Pure Endowments: Practice Problems and Solutions

As in Section 21, the following is defined to be the present-value function.

z_t = Z = b_tv_t

z_t = Z is the present value, at policy issue, of the benefit payment.

b_tis the benefit function.

v_tis the discount function. v is the one-year discount factor by which a sum of money payable one year from now is multiplied to get its present value today. If the annual effective interest rate is r, then v = 1/(1+r).

An n-year pure endowment makes a payment at the conclusion of n years if and only if the insured person is alive n years after the policy has been issued. An endowment that pays one unit in benefits has the following functions associated with it:

b_t = 0 if t ≤ n;

b_t = 1 if t > n.

v_t = vⁿ for t ≥ 0;

Z = 0if T ≤ n;

Z = vⁿ if T > n.

The actuarial present value of an n-year pure endowment entered into by life (x) and paying a unit benefit is denoted as A¹_x:n¬ and has the following formulas associated with it.

E[Z] = A¹_x:n¬ = vⁿ*_np_x

Var(Z) = v²ⁿ*_np_x*_nq_x

Var(Z) = ²A¹_x:n¬ - (A¹_x:n¬) ²

Meaning of Variables:

²A¹_x:n¬ = E[Z²] = the actuarial present value of n-year pure endowment which pays a benefit of 1 upon the death of life (x). Important: The force of interest for ²Ā¹_x:n¬is assumed to be 2δ.

_np_x = probability that life (x) will survive to age x+n.

_nq_x = probability that life (x) will not survive to age x+n.

Source: Bowers, Gerber, et. al. Actuarial Mathematics. 1997. Second Edition. Society of Actuaries: Itasca, Illinois. p. 101.

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

Problem S3L27-1. The life of a triceratops has the following survival function associated with it: s(x) = e^-0.34x. The annual effective interest rate in Triceratopsland is 0.07. Jugurtha the Triceratops is currently 8 years old has a 3-year pure endowment, which will pay him 1 Triceratops Currency Unit (TCU) if he survives to age 11. Find the actuarial present value of this policy.

Solution S3L27-1. We use the formula A¹_x:n¬ = vⁿ*_np_x. We know that x = 8, n = 3, and v = (1/1.07). Thus, v³ = (1/1.07)³.

We find ₃p₈ = s(11)/s(8) = e^-0.34*3. Hence, A¹_8:3¬ = (1/1.07)³e^-0.34*3 =

A¹_8:3¬ = about 0.2943528841.

Problem S3L27-2. The life of a triceratops has the following survival function associated with it: s(x) = e^-0.34x. The annual effective interest rate in Triceratopsland is 0.07. Jugurtha the Triceratops is currently 8 years old has a 3-year pure endowment, which will pay him 1 Triceratops Currency Unit (TCU) if he survives to age 11. Find the variance of the present-value random variable for this policy.

Solution S3L27-2. We use the formula Var(Z) = v²ⁿ*_np_x*_nq_x.

We know from Solution S3L27-1 that ₃p₈ = e^-0.34*3. _nq_x = 1 - _np_x. Thus, ₃q₈ = 1 - e^-0.34*3.

v²ⁿ = v⁶ = (1/1.07)⁶.

Hence, Var(Z) = (1/1.07)⁶*e^-0.34*3*(1 - e^-0.34*3) = Var(Z) = about 0.153636014.

Problem S3L27-3. The life of a triceratops has the following survival function associated with it: s(x) = e^-0.34x. The annual effective interest rate in Triceratopsland is 0.07. Jugurtha the Triceratops is currently 8 years old has a 3-year pure endowment, which will pay him 1 Triceratops Currency Unit (TCU) if he survives to age 11. Find the second moment of the present-value random variable for this policy.

Solution S3L27-3. We use the formula Var(Z) = ²A¹_x:n¬ - (A¹_x:n¬) ², rearranging it thus:

²A¹_x:n¬ = Var(Z) + (A¹_x:n¬) ². We want to find ²A¹_8:3¬. From Solutions S3L27-1 and S3L27-2, we know that Var(Z) = 0.153636014 and A¹_8:3¬ = 0.2943528841. Thus,

²A¹_8:3¬ = 0.153636014 + 0.2943528841² = ²A¹_8:3¬ = about 0.2402796343.

Problem S3L27-4. The life of a giant pin-striped cockroach has the following survival function associated with it: s(x) = 1 - x/94, for 0 ≤ x ≤ 94 and 0 otherwise. Odoacer the Giant Pin-Striped Cockroach is currently 44 years old and has a 13-year endowment which will pay 1 Golden Hexagon (GH) if he reaches age 57. The annual force of interest is 0.02. Find the actuarial present value of Odoacer's policy.

Solution S3L27-4. We use the formula A¹_x:n¬ = vⁿ*_np_x to find A¹_44:13¬. We know that x = 44, n = 13, and

v = e^-0.02. Thus, v¹³ = e^-0.02*13 = e^-0.26.

We find _np_x = ₁₃p₄₄ = s(57)/s(44) = (37/94)/(50/94) = 37/50.

Thus, A¹_44:13¬ = e^-0.26(37/50) = A¹_44:13¬ = about 0.5705781735.

Problem S3L27-5. The life of a giant pin-striped cockroach has the following survival function associated with it: s(x) = 1 - x/94, for 0 ≤ x ≤ 94 and 0 otherwise. Odoacer the Giant Pin-Striped Cockroach is currently 44 years old and has a 13-year endowment which will pay 1 Golden Hexagon (GH) if he reaches age 57. The annual force of interest is 0.02. Find the variance of the present-value random variable for this policy.

Solution S3L27-5. We use the formula Var(Z) = v²ⁿ*_np_x*_nq_x. We know that x = 44, n = 13, and

v = e^-0.02. Thus, v²ⁿ = v²⁶ = e^-0.02*26 = e^-0.52.

We know from Solution S3L27-4 that ₁₃p₄₄ = 37/50.

We find ₁₃q₄₄ = 1 - ₁₃p₄₄ = 13/50.

Thus, Var(Z) = e^-0.52(37/50)(13/50) = Var(Z) = about 0.1143857534.

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