Building a Single-Period Indication
A rate indication is only as useful as the inputs assembled behind it. For one experience period, the calculation must put the premium and projected losses on compatible bases: the premium is adjusted to the current rate level, while the selected losses are developed and trended to the prospective period. Exposure counts help describe and reconcile the experience, and can be needed to calculate fixed expenses, but they are not a substitute for premium in the indication.
This tutorial assembles one hypothetical period from those inputs through an indicated factor. It assumes the selected losses already include loss adjustment expenses (LAE), measured on a basis consistent with the expense provision. The calculation is a simplified overall indication: it does not address classification changes, multiple experience periods, or other adjustments that are not specified in the example data.
Label the Inputs Before Calculating
Use the following labels to keep the components distinct. Let \(E\) be the number of earned exposure units, \(P^{H}\) the historical earned premium, \(O\) the on-level factor, \(P^{CL}\) the current-level premium, \(L^{S}\) the selected losses on the stated loss basis, \(D\) the development factor, and \(T\) the trend factor. In this tutorial, a factor is applied multiplicatively, and \(D\) and \(T\) are assumed to be appropriate for the selected loss measure.
Here \(L^{P}\) denotes projected losses. If the selected losses are already ultimate, use \(D=1\). If they are already at the prospective trend level, use \(T=1\). Do not apply an adjustment twice just because a column or label in a data extract is ambiguous: first establish what the selected amount represents.
For the expense treatment below, let \(f\) be fixed expense per exposure, \(v\) the variable expense ratio, and \(q\) the profit and contingencies provision as a proportion of premium. The stated form assumes these provisions are the only additional components needed: fixed expenses are \(fE\), and variable expenses and the profit provision are proportional to the required premium. No investment income, taxes, or other provisions are included unless they have been explicitly incorporated into these assumptions.
\(P^{*}\) is the target premium for the experience under the stated assumptions, and \(I\) is the indicated factor. The indicated percentage change is \((I-1)\times 100\%\). The formula requires \(1-v-q>0\); if the remaining premium share is zero or negative, this setup does not yield a finite, meaningful target premium.
Why the Indication Formula Holds
The target premium must fund projected losses and fixed expenses. It must also leave enough premium after variable expenses and the profit provision to fund those amounts. If the target premium is \(P^{*}\), the available share is \(P^{*}(1-v-q)\). Equating that amount to projected losses plus fixed expenses gives the first equation below. Dividing by the positive share \(1-v-q\), then comparing target premium with current-level premium, gives the indication.
This derivation also explains why it is important to label the fixed expense basis. If the fixed expense is supplied per exposure, multiply by \(E\) before solving. If instead a problem supplies a total fixed expense for the period, use that total directly; multiplying it by exposure again would count it twice.
Worked Example: One Period With Ultimate Losses
A hypothetical commercial equipment policy has 2,000 earned exposure units. Historical earned premium is $1,080,000, and the applicable on-level factor is 1.05. Selected ultimate losses and LAE are $720,000. They require two years of trend at 4% per year. Fixed expense is $18 per exposure; variable expenses are 20% of premium, and the profit and contingencies provision is 5% of premium. Calculate the indicated factor and percentage change.
Method and assumptions. The selected loss amount is already ultimate, so \(D=1\). Trend is compounded annually for two years, so \(T=(1.04)^2\). The period’s fixed expenses are based on its 2,000 exposures. The premium is adjusted by the given on-level factor.
Step 1: Record and on-level the premium. The exposure count is \(E=2{,}000\). Current-level premium is
Step 2: Project losses. The two-year trend factor is \(T=1.04^2=1.0816\). Therefore,
Step 3: Calculate fixed expenses and the remaining premium share.
Step 4: Calculate target premium and factor. The amount to fund before variable expenses and profit is \(\$778{,}752+\$36{,}000=\$814{,}752\). Thus,
Rounded only at the final step, the indicated factor is 0.9580 to four decimal places, or an indicated change of \((0.957968\ldots-1)\times100\%=\mathbf{-4.20\%}\) to two decimal places. Check: the target premium is less than current-level premium, so the factor is below 1 and the indicated change is a decrease.
Keep Development, Trend, and Premium Adjustments Separate
Development and trend address different changes in the loss amount. Development estimates the eventual value of losses from a less mature measure; trend projects the loss level forward in time. On-leveling adjusts premium for rate changes, not losses. A clear ledger makes it easier to see which inputs have been adjusted and which have not.
Record the exposure count and earned premium for the same experience period. Use the stated exposure unit, such as vehicle-years or insured locations.
Apply the supplied on-level adjustment to historical earned premium to obtain current-level premium. Do not apply this factor to losses.
Determine whether losses are paid, reported, or already selected to ultimate, and confirm whether LAE is included. Use a development adjustment only when needed to reach the selected ultimate basis.
Apply the stated trend over the stated time interval. The period must be measured from the loss data’s applicable average date to the prospective date.
Calculate any fixed expense from its stated basis, use the remaining premium share \(1-v-q\), and divide target premium by current-level premium.
Worked Example: Reported Losses Requiring Development
A hypothetical inland-watercraft program has 1,500 earned exposure units and $900,000 of historical earned premium. Its on-level factor is 1.10. Selected reported losses and LAE are $500,000; the selected development factor to ultimate is 1.20. Trend is 5% for one year. Fixed expense is $12 per exposure, variable expenses are 22% of premium, and profit and contingencies are 4% of premium. Calculate the indicated factor.
Method and assumptions. Apply development to reported losses to estimate ultimate losses, then apply one year of trend. The development and trend factors are both supplied and apply to the same selected loss basis. They are not premium adjustments.
Step 1: Current-level premium.
Step 2: Develop and trend the selected losses. Development gives \(\$500{,}000(1.20)=\$600{,}000\) at ultimate. The one-year trend factor is \(1.05\), so
Step 3: Include fixed expenses and calculate the target premium.
The amount to be funded is \(\$630{,}000+\$18{,}000=\$648{,}000\), and hence
The indicated factor is 0.8845 to four decimal places, or an indicated decrease of 11.55% to two decimal places. Check: the indicated target premium is below the current-level premium, consistent with a factor below 1.
Common mistake avoided. Trending the reported amount but omitting development would give \(\$500{,}000(1.05)=\$525{,}000\), an incorrect projected loss amount. The data explicitly require development to ultimate, so the correct sequence includes both factors: \(\$500{,}000(1.20)(1.05)=\$630{,}000\).
A Full Input Ledger and Final Checks
A useful single-period calculation shows each input and each adjustment separately. The ledger below is not an additional actuarial method; it is a way to make the chosen method auditable. It helps distinguish a source amount from a selected amount and a selected amount from a projected amount.
Worked Example: Reconciling the Inputs Before Indicating
A hypothetical specialty property portfolio has 4,000 earned exposure units, $1,600,000 of historical earned premium, and an on-level factor of 1.08. Selected losses and LAE are $950,000 before development to ultimate. The selected development factor is 1.10, and the selected annual trend rate is 3% for two years. Fixed expense is $25 per exposure; variable expenses are 18% of premium, and profit and contingencies are 5% of premium. Calculate the indication and show the ledger.
Method and assumptions. The selected loss amount is before development, so apply the 1.10 development factor. Then trend for two years using compound annual trend, \(T=1.03^2\). The premium is on-leveled separately.
| Item | Calculation | Result |
|---|---|---|
| Earned exposure units | Given | 4,000 |
| Historical earned premium | Given | $1,600,000 |
| Current-level premium | $1,600,000 × 1.08 | $1,728,000 |
| Selected losses before development | Given | $950,000 |
| Ultimate losses | $950,000 × 1.10 | $1,045,000 |
| Trend factor | 1.03² | 1.0609 |
| Projected losses | $1,045,000 × 1.0609 | $1,108,640.50 |
| Fixed expenses | $25 × 4,000 | $100,000 |
The remaining premium share is \(1-0.18-0.05=0.77\). Target premium and indicated factor are
The indicated factor is 0.9084 to four decimal places, or an indicated decrease of 9.16% to two decimal places. Check: the projected losses plus fixed expenses are below the premium available at current rates after the stated provisions, and the calculated target premium is also below current-level premium.
Common Traps in a Single-Period Setup
- Using historical premium without on-leveling it. The denominator is current-level premium, not automatically the historical earned premium. Apply the supplied on-level factor once.
- Applying on-level factors to losses. On-leveling corrects premium for rate changes. Development and trend are loss adjustments in this setup.
- Leaving out a required adjustment. If selected reported losses need development to ultimate, trend alone does not make them ultimate. Conversely, do not develop losses already selected to ultimate.
- Using the wrong trend interval. A trend rate is applied over the stated time between the relevant loss date and prospective date. For \(n\) years at an annual compound rate \(r\), the factor is \((1+r)^n\), not \(1+nr\) unless a simple-trend approximation is specifically prescribed.
- Misapplying fixed expenses. Multiply a per-exposure amount by exposure count. Add a supplied total expense once; do not multiply that total by exposures.
- Confusing factor and percentage change. A factor of 0.95 means a 5% decrease, not a 95% decrease. Convert using \((I-1)\times100\%\).
A final reasonableness check should follow the arithmetic: confirm that the premium and losses refer to the same experience period, each adjustment is applied to the intended item, the expense denominator is positive, and the direction of the result agrees with the comparison between target and current-level premium. This disciplined setup is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.
Check Your Understanding
For each hypothetical single-period indication, calculate the requested result. Unless stated otherwise, losses include LAE, fixed expenses are per exposure, and the indicated factor is target premium divided by current-level premium.
- A program has 800 earned exposures, historical earned premium of $480,000, an on-level factor of 1.05, and selected ultimate losses of $300,000. Trend is 4% annually for two years. Fixed expense is $10 per exposure; variable expenses are 20% and profit is 5% of premium. Find the indicated factor. (Answer: 0.8796, a 12.04% decrease.)
- A portfolio has 1,000 exposures, $750,000 historical earned premium, and an on-level factor of 1.08. Reported losses of $420,000 require a 1.25 development factor and one year of 4% trend. Fixed expense is $15 per exposure; variable expenses are 25% and profit is 5%. Find the indicated factor. (Answer: 0.9894, a 1.06% decrease.)
- Current-level premium is $1,375,000 for 2,500 earned exposures. Selected ultimate losses are $880,000 and require a 5% trend adjustment. Fixed expense is $20 per exposure; variable expenses are 20% and profit is 5%. Find target premium and the indicated factor. (Answer: target premium $1,298,666.67; factor 0.9445, a 5.55% decrease.)
- A portfolio has 1,200 exposures, $600,000 historical earned premium, and an on-level factor of 1.12. Reported losses of $360,000 require development by 1.15 and two years of 3% annual trend. Fixed expense is $8 per exposure; variable expenses are 18% and profit is 7%. Find the indicated factor. (Answer: 0.8905, a 10.95% decrease.)