When the death benefit equals the policy value, which technique helps compute the premium P?

• A. Solve a quadratic equation
• B. Use the recursive formula to derive two expressions for P
• C. Use a numerical root-finding method
• D. Use direct substitution of P into a closed form

Answer: (B)

The happening idea is to use how the policy value evolves over time, with a fixed premium, to solve for the premium directly. When the death benefit equals the policy value, you can write a recurrence that expresses the current policy value in terms of the next period’s value and the premium. That recurrence lets you generate an expression for P in two different ways: once by looking at the value progression forward from today, and again by enforcing the boundary condition at the moment death occurs (or by using the initial boundary where the value must equal the death benefit). Since both expressions come from the same dynamic model of the policy, they must agree, so you can set them equal and solve for P algebraically. This approach avoids guessing or iterative methods and provides a clean, direct calculation of the premium.

The other techniques are less natural here: solving a quadratic would require the relationship to be explicitly quadratic in P, which isn’t guaranteed by the setup; numerical root-finding would be unnecessary once you have two consistent expressions for P; and substituting into a closed form presumes a closed-form for P already exists, which the recursive derivation typically provides.