Continuous Life Annuity¶

Definition¶

A continuous life annuity pays at rate one per year while the annuitant is alive. Its present value is $Y=\int_0^{T_x}\mathrm{e}^{-\delta t}\,\mathrm{d}t$ $\int_{0}^{T_{x}} e^{- δ t} d t$ , and its actuarial present value (APV) is $\bar a_x=\mathbb{E}[Y]$ $E [Y]$ . A temporary version stops at $\min(T_x,n)$ instead.

Interpretation¶

Continuous payments provide a tractable idealization of a frequent payment stream. Payment rate one means an amount $\mathrm{d}t$ over an interval of length $\mathrm{d}t$ , not a unit payment at every instant.

Decaying curve 100 times exp(-0.05t) for the survival-weighted, discounted benefit rate of a continuous life annuity, shown to time 100; the full area to infinity gives APV 2000.

The figure uses the Examples values $\lambda=0.02$ and $\delta=0.03$ $0.03$ . At time $t$ the benefit rate 100 is discounted by $v^t=\mathrm{e}^{-0.03t}$ and is paid only if the annuitant is alive, which has probability ${}_tp_x=\mathrm{e}^{-0.02t}$ $e^{- 0.02 t}$ , so the expected discounted rate is $100\mathrm{e}^{-0.05t}$ . The full integral to infinity is $100\bar a_x=100/(\lambda+\delta)=2000$ $2000$ . The visible shading covers only $0\le t\le100$ $100$ , contributing $2000(1-\mathrm{e}^{-5})\approx1986.52$ $1986.52$ ; the omitted tail contributes $2000\mathrm{e}^{-5}\approx13.48$ $13.48$ .

Change $\lambda$ and $\delta$ : the APV is the area under $100v^t{}_tp_x$ , $100/(\lambda+\delta)=2000$ at the defaults, and $\bar A_x+\delta\bar a_x=1$ $1$ . The left panel uses benefit rate $100$ ; the right panel shows unit-rate present value $Y$ . Drag the probe to see it rise with the lifetime: $Y=(1-v^{T_x})/\delta$ for $\delta\ne0$ $0$ , and $Y=T_x$ at $\delta=0$ $0$ . The displayed window is finite; compare its shown contribution with the full lifetime value and omitted tail in the readout.

The whole life annuity-due values annual survival-contingent payments by a sum. Here a payment rate replaces those individual payments, so survival-weighted discount factors are integrated over time. The force of mortality determines ${}_tp_x$ ; a constant force $\lambda$ gives $\mathrm{e}^{-\lambda t}$ , producing the closed form below.

Key Results¶

$\bar a_x=\int_0^\infty v^t{}_tp_x\,\mathrm{d}t$ and $\bar a_{x:\actuarialterm{n}}=\int_0^n v^t{}_tp_x\,\mathrm{d}t$ $\int_{0}^{n} v^{t}_{t} p_{x} d t$ .
For $\delta\ne0$ $0$ , $Y=(1-v^{T_x})/\delta$ and $\bar A_x+\delta\bar a_x=1$ $1$ .
At constant force of mortality $\lambda>0$ and force of interest $\delta\ge0$ $0$ , $\bar a_x=1/(\lambda+\delta)$ $δ)$ .
At zero interest, $Y=T_x$ and the APV is the complete expectation of life.

Examples¶

At $\lambda=0.02$ and $\delta=0.03$ $0.03$ , a continuous benefit rate of 100 per year has APV $100/0.05=2000$ $2000$ .

Prerequisites¶

Sources¶

Life Insurance · Lecture 5

Aliases: life annuity payable continuously, continuous temporary annuity, continuous whole life annuity, continuously payable life annuity Keywords: a bar x, A bar x + delta a bar x = 1, 1/(lambda + delta) at constant force, Y = (1 - v^T)/delta, area under the discounted survival curve Symbols: delta, infinity, integral, lambda

Last updated: 27 October 2026 11:52