Varying Life Insurance Benefits¶

Definition¶

A varying death benefit is specified by a deterministic payment rule $b(t)\ge0$ $0$ . For payment at death, $Z=b(T_x)v^{T_x}$ $b (T_{x}) v^{T_{x}}$ ; for annual settlement, $Z=b(K_x+1)v^{K_x+1}$ $1) v^{K_{x} + 1}$ . A compound reversionary bonus increases the insured benefit by a fixed proportion at specified vesting dates (the dates on which each bonus becomes a guaranteed part of the benefit).

Interpretation¶

The mortality model determines whether and when payment occurs; the benefit rule determines its amount. Write the vesting convention explicitly before valuing a bonus.

Timeline with a death benefit of 1 that rises by 2% at each completed anniversary; the life dies at time 3.4 and the benefit 1.02 cubed is paid at time 4.

The figure draws the example: a compound reversionary bonus of 2% vests at each completed anniversary, so the benefit in force is 1 in the first year, $1.02$ in the second and $(1.02)^k$ after $k$ anniversaries. Death at $T_x=3.4$ comes after three anniversaries, so the benefit is $(1.02)^3$ ; with end-of-year settlement it is paid at $K_x+1=4$ $4$ , giving $Z=(1.02)^3v^4$ $(1.02)^{3} v^{4}$ . This is the course formula $Z=v^{K_x+1}(1+j)^{K_x}$ with $j=0.02$ and $K_x=3$ $3$ : the mortality model decides when the payment is made, and the bonus rule decides its size.

Whole life insurance provides the death time and its payment convention. Replacing the unit benefit by $b(T_x)$ , or by $b(K_x+1)$ for annual settlement, weights the same death-payment probabilities by the specified benefit schedule.

Key Results¶

The immediate-settlement actuarial present value (APV) is $\int_0^\infty b(t)v^tf_x(t)\,\mathrm{d}t$ .
Annual APV is $\displaystyle \sum_{k=0}^\infty b(k+1)v^{k+1}{}_kp_xq_{x+k}$ $1) v^{k + 1}_{k} p_{x} q_{x + k}$ .
For annual anniversary bonuses at rate $j\ge0$ $0$ , base benefit 1 and end-of-year death settlement, the course uses $Z=v^{K_x+1}(1+j)^{K_x}$ $j)^{K_{x}}$ .
At-death settlement needs its own rule for partial-year bonuses; do not infer one from annual settlement.

Examples¶

With a 2% bonus vested only on completed anniversaries, death at time 3.4 gives benefit $(1.02)^3$ . If settlement is at the end of the death year, discount this amount by $v^4$ .

Prerequisites¶

Whole Life Insurance

Sources¶

Life Insurance · Lecture 4

Aliases: non-standard death benefits, compound reversionary bonus, varying insurance, increasing whole life insurance, reversionary bonus, variable death benefit Keywords: (IA) x, bonus rate j, b(K+1) v^(K+1), vesting dates Symbols: infinity, integral, sum

Last updated: 13 October 2026 13:48