Actuarial Present Value of Life Insurance Benefits

The Concept of Actuarial Present Value

The actuarial present value (APV) is the expected present value of a future contingent payment, where the contingency is typically survival or death of an insured life. Unlike a certain payment that can be discounted using standard financial mathematics, a contingent payment requires accounting for both the time value of money and the probability that the payment will actually be made. The APV combines these two elements into a single number that represents the fair value of the contingent obligation today.

For a life insurance benefit of amount b payable at some future time that depends on when the insured dies, the present value is a random variable because the payment timing is uncertain. If death occurs at time T_x (the future lifetime of the insured), the present value of the benefit is b * v^{T_x}, where v = 1/(1+i) is the discount factor. The APV is the expected value of this random variable: E[b * v^{T_x}].

Continuous Insurance APV

In the continuous model, the benefit is paid at the exact moment of death. The present value random variable for a unit benefit is Z = v^{T_x} = exp(-delta * T_x), where delta = ln(1+i) is the force of interest. The APV in continuous form is:

A-bar_x = E[v^{T_x}] = integral from 0 to infinity of v^t * f_{T_x}(t) dt = integral from 0 to infinity of v^t * t_p_x * mu_{x+t} dt.

This integral weights each possible death time t by the discount factor v^t (time value of money) and the density t_p_x * mu_{x+t} (probability of dying at exactly time t). The integral sums these weighted contributions over all possible death times from 0 to infinity.

Under a constant force of mortality mu, the integral simplifies beautifully: A-bar_x = integral from 0 to infinity of exp(-delta*t) * exp(-mu*t) * mu dt = mu * integral from 0 to infinity of exp(-(mu+delta)*t) dt = mu/(mu+delta). This formula shows that the APV increases with mortality (higher mu means earlier expected payment, less discounting) and decreases with the interest rate (higher delta means more discounting).

Discrete Insurance APV

In the discrete (annual) model, the benefit is paid at the end of the year of death. The present value random variable is Z = v^{K_x+1}, where K_x is the curtate future lifetime (the number of complete years survived). The APV is:

A_x = E[v^{K_x+1}] = sum from k=0 to infinity of v^{k+1} * P(K_x = k) = sum from k=0 to infinity of v^{k+1} * k_p_x * q_{x+k}.

Each term in the sum represents the contribution from death in year k+1: the discount factor v^{k+1} (payment at end of year k+1) multiplied by the probability of dying in that year (surviving k years then dying within the next year). In practice, the sum terminates at the limiting age omega of the mortality table.

For a life table with l_{65} = 85000, q_{65} = 0.015, q_{66} = 0.017, q_{67} = 0.019, and i = 5%, the first three terms of A_{65} are: v * q_{65} + v^2 * p_{65} * q_{66} + v^3 * 2_p_{65} * q_{67} = (1/1.05)*0.015 + (1/1.05)^2*0.985*0.017 + (1/1.05)^3*0.985*0.983*0.019 = 0.01429 + 0.01513 + 0.01590 = 0.04532 for just these three years.

Actuarial Notation System

Standard actuarial notation uses the letter A with various decorations to denote different types of insurance APVs. The basic symbol A_x represents a discrete whole life insurance. A bar over A (written A-bar_x) indicates a continuous insurance where the benefit is paid at the moment of death. A superscript (1) as in A_{x:n}^1 denotes a term insurance (paying only if death occurs within n years). The symbol A_{x:n} without a superscript denotes an endowment insurance (paying on death within n years or on survival to n years).

The 1/m-thly insurance pays at the end of the 1/m-th of a year of death. For example, a monthly insurance (m=12) pays at the end of the month of death. The APV is denoted A_x^{(m)}. The relationship between continuous, annual, and 1/m-thly models is: A_x^{(m)} is approximately (i/i^{(m)}) * A_x, where i^{(m)} = m*((1+i)^{1/m} - 1). As m approaches infinity, A_x^{(m)} approaches A-bar_x, and i^{(m)} approaches delta.

Second Moments and Variance

The variance of the present value random variable Z measures the risk in an insurance contract. For continuous insurance: Var(Z) = E[v^{2T_x}] - (E[v^{T_x}])^2 = ^2A-bar_x - (A-bar_x)^2. The notation ^2A-bar_x denotes the second moment, which is computed using the same integral formula as A-bar_x but with the force of interest doubled: ^2A-bar_x is A-bar_x evaluated at force 2*delta (or equivalently at interest rate (1+i)^2 - 1).

This "rule of double the force" is one of the most useful computational shortcuts in life contingencies. To compute the second moment of any insurance APV, simply recompute the APV at twice the force of interest (which corresponds to squaring the one-year discount factor: v^2 = (v^2)^1). For example, under constant force mu: ^2A-bar_x = mu/(mu + 2*delta), and Var(Z) = mu/(mu+2*delta) - (mu/(mu+delta))^2.

Variable Benefit Insurance

Insurance benefits need not be constant. An increasing whole life insurance pays benefit k+1 if death occurs in year k+1 (for the discrete case). The APV is (IA)_x = sum from k=0 to infinity of (k+1)*v^{k+1}*k_p_x*q_{x+k}. A decreasing n-year term insurance pays n-k if death occurs in year k+1: (DA)_{x:n}^1 = sum from k=0 to n-1 of (n-k)*v^{k+1}*k_p_x*q_{x+k}.

The relationship (IA)_{x:n}^1 + (DA)_{x:n}^1 = (n+1)*A_{x:n}^1 provides a useful shortcut when one of the three quantities is known. These variable benefit formulas are used to model policies with changing face amounts and to derive reserves for policies with level premiums.

Common Pitfalls

Students often confuse the timing of the benefit payment. In the discrete model, the benefit is paid at the end of the year of death (time k+1 for death in year k+1), not at the beginning. This means the discount factor is v^{k+1}, not v^k. Another common error is forgetting to double the force of interest (not the interest rate) when computing second moments. Doubling i instead of delta produces incorrect results because exp(-2*delta) = v^2, not (1+2i)^{-1}.

When converting between continuous and discrete models, remember the approximation A-bar_x is approximately (i/delta) * A_x under UDD. This ratio i/delta is greater than 1, reflecting that the continuous insurance pays slightly earlier on average (at the moment of death rather than end of year), making it more valuable.

Study Tips

Exam LTAM requires you to compute APVs from life tables, from parametric mortality laws, and from given relationships between insurance and annuity values. Practice all three approaches. Memorize the constant force formulas as they provide quick checks on more complex calculations. Build fluency with the notation system so you can read problems quickly. When a problem gives you A_x and asks for ^2A_x, immediately think "evaluate at 2*delta" rather than trying to derive the second moment from scratch.