Term Life Insurance: Pricing and Reserving for Exam LTAM

Term Insurance Fundamentals

An n-year term life insurance pays a death benefit only if the insured dies within the n-year term. If the insured survives to the end of the term, no benefit is paid and the contract expires worthless. Term insurance provides pure death protection without any savings component, making it the simplest and most affordable form of life insurance coverage.

From an actuarial perspective, term insurance involves a contingent payment that may or may not be made. The present value random variable for a continuous n-year term insurance with unit benefit is Z = v^{T_x} if T_x <= n and Z = 0 if T_x > n. The probability of a payment being made equals n_q_x (the n-year mortality probability), and the expected timing and discounting of that payment determine the APV.

Computing the Term Insurance APV

The continuous APV is A-bar_{x:n}^1 = integral from 0 to n of v^t * t_p_x * mu_{x+t} dt. The integration limit is n (not infinity as in whole life) because benefits are only paid for deaths within the term. The discrete APV is A_{x:n}^1 = sum from k=0 to n-1 of v^{k+1} * k_p_x * q_{x+k}. Each term represents the discounted probability of dying in year k+1 (surviving k years then dying within the next year).

A powerful decomposition connects term insurance to whole life: A_x = A_{x:n}^1 + n_E_x * A_{x+n}, where n_E_x = v^n * n_p_x is the pure endowment factor. This says that whole life insurance is the sum of n-year term insurance (covering death within n years) plus deferred whole life insurance starting at age x+n (covering death after n years). Rearranging gives A_{x:n}^1 = A_x - n_E_x * A_{x+n}, which is often the most efficient way to compute a term APV when A_x and A_{x+n} are available from tables.

For example, suppose A_{40} = 0.1214, A_{60} = 0.3690, and 20_E_{40} = v^{20} * 20_p_{40} = 0.3120 at i = 5%. Then A_{40:20}^1 = 0.1214 - 0.3120 * 0.3690 = 0.1214 - 0.1151 = 0.0063. This relatively small APV reflects the low mortality risk for ages 40 to 60.

Net Premium for Term Insurance

The net premium for an n-year term insurance with level annual premiums is determined by the equivalence principle: the APV of premiums equals the APV of benefits at policy inception. If premiums are paid as an annuity-due for n years, P_{x:n}^1 = A_{x:n}^1 / a-ddot_{x:n}, where a-ddot_{x:n} is the n-year temporary life annuity-due.

Using the example above, if a-ddot_{40:20} = 13.25, then P_{40:20}^1 = 0.0063 / 13.25 = 0.000475 per unit of insurance. For a policy with a death benefit of 500,000, the annual net premium would be 500,000 * 0.000475 = 237.50. This low premium reflects the high probability of the insured surviving the 20-year term.

The loss-at-issue random variable for the insured is L_0 = v^{K+1} - P * a-ddot_{K+1} if K < n, and L_0 = -P * a-ddot_{n} if K >= n. Setting E[L_0] = 0 yields the equivalence principle premium. The variance of L_0 is needed for computing probabilities of loss and for understanding the risk borne by the insurer on a single policy.

Term Insurance Reserves

The prospective net premium reserve at time t (for 0 < t < n) is the expected present value of future benefits minus the expected present value of future premiums, given survival to time t: t_V = A_{x+t:n-t}^1 - P_{x:n}^1 * a-ddot_{x+t:n-t}. This reserve represents the amount the insurer should hold at time t to meet future obligations on the policy.

Term insurance reserves have a distinctive pattern. They start at zero (by the equivalence principle), increase during the early policy years as the insured ages and mortality risk increases, reach a peak somewhere during the middle of the term, and then decrease back toward zero as the term approaches expiration. At duration n, the reserve is exactly zero because no future benefit is payable and no future premiums are due.

This pattern contrasts sharply with whole life and endowment insurance reserves, which increase monotonically over time. The term reserve decreases late in the term because the insured has survived most of the risk period and few premium payments remain. The peak reserve is typically small relative to the face amount, which reflects the low overall probability of claim on a term policy.

Recursion for Term Reserves

The reserve recursion for term insurance is (t_V + P) * (1+i) = q_{x+t} * 1 + p_{x+t} * (t+1)_V for 0 <= t < n-1. At the final year, (n-1_V + P) * (1+i) = q_{x+n-1} * 1 + p_{x+n-1} * 0, since the reserve at time n is zero. This recursion states that the reserve plus premium, accumulated with interest, funds the expected death benefit and the next reserve.

Working backward from n_V = 0, you can compute the reserve at each prior duration. This is particularly useful for building a complete reserve schedule. For n-1: (n-1)_V = (q_{x+n-1} * v - P) / (1 + i... actually, rearranging: (n-1)_V = v * q_{x+n-1} - P * (1 - v * p_{x+n-1}) is more complex. The direct prospective formula is generally easier for individual duration calculations.

Comparing Term and Whole Life

Understanding the differences between term and whole life insurance is important for Exam LTAM. Term insurance has a smaller APV (A_{x:n}^1 < A_x) because it only covers a portion of the mortality risk. Term premiums are lower because the probability of paying a claim is lower. Term reserves are smaller and non-monotonic, while whole life reserves increase steadily toward the face amount.

The relationship A_x = A_{x:n}^1 + n_E_x * A_{x+n} shows exactly how term insurance relates to whole life. The pure endowment factor n_E_x represents the probability-weighted discount factor for deferring insurance from age x to age x+n. When n is small or when mortality between ages x and x+n is low, A_{x:n}^1 is a small fraction of A_x.

Common Pitfalls

Students sometimes forget that the term insurance superscript (1) in A_{x:n}^1 indicates that only the death benefit is contingent on the term, distinguishing it from endowment insurance A_{x:n} (which also pays on survival). Confusing these two symbols leads to fundamentally different calculations. Another common error is computing the reserve at the wrong duration or using the wrong remaining term. At duration t, the remaining term is n-t, and the attained age is x+t.

Study Tips

Exam LTAM frequently tests term insurance in combination with other products through decomposition formulas. Practice using A_x = A_{x:n}^1 + n_E_x * A_{x+n} in both directions: computing term APVs from whole life values and computing whole life values from term and deferred components. Build a reserve table for a sample term policy to develop intuition for the reserve pattern. Know the net premium formula and be able to compute it quickly from given annuity and insurance values.