A Worked Prospective Rate Indication

Purpose and Structure of a Prospective Indication

A prospective rate indication estimates the premium needed for a future coverage period using adjusted historical experience and the insurer’s expense and profit assumptions. The calculation is a sequence of adjustments: losses are brought to ultimate and trended to the prospective period, premiums are adjusted to the current rate level, and the resulting experience may be blended with an expected value using credibility. The selected loss cost is then converted into a premium.

The adjustments have different purposes and should not be confused. A loss development factor estimates the relationship between losses observed at a particular maturity and their ultimate value. A loss trend factor adjusts losses from their historical cost level to the cost level of the prospective period. An on-level premium factor adjusts historical premium to the current rate level. In particular, loss trend applies to losses, while on-level factors apply to premiums.

Definition: A prospective rate indication is an estimate of the rate level needed for a future period, based on experience adjusted to the appropriate loss and premium levels and on assumptions for expenses and profit. For a single homogeneous exposure group, the indicated premium per exposure can be compared with the current on-level premium per exposure to calculate an indicated percentage change.

Adjustment and Selection Methods

For accident year $j$, let $L_j$ be the observed loss, $D_j$ the loss development factor to ultimate, $T_j$ the trend factor to the prospective cost level, and $E_j$ the earned exposures. The adjusted loss cost per exposure is calculated by summing adjusted losses and dividing by total exposures:

\text{Adjusted loss cost} = \frac{\sum_j L_jD_jT_j}{\sum_j E_j}.

This calculation assumes that the reported losses and factors refer to the same coverage and loss basis, that the development and trend factors are appropriate for the periods being adjusted, and that the exposures are suitable denominators for the indicated risk. The factors are multiplied because each adjusts the preceding loss amount for a separate effect. A loss that is first developed and then trended is therefore $L_jD_jT_j$, not $L_j(D_j+T_j)$.

Let $P_j$ denote historical earned premium and $O_j$ its on-level factor. Total current on-level premium is $\sum_j P_jO_j$. This premium should generally be aggregated before calculating a premium per exposure or a weighted overall factor. A simple average of on-level factors is only appropriate when the underlying premiums receive equal weight.

Credibility can be applied to the observed loss cost $LC_{\text{obs}}$ and an expected loss cost $LC_{\text{exp}}$. If $Z$ is the credibility assigned to the experience, with $0 \leq Z \leq 1$, the selected loss cost is

LC_{\text{sel}}=ZLC_{\text{obs}}+(1-Z)LC_{\text{exp}}.

The formula assumes both values are expressed on the same coverage, exposure, and prospective cost basis. Alternatively, credibility may be applied to a loss ratio and an expected loss ratio; the selected loss ratio can then be converted to a loss cost using current on-level premium per exposure.

From Selected Loss Cost to Indicated Premium

Suppose the selected loss cost per exposure is $LC_{\text{sel}}$, the fixed expense per exposure is $F$, the variable expense provision is $e$ of premium, and the profit and contingency provision is $q$ of premium. Let $R$ be the indicated premium per exposure. Under this model, expected premium must cover losses, fixed expenses, variable expenses, and the profit provision:

R=LC_{\text{sel}}+F+eR+qR.

The variable expense and profit provisions depend on premium, so moving those terms to the left and solving gives:

R(1-e-q)=LC_{\text{sel}}+F, \qquad R=\frac{LC_{\text{sel}}+F}{1-e-q}.

The derivation is a balance equation: the fraction $e+q$ of premium is reserved for the stated proportional provisions, leaving $(1-e-q)R$ to fund loss cost and fixed expense. The formula applies when the proportions are defined as percentages of premium, the provisions are additive as specified, and $e+q<1$. If a provision is defined as a percentage of losses or of some other base, the equation must be changed. This model also assumes the fixed expense is stated per exposure; it should not be treated as a percentage of premium.

If the current on-level premium per exposure is $R_{\text{current}}$, the indicated overall rate change is:

\text{Indicated rate change} = \frac{R}{R_{\text{current}}}-1.

This comparison assumes the current premium and indicated premium refer to the same exposure basis and coverage. It compares an indicated premium with a current on-level premium, not with unadjusted historical premium.

Worked Example: A Full Credibility-Adjusted Indication

A hypothetical insurer reviews two accident years for a coverage with unchanged terms. The first year has reported losses of $420,000, a development factor of 1.10, a trend factor of 1.05, 2,000 earned exposures, and $640,000 of earned premium. The second year has reported losses of $520,000, a development factor of 1.02, a trend factor of 1.02, 2,500 earned exposures, and $900,000 of earned premium. The premium on-level factors are 1.10 and 1.04, respectively. The expected loss cost is $210 per exposure and the selected credibility is $Z=0.60$. Fixed expense is $30 per exposure, variable expense is 25% of premium, and profit and contingency provision is 5% of premium. Find the indicated overall rate change from the current on-level premium level.

Method and assumptions. Develop and trend each year’s losses separately, then aggregate them. Treat the stated factors as applicable to each year’s losses and premiums, and assume both loss costs are on the prospective basis. Apply credibility to loss cost; then use the stated expense and profit provisions.

Step 1: Develop and trend the losses. For the first year, the ultimate estimate is $420{,}000(1.10)=462{,}000$, and its prospective-cost-level estimate is $462{,}000(1.05)=485{,}100$. For the second year, the ultimate estimate is $520{,}000(1.02)=530{,}400$, and its prospective-cost-level estimate is $530{,}400(1.02)=541{,}008$. Total adjusted losses are $485{,}100+541{,}008=1{,}026{,}108$. Total exposures are $2{,}000+2{,}500=4{,}500$, so the observed loss cost is $1{,}026{,}108/4{,}500=\$228.024$ per exposure.

Step 2: Apply credibility. The experience receives 60% weight and the expected loss cost receives 40% weight: $LC_{\text{sel}}=0.60(228.024)+0.40(210)=136.8144+84=\$220.8144$ per exposure.

Step 3: Calculate current on-level premium per exposure. The first year’s on-level premium is $640{,}000(1.10)=\$704{,}000$. The second year’s on-level premium is $900{,}000(1.04)=\$936{,}000$. Total current on-level premium is $704{,}000+936{,}000=\$1{,}640{,}000$. Dividing by 4,500 exposures gives $R_{\text{current}}=1{,}640{,}000/4{,}500=\$364.4444$ per exposure.

Step 4: Calculate the indicated premium. Here $F=30$, $e=0.25$, and $q=0.05$. The proportional provisions total 0.30 of premium, so: $R=(220.8144+30)/(1-0.25-0.05)=250.8144/0.70=\$358.3063$ per exposure.

Step 5: Calculate the overall change. The ratio is $358.3063/364.4444-1$, which is approximately $-0.016842$. Rounded to two decimal places, the indicated overall rate change is a 1.68% decrease.

Check. The indicated premium is below the current on-level premium, so a decrease is consistent with the calculated ratio. The dollar difference is approximately $\$6.14$ per exposure, and $\$6.14/\$364.4444$ is approximately 1.68%.

Common mistake avoided: Averaging the two on-level factors to get 1.07 would ignore the different premium volumes. The correct total is obtained by adjusting each premium separately and adding the results.

Worked Example: An Indication Without a Credibility Adjustment

A hypothetical program has 5,000 earned exposures. Historical earned premiums are $480,000 and $720,000, with on-level factors of 1.05 and 0.95. The corresponding losses are combined at $700,000; assume a development factor of 1.08 and a trend factor of 1.04 apply to those losses. Fixed expense is $22 per exposure, variable expense is 28% of premium, and profit and contingency provision is 4% of premium. Calculate the indicated rate change from the current on-level premium per exposure.

Method and assumptions. Adjust the combined losses to ultimate and to the prospective cost level. Adjust each historical premium separately to the current rate level. No credibility adjustment is specified, so use the adjusted experience loss cost directly.

Step 1: Adjust losses. Developed losses are $700{,}000(1.08)=\$756{,}000$. Trended losses are $756{,}000(1.04)=\$786{,}240$. The loss cost is $786{,}240/5{,}000=\$157.248$ per exposure.

Step 2: Adjust premiums. On-level premiums are $480{,}000(1.05)=\$504{,}000$ and $720{,}000(0.95)=\$684{,}000$. Their total is $\$1{,}188{,}000$, so current on-level premium per exposure is $1{,}188{,}000/5{,}000=\$237.60$.

Step 3: Calculate indicated premium. The total variable and profit provisions are $0.28+0.04=0.32$ of premium. Therefore, $R=(157.248+22)/(1-0.28-0.04)=179.248/0.68=\$263.60$ per exposure.

Step 4: Calculate the change. The indicated change is $263.60/237.60-1=0.1094276$, or a 10.94% increase rounded to two decimal places. As a check, the indicated premium exceeds the current on-level premium by $\$26.00$, and $26/237.60$ is approximately 10.94%.

Common mistake avoided: Do not apply the on-level factor to losses or the trend factor to premiums. Those adjustments represent different effects and are applied to different quantities.

Worked Example: Credibility Applied to a Loss Ratio

A hypothetical portfolio has 4,000 exposures, $640,000 and $800,000 of earned premium in two periods, and on-level factors of 1.05 and 1.00. Aggregate losses of $600,000 are subject to a development factor of 1.10 and a trend factor of 1.05. The expected loss ratio is 50% and experience credibility is $Z=0.40$. Fixed expense is $25 per exposure, variable expense is 22% of premium, and profit and contingency provision is 4% of premium. Find the indicated rate change.

Method and assumptions. First calculate the experience loss ratio using developed and trended losses over on-level premiums. Blend that ratio with the expected loss ratio, then convert the selected ratio to a loss cost using current on-level premium per exposure. This is equivalent to blending loss costs when both are expressed on the same premium and exposure basis.

Step 1: Calculate adjusted losses and premiums. Adjusted losses are $600{,}000(1.10)(1.05)=\$693{,}000$. On-level premiums are $640{,}000(1.05)=\$672{,}000$ and $800{,}000(1.00)=\$800{,}000$, totaling $\$1{,}472{,}000$.

Step 2: Calculate and select the loss ratio. The experience loss ratio is $693{,}000/1{,}472{,}000=0.470788$, or approximately 47.0788%. The selected loss ratio is $0.40(0.470788)+0.60(0.50)=0.488315$, or approximately 48.8315%.

Step 3: Convert to a selected loss cost. Current on-level premium per exposure is $1{,}472{,}000/4{,}000=\$368$. Thus the selected loss cost is $0.488315(368)=\$179.70$ per exposure (unrounded value $179.7000$).

Step 4: Calculate the premium and indicated change. The proportional provisions total $0.22+0.04=0.26$. The indicated premium is $(179.70+25)/(1-0.22-0.04)=204.70/0.74=\$276.6216$ per exposure. The rate change is $276.6216/368-1=-0.248311$, so the indication is a 24.83% decrease rounded to two decimal places. The direction is consistent because the indicated premium is below the current on-level premium.

Common mistake avoided: The 50% expected loss ratio is not a loss cost of $50. It is a proportion of premium; convert the selected ratio using the current on-level premium per exposure before applying the premium formula.

Checks Before Finalizing an Indication

A reliable indication is not just a sequence of arithmetic operations. It also depends on aligning the experience and assumptions. Before interpreting the result, confirm that the losses and premiums relate to the same exposures and coverage, that the loss adjustments target the prospective period, and that the on-level premium factors target the current rate level. When credibility is used, verify that the experience and complement have the same basis.

The expense formula is another important check. Fixed expense per exposure is added to loss cost before dividing by the proportion of premium left after variable expenses and profit. It is not itself grossed up as though it were a percentage of premium. Finally, compare the indicated premium with current on-level premium. Comparing against unadjusted historical premium can mix rate-level changes into the indicated change a second time.

Key takeaway: Adjust losses for development and trend, adjust premiums for rate level, apply credibility on a consistent basis, and then solve the premium balance equation. Each adjustment has its own data and purpose; keeping those roles separate is essential to a defensible prospective indication.

Check Your Understanding

Work each problem in sequence and retain full precision until the final answer.

A hypothetical coverage has $180,000 of losses, a development factor of 1.05, a trend factor of 1.04, and 1,200 earned exposures. Find the adjusted loss cost per exposure. (Answer: $163.80)
Historical earned premiums are $300,000 and $500,000, with on-level factors 1.08 and 0.96. There are 3,600 exposures in total. Find total current on-level premium and current on-level premium per exposure. (Answer: $804,000 total; $223.33 per exposure)
The observed prospective loss cost is $150, the expected loss cost is $170, and $Z=0.25$. Find the selected loss cost. (Answer: $165.00)
A selected loss cost is $190 per exposure. Fixed expense is $20 per exposure, variable expense is 24% of premium, and profit provision is 6% of premium. Find the indicated premium per exposure. (Answer: $300.00)
Current on-level premium is $325 per exposure and the indicated premium is $300 per exposure. Calculate the indicated overall rate change, rounded to two decimal places. (Answer: 7.69% decrease)

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