Discrete Temporary Life Annuities: Practice Problems and Solutions

It is possible to find the actuarial present value of a discrete whole life annuity-immediate as follows:

a_x = (1 - (1+i)A_x)/i, where i is the annual effective interest rate and A_x is the actuarial present value of a whole life insurance policy with benefits payable at the end of the year of death.

The actuarial present value of a discrete n-year temporary life annuity-due is denoted as ä_x:n¬and can be found via the following formula.

ä_x:n¬= _k=0^n-1Σv^k*_kp_x

The actuarial present value of a discrete n-year temporary life annuity-immediate is denoted as a_x:n¬and can be found via the following formula.

a_x:n¬= _k=1ⁿΣv^k*_kp_x

The following relationships hold:

ä_x:n¬= 1 + a_x:n-1¬

If Y is the present-value random variable for a discrete n-year temporary annuity-due, then

Var(Y) = [²A_x:n¬ - (A_x:n¬)²]/d², where d = r/(1+r) and A_x:n¬is the actuarial present value of an n-year endowment insurance policy paying one unit in benefits at the end of the year of death.

Source: Bowers, Gerber, et. al. Actuarial Mathematics. 1997. Second Edition. Society of Actuaries: Itasca, Illinois. pp. 145-148.

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

Problem S3L42-1. 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. Eduardo the Triceratops is currently 14 years old and takes out a discrete 5-year life annuity-due paying a benefit of 1 Golden Hexagon (GH) at the beginning of each year. Find the actuarial present value of this annuity.

Solution S3L42-1. We use the formula ä_x:n¬ = _k=0^n-1Σv^k*_kp_x. Since the lifetimes of triceratopses are exponentially distributed, _kp_x = e^-0.34k for all x. Moreover, v^k = e^-0.06k. Thus, for n = 5,

ä_14:5¬ = _k=0⁴Σe^-0.06k*e^-0.34k = _k=0⁴Σe^-0.4k = 1 + e^-0.4 + e^-0.8 + e^-1.2 + e^-1.6 = about 2.62273974 GH.

Problem S3L42-2. Odaroloc the Colorado Beetle has a whole life insurance policy with benefits payable at the end of the year of death, whose actuarial present value is 0.92 Golden Hexagons (GH). The annual effective interest rate is 0.03. Odaroloc wishes to purchase a discrete whole life annuity-immediate from an actuarially fair insurance provider. How much will he have to pay for the annuity?

Solution S3L42-2. We use the formula a_x = (1 - (1+i)A_x)/i, where we are given i = 0.03 and A_x = 0.92. Thus, a_x = (1 - 1.03*0.92)/0.03 = a_x = about 1.746666666667 GH.

Problem S3L42-3. The life of a yellow whale has the following survival function associated with it: s(x) = e^-0.07x. The annual force of interest is currently 0.02. Wolley the Yellow Whale is currently 34 years old and takes out a discrete 17-year temporary life annuity-immediate paying a benefit of 1 Golden Hexagon (GH) at the end of each year. Find the actuarial present value of this annuity.

Solution S3L42-3. We use the formula a_x:n¬ = _k=1ⁿΣv^k*_kp_xfor n = 17. Since the lifetimes of yellow whales are exponentially distributed, _kp_x = e^-0.07k for all x. Moreover, v^k = e^-0.02k. Thus,

a_34:17¬ = _k=1¹⁷Σe^-0.02k*e^-0.07k = _k=1¹⁷Σe^-0.09k = e^-0.09(1 - e^-0.09*17)/(1 - e^-0.09) = a_34:17¬ = about 8.319302275 GH.

Problem S3L42-4. A 12-year-old swimming squirrel can get a 4-year discrete temporary life annuity-immediate for 34 Golden Hexagons (GH). It can also get a 15-year discrete temporary life annuity-due for twice the actuarial present value of a 5-year discrete temporary life annuity-due. How much will the swimming squirrel have to pay for the 15-year annuity?

Solution S3L42-4. We use the formula ä_x:n¬ = 1 + a_x:n-1¬ to find ä_12:5¬ = 1 + a_12:4¬ = 1 + 34 = 35.

We are also given ä_12:15¬ = 2ä_12:5¬ = 2*35 = ä_12:15¬ =70 GH.

Problem S3L42-5. Elteeb the Colorado Beetle can get a 6-year endowment insurance policy paying one unit in benefits at the end of the year of death for 0.67 Golden Hexagons (GH). The second moment of the present value of this policy is 0.55. The annual effective interest rate is 0.03. Find the variance of the present value of a discrete 6-year temporary annuity-due available to Elteeb.

Solution S3L42-5. We use the formula Var(Y) = [²A_x:n¬ - (A_x:n¬)²]/d² =

[²A_x:n¬ - (A_x:n¬)²]/(i/(1+i))²

We are given i = 0.03, ²A_x:7¬ = 0.55, and A_x:7¬ = 0.67.

Thus, Var(Y) = (0.55 - 0.67²)/(0.03/1.03)² = Var(Y) = about 119.1744333.

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