10% provision is 10% of whatever premium is ultimately needed. Starting with costs \(L+F\), the premium must also fund the provision on itself. Subtracting that provision from both sides gives
Dividing by \(1-r\) yields the formula. A useful check is to substitute the result back into the balance: \(L+F+rP_{\mathrm{req}}\) must equal \(P_{\mathrm{req}}\), apart from any stated rounding. This derivation assumes each listed percentage is applied to the same premium base. If a provision is instead explicitly defined on a different base, it must be modeled on that base rather than automatically included in \(r\).
Worked Example: Combining Four Premium-Based Provisions
A hypothetical book has 20,000 projected exposures. Projected losses and loss adjustment expenses are $1,080,000. Fixed expenses are $12 per exposure. The proposed provisions are 12% variable expense, 3% taxes, 5% profit, and 2% contingency, each as a percentage of premium. Current premium for this exposure base is $1,500,000. Calculate the required premium and indicated rate change.
Method. Convert fixed expense to a total dollar amount, add it to projected losses, combine the premium-based percentages, and solve for required premium.
Step 1: Calculate fixed expenses.
The dollar cost to be funded before premium-based provisions is
Step 2: Combine the premium-based provisions.
Thus, 78% of required premium remains to fund the dollar costs:
Step 3: Calculate required premium.
Rounded to cents, required premium is $1,692,307.69.
Step 4: Calculate the indicated change.
The indicated change is 12.82% upward, rounded to two decimal places.
Check and common mistake. Using the unrounded required premium, the premium-based provisions are \(0.22\times \$1{,}692{,}307.692307\ldots=\$372{,}307.692307\ldots\). Adding these to $1,320,000 gives $1,692,307.692307… as required. A common mistake is to add 22% of the dollar costs, or $290,400, to $1,320,000. That produces $1,610,400 and does not fund a provision equal to 22% of the resulting premium.
Worked Indications With Additional Fixed Costs
Not every provision is a percentage of premium. A fixed contingency amount, for example, belongs with other fixed dollar costs in \(F\), while a contingency stated as a percentage of premium belongs in \(r\). The wording of the provision determines its treatment. The next example includes a fixed dollar contingency as well as premium-based provisions.
Worked Example: Including a Fixed Contingency Amount
A hypothetical program has 6,000 projected exposures and projected losses and loss adjustment expenses of $510,000. Fixed operating expenses are $20 per exposure. A separate contingency provision is $30,000 in dollars. Variable expenses are 10% of premium, taxes are 2.5%, and profit is 3%. Current premium is $800,000. Find the overall indication.
Method. Include both fixed operating expenses and the fixed contingency in dollar costs. Include only the premium-based percentages in \(r\).
Step 1: Total the fixed dollar provisions.
so
Step 2: Total the percentages of premium.
Step 3: Solve for required premium.
Required premium is $781,065.09, rounded to cents.
Step 4: Compare with current premium.
The indicated change is 2.37% downward, rounded to two decimal places.
Check and common mistake. The premium-based provisions equal \(0.155\times \$781{,}065.088757\ldots=\$121{,}065.088757\ldots\). Adding that amount to $660,000 recovers required premium. Treating the $30,000 fixed contingency as a percentage provision would be incorrect: it is a stated dollar cost, so it belongs in the numerator with other fixed costs.
Worked Example: Reading a Complete Indication as a Rate Change
A hypothetical specialty portfolio has projected losses and loss adjustment expenses of $2,400,000. Fixed expenses are $350,000. The proposed premium-based provisions are 8% variable expense, 2% taxes, 4% profit, and 3% contingency. Current premium for the same exposure base is $3,000,000. Calculate required premium, the indicated rate change, and a reconciliation of the provision dollars.
Method. Treat all stated provisions as percentages of premium, add them once, solve the premium balance, and then compare with current premium.
Step 1: Calculate dollar costs and the premium-based ratio.
and
Step 2: Calculate required premium.
Rounded to cents, required premium is $3,313,253.01.
Step 3: Calculate the rate change.
The indicated rate change is 10.44% upward, rounded to two decimal places.
Step 4: Reconcile the provisions. The combined premium-based provisions are
Adding the dollar costs gives $2,750,000 + $563,253.012048… = $3,313,253.012048…, which matches the unrounded required premium. To confirm the rate-change interpretation, current premium multiplied by one plus the unrounded indicated change also gives $3,313,253.012048…. The comparison must use the specified current premium for the same exposure base; comparing with a different premium volume would not measure the same overall rate change.
Interpretation and Common Traps
The indication is a comparison of required premium with current premium, not a comparison of required premium with projected losses alone. Losses are only one component of the premium need. Fixed expenses and premium-based provisions affect the indicated level, and their treatment depends on how each is stated.
| Provision form | Where it enters | Example |
|---|---|---|
| Dollar amount for the exposure base | In \(F\), with other fixed costs | $30,000 fixed contingency |
| Dollar amount per exposure | Multiply by exposures, then include in \(F\) | $20 per exposure for 6,000 exposures |
| Percentage of premium | Include in \(r\), then use \(1-r\) in the denominator | 3% profit provision |
The formula assumes that all percentages in \(r\) are defined on the same premium base and that the costs and current premium match in period and exposure basis. If the data do not meet those conditions, first make the bases consistent or model the differently defined provision explicitly. Do not round intermediate calculations unnecessarily: retain the unrounded required premium for the percentage-provision check and rate-change calculation, then round the reported dollar amounts and rate as instructed.
- Adding percentage provisions to dollar costs. A percentage of premium is not a fixed dollar amount known before solving for premium; use the denominator \(1-r\).
- Mixing fixed and percentage contingencies. A fixed dollar contingency belongs in \(F\); a contingency defined as a percentage of premium belongs in \(r\).
- Using inconsistent comparison bases. Required and current premium must refer to the same prospective exposure base for their ratio to represent the overall rate change.
- Reporting required premium as the rate change. Convert the premium ratio to a change by subtracting 1, and state whether the result is an increase or decrease.
Check Your Understanding
For each hypothetical indication, identify the fixed costs and premium-based provisions before calculating. Round dollar answers to cents and rate changes to two decimal places.
- A program has projected losses and loss adjustment expenses of $950,000, fixed expenses of $150,000, premium-based provisions totaling 20%, and current premium of $1,250,000. Find required premium and the indicated rate change. (Answer: Required premium $1,375,000.00; indicated change 10.00% upward.)
- A hypothetical book has 6,000 exposures, projected losses of $510,000, fixed expenses of $20 per exposure, and a fixed dollar contingency of $30,000. Premium-based provisions total 15%, and current premium is $800,000. Find required premium and the indicated change. (Answer: Required premium $776,470.59; indicated change 2.94% downward.)
- Projected losses are $1,800,000 and fixed expenses are $270,000. Premium-based provisions are 11% variable expense, 2% taxes, 4% profit, and 1% contingency. Current premium is $2,300,000. Calculate the required premium and indicated change. (Answer: Required premium $2,524,390.24; indicated change 9.76% upward.)
- A portfolio has projected losses of $2,760,000, fixed expenses of $420,000, and a fixed dollar contingency of $80,000. Its premium-based provisions are 9% variable expense, 2% taxes, 5% profit, and 1% contingency. Current premium is $4,000,000. Calculate required premium and the indicated change. (Answer: Required premium $3,927,710.84; indicated change 1.81% downward.)