Gross Premium Calculation: Expenses and Profit Loading

Why Gross Premiums Differ from Net Premiums

The net premium covers only the expected cost of benefits. In reality, insurers incur substantial expenses to sell, issue, and administer policies, and they need to earn a profit for shareholders. The gross premium must cover all of these: benefits, expenses, and a profit margin. The difference between gross and net premiums reflects the total loading for expenses and profit.

Understanding how to compute gross premiums is essential for Exam LTAM because it mirrors actual insurance pricing. While net premiums are a theoretical construct useful for reserve calculations, gross premiums are what policyholders actually pay. The gross premium calculation uses the same equivalence principle as net premiums but expands the equation to include expense cash flows.

Categories of Expenses

Life insurance expenses fall into three categories based on how they scale. Percentage of premium expenses are proportional to the gross premium itself. Common examples include agent commissions (often 50% or more of the first-year premium, declining to 5-10% in renewal years) and premium taxes (typically 2-3% of premium). These create a circular dependency: the gross premium depends on the expenses, but the percentage expenses depend on the gross premium.

Per-policy expenses are a fixed dollar amount per policy, regardless of face amount or premium. These include the cost of issuing a policy, maintaining records, sending annual statements, and processing claims. They are the same whether the policy face amount is 10,000 or 1,000,000.

Per-unit expenses are proportional to the face amount (or sum insured). These include the cost of underwriting (which increases with the amount at risk) and certain regulatory fees. Per-unit expenses are less common on Exam LTAM but appear occasionally.

First-Year vs. Renewal Expenses

Expenses vary significantly between the first policy year and subsequent (renewal) years. First-year expenses are substantially higher due to acquisition costs: agent commissions for selling the policy, underwriting expenses for evaluating the risk, medical exam costs, and policy issue expenses. These front-loaded costs create "new business strain," where the insurer invests heavily in the first year and recovers those costs gradually through renewal premiums.

A typical expense structure might be: first-year commission of 55% of premium, renewal commission of 5% of premium, first-year per-policy expense of 200, renewal per-policy expense of 50, and premium tax of 2% of premium in all years. The large gap between first-year and renewal expenses has important implications for reserves, as we will see.

The Gross Premium Equivalence Equation

The equivalence principle for gross premiums sets the APV of gross premium income equal to the APV of all outflows (benefits plus expenses): G * a-ddot_{x:n} = APV(Benefits) + APV(Expenses). The challenge is that some expenses depend on G itself, creating an equation that must be solved for G.

For a concrete setup, suppose the expenses are: first-year commission c_1 * G, renewal commission c_r * G (years 2 through n), per-policy expense e_1 in year 1 and e_r in renewal years, premium tax f * G in all years, and settlement expense E_s at the time of claim. The APV of expenses is:

APV(Expenses) = c_1*G + c_r*G*(a-ddot_{x:n} - 1) + e_1 + e_r*(a-ddot_{x:n} - 1) + f*G*a-ddot_{x:n} + E_s*A_{x:n}.

Grouping terms involving G on the left side: G*a-ddot_{x:n} - c_1*G - c_r*G*(a-ddot_{x:n}-1) - f*G*a-ddot_{x:n} = A_{x:n} + e_1 + e_r*(a-ddot_{x:n}-1) + E_s*A_{x:n}.

Factoring G: G*[a-ddot_{x:n}*(1-f) - c_1 - c_r*(a-ddot_{x:n}-1)] = (1+E_s)*A_{x:n} + e_1 + e_r*(a-ddot_{x:n}-1).

Solving for G gives the gross premium. The denominator simplifies the premium-dependent expense terms, while the numerator collects the fixed expenses and benefit costs.

Worked Example

Consider a 20-year endowment insurance for (40) with face amount 100,000. Given: A_{40:20} = 0.32, a-ddot_{40:20} = 13.25, i = 5%. Expenses: first-year commission 50% of G, renewal commission 5% of G, premium tax 2.5% of G all years, per-policy expense 300 in year 1 and 50 in renewal years, settlement expense 100 at claim.

APV of benefits = 100,000 * 0.32 = 32,000. Adding settlement expense: (100,000 + 100) * 0.32 = 32,032. APV of per-policy expenses = 300 + 50*(13.25 - 1) = 300 + 612.50 = 912.50.

The premium-dependent terms on the denominator side: a-ddot*(1 - 0.025) - 0.50 - 0.05*(a-ddot - 1) = 13.25*0.975 - 0.50 - 0.05*12.25 = 12.919 - 0.50 - 0.6125 = 11.806.

Therefore G = (32,032 + 912.50) / 11.806 = 32,944.50 / 11.806 = 2,790 per year. This gross premium of 2,790 significantly exceeds the net premium P_{40:20} = 32,000/13.25 = 2,415, with the difference funding expenses.

Expense Reserves and Modified Reserves

When gross premiums are used to compute reserves (gross premium reserves), the reserve reflects both future benefit obligations and future expense obligations, offset by future gross premium income. The gross premium reserve at time t is: t_V^g = APV(future benefits at t) + APV(future expenses at t) - G * a-ddot_{x+t:n-t}.

Because first-year expenses are high but spread over all years through level premiums, the gross premium reserve can be negative in early years. A negative reserve means the insurer has not yet recovered its acquisition costs. This "new business strain" is a fundamental feature of life insurance economics and is the motivation for modified reserve methods like the Full Preliminary Term method.

Common Pitfalls

The most common error is mishandling the first-year vs. renewal expense split. Remember that the first payment of the annuity-due is at time 0, so the first-year expense applies to the payment at time 0, and renewal expenses apply to payments at times 1 through n-1. The APV of renewal-year premiums is G*(a-ddot_{x:n} - 1), not G*a-ddot_{x:n}. Similarly, per-policy renewal expenses have APV e_r*(a-ddot_{x:n} - 1).

Another common mistake is forgetting to include the settlement expense in the benefit APV, or adding percentage-of-premium expenses to both sides of the equation instead of collecting them on the G side. Careful algebraic organization prevents these errors.

Study Tips

Exam LTAM problems specify various expense structures and require solving for G. Practice setting up the equivalence equation methodically: first write out all expense components, then separate G-dependent and G-independent terms, and finally solve for G. Use a consistent approach every time to avoid errors under exam pressure. Double-check your answer by verifying that G * a-ddot_{x:n} approximately equals the benefit APV plus a reasonable expense loading.