Gross Premium Calculation: Expenses and Profit Loading

Expense Types

Gross premiums must cover benefits, expenses, and profit. Expenses fall into three categories: percentage of premium expenses (commissions, premium taxes), per-policy expenses (administration costs), and per-unit expenses (proportional to face amount). First-year expenses are typically higher due to acquisition costs (agent commissions, underwriting, policy issue). Renewal expenses cover ongoing administration. The timing of expenses matters: some occur at policy inception, others throughout the premium-paying period.

Gross Premium Formula

The equivalence principle for gross premiums sets the APV of gross premiums equal to the APV of benefits plus the APV of all expenses. If G is the gross premium, e is the per-policy expense, c_1 is the first-year commission rate, c_r is the renewal commission rate, and f is the fraction-of-premium expense, then G * a-ddot_{x:n} = A_{x:n} + e * a-ddot_{x:n} + (c_1 - c_r)*1 + c_r * G * a-ddot_{x:n} + f * G * a-ddot_{x:n}. Solving for G gives the expense-loaded gross premium. Exam LTAM problems specify various expense structures and require solving for G.