Term Insurance and Endowment Profit-Test Comparison

Why Compare Term and Endowment Profit Tests?

Term insurance and endowment insurance can share the same insured population, death benefit, premium timing, expenses, and mortality assumptions. Yet their profit vectors and reserves can differ substantially: an endowment also promises a maturity benefit if the insured is alive at the end of the term. To compare the contracts correctly, use the same state transitions and cash-flow timing for both, then identify the effect of that additional benefit.

The hypothetical contracts below have two living states, healthy and disabled, and an absorbing dead state. This simple multi-state structure captures an important point: the chance of death during the next year depends on the insured's current state. The example uses annual transitions, so it does not require continuous-time transition equations.

Assume the same transition probabilities in each year. From H, the probabilities of moving to H, D, and X are 0.8, 0.1, and 0.1. From D, the corresponding probabilities are 0.2, 0.6, and 0.2. State X is absorbing. Thus, a healthy insured has a 0.1 probability of dying during the next year, while a disabled insured has a 0.2 probability. We use an effective annual interest rate of 5%, an annual gross premium of $15, expenses of $2 in H and $3 in D, a death benefit of $100, and, for the endowment, a maturity benefit of $100. There are no lapses, taxes, or expenses on a policy anniversary after death.

Reserves and Transition-Contingent Profits

Let \(V_t(s)\) be the prospective net reserve at time \(t\) in living state \(s\), immediately before the premium and expense. Let \(G\) denote the annual premium, \(e_s\) the expense in state \(s\), and \(b_{sr,t+1}\) the benefit paid at the end of the year if the transition from state \(s\) to state \(r\) occurs. For a two-year contract, \(V_2(s)=0\) after all benefits at maturity have been paid. The prospective reserve is the present value of future benefits and expenses less future premiums, conditional on the current state.

Here \(p_{sr}\) is the one-year transition probability from \(s\) to \(r\). In the sum, a death transition has no continuing reserve after the claim, so its \(V_{t+1}(X)\) is zero. For the endowment's final year, \(b_{sr,2}=100\) on every transition out of a living state: the benefit is paid on death or on survival to maturity. For term insurance, the final-year benefit is paid only on a transition to death.

The profit on a particular transition is the accumulated opening reserve, premium, and expense cash flow, less any benefit paid and the reserve needed for the next duration. Under this convention, a positive expense is an outflow.

For a death transition, take the next reserve as zero. For a living transition in the final year, the maturity benefit is included in \(b_{sr,2}\), and the post-payment reserve is also zero. This convention keeps the benefit and reserve timing consistent.

Worked Example: Calculate Both Reserve Bases

Why the Expected Profit Is Zero on the Reserve Basis

The reserve recurrence is not merely a valuation formula: it also explains why expected profit for the next year is zero when the reserve assumptions, transition assumptions, and cash flows all agree. Starting from the profit on each transition, take the conditional expectation given the current state \(s\):

The second equality follows by rearranging the reserve recurrence and using \(1/v=1+i\). The result is conditional on the specified basis; it does not say that every realized transition produces zero profit. Death and survival can produce very different profits, even when their probability-weighted average is zero.

NPV Comparison on the Same Assumptions

The issue-date net present value (NPV) of expected profit is the present value of premiums less expenses and benefits. On the same basis used for the prospective reserve, it equals \(-V_0(H)\). A positive NPV therefore corresponds to a negative net reserve at issue, and vice versa. This equality requires consistent timing and assumptions; an NPV based on a different experience basis need not equal the negative of a valuation reserve.

Isolating the Effect of the Maturity Benefit

Because the contracts have the same death benefit, premiums, expenses, and transition assumptions, their reserve difference isolates the maturity benefit. At issue, the probability of surviving to time 2 is 0.8. The present value of the $100 maturity benefit, conditional on issue, is consequently \(v^2(0.8)(100)\). This is also the difference between the issue-date endowment and term reserves:

The same comparison depends on the state at time 1. From H, the chance of surviving the final year is \(0.9\), so the time-1 reserve difference is \(v(0.9)(100)=85.714286\). From D, the chance is \(0.8\), giving a difference of \(v(0.8)(100)=76.190476\). The endowment thus has a larger continuation liability from H than from D, despite its higher death probability in D: survival, not death, triggers the additional maturity benefit.

Common Traps in the Comparison

• Using one death probability for every living state. The model assigns a 0.1 annual death probability in H and 0.2 in D. The state at the anniversary determines which transition row applies.
• Using the first-year survival probability as the maturity probability. The probability of surviving the first year is 0.9, but the probability of reaching the end of year 2 alive is 0.8. The maturity benefit's expected value at time 2 is $80, not $90.
• Discounting an end-of-year benefit to the wrong time. First-year death benefits are paid at time 1 and receive one year of discounting from issue; second-year benefits are paid at time 2 and receive two years.
• Mixing reserve timing conventions. These reserves are immediately before the premium and expense. A reserve defined just after those cash flows would differ by the net amount \(G-e_s\).
• Treating zero expected profit as zero profit on every transition. The profit vectors show large differences by outcome. Only their conditional probability-weighted means are zero on this matching basis.