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Lecture Notes-AM Note-Chapter 2 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. 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) 2 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%. 3 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. 5 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+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) 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)) 8 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 9 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 payments for the same NSP), unless unisex mortality tables are used Define ax= APV of an ordinary life annuity to a person age x where payments are $1 at the end of each year (as long as the person is living) 10

AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2 CFs: expected CFS: __ $1 $1px $1 $12px ? x+2 $1 …. $1 $1? -1-xpx ? ?-1 $13px …. ? x+3 ______________________________________________________________ ? ? x x+1 ^ NSP=APV= ax ….. To find the NSP(APV), just discount the expected cashflows; NSP = ax = vpx + v22px + v33px + NSP = ax = 1Ex + 2 …. + v? -1-x? -1-xpx or + ?-1-xEx Ex +3Ex + …. && a x ? APV of a life annuity due to a person age x where payments are $1 at beginning of each year (as long as the person is living) && a x = 1 + ax 11 AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2

Example 1 Find the NSP for an ordinary life annuity (i. e. payable at the end of the year) to a person aged 65 which pay $1,000 per year using the study note Table2(attached)—which is assuming 2001 CSO mortality and i=4. 5% 12 AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2 Example 2 Using Table 2, find the NSP for a life annuity to a person age 70 if payments of $10,000 per year are made; (a) (b) starting immediately starting 1 year from now 13 AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2 Example 3 A person aged 65 has accumulated $250,000 in his RRSP.

What regular annual payments from a life annuity starting one year from now will this amount provide? (Use text tables for values) 14 AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2 Example 4 Find the NSP of a whole life annuity due paying $20,000 per year issued to a person aged 60 given i=5%, p60= 0. 9975 and && a61 =10. & & In general a& x= 1 + vpx a& x+1 = 1 + ax (consider time diagram argument (board) 15 AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2 2. 5 Temporary Life Annuities • temporary life annuities are annuities that provide payments for a maximum number of years(n years).

The payments end at the earlier of when (x) dies, or when n-payments have been made. • as with whole life annuities, there are temporary life annuities due and temporary ordinary(immediate) life annuities (i) Temporary Life Annuity Due An n-year temporary life annuity due is issued to (x), and provides a payment of $1 at the beginning of every year that (x) is alive for a maximum of n-payments Define & a& x : n | = APV (NSP) of a temporary life Annuity due 16 AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2 (i) Temporary Life Annuity Due(continued) $1 $1 $1 $1 ….. $1 ____________________________________________________ ? ? ? ? ? x x+1 x+2 x+3 ….. x+n-1 x+n ^ && NSP=APV= ax:n| && (if payment is $R then NSP =$R ax:n| ) & a& x : n | = 1+ vpx + v22px + v33px + …. + n-1n-1px or & a& x : n | = 0Ex + 1Ex +2Ex + …. + n-1Ex 17 AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2 (ii) Temporary Immediate(Ordinary)Life Annuity • An n-year temporary life annuity immediate is issued to (x), and provides a payment of $1 at the end of every year that (x) is alive for a maximum of n-payments a x : n | ? APV(NSP) of a temporary life Annuity immediate $1 $1 $1 $1 $1 ________________________________________________ ? ? ? ? ? x+1 x+2 x+3 ….. x+n-1 x+n x ^ (if payment is $R then NSP =$R a x : n | ) NSP=APV= a x : n | ax : n| = ax : n| = vpx + v22px + v33px + …. + nnpx or 1Ex + 2Ex +3Ex + …. + nEx 18 AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2 • problems you should be able to solve include i) determining NSP(APV) for different categories of temporary life annuities ii) determining a level payment ($R) for a temporary life annuity iii) understanding of identities that link different types of life annuities • actually, problem types listed above apply to any of the Ch. annuity topics we have/will cover • for any of the above problem types, you can be directed to assume 2001 CSO mortality and i=4. 5%, which means using text tables, or solutions may be done using other assumptions or given values. • Text tables (2001 CSO Mortality,i=4. 5% assumption) used to date(see pages at end of these notes for copies); Table 1-2001 CSO mortality table Table 3- NSP for 1,000 Pure Endowment for various n values • Table 2- Life Annuity Due NSPs & Whole Life Insurance NSPs • Table 4-Temporary Life Annuity Due NSPs (attached & will use with examples below) 19

AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2 Example1 What is the NSP for a temporary life annuity issue to a person aged 50 with payments of $10,000 per year for four years if: a) it is an annuity due b) it is an ordinary (immediate) annuity Use Table 3 and 4. 20 AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2 • Types of identities you may need to work with and/or should be able to verify(time diagram argument best approach) && a x :n | = a x:n +1| ? 1 & a& x :n | = 1 + a x:n ? 1| a x :n | = a& x :n | ? 1 + nEx & 21 AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2

Example 2 An inheritance of $100,000 is used to buy a 10-year life annuity for a person aged 68. What payments will she receive if(use text tablesie. assume 2001 CSO mortality and i=4. 5%): a) it is an annuity due b) it is an immediate annuity 22 AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2 Example 3 Find the NSP for a temporary life annuity due to a person aged 65 with payments of $25,000 per year for 10 years using Table 2. You are also given 1,000 10E65= $498. 9973(from Table 3) Identity used in above problem that you may need to work with and/or should be able to verify(time diagram argument best approach) a& x :n | = a x ? nEx ax+n && && 23 AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2 Example 4 Find the NSP for a life annuity due issued to a person aged 35 with 15 annual payments of $15,000. (a) Use Table 4 (b) Now assume you only had the 1st page of table 4(ie. no 15 year terms). How would you now solve this problem. For this part, you are given that that 1,000 5E35= $797. 0012(from Table 3) Identity used in above problem that you may need to work with and/or should be able to verify (time diagram argument best approach) && & & a& x : n + m | = a& x :n | + E a x + n:m | n x 24

AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2 2. 6 Deferred Life Annuities • deferred life annuities are annuities whose payments being some time into the future • deferred life annuity types include deferred whole life annuities(due or ordinary) and deferred temporary life annuities(due or ordinary(immediate)) (i) Deferred An n-year deferred whole life annuity is issued to (x) and provides payments of $1 at the beginning of every year that (x) is alive, with first payment made n-years from now(at age x+n). n|ax =NSP(APV) of a Deferred Whole Life Annuity (with annual payments of $1) 25

AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2 $1 $1 $1 ……… ____________________________________________________ ? ? ? ? ? ? ? x x+1 +2 ….. x+n-1 x+n x+n+1 x+n+2 …….. ^ NSP=APV= n|ax (if payment is $R then NSP =$R n|ax) |ax = vnpx + vn+1n+1px + vn+2n+2px + …. + v? -x-1? -x–1px or n n|ax = nEx + n+1Ex +n+2Ex + …. + ? -x-1Ex This can be re-written as n|ax =nEx[1 +1Ex+n +2Ex+n +…] n|ax= nEx ax+n (see also time diagram argument) Also, can show(using a time diagram or algebraically) n|ax = & a& x ? a& x : n | & 26 AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2 Example 1 (assume i=4. % and 2001 CSO mortality) (a) Find the NSP for a life annuity due deferred 10 years issued to a person age 55 with payments of $10,000 per year. (b) Find the NSP for a life annuity due deferred 10 years issues to (55) with payments of 10,000 per year, with the first 5 year’s payments guaranteed(repeat of (a) with 5 year guarantee period) 27 AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2 Example 2 Find the NSP for a life annuity due deferred 5 years issued to a person aged x with payments of $1,000 per year if: & a& x =16 & a& x + 5 =14 1,000 5Ex= $857. 14 && a x:5| = 4. 28 AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2 ii) Deferred Temporary Life Annuities An n-year deferred, m-year temporary life annuity due is issued to (x) and provides a payment of $1 at the beginning of every year (x) is alive, for a maximum of m-payments, with the first payment made n-years from now (at age x+n) n & | a& x : m | ? NSP(APV) of a Deferred Temporary Life Annuity Due $1 $1 $1 ……… $1 _______________________________________________________ ? ? ? ? ? ? ? ? x x+1 x+2 ….. x+n x+n +1 x+n+2 …. x+n+m-1 x+n+m ^ NSP=APV= n n n & | a& x : m | & (if payment is $R then NSP = $R n | a& x : m | ) & | a& x : m | = & | a& x : m | vnpx + vn+1n+1px + vn+2n+2px + …. vn+m-1n+m-1px or = nEx + n+1Ex +n+2Ex + …. + n+m-1Ex n Also, can show & && | a& x : m | = nEx a x + n:m | and that n & | a& x : m | = 29 && a x:n+ m| ? && a x:n| AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2 Example 2 Find the NSP for a life annuity due deferred 5 years issued to a person aged x with payments of $1,000 per year if: && ax =16 && a x +5 =14 1,000 5Ex= $857. 14 && a x:5| = 4. 30 AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2 Example 3 Find the NSP for a temporary deferred life annuity of $20,000 per year issued to a person age 40 with a maximum of 30 payments starting at age 60. Assume i=4. 5% and 2001 CSO mortality. 1 AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2 Example 4 A person age 35 has $50,000 to purchase a 10 year life annuity of $R per year with payments beginning at age 50. Find $R if && a35:25| = 13. 07 and && a35:15| = 10. 11 32 AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2 2. 7 Varying Benefits • so far, we’ve only worked with life annuities whereby we’ve assumed a level annual benefit • varying life annuities(can be whole life, temporary, or deferred) due exist and these are life annuities whereby the annual payment can vary each year • the NSP for any varying life annuity can always be calculated by using NSP=?

Rtvttpx • when life annuity payments varying in a step-rate fashion(ie. payments increase or decrease after a stipulated number of years), NSP can be determined using the life annuity factors that we’ve covered so far. Problems can be approached using, either (a) temporary and deferred life annuity factors (b) without using deferred life annuity factors 33 AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2

Example 1 Write an expression for the NSP for a life annuity issued to a person age 30 if the benefits are: $10,000 per year from age 30 to 39 $25,000 per year from age 40 to 49 and $40,000 thereafter a) write your expression in terms of temporary and deferred annuities b) write your expression without using deferred annuities 34 AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2 2. 8 Life Annuities Payable More than Once a Year • often life annuity payments are more frequent than annually(ie. monthly is common) • Consider the following general case where a life annuity pays $1 in total per year, as 1/m payable m times per year •a (m) x NSP(APV) of a Life annuity due payable m times per year with each payment equal to $1/m 35 AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2 1/m 1/m 1/m 1/m … 1/m 1/m 1/m …. ______________________________________________________ ? ? ? ? ? ? ? x x+1/m x+2/m x+3/m … x+m/m x+1+1/m x+1+2/m …. ^ NSP=APV= a(m)x (if total amount paid each year is $R , then NSP =$R n|a(m)x ) Aproximation: a(m)x ? a x – (m-1)/2m • All other “mthly’ annuities can be derived from this relationship. For example, it can be shown that (board) a(m)x ? a x – (m+1)/2m (1st payment is at the end of the “mthly period”) Can show that: ax < a(12)x < a(52)x < a(52)x < a(12)x < ax • Some examples of Other life annuities payable m-thly include; |a (m) x = nEx a(m)x+n ? n &&( n a x:m|) nEx (ax+n-(m-1)/2m) = a(m)x – n|a(m)x ? && a x:n| ? (m ? 1) 2m ( 1 ? nEx ) 36 AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2 Example 1 (use text tables for calculations) What is the NSP of a whole life annuity issued to (65) with monthly payments of $1,000 if payments are a) payable at the beginning of each month b) Payable at the end of each month? 37 AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2

Example 2 (use text tables for calculations) (40) wants to buy an annuity with monthly payments for 20 years starting today. If he has $100,000 to invest, what are the monthly payments? 38 AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2 39 AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2 40 AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2 41 AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2 42 AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2 43 AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2 44 AS2053 Feb. 2012 Lecture Notes-AM Note-Chapter 2 45

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