Profit Testing for Life Insurance Products
Conduct profit tests for life insurance products using cash flow analysis for Exam LTAM.
Profit Vector and Profit Signature
Profit testing projects a policy's expected cash flows year by year. The profit vector Pr_t is the expected profit emerging in year t per policy in force at the start of year t. It includes premium income, investment income on reserves, less benefit payments, expense outgo, and the increase in reserves. The profit signature pi_t = Pr_t * product of (1 minus q_{x+k-1}^(tau)) for k=1 to t-1 converts per-in-force profits to per-policy-issued profits by multiplying by the probability of surviving to the start of year t in the multiple decrement environment.
To understand the difference between the profit vector and the profit signature, consider a 3-year term insurance policy. The profit vector measures profit per survivor at the start of each year. The profit signature adjusts these values by the probability that the policy is still in force, giving the expected profit contribution per policy originally issued. The profit signature is what matters for assessing overall product profitability because it accounts for the attrition of policies over time.
Components of the Profit Test
Each year's profit vector entry combines several cash flow components. Premium income G_t is the gross premium collected at the start of year t. Investment income is earned on the reserve held at the start of the year plus the premium received minus expenses paid at policy inception: (tV + G_t - e_t) * i, where i is the earned rate. The expected death benefit outgo is q_{x+t-1}^d * b_{t}, where b_t is the death benefit. Expected surrender outgo is q_{x+t-1}^w * CV_t, where CV_t is the cash value paid on withdrawal. The increase in reserve, weighted by survival, is p_{x+t-1}^(tau) * (t+1)V.
Putting these together, the profit vector for year t is: Pr_t = G_t - e_t + (tV + G_t - e_t) * i - q_{x+t-1}^d * b_t - q_{x+t-1}^w * CV_t - p_{x+t-1}^(tau) * (t+1)V + tV. This formula can also be written as Pr_t = (tV + G_t - e_t)(1+i) - q^d * b - q^w * CV - p^(tau) * (t+1)V, which shows that the opening fund accumulates with interest and then pays for mortality, withdrawals, and the continuing reserve.
Profitability Measures
The net present value (NPV) discounts the profit signature at the risk discount rate r: NPV = sum of pi_t * v_r^t where v_r = 1/(1+r). A positive NPV means the product adds value for shareholders at the chosen hurdle rate. The internal rate of return (IRR) is the discount rate that sets the NPV to zero. The IRR provides a single rate-of-return measure that can be compared to the company's cost of capital.
The discounted payback period is the earliest time T such that the cumulative sum of pi_1*v_r + pi_2*v_r^2 + ... + pi_T*v_r^T turns positive. This measures how quickly the product recovers its initial investment. A shorter payback period indicates lower risk because capital is tied up for less time. The profit margin is the NPV divided by the APV of premiums, expressing profitability as a percentage of revenue.
Constructing a Profit Test Table
A systematic approach to profit testing involves building a table with one row per policy year. The columns typically include: year t, premium, expenses, opening reserve, interest earned, expected death claims, expected surrender payments, closing reserve contribution, profit vector Pr_t, survival probability to year t, and profit signature pi_t. Working through this table row by row ensures that all cash flow components are captured and properly timed.
For example, consider a 3-year endowment insurance issued to age 60 with annual premium 1,000, death benefit 10,000, maturity benefit 10,000, and first-year expenses of 400 with renewal expenses of 50. If the reserves are 0V = 0, 1V = 3,100, 2V = 6,400, and the interest rate is 5%, you would compute each year's profit vector by accumulating the opening fund with interest and subtracting the expected outflows. The profit signature then multiplies each Pr_t by the probability of the policy being in force at the start of that year.
Setting Premiums Using Profit Tests
One powerful application of profit testing is determining the premium that achieves a target profitability level. If the company requires an NPV of zero at a risk discount rate of 12%, you can set up the profit test with the premium as an unknown variable and solve for the premium that satisfies this constraint. This approach is more flexible than traditional equivalence principle pricing because it can incorporate realistic expense patterns, withdrawal assumptions, and multiple decrement rates.
The target can also be expressed as a target profit margin (for example, NPV equal to 5% of APV of premiums) or a target IRR. Iterative methods or spreadsheet solver tools are typically used to find the premium that meets the specified target. This method is widely used in product development to ensure that new products meet shareholder return requirements.
Common Pitfalls
A frequent mistake is confusing the profit vector with the profit signature. The profit vector is per policy in force at the start of the year, while the profit signature is per policy issued at time zero. Another common error involves the timing of cash flows: premiums and expenses at the start of the year, death benefits at the end. Misplacing these in the accumulation formula leads to incorrect profit calculations.
Students also sometimes forget to include the reserve release on death or withdrawal. When a policyholder dies or withdraws, the reserve held for that policyholder is no longer needed, which creates a gain (or loss) depending on the relationship between the reserve and the benefit paid. Make sure your profit vector formula properly accounts for all decrements and their financial effects.
Study Tips
Exam LTAM frequently presents profit testing problems that require constructing a full profit test table and computing NPV or IRR. Practice building these tables from scratch, paying careful attention to the timing of each cash flow. Memorize the profit vector formula and understand each component. Be comfortable computing the profit signature from the profit vector using survival probabilities. Work through problems that ask you to find the premium achieving a specified profitability target, as this integrates all the concepts into a single calculation.