Life Insurance Valuation Recursions¶

Proposition¶

For annual unit whole life insurance with actuarial present value (APV) $A_x$ , conditioning on the first year gives the valuation recursion

A_x=vq_x+vp_xA_{x+1},

where $v$ is the annual discount factor and $p_x=1-q_x$ $q_{x}$ . The first term represents death during the year; the second represents survival followed by the remaining contract.

Interpretation¶

A long contract can be valued one year at a time. This also explains why product decompositions have both financial and survival discount factors.

$One-year tree from age x: death with probability q_x = 0.02 pays 1 at time 1, survival with p_x = 0.98 leaves a contract worth A_{x+1} = 0.4, both discounted by v = 0.95.$

The figure follows the Examples. From age $x$ the life either dies within the year with probability $q_x=0.02$ $0.02$ , and the benefit 1 is paid at time 1, or survives with probability $p_x=0.98$ and then holds a contract worth $A_{x+1}=0.4$ at time 1. Weighting the two time-1 values by their probabilities and discounting by $v=0.95$ gives $A_x=0.95(0.02\times1+0.98\times0.4)=0.3914$ $0.3914$ .

For endowment insurance, survival through the first year leaves an endowment contract with one fewer year. At zero remaining term the survival payment is due immediately, explaining the terminal value $A_{x:\actuarialterm{0}}=1$ $1$ .

Key Results¶

Let $A_{x:\actuarialterm{n}}$ denote annual-settlement unit $n$ -year endowment APV. Then $A_{x:\actuarialterm{n}}=v[q_x+p_xA_{x+1:\actuarialterm{n-1}}]$ $p_{x} A_{x + 1 : n - 1}]$ , with $A_{x:\actuarialterm{0}}=1$ $1$ .
Annual term insurance uses the same recursion but a terminal value of zero.
Backward numerical valuation requires a terminal age or a justified tail treatment.
The recursion concerns conditional values at successively older ages, not independent new policies.

Examples¶

If $v=0.95$ $0.95$ , $q_x=0.02$ and $A_{x+1}=0.4$ $0.4$ , then $A_x=0.95(0.02+0.98\times0.4)=0.3914$ $0.3914$ .

Prerequisites¶

Whole Life Annuity Due

Sources¶

Life Insurance · Lecture 4

Aliases: one-year insurance recursion, backward insurance valuation, insurance recursion, backward recursion for insurance values Keywords: A x = v q x + v p x A x+1, recursion for endowment insurance, terminal value, conditioning on the first year

Last updated: 14 October 2026 08:21