Why Gross Rates Must Be Grossed Up
A pure premium covers expected losses per exposure, but a charged rate may also need to fund expenses and a target profit provision. When some expenses and the profit provision are specified as percentages of the gross premium itself, adding them directly to the pure premium does not produce a rate that funds all components. The rate must be solved as a balance equation.
This calculation is distinct from estimating the pure premium. Treat the expected loss cost per exposure as an input, then add the expense and profit provisions on their stated bases. The central exam skill is to identify which amounts are fixed per exposure and which amounts vary with the gross rate.
The variables \(a\) and \(b\) are proportions, not percentage numbers: for example, 12% is \(0.12\). A fixed expense of $30 per exposure is an amount, not a proportion. The exposure unit must also be consistent throughout: if \(L\) is per exposure-year, then fixed expenses and the indicated rate must also be per exposure-year.
Deriving the Gross-Up Formula
Start with the statement that the gross premium funds all expected components. Losses and fixed amounts do not vary with \(G\); percentage provisions do.
Collect the terms involving \(G\) on the left. This step explains why premium-based provisions belong in the denominator rather than being added as fixed dollars.
If \(a+b<1\), divide by the positive retained-premium proportion \(1-a-b\):
The condition \(a+b<1\) is necessary for a finite positive solution when \(L+e+k>0\). If \(a+b=1\), the entire premium is allocated to percentage provisions, leaving nothing to fund a positive loss and fixed-cost amount. If \(a+b>1\), those provisions alone exceed the premium, so this balance cannot produce a positive gross rate.
Use the supplied pure premium and express each fixed expense and fixed profit amount per the same exposure unit.
Add \(L\), \(e\), and \(k\) in the numerator. Add the percentages that apply to gross premium to obtain \(a+b\).
Calculate \(G=(L+e+k)/(1-a-b)\), and verify that the resulting dollar allocations sum to \(G\).
Worked Rate Indications
Worked Example: Fixed Costs and Two Premium Ratios
A hypothetical coverage has a pure premium of $360 per exposure-year. Fixed expenses are $30 per exposure-year. Variable expenses are 12% of gross premium, and the profit provision is 8% of gross premium. Find and check the indicated gross rate.
Step 1: Identify the bases. The loss cost and fixed expense are dollar amounts per exposure-year. The variable expense and profit provisions are proportions of the unknown gross rate. Thus \(L=360\), \(e=30\), \(a=0.12\), and \(b=0.08\); there is no separate fixed profit amount, so \(k=0\).
Step 2: Check the retained proportion. The combined premium ratios are \(0.12+0.08=0.20\). The retained proportion is \(1-0.20=0.80\), which is positive.
Step 3: Calculate the gross rate.
Step 4: Check each allocation. Variable expenses are \(0.12(487.50)=\$58.50\). The profit provision is \(0.08(487.50)=\$39.00\). The full balance is \(360+30+58.50+39.00=\$487.50\), as required.
Method and trap avoided. This is a gross-up using the retained premium proportion. Adding 12% and 8% of the pure premium to $390 would calculate the ratios on the wrong base and would not balance to the resulting gross rate.
Worked Example: Converting Frequency and Severity to a Gross Rate
A hypothetical portfolio is expected to have 0.15 claims per exposure-year, with an average covered claim of $3,500. Fixed expenses are $40 per exposure-year, and an additional fixed profit amount is $10 per exposure-year. Variable expenses equal 9% of gross premium, and the profit provision equals 6% of gross premium. Find the indicated gross rate and verify it.
Step 1: Establish the pure premium. Expected loss cost per exposure-year is expected claim frequency times expected severity:
Step 2: Classify all provisions. The fixed amounts are $40 of expenses and $10 of profit, so \(e+k=50\). The premium-based ratios are \(a=0.09\) and \(b=0.06\). Their sum is \(0.15\), leaving \(1-0.15=0.85\) of gross premium for losses and fixed amounts.
Step 3: Gross up.
Rounded to the nearest cent, the indicated rate is $676.47 per exposure-year.
Step 4: Check at the unrounded rate. The variable expense is \(0.09(676.470588\ldots)=\$60.882353\ldots\), and the percentage profit provision is \(0.06(676.470588\ldots)=\$40.588235\ldots\). Then losses, fixed expense, fixed profit, variable expense, and percentage profit sum to \(525+40+10+60.882353\ldots+40.588235\ldots=\$676.470588\ldots\). The rounded rate is $676.47.
Method and trap avoided. First form the pure premium from frequency and severity, then gross it up. The $10 fixed profit amount belongs in the numerator; it is not part of the 6% ratio.
Worked Example: Multiple Variable Expenses and a Profit Target
A hypothetical policy has an expected loss cost of $465 per exposure and fixed expenses of $35 per exposure. A commission of 8%, a premium tax of 2%, and other variable expenses of 5% are each calculated as proportions of gross premium. The target profit provision is 5% of gross premium. Determine the gross rate and show the dollar allocation.
Step 1: Combine only ratios with the same stated base. All four provisions are percentages of gross premium:
The fixed dollar amount in the numerator is \(L+e=465+35=500\).
Step 2: Solve the balance. The premium proportion available after these ratios is \(1-0.20=0.80\). Therefore,
Step 3: Allocate and check. Commission is \(0.08(625)=\$50.00\), premium tax is \(0.02(625)=\$12.50\), other variable expenses are \(0.05(625)=\$31.25\), and profit is \(0.05(625)=\$31.25\). Total funding is \(465+35+50+12.50+31.25+31.25=\$625.00\).
Method and trap avoided. Combine provisions only after confirming they share the gross-premium base. Adding the four ratios to $500 as if they were fixed dollar charges would be dimensionally incorrect.
Keep the Percentage Base Explicit
The formula above applies only when the specified variable expense and profit ratios are percentages of gross premium. A ratio based on expected losses, claims, or another amount must not be inserted into \(a+b\) without first expressing its dollar amount in the balance. For instance, an expense equal to \(c\) times expected losses contributes \(cL\), which is a fixed dollar amount once \(L\) is known; it does not contribute \(cG\).
Likewise, a profit target described as a percentage of costs is not the same as a percentage of gross premium. If profit is \(r\) times the non-profit funding amount \(C\), then the required premium is \(C+rC=(1+r)C\). That is a markup on costs. If the profit is instead \(b\) times gross premium, then the required premium is \(C+bG\), giving \(G=C/(1-b)\), before accounting for any other premium-based ratios.
Common Exam Trap: Adding Premium Ratios to Costs
Suppose pure premium is $400 per exposure-year, fixed expenses are $25 per exposure-year, variable expenses are 10% of gross premium, profit is 7% of gross premium, and there is an additional fixed profit amount of $15 per exposure-year.
A common incorrect calculation is to treat the 17% as an addition to the fixed-dollar amount: \((400+25+15)(1.17)=\$514.80\). This does not fund 17% of the resulting gross premium. At $514.80, the stated premium-based expense and profit would be \(0.17(514.80)=\$87.516\), while the rate has only added \(514.80-440=\$74.80\) above the fixed-dollar total.
The correction is to put the known amounts in the numerator and the premium ratios in the denominator:
Rounded to the nearest cent, the indicated rate is $530.12 per exposure-year. At the unrounded rate, 10% variable expenses are $53.012048..., 7% profit is $37.108434..., and \(440+53.012048\ldots+37.108434\ldots=530.120481\ldots\). The balance checks.
Interpretation and Final Checks
A gross-up is a rate-level calculation under specified assumptions; it is not, by itself, a rate change indication. Its usefulness depends on matching the loss cost, expenses, profit provision, and exposure unit. Before reporting a result, check the following:
- Is the loss cost stated per the same exposure unit as the rate?
- Are fixed expenses and fixed profit amounts included once, in dollars?
- Does each ratio apply to gross premium, losses, or another clearly identified base?
- Is the sum of gross-premium ratios less than 1?
- Do the dollar allocations at the calculated rate add back to the gross rate?
These checks catch the most consequential errors: treating a premium percentage as a fixed loading, mixing bases, and overlooking a fixed profit amount. A short reconciliation is often the clearest way to demonstrate that an indicated rate satisfies the stated pricing balance.
Check Your Understanding
For each problem, identify the percentage bases before calculating. Round dollar rates to the nearest cent.
- A hypothetical coverage has pure premium $240 per exposure, fixed expenses $20 per exposure, variable expenses of 10% of gross premium, and a profit provision of 5% of gross premium. Find the gross rate. (Answer: $305.88 per exposure)
- Expected frequency is 0.08 claims per exposure-year and expected severity is $5,000 per claim. Fixed expenses are $32 per exposure-year; variable expenses are 7% and profit is 8% of gross premium. Find the gross rate. (Answer: $508.24 per exposure-year)
- A pure premium is $610 per exposure. Fixed expenses are $25 and fixed profit is $15 per exposure. Variable expenses are 11% of gross premium, premium tax is 2% of gross premium, and profit is 6% of gross premium. Find the gross rate and state the combined percentage provision. (Answer: $802.47 per exposure; combined percentage provision is 19%)
- A gross rate is $720 per exposure. Pure premium is $540, fixed expenses are $30, and variable expenses are 10% of gross premium. No other provision applies. What percentage of gross premium remains as profit? Verify the dollar profit. (Answer: 10.8333% of gross premium; $78.00 profit per exposure)
- A hypothetical rate has pure premium $700 per exposure-year, fixed expenses $40 per exposure-year, variable expenses 12% of gross premium, and profit 8% of gross premium. Find the gross rate and check that the total percentage provisions and retained proportion sum to 100%. (Answer: $925.00 per exposure-year; percentage provisions are 20% and the retained proportion is 80%)