Actuarial Present Value¶

Definition¶

The actuarial present value (APV) of an insurance cash flow is the expectation of its random present value $Z$ : $\operatorname{APV}=\mathbb{E}[Z]$ $E [Z]$ , when integrable. For a benefit $b(T_x)$ paid at death under constant force of interest $\delta$ , $Z=b(T_x)\mathrm{e}^{-\delta T_x}$ $b (T_{x}) e^{- δ T_{x}}$ . Define $v=(1+i)^{-1}$ and $\delta=\log(1+i)$ for annual effective interest $i>-1$ $- 1$ .

Interpretation¶

The payment timing is uncertain even when the benefit amount is fixed. Discount each possible payment first, then average over the mortality model.

Timeline from 0 to 5 with a payment of 100 at time 5 made only if the life survives to time 5, carried back to time 0 by discounting and survival, giving an APV of about 73.97.

The figure uses the Examples case. The payment of 100 at time 5 is made only if $T_x>5$ $5$ ; carrying it back to time 0 multiplies it by the discount factor $v^5=(1.04)^{-5}$ and by the survival probability ${}_5p_x=0.9$ $0.9$ . The result, $100(1.04)^{-5}(0.9)\approx73.97$ $73.97$ , is the expectation of a present value that equals $100v^5$ with probability $0.9$ and $0$ otherwise.

The future lifetime $T_x$ determines when a death benefit is paid. Its density weights the discounted benefit in the integral below; for a survival payment, the event $T_x>t_k$ determines whether that payment occurs.

Key Results¶

A nonnegative benefit paid at death has APV $\int_0^\infty b(t)v^tf_x(t)\,\mathrm{d}t$ .
For payments $b_k$ at fixed times $t_k$ conditional on survival, APV is $\displaystyle \sum_k b_kv^{t_k}{}_{t_k}p_x$ .
Scaling benefits by $B$ scales APV by $B$ and variance by $B^2$ .
A net single premium covers expected discounted benefits under the chosen basis; expenses and risk loading require additional treatment.

Examples¶

A payment of 100 at time 5 with survival probability $0.9$ and $i=0.04$ has APV $100(1.04)^{-5}(0.9)\approx73.97$ $73.97$ .

Prerequisites¶

Sources¶

Life Insurance · Lecture 3

Aliases: APV, expected present value, net single premium, EPV, actuarial value, expected discounted value Keywords: expected discounted benefit, E[Z] of the random present value, discount then average over survival, discount factor v = 1/(1+i), force of interest delta = log(1+i) Symbols: delta, infinity, integral, sum

Last updated: 17 October 2026 21:53