Calculating an Indicated Catastrophe Premium

From Modeled Loss to an Indicated Premium

A catastrophe model can estimate the average annual loss for a defined coverage, but that estimate alone is not an indicated premium. A pricing indication must also reflect the selected catastrophe risk load and the applicable expense and profit provisions. The calculation depends on how each provision is specified: a fixed dollar amount is added to the required dollars, while a percentage of premium must be allowed for in the premium equation.

The calculation is meaningful only when its components refer to the same coverage, exposure, currency, and annual period. For example, an annual modeled loss should not be combined with a monthly expense amount unless that expense has first been converted to the same annual basis. The loss estimate and risk load also need to be consistent with the coverage being priced; a load selected for one limit or portfolio should not be carried over without justification to a different one.

Definition: For a defined coverage and rating period, an indicated premium is the premium that covers the modeled average annual loss, the selected catastrophe risk load, fixed-dollar expense and profit provisions, and the portions of premium required for variable expenses, taxes, and profit. The indication is an output of the stated assumptions; it is not, by itself, a statement that the premium is approved or commercially appropriate.

Let $A$ denote modeled average annual loss, $R$ the selected dollar catastrophe risk load, $F_E$ fixed-dollar expenses, and $F_P$ a fixed-dollar profit provision. Let $e$, $t$, and $p$ be the applicable expense, tax, and profit rates expressed as fractions of the indicated premium $P$. When these provisions are all percentages of the same premium base, the required dollars satisfy

$$ P = A+R+F_E+F_P+(e+t+p)P. $$

Solving for $P$ gives

P = \frac{A+R+F_E+F_P}{1-e-t-p}, \qquad e+t+p<1.

The condition $e+t+p<1$ is necessary for a finite positive premium when the numerator is positive. If all provisions are fixed-dollar amounts, there is no premium-based percentage to solve for: the indication is simply the sum of the required dollars. If a provision uses a different base—for example, a percentage of loss rather than a percentage of premium—it must be calculated using that base instead of being placed automatically in the premium-rate denominator.

Why the Premium Denominator Appears

The denominator is not an extra load chosen independently. It follows from the fact that premium-based provisions increase when premium increases. Starting with the required-dollar equation, gather the terms proportional to premium on the left:

$$ P-(e+t+p)P=A+R+F_E+F_P. $$

Factor $P$ and divide by the remaining premium share:

P\bigl(1-e-t-p\bigr)=A+R+F_E+F_P, \qquad P=\frac{A+R+F_E+F_P}{1-e-t-p}.

Thus, if the variable provisions total 20% of premium, only 80% of each premium dollar remains to fund the stated dollar requirements. Dividing by 0.80 grosses up those requirements so the premium can fund both the requirements and the premium-based provisions. Adding 20% to the dollar requirements would not do that, because the added amount itself would also be subject to the 20% provision.

Worked Indications

Worked Example: Percentage-Based Expense and Profit Provisions

A hypothetical annual catastrophe coverage has modeled average annual loss of $A=\$1{,}200{,}000$. The selected catastrophe risk load is 30% of modeled average annual loss. Fixed expenses are $\$45{,}000$. Acquisition expenses are 12% of premium, taxes are 3% of premium, and the profit provision is 5% of premium. Calculate the indicated premium.

Assumptions and method. Use the stated annual loss estimate and the selected load as the loss-related dollar requirement. No parametric loss distribution is assumed; the model output needed here is its average annual loss. This is not a life-contingent calculation, so no mortality assumption applies, and no interest discounting is used. Apply the premium-equation gross-up because the three stated rates are all percentages of premium.

Step 1: Calculate the selected risk load.

R=0.30(1{,}200{,}000)=\$360{,}000.

Step 2: Add the dollar requirements. The dollar amount to be funded before premium-based provisions is

A+R+F_E =\$1{,}200{,}000+\$360{,}000+\$45{,}000 =\$1{,}605{,}000.

Step 3: Add the premium-based rates and solve. Their combined rate is $0.12+0.03+0.05=0.20$, so 80% of premium is available for the dollar requirements.

P=\frac{\$1{,}605{,}000}{1-0.20} =\frac{\$1{,}605{,}000}{0.80} =\$2{,}006{,}250.00.

Check. At this premium, acquisition expense is $0.12(\$2{,}006{,}250)=\$240{,}750$, taxes are $0.03(\$2{,}006{,}250)=\$60{,}187.50$, and profit is $0.05(\$2{,}006{,}250)=\$100{,}312.50$. Together, the three provisions equal $\$401{,}250$. Adding the $\$1{,}605{,}000$ dollar requirement gives $\$2{,}006{,}250$, as required. The method avoids the mistake of adding 20% to the dollar requirement rather than grossing it up. Money is rounded to the nearest cent; the displayed result is exact to cents.

Worked Example: Fixed-Dollar Profit Must Be Included Before Gross-Up

A hypothetical annual catastrophe layer has modeled average annual loss of $\$640{,}000$, a selected catastrophe risk load of $\$140{,}800$, and fixed expenses of $\$18{,}500$. The plan also requires a fixed-dollar profit provision of $\$35{,}000$, while variable expenses are 12% of premium. Find the indicated premium.

Assumptions and method. Treat the loss and risk load as annual dollar amounts for this layer. No parametric loss distribution is needed or assumed, because the modeled average annual loss and selected dollar load are supplied. No mortality assumption or interest discounting applies. Include the fixed profit in the dollar requirement, then gross up only for the expense rate stated as a percentage of premium.

Step 1: Add all fixed-dollar requirements.

\$640{,}000+\$140{,}800+\$18{,}500+\$35{,}000 =\$834{,}300.

Step 2: Solve for premium. Variable expenses use 12% of premium, leaving 88% to fund the fixed-dollar requirements.

P=\frac{\$834{,}300}{1-0.12} =\frac{\$834{,}300}{0.88} =\$948{,}068.1818\ldots \approx \$948{,}068.18.

Check. Variable expenses are $0.12(\$948{,}068.1818\ldots)=\$113{,}768.1818\ldots$. Adding them to the $\$834{,}300$ fixed-dollar requirement gives $\$948{,}068.1818\ldots$, the unrounded premium.

Common mistake and correction. Omitting fixed profit would give $(\$640{,}000+\$140{,}800+\$18{,}500)/0.88=\$908{,}295.45$, which is too low. The omitted $\$35{,}000$ must itself be funded from the 88% of premium remaining after variable expenses. The correct indication is $\$948{,}068.18$, not the result of the calculation that omits the fixed provision. Money is rounded to the nearest cent.

Worked Example: Several Premium-Based Provisions

A hypothetical annual catastrophe program has modeled average annual loss of $\$2{,}400{,}000$. The selected risk load is 15% of that loss. Fixed expenses are $\$72{,}000$. The applicable variable expense rate is 6% of premium, the tax rate is 10% of premium, and the profit provision is 8% of premium. Calculate the indicated premium.

Assumptions and method. Use the modeled annual average loss and the stated load; no additional loss distribution or parameters are specified or needed. No mortality assumption or interest discounting applies. All three percentages use the same premium base, so add them before solving the premium equation.

Step 1: Calculate the load and dollar requirement.

R=0.15(\$2{,}400{,}000)=\$360{,}000,

A+R+F_E =\$2{,}400{,}000+\$360{,}000+\$72{,}000 =\$2{,}832{,}000.

Step 2: Calculate the total premium-based rate.

$$ e+t+p=0.06+0.10+0.08=0.24. $$

Step 3: Gross up the dollar requirement. The share remaining after these provisions is $1-0.24=0.76$.

P=\frac{\$2{,}832{,}000}{0.76} =\$3{,}726{,}315.7894\ldots \approx \$3{,}726{,}315.79.

Check. The variable expense, tax, and profit provisions total $0.24(\$3{,}726{,}315.7894\ldots)=\$894{,}315.7894\ldots$. Adding $\$2{,}832{,}000$ gives the unrounded indicated premium. The sum of the rates is below 100%, so the denominator is positive. The method avoids using one provision rate as though it replaced the other premium-based rates. Money is rounded to the nearest cent.

Practical Checks Before Selecting an Indication

The arithmetic is straightforward once the coverage basis and provision bases are clear. The most consequential errors usually arise before the division: a loss estimate may not match the coverage, a load may be applied to the wrong base, or a fixed-dollar provision may be treated as a percentage. Before accepting a result, confirm the following:

Match the coverage. Confirm that modeled average annual loss and the selected risk load refer to the same layer, limit, exposure basis, and rating period as the premium being indicated.
Identify each provision's base. Label provisions as fixed dollars, percentages of premium, or amounts based on another quantity. Only percentages of the same premium base belong together in the denominator.
Keep the risk load interpretation explicit. If the selected load is a percentage of average annual loss, calculate it as a dollar amount before adding it to the dollar requirement. Do not mistake a load percentage of loss for a percentage of premium.
Check the premium share. The sum of premium-based rates must be less than 1 for this gross-up formula to yield a finite positive premium when required dollars are positive.
Reconcile the final result. Multiply the unrounded premium by each premium-based rate, then add those provisions to the dollar requirements. The total should equal the unrounded premium.

An indicated catastrophe premium is conditional on its inputs and the chosen pricing structure. It should not be read as a direct output of the catastrophe model: modeled average annual loss is one component, and the selected risk load and expense and profit provisions complete the indication. A clearly stated calculation makes it possible to distinguish a change in modeled loss from a change in pricing provisions.

Key takeaway: Add modeled average annual loss, the selected dollar catastrophe risk load, and fixed-dollar provisions. Then divide by one minus the sum of premium-based provision rates, provided those rates use the same premium base and sum to less than 1.

Check Your Understanding

For each hypothetical annual coverage, calculate the requested indication. Round money to the nearest cent unless the answer is a rate.

A coverage has average annual loss of $\$500{,}000$, a selected load equal to 20% of average annual loss, fixed expenses of $\$10{,}000$, variable expenses of 10% of premium, and profit of 5% of premium. What is the indicated premium? (Answer: $\$717{,}647.06$)
A catastrophe layer has average annual loss of $\$800{,}000$, a selected dollar risk load of $\$160{,}000$, fixed expenses of $\$25{,}000$, fixed profit of $\$30{,}000$, and variable expenses of 12% of premium. What is the indicated premium? (Answer: $\$1{,}153{,}409.09$)
A program has average annual loss of $\$1{,}500{,}000$, a selected dollar risk load of $\$275{,}000$, fixed expenses of $\$40{,}000$, variable expenses of 7% of premium, taxes of 2% of premium, and profit of 6% of premium. Calculate the indicated premium. (Answer: $\$2{,}135{,}294.12$)
A coverage has average annual loss of $\$900{,}000$, a selected risk load of $\$180{,}000$, and fixed expenses of $\$20{,}000$. At an indicated premium of $\$1{,}400{,}000$, variable expenses are 10% of premium. What profit provision, as a percentage of premium, is implied by the indication? (Answer: 11.43%)
A hypothetical catastrophe contract has average annual loss of $\$1{,}050{,}000$, a selected dollar load of $\$210{,}000$, fixed expenses of $\$40{,}000$, variable expenses of 8% of premium, taxes of 2% of premium, and a profit target of 7% of premium. Is a proposed premium of $\$1{,}500{,}000$ sufficient? State the indicated premium and the shortfall of the proposal. (Answer: Indicated premium $\$1{,}566{,}265.06$; the proposal is short by $\$66{,}265.06$.)

What You'll Learn