Endowment Insurance: Pure and Modified for Actuaries
Understand pure endowment and endowment insurance products and their valuation for Exam LTAM.
Pure Endowment
A pure endowment is a contract that pays a benefit of 1 if and only if the insured survives to the end of n years. If the insured dies before the n years are up, nothing is paid. The APV of a pure endowment is n_E_x = v^n * n_p_x, the product of the n-year discount factor and the n-year survival probability. This represents the present value of a certain payment discounted for both the time value of money and the probability that the payment will never be made (due to death before maturity).
Pure endowments are fundamental building blocks in life contingencies. A whole life annuity-due paying 1 at the start of each year while the insured survives can be decomposed as a sum of pure endowments: a-ddot_x = sum from k=0 to infinity of k_E_x = sum from k=0 to infinity of v^k * k_p_x. Similarly, deferred annuities and deferred insurance products are expressed using pure endowments as multipliers that combine discounting with survival conditioning.
Pure endowments also appear prominently in pension mathematics. The projected benefit obligation for a pension plan involves the present value of future pension payments, each weighted by the probability of the employee surviving to retirement and beyond. These probability-weighted discount factors are essentially pure endowments in a multiple-decrement environment.
Properties of Pure Endowments
Several properties of pure endowments are worth noting. First, n_E_x is always between 0 and v^n, since 0 <= n_p_x <= 1. The pure endowment factor decreases with n (longer deferral means more discounting and more mortality risk) and increases with the survival probability n_p_x (healthier lives have higher pure endowment values).
The recursion k_E_x = v * p_x * (k-1)_E_{x+1} allows efficient computation from a life table. Starting from 0_E_x = 1 (surviving zero years is certain), each successive pure endowment is obtained by multiplying by the one-year discount and survival factors.
The present value random variable for a pure endowment is Z = v^n if T_x > n and Z = 0 if T_x <= n. Since Z takes only two values, its variance is straightforward: Var(Z) = v^{2n} * n_p_x * (1 - n_p_x) = (n_E_x)^2 * (1 - n_p_x) / n_p_x. This variance depends on the n_p_x term reflecting uncertainty about whether the benefit will be paid.
Endowment Insurance
An n-year endowment insurance combines term insurance and a pure endowment. It pays 1 upon death within n years (the term insurance component) or upon survival to n years (the pure endowment component), whichever comes first. Since one of these events is certain to occur (the insured either dies within n years or survives), the benefit of 1 is always paid. The only uncertainty is the timing.
The APV is A_{x:n} = A_{x:n}^1 + n_E_x. This decomposition into term insurance plus pure endowment is both conceptually clear and computationally useful. Since the benefit is always paid, A_{x:n} is always greater than either component and always less than 1 (because the benefit is discounted).
For example, with A_{40:20}^1 = 0.0063 and 20_E_{40} = 0.3120 (using earlier values): A_{40:20} = 0.0063 + 0.3120 = 0.3183. The pure endowment component dominates because the mortality risk at ages 40 to 60 is relatively low and most of the APV comes from the survival benefit.
Endowment Insurance Premiums
The net level premium for an n-year endowment insurance with premiums paid over the full n-year term is P_{x:n} = A_{x:n} / a-ddot_{x:n}. Using the identity A_{x:n} = 1 - d * a-ddot_{x:n}, this simplifies to P_{x:n} = 1/a-ddot_{x:n} - d, where d = i/(1+i) is the annual discount rate. This elegant formula shows that the premium depends only on the annuity value and the discount rate.
Continuing the example: if a-ddot_{40:20} = 13.25, then P_{40:20} = 0.3183 / 13.25 = 0.02402 per unit. Alternatively, using the simplified formula with i = 5% so d = 0.04762: P_{40:20} = 1/13.25 - 0.04762 = 0.07547 - 0.04762 = 0.02785. (The small difference is due to rounding in the example values.)
Endowment premiums are higher than term premiums because the benefit is always paid. The premium funds both the death protection and the savings component. Limited-payment endowment insurance, where premiums are paid for h < n years, has even higher premiums: h_P_{x:n} = A_{x:n} / a-ddot_{x:h}.
Endowment Insurance Reserves
Reserves for endowment insurance build substantially over the policy term and reach the face amount at maturity. The prospective reserve at time t is t_V_{x:n} = A_{x+t:n-t} - P_{x:n} * a-ddot_{x+t:n-t}. Using the identity A_{x+t:n-t} = 1 - d*a-ddot_{x+t:n-t}, this becomes t_V_{x:n} = 1 - a-ddot_{x+t:n-t} / a-ddot_{x:n}.
This formula reveals the reserve pattern: as the insured ages, the remaining annuity a-ddot_{x+t:n-t} decreases (fewer remaining premium payments and higher mortality), so the ratio a-ddot_{x+t:n-t} / a-ddot_{x:n} decreases and the reserve increases. At maturity (t = n), a-ddot_{x+n:0} = 0 and t_V = 1, confirming that the reserve equals the face amount.
The reserve pattern for endowment insurance is monotonically increasing, starting at 0_V = 0 and ending at n_V = 1. This contrasts with term insurance where reserves return to zero at expiration. The substantial reserve buildup in endowment insurance reflects the savings component of the product.
Relationship to Other Products
The endowment insurance occupies a central position in the taxonomy of life insurance products. Whole life insurance is the limiting case as n approaches omega - x (the remaining lifetime span). Term insurance is the death benefit component only. The pure endowment is the survival benefit component only. Many premium and reserve formulas for these products are special cases of the endowment formulas.
The identity 1 = d * a-ddot_{x:n} + A_{x:n} connects endowment insurance and temporary annuities through the discount rate. Rearranging gives a-ddot_{x:n} = (1 - A_{x:n})/d, which allows computation of annuity values from insurance values and vice versa. This relationship is one of the most useful identities on Exam LTAM.
Common Pitfalls
Students sometimes confuse A_{x:n} (endowment insurance, which always pays) with A_{x:n}^1 (term insurance, which pays only on death). The notation is critical: the superscript (1) restricts the insurance to the death benefit only. Without the superscript, both death and survival benefits are included. Another common error is forgetting that the reserve at maturity for an endowment equals the face amount, not zero. This is the defining feature that distinguishes endowment reserve patterns from term reserve patterns.
Study Tips
Exam LTAM tests endowment insurance extensively in premium and reserve calculations. Practice computing APVs using the decomposition A_{x:n} = A_{x:n}^1 + n_E_x. Master the simplified premium formula P_{x:n} = 1/a-ddot_{x:n} - d and the reserve formula t_V = 1 - a-ddot_{x+t:n-t}/a-ddot_{x:n}. These formulas avoid the need to compute A values directly and are faster on the exam. Work through complete reserve schedules for sample endowment policies to build intuition for the reserve pattern.