Using Projected Ultimate Losses as Pure Premium Inputs

What the Pure Premium Calculation Uses

A pure premium calculation turns a total amount of loss into an average loss cost per exposure unit. For this topic, the key input is a set of supplied projected ultimate losses: estimates of the total losses expected to emerge from the relevant exposures, after the applicable ultimate-loss selection and any specified projection have already been made. The calculation uses those loss estimates; it does not determine them.

That distinction matters in practice. A loss estimate may have been developed from incomplete experience and adjusted to a target period before it reaches the pure premium calculation. If the instructions say that the amounts are already projected ultimate losses, applying development or trend again would alter the given basis and could count an adjustment twice.

Definition: For a group of risks with projected ultimate losses $L$ and $E$ matching exposure units, the pure premium, also called the loss cost per exposure unit, is $$ \text{Pure premium}=\frac{L}{E}. $$ The result is an average projected loss amount per exposure unit. It is not, by itself, a complete rate: expense provisions, profit and other rate components are not included in this ratio.

A projected ultimate loss amount and its exposure denominator must describe the same risks and coverage. They must also be on compatible time and projection bases. For example, if losses relate to a specified coverage and period, the exposures should represent that coverage and the corresponding period. Before dividing, confirm what an exposure unit means in the problem: it might be a policy-year, vehicle-year, payroll unit, or another defined measure.

If several periods or groups belong in one indication, pool their losses and exposures before taking the ratio. The pooled result gives more weight to a group with more exposure; it is not generally the simple average of the individual groups’ pure premiums.

Why the Ratio Gives an Average Loss Cost

Suppose group $j$ has $E_j$ exposure units and projected ultimate losses $L_j$. Its average loss amount per exposure unit is $L_j/E_j$. Across $k$ groups, the total loss is $\sum_{j=1}^{k} L_j$, while the total exposure is $\sum_{j=1}^{k} E_j$. Therefore, the average across all exposure units is total loss divided by total exposure:

\frac{\sum_{j=1}^{k}L_j}{\sum_{j=1}^{k}E_j} = \sum_{j=1}^{k} \left( \frac{E_j}{\sum_{r=1}^{k}E_r} \right) \frac{L_j}{E_j}.

The right side shows why pooled pure premium is an exposure-weighted average of the group pure premiums. Each group’s weight is its share of the total exposure, and the weights sum to one. This identity holds when the groups’ exposure units and loss bases are compatible. It justifies pooling the totals rather than averaging group results without weights.

Calculate From the Supplied Ultimate Losses

Worked Example: Pooling Two Experience Periods

A hypothetical insurer is given projected ultimate losses for two periods on the same coverage and target cost basis. The first period has $4,320,000 of projected ultimate losses and 24,000 policy-years of exposure. The second has $4,800,000 and 25,000 policy-years. Calculate the pooled pure premium per policy-year. Treat the supplied losses as already projected; no further loss adjustment is requested.

Method. Pool losses and exposures on their common basis, then divide. Do not calculate a simple average of the two period pure premiums.

First total the projected ultimate losses:

L_{\text{total}}=\$4{,}320{,}000+\$4{,}800{,}000=\$9{,}120{,}000.

Next total the exposures:

E_{\text{total}}=24{,}000+25{,}000=49{,}000\text{ policy-years}.

Divide total loss by total exposure:

\frac{\$9{,}120{,}000}{49{,}000} =\$186.122448\ldots\text{ per policy-year}.

Rounded to the nearest cent, the pooled pure premium is $186.12 per policy-year. As a check, the period pure premiums are $4,320,000 / 24,000 = $180.00 and $4,800,000 / 25,000 = $192.00. The pooled answer, $186.12, lies between them and is weighted toward the second period, which has slightly more exposure. No trend or development factor was applied because the problem supplied losses already projected to the target basis.

Pooling Groups Without Losing Their Weights

A common setup gives separate loss and exposure information for different risk groups. Calculate each group’s loss cost if useful for interpretation, but use the total-loss-to-total-exposure ratio for the combined result. The combined pure premium represents the average loss per exposure over the mix in the supplied data; a change in that mix could change the combined result even when the group pure premiums stay the same.

Worked Example: Combining Two Coverage Tiers

A hypothetical insurer is reviewing two tiers of a specialty equipment policy. On a common projected ultimate loss basis, the basic tier has $2,970,000 of losses across 18,000 policy-years. The expanded tier has $1,650,000 across 7,500 policy-years. Calculate the combined pure premium and check why a simple average of the tier results is not appropriate.

Method. Calculate the tier values, then pool loss and exposure totals. The exposure units are policy-years for both tiers, and the losses are on the same stated basis.

For the basic tier:

\frac{\$2{,}970{,}000}{18{,}000} =\$165.00\text{ per policy-year}.

For the expanded tier:

\frac{\$1{,}650{,}000}{7{,}500} =\$220.00\text{ per policy-year}.

The combined loss and exposure totals are:

L_{\text{total}}=\$2{,}970{,}000+\$1{,}650{,}000=\$4{,}620{,}000, $$ $$ E_{\text{total}}=18{,}000+7{,}500=25{,}500\text{ policy-years}.

Thus:

\text{Combined pure premium} =\frac{\$4{,}620{,}000}{25{,}500} =\$181.176470\ldots =\$181.18\text{ per policy-year}.

The unweighted average of the two tier pure premiums would be $(\$165+\$220)/2=\$192.50$, which incorrectly gives each tier equal weight despite their different exposure volumes. The exposure-weighted check is $(18{,}000/25{,}500)(\$165)+(7{,}500/25{,}500)(\$220)=\$181.176470\ldots$, matching the pooled calculation. The common mistake is treating the two group results as equally representative of all policies; the correction is to weight each by its share of exposure.

Check the Exposure Unit and Projection Basis

If exposure is reported in different units, convert it before pooling. A loss amount divided by an exposure count in the wrong unit can be numerically plausible but materially wrong. Conversion must reflect the exposure definition in the problem; for example, 12 vehicle-months equal one vehicle-year when the units refer to the same vehicle and coverage.

Likewise, the word “projected” does not alone specify which adjustments are already included. Read the data description carefully. If losses have already been selected as ultimate and projected to the target period, use them as supplied. If a problem instead gives unprojected historical loss amounts and requests an adjustment, follow that stated instruction. Do not infer an additional adjustment merely because the calculation is called a pure premium indication.

Worked Example: Converting Exposure Units Before Pooling

A hypothetical insurer is given projected ultimate losses of $1,080,000 for 3,600 vehicle-years and $450,000 for 5,400 vehicle-months. The two loss amounts relate to the same coverage and target cost basis. Calculate the pooled pure premium per vehicle-year. Use 12 vehicle-months per vehicle-year.

Method. Convert the second exposure count to vehicle-years, then combine both loss amounts and both exposure amounts. The supplied losses are already projected ultimate losses, so the conversion applies to exposure only.

Convert 5,400 vehicle-months to vehicle-years:

5{,}400\text{ vehicle-months}\div12 =450\text{ vehicle-years}.

The total exposure is therefore:

E_{\text{total}}=3{,}600+450=4{,}050\text{ vehicle-years}.

The total projected ultimate losses are:

L_{\text{total}}=\$1{,}080{,}000+\$450{,}000=\$1{,}530{,}000.

Divide the totals:

\frac{\$1{,}530{,}000}{4{,}050} =\$377.777777\ldots =\$377.78\text{ per vehicle-year}.

The check is dimensional: the numerator is dollars and the denominator is vehicle-years, so the answer is dollars per vehicle-year. Dividing by 9,000 instead would mix vehicle-years and vehicle-months, producing a result with no single consistent exposure-unit interpretation.

Common Errors and Their Corrections

Re-selecting the loss amount. If losses are supplied as projected ultimate losses, do not apply a loss development factor or another ultimate selection unless the problem explicitly asks for it.
Applying projection twice. If the stated losses are already on the target period’s basis, applying a further trend adjustment changes the specified input basis.
Averaging group pure premiums equally. Use the exposure-weighted result, equivalently total losses divided by total exposures, unless equal weighting is specifically requested for a different purpose.
Mixing exposure units. Convert all exposures to one common unit before adding them or interpreting the final ratio.
Calling the result a complete rate. Pure premium is the loss cost per exposure unit. It does not automatically include expenses, profit, or other rate components.

Key takeaway: When projected ultimate losses and matching exposures are supplied, calculate the pure premium as total projected ultimate losses divided by total compatible exposure. The calculation is separate from selecting the ultimate losses; preserve the basis of the supplied data and do not apply adjustments twice.

Check Your Understanding

For each problem, calculate from the data as stated and keep the loss and exposure bases consistent.

A hypothetical coverage has $2,640,000 of projected ultimate losses over 12,000 policy-years. What is the pure premium per policy-year? (Answer: $220.00 per policy-year)
Two compatible groups have projected ultimate losses of $5,200,000 over 40,000 exposure units and $1,800,000 over 10,000 exposure units. Find the pooled pure premium. (Answer: $140.00 per exposure unit)
A hypothetical insurer supplies $960,000 in projected ultimate losses for 4,800 matching exposure units. Calculate the loss cost per exposure unit, rounded to cents. (Answer: $200.00 per exposure unit)
A portfolio has $1,260,000 in projected ultimate losses over 3,000 vehicle-years and 9,000 vehicle-months. Convert the exposures and calculate the pooled pure premium per vehicle-year, using 12 months per year. (Answer: $336.00 per vehicle-year)
Projected ultimate losses of $1,440,000 are already adjusted to the target period. They match 8,000 exposure units. Calculate the pure premium and state whether another trend adjustment should be applied based on these facts. (Answer: $180.00 per exposure unit; no additional trend adjustment)

What You'll Learn