# Actuarial Science Study Note - Life Annuity

**Topics:**Life insurance, Life annuity, Actuarial present value

**Pages:**18 (3194 words)

**Published:**November 7, 2012

Chapter 2:

(Contingent) Life Annuities

2.1 Annuity Certain Review 2.2 Net Single Premium(NSP) 2.3 Pure Endowment 2.4 Whole Life annuities 2.5 Temporary Life Annuities 2.6 Deferred Life Annuities 2.7 Varying Life Annuities 1

AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2

Chapter 2- Life Annuities

Note: There are copies of the text tables at the end of the lecture note handout. You should print them off and bring them to class (for examples)

2.1 Annuity Certain Review • most of this material you have seen before • the following additional notation/formulas(to Zima material) is and/or will be referenced in the Actuarial Mathematics note; Define v=(1+i) -1 then d= 1 – v && Also, a n

|i

(d= 1- 1/(1+i)= i/(1+i))

&& (an

|i

= (1-vn)/d

=

a

n|i (1+i)

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AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2

Example 1 Find the present value of an annuity certain with payments of $1,000 per year for 10 years, first payment made immediately. You are given that d=5%.

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AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2

2.2

Net Single Premiums(NSP)

NSP is the single sum of money paid at time 0 for a benefit • NSP is the net amount and therefore excludes expenses, profit loadings and taxes o the customer would actually pay a gross single premium(GSP) as opposed to a NSP o GSP calculations are similar in approach to NSP calculations (expenses are considered in Ch. 4 of the AM note) • Life annuities and Life Insurance benefits are not guaranteed to

be paid , so NSP for these benefits depends on probability of receiving the benefit • NSP and APV (Actuarial Present Value) mean the same

thing. APV terminology often used to denote that probabilities and interest are involved. 4

AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2

Example 1 If j1=8%, compare the net single premiums of two products both paying $1,000 at the end of 3 years to a person currently age 30. The first product guarantees the $1,000 payment and the second product pays $1,000 only if the person is alive at age 33.

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AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2

2.3 Endowments

• Consider a contract that promises to pay a person age x today at age (x+n) if the person is alive and $0 otherwise $1 ________________________ x x+1 …. x+n ↑ NSP=APV To find the NSP(APV); 1. determine the expected cashflow(CF) = $1npx= npx 2. Discount expected CF back to time 0(age x) = vnnpx = vnlx+n/lx

Ex=vnnpx = vnlx+n/lx n

nEx

(nEx≡APV of a $1 due at age (x+n) if (x) is alive at age x+n)

is the discount factor that reflects “interest and survivorship” (whereas vn is the discount factor reflecting interest only) 6

AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2

Example 1 Find the APV(NSP) of a pure endowment of $10,000 in 20 years to a person now aged 50 using: a) b) probabilities from Table 1 and j1=4.5% using Table 3(which is attached)

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AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2

Example 2 A man aged 30 has $15,000 to invest. What is the amount of the pure endowment that can be paid to him at age 65? Use the text tables to solve this question (i.e. assume i=4.5%,& 2001 mortality(table 3))

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AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2

Example 3 Given 20E50=0.215 and 5E70 =0.576, find the APV of a pure endowment of $30,000 payable in 25 years to a person now aged 50.

In general,

Ex nEx+m= (vmmpx)(vnnpx+m) m vm+nn+mpx= m+nEx

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AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2

2.4 Whole Life Annuities

• a Whole Life annuity is a series of (level) payments made to a

person age x as long as they are living (payments can be annual or other frequencies(e.g. monthly) • whole life annuities can be viewed as a series of pure endowments

and the NSP(APV) is therefore the sum of a series of pure endowments • in practice NSPs are higher for a female age x than for a male age

x(or the female receives smaller...

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