Why a Loss Cost Is Not the Rate
A pure premium indication begins with an estimate of losses per exposure. That amount is a loss cost, not automatically the rate to charge. The rate must also provide for expenses and, when specified, a profit or contingency provision. Confusing the two can understate the rate even when the projected loss cost itself is correct.
For this tutorial, assume the projected loss cost is already expressed per unit of exposure and is on the basis intended for the future rate period. The focus is the conversion from that loss cost to a rate, not the selection of the loss projection. This distinction is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.
The exposure unit must be the same for the loss cost and the rate. For example, if a hypothetical product is rated per insured vehicle-year, a loss cost of $420 means $420 of projected losses per vehicle-year. It does not mean that the rate is $420: part of the premium may need to fund expenses and other provisions.
Rate Components and the Basic Equation
Separate the provisions according to how they are measured. A fixed expense per exposure is a dollar amount that does not vary with the premium. A variable expense provision is often stated as a percentage of premium. A profit or contingency provision may also be specified as a percentage of premium. These percentages must be interpreted on the stated basis: the formulas below assume they are percentages of the indicated rate.
The two percentage provisions depend on the rate itself. Therefore, they belong on the right-hand side as \(eR\) and \(pR\), rather than as fixed dollar amounts calculated from \(L\). Rearranging gives the indicated-rate formula.
This formula requires \(1-e-p>0\) for a finite positive rate when \(L+F>0\). If the denominator is zero or negative, the stated percentage provisions leave no positive share of premium to fund losses and fixed expenses. A calculation that produces such a denominator is not a usable positive indicated rate under this model; the assumptions or provisions must be reviewed.
Why the Denominator Appears
The denominator follows directly from the funding requirement. The indicated premium per exposure is \(R\). The variable expense and profit or contingency provisions together use \((e+p)R\). The share left for losses and fixed expenses is therefore \(R-(e+p)R=(1-e-p)R\). To fund the required \(L+F\), that remaining share must equal \(L+F\):
Dividing both sides by \(1-e-p\), provided it is positive, yields \(R=(L+F)/(1-e-p)\). This also explains why percentage-of-premium provisions increase the rate by more than simply adding those percentages to the loss cost: the provisions are shares of the final premium, including the premium needed to fund the provisions themselves.
Calculating and Checking an Indicated Rate
A reliable calculation identifies the basis of each input, forms the dollar amount that must remain after percentage provisions, divides by the remaining premium share, and then checks the result by reconstructing each component. The following examples use hypothetical insurance products and invented values.
Worked Example: Converting a Loss Cost With Percentage Provisions
A hypothetical policy is rated per exposure unit. Its projected loss cost is $420 per exposure. Variable expenses are 18% of premium, and the profit provision is 5% of premium. There is no separate fixed expense. Find the indicated rate and verify its components.
Method and inputs. Use \(L=\$420\), \(F=\$0\), \(e=0.18\), and \(p=0.05\). The total percentage provision is \(0.18+0.05=0.23\), so the share of premium available for losses is \(1-0.23=0.77\).
Calculate the rate.
Rounded to the nearest cent, the indicated rate is $545.45 per exposure.
Check the allocation. Using the unrounded rate, variable expenses are \(0.18(545.4545\ldots)=\$98.1818\ldots\), and the profit provision is \(0.05(545.4545\ldots)=\$27.2727\ldots\). The remainder is \(545.4545\ldots-98.1818\ldots-27.2727\ldots=\$420\), exactly the loss cost. Rounded to cents, the components are $420.00, $98.18, and $27.27; their rounded sum is $545.45.
Method and mistake avoided. This is a percentage-of-premium gross-up. It avoids adding 18% and 5% directly to the $420 loss cost, which would use the wrong base.
Worked Example: Including a Fixed Expense Per Exposure
A hypothetical coverage has a projected loss cost of $615 per exposure and a fixed expense of $24 per exposure. Variable expenses are 15% of premium, and the profit and contingency provision is 5% of premium. Calculate and check the indicated rate.
Method and inputs. The loss cost and fixed expense are both per exposure, so the amount that must be funded before percentage provisions is \(615+24=\$639\). Here \(e=0.15\), \(p=0.05\), and \(1-e-p=0.80\).
Calculate the rate.
Check the allocation. Variable expenses are \(0.15(798.75)=\$119.8125\), and the profit and contingency provision is \(0.05(798.75)=\$39.9375\). The premium remaining after those provisions is \(798.75-119.8125-39.9375=\$639\), which funds the $615 loss cost and $24 fixed expense. Rounded to cents, the components are $615.00, $24.00, $119.81, and $39.94; they sum to $798.75.
Method and mistake avoided. The fixed expense is added to the loss cost before dividing by the remaining premium share. It is not multiplied by the variable expense or profit percentage, because those percentages apply to the indicated premium.
Worked Example: Combining Fixed and Percentage Loadings
A hypothetical product has a projected loss cost of $280 per exposure and a fixed expense of $12 per exposure. Variable expenses are 22% of premium, and the profit provision is 3% of premium. Determine the indicated rate to the nearest cent and verify it.
Method and inputs. First add the per-exposure dollar amounts: \(L+F=280+12=\$292\). The percentage provisions total \(0.22+0.03=0.25\), leaving \(1-0.25=0.75\) of premium to fund the $292.
Calculate the rate.
The indicated rate, rounded to the nearest cent, is $389.33 per exposure.
Check the allocation. On the unrounded rate, variable expenses are \(0.22(389.3333\ldots)=\$85.6533\ldots\), and the profit provision is \(0.03(389.3333\ldots)=\$11.68\). Subtracting both from the unrounded rate leaves \(389.3333\ldots-85.6533\ldots-11.68=\$292\), equal to the loss cost plus fixed expense. To cents, the amounts are $280.00 for losses, $12.00 for fixed expense, $85.65 for variable expenses, and $11.68 for profit; they sum to $389.33.
Method and mistake avoided. Keep dollar amounts per exposure separate from percentages of premium until the rate equation is formed. Treating all three provisions as a single “loading percentage” would lose the distinction between the $12 fixed amount and the percentage provisions.
Common Exam Trap: Loading the Loss Cost Directly
A frequent error is to multiply the loss cost by \(1+e+p\), as though the expense and profit percentages were markups on losses. That is not the model when the provisions are percentages of premium. For the first example, the incorrect calculation would be:
But 18% and 5% of this incorrect rate are \(0.18(516.60)=\$92.988\) and \(0.05(516.60)=\$25.83\). Only \(516.60-92.988-25.83=\$397.782\) remains for losses, which is less than the required $420. The calculation fails the funding check. The correct calculation divides by \(1-0.18-0.05=0.77\), giving $545.45 to the nearest cent. Always verify that the stated loss cost and fixed expenses are fully funded after subtracting percentage-of-premium provisions.
Interpreting the Result
The indicated rate is a required premium per exposure under the specified assumptions; it is not another name for the loss cost. A rate greater than the loss cost is expected when the premium must also fund positive expenses or a profit and contingency provision. The difference between rate and loss cost is not necessarily one simple “loading”: it can include both fixed dollar expenses and percentages of premium.
The calculation is valid only when all inputs refer to a consistent exposure and coverage basis, and when the percentages are applied to the indicated rate as stated. If an input is instead defined as a percentage of losses, or as a fixed amount per policy rather than per exposure, it must be translated to the appropriate basis before using this equation. If the percentage provisions leave a nonpositive share of premium, the formula does not produce a finite positive rate for a positive amount to fund.
For an exam calculation, write the funding equation before substituting values. Label dollar amounts “per exposure,” label percentage provisions “of premium,” and check the final rate by subtracting its percentage provisions. That check makes the distinction between a projected loss cost and the final indicated rate explicit.
Check Your Understanding
For each hypothetical situation, calculate the requested amount and use the stated basis for every provision.
- A projected loss cost is $525 per exposure. Variable expenses are 16% of premium and the profit provision is 4% of premium; there is no fixed expense. Find the indicated rate and state the dollar amounts for variable expenses and profit, rounded to cents. (Answer: indicated rate $656.25; variable expenses $105.00; profit $26.25.)
- A coverage has a projected loss cost of $440 per exposure and a fixed expense of $28 per exposure. Variable expenses are 12% of premium and the profit provision is 8% of premium. Find the indicated rate to the nearest cent. (Answer: $585.00.)
- A hypothetical policy has a projected loss cost of $390 per exposure and a fixed expense of $18 per exposure. Variable expenses are 21% of premium and the contingency provision is 4% of premium. Calculate the indicated rate and verify the amount left after the percentage provisions. (Answer: indicated rate $544.00; amount left $408.00.)
- A projected loss cost consists of two stated components, $260 and $190 per exposure, which both apply to the same coverage and exposure basis. Fixed expense is $22 per exposure, variable expenses are 17% of premium, and profit is 3% of premium. Find the indicated rate. (Answer: $590.00.)
- A hypothetical rate is $750 per exposure. Variable expenses are 16% of premium, profit is 4% of premium, and fixed expense is $30 per exposure. Under the model in this tutorial, determine the projected loss cost funded by this rate. (Answer: $570.00 per exposure.)