Continuous Whole and Temporary Life Annuities: Practice Problems and Solutions

A life annuity is "a series of payments made continuously or at equal intervals... while a given life survives" (Bowers et. al, p. 133). Life annuities are different from annuities-certain, the subject of Exam 2/FM, in that payment is condition on the life's survival.

A whole life annuity makes payments to the annuitant until death. The present value of payments to be made for a whole life annuity is denoted by the symbol Y = ā_T¬ and has the following distribution function

(c. d. f.):

F_Y(y) = F_T(-ln(1 - δy)/δ) for 0 < y < 1/δ, where δ is the annual force of interest and T is the future-lifetime random variable for the annuitant.

Y = ā_T¬ also has the following probability density function (p. d. f.):

f_Y(y) = f_T(-ln(1 - δy)/δ)/(1 - δy) for 0 < y < 1/δ.

The actuarial present value of a continuous whole life annuity (where payments are made continuously, with a momentary infinitesimal payment of dt made at every time t) is denoted by the symbol E[Y] = ā_x and has the following actuarial present value:

E[Y] = ā_x = ₀^∞∫v^t*_tp_x*dt = ₀^∞∫_tE_x*dt

The symbol _tE_x is another way to express the actuarial present value of a t-year pure endowment, and we recall also that _tE_x = v^t*_tp_x.

The actuarial present value for a continuous n-yeartemporary life annuity that only makes payments at most up to time n (if the annuitant survives until time n) is denoted as ā_x:n¬ and can be found as follows:

ā_x:n¬ = ₀ⁿ∫v^t*_tp_x*dt = ₀ⁿ∫_tE_x*dt

A useful relationship holds between ā_x and ā_x+1.

ā_x = ā_x:1¬ + vp_x*ā_x+1

Source: Bowers, Gerber, et. al. Actuarial Mathematics. 1997. Second Edition. Society of Actuaries: Itasca, Illinois. pp. 133-137.

Original Problems and Solutions from The Actuary's Free Study Guide

Problem S3L37-1. The life of a triceratops has the following survival function associated with it: s(x) = e^-0.34x. The annual force of interest in Triceratopsland is currently 0.06. Charlie the Triceratops has a continuous whole life annuity. What is the distribution function F_Y(y)of the present value payments to be made for such an annuity?

Solution S3L37-1. The lives of triceratopses are exponentially distributed.

Thus, F_T(t) = 1 - e^-0.34t. We use the formula F_Y(y) = F_T(-ln(1 - δy)/δ) for 0 < y < 1/δ, where δ = 0.06. Thus, F_Y(y) = 1 - e^{0.34ln(1 - δy)/δ} = 1 - e^{0.34ln(1 - 0.06y)/0.06} = 1 - e^{(17/3)ln(1 - 0.06y)} =

1 - e^{ln((1 - 0.06y)^(17/3))} = F_Y(y) = 1 - (1 - 0.06y)^17/3 for 0 < y < 16.666666667.

Problem S3L37-2. The life of a triceratops has the following survival function associated with it: s(x) = e^-0.34x. The annual force of interest in Triceratopsland is currently 0.06. Charlie the Triceratops has a continuous whole life annuity. What is the probability density function f_Y(y)of the present value payments to be made for such an annuity?

Solution S3L37-2. We use the formula f_Y(y) = f_T(-ln(1 - δy)/δ)/(1 - δy) for 0 < y < 1/δ.

The lives of triceratopses are exponentially distributed. Thus, f_T(t) = 0.34e^-0.34t and so

f_Y(y) = 0.34e^{-0.34(-ln(1 - δy)/δ)/(1 - δy)} = 0.34e^{(17/3)ln(1 - 0.06y)}/(1 - 0.06y) =

0.34e^{ln((1 - 0.06y)^(17/3))}/(1 - 0.06y) = 0.34*(1 - 0.06y)^17/3/(1 - 0.06y) =

f_Y(y) =0.34*(1 - 0.06y)^14/3 for 0 < y < 16.666666667.

Problem S3L37-3. The life of a triceratops has the following survival function associated with it: s(x) = e^-0.34x. The annual force of interest in Triceratopsland is currently 0.06. Charlie the Triceratops has a continuous whole life annuity. What is the actuarial present value ā_x of Charlie's annuity?

Solution S3L37-3. We use the formula ā_x = ₀^∞∫v^t*_tp_x*dt. Since the lifetimes of triceratops are exponentially distributed, _tp_x = e^-0.34t for all x. Moreover, v^t = e^-0.06t. Thus,

ā_x = ₀^∞∫e^-0.06t*e^-0.34t*dt = ₀^∞∫e^-0.4t*dt = (-5/2)e^-0.4t│₀^∞ = ā_x = 5/2 = 2.5.

Problem S3L37-4. The life of a triceratops has the following survival function associated with it: s(x) = e^-0.34x. The annual force of interest in Triceratopsland is currently 0.06. Desiderius the Triceratops has a continuous 4-year temporary life annuity. What is the actuarial present value ā_x:4¬of Desiderius's annuity?

Solution S3L37-4. We use the formula ā_x:n¬ = ₀ⁿ∫v^t*_tp_x*dt. Since the lifetimes of triceratops are exponentially distributed, _tp_x = e^-0.34t for all x. Moreover, v^t = e^-0.06t. Thus,

ā_x:4¬ = ₀⁴∫e^-0.06t*e^-0.34t*dt = ₀⁴∫e^-0.4t*dt = (-5/2)e^-0.4t│₀⁴ = (5/2)(1 - e^-1.6) = ā_x:4¬ = about 1.99525871.

Problem S3L37-5. Seven-finned jumping fish can take out continuous whole and temporary life annuities. Each annuity pays 1*dt in benefits at each time t. A 5-year-old seven-finned jumping fish can get either a one-year temporary life annuity with present value 0.67 Golden Hexagons (GH) or a whole life annuity with present value 5.6 GH. The annual force of interest among seven-finned jumping fish is 0.2, and a 5-year-old seven-finned jumping fish has a probability of 0.77 of surviving to age 6. Find the actuarial present value of a continuous whole life annuity available to a 6-year-old seven-finned jumping fish.

Solution S3L37-5. We use the formula ā_x = ā_x:1¬ + vp_x*ā_x+1, which we rearrange thus:

vp_x*ā_x+1 = ā_x - ā_x:1¬

ā_x+1 = (ā_x - ā_x:1¬)/(vp_x)

We are given that ā₅= 5.6 and ā_5:1¬ = 0.67. Moreover, we know that v = e^-0.2 and p₅ = 0.77.

Thus, ā₆ = (5.6 - 0.67)/(0.77e^-0.2) = 4.93/(0.77e^-0.2) = about 7.82015013 GH.

(Note: This result is a genuine possibility. It is possible for the actuarial present values of life annuities to increase for older annuitants, if the rate at which older annuitants die for some reason happens to be less than the rate at which younger annuitants die. Once a seven-finned jumping fish has gotten over the mortality spike during the sixth year of its life, its life expectancy can actually increase.)

See other sections of The Actuary's Free Study Guide for Exam 3L.