An annuity is a series of payments made at equal intervals. There are many practical examples of financial transactions involving annuities, such as

Transcription

1 2 Annutes An annuty s a seres of payments made at equal ntervals. There are many practcal examples of fnancal transactons nvolvng annutes, such as a car loan beng repad wth equal monthly nstallments a retree purchasng an annuty from an nsurance company upon hs retrement a lfe nsurance polcy beng purchased wth monthly premums a bondholder recevng an annuty n the form of semannual coupon payments In ths chapter we consder annutes of level or varyng amounts, the payments of whch are certan. Thus, we assume there s no default rsk n the case of a bond or an annuty purchased from an nsurance company. We shall dscuss the calculaton of the present and future values of these annutes. When there s uncertanty n the annuty payments, as n the case of the default of a car loan, the payments are contngent upon some random events. Such annutes wll not be dscussed n ths book.

2 40 CHAPTER 2 Learnng Objectves Annuty-mmedate and annuty-due Present and future values of annutes Perpetutes and deferred annutes Other accumulaton methods Payment perods and compoundng perods Varyng annutes 2.1 Annuty-Immedate Consder an annuty wth payments of 1 unt each, made at the end of every year for n years. Ths knd of annuty s called an annuty-mmedate (also called an ordnary annuty or an annuty n arrears). The present value of an annuty s the sum of the present values of each payment. The computaton of the present value of an annuty can be explaned n tabular form, as llustrated by the followng example. Example 2.1: Calculate the present value of an annuty-mmedate of amount $100 pad annually for 5 years at the rate of nterest of 9% per annum. Soluton: ther total. Table 2.1 summarzes the present values of the payments as well as Table 2.1: Present value of annuty Year Payment ($) Present value ($) (1.09) 1 = (1.09) 2 = (1.09) 3 = (1.09) 4 = (1.09) 5 =64.99 Total

3 Annutes 41 Fgure 2.1 llustrates the tme dagram of an annuty-mmedate of payments of 1 unt at the end of each perod for n perods. As the payments occur at dfferent tmes, ther tme values are dfferent. We are nterested n the value of the annuty at tme 0, called the present value, and the accumulated value of the annuty at tme n, called the future value. Fgure 2.1: Tme dagram for an n-payment annuty-mmedate Cash flow Tme n 1 n Suppose the rate of nterest per perod s, and we assume the compoundnterest method apples. Let a n denote the present value of the annuty, whch s sometmes denoted as a n when the rate of nterest s understood. As the present value of the jth payment s v j,wherev = 1 1+ s the dscount factor, the present value of the annuty s (see Appendx A.5 for the sum of a geometrc progresson) a n = v + v 2 + v v n [ 1 v n ] = v 1 v = 1 vn 1 (1 + ) n =. (2.1) The accumulated value of the annuty at tme n s denoted by s n or s n. Ths s the future value of a n at tme n. Thus, we have s n = a n (1 + ) n = (1 + )n 1. (2.2) s n wll be referred to as the future value of the annuty. If the annuty s of level payments of P, the present and future values of the annuty are Pa n and Ps n, respectvely.

4 42 CHAPTER 2 Example 2.2: Calculate the present value of an annuty-mmedate of amount $100 pad annually for 5 years at the rate of nterest of 9% per annum usng formula (2.1). Also calculate ts future value at the end of 5 years. Soluton: From (2.1), the present value of the annuty s [ 1 (1.09) 5 ] 100 a 5 = 100 = $388.97, 0.09 whch agrees wth the soluton of Example 2.1. The future value of the annuty s (1.09) 5 (100 a 5 )=(1.09) = $ Alternatvely, the future value can be calculated as [ (1.09) 5 ] s 5 = 100 = $ Example 2.3: Calculate the present value of an annuty-mmedate of amount $100 payable quarterly for 10 years at the annual rate of nterest of 8% convertble quarterly. Also calculate ts future value at the end of 10 years. Soluton: Note that the rate of nterest per payment perod (quarter) s 8 4 %=2%, and there are 4 10 = 40 payments. Thus, from (2.1) the present value of the annuty-mmedate s [ ] 1 (1.02) a = 100 =$2,735.55, 0.02 and the future value of the annuty-mmedate s 2, (1.02) 40 =$6, A common problem n fnancal management s to determne the nstallments requred to pay back a loan. Note that we may use (2.1) to calculate the amount of level nstallments requred. The example below llustrates ths. Example 2.4: A man borrows a loan of $20,000 to purchase a car at annual rate of nterest of 6%. He wll pay back the loan through monthly nstallments over 5 years, wth the frst nstallment to be made one month after the release of the loan. What s the monthly nstallment he needs to pay?

5 Annutes 43 Soluton: The rate of nterest per payment perod s 6 12 %=0.5%. LetP be the monthly nstallment. As there are 5 12 = 60 payments, from (2.1) we have 20,000 = Pa [ ] 1 (1.005) 60 = P = P , so that P = 20, = $ In the example above, we have assumed that the perod of converson of nterest s equal to the payment perod,.e., both are monthly. We shall consder later the case when the two are not equal. Note that the quoted rate of nterest of 6% n the above example s nomnal and not the effectve rate of nterest. The example below llustrates the calculaton of the requred nstallment for a targeted future value. Example 2.5: A man wants to save $100,000 to pay for hs son s educaton n 10 years tme. An educaton fund requres the nvestors to depost equal nstallments annually at the end of each year. If nterest of 7.5% per annum s pad, how much does the man need to save each year n order to meet hs target? Soluton: We frst calculate s 10, whch s equal to (1.075) Then the requred amount of nstallment s = P = 100,000 s 10 = 100, =$7, Annuty-Due An annuty-due s an annuty for whch the payments are made at the begnnng of the payment perods. The tme dagram n Fgure 2.2 llustrates the payments of an annuty-due of 1 unt n each perod for n perods.

6 44 CHAPTER 2 Fgure 2.2: Tme dagram for an n-payment annuty-due Cash flow Tme n 1 n Note that the frst payment s made at tme 0, and the last payment s made at tme n 1. We denote the present value of the annuty-due at tme 0 by ä n (or ä n f the rate of nterest per payment perod s understood), and the future value of the annuty at tme n by s n (or s n f the rate of nterest per payment perod s understood). The formula for ä n can be derved as follows Also, we have ä n = 1+v + + v n 1 = 1 vn 1 v = 1 vn. (2.3) d s n = ä n (1 + ) n = (1 + )n 1. (2.4) d As each payment n an annuty-due s pad one perod ahead of the correspondng payment of an annuty-mmedate, the present value of each payment n an annuty-due s (1 + ) tmes the present value of the correspondng payment n an annuty-mmedate. Thus, we conclude ä n =(1+) a n (2.5) and, smlarly, s n =(1+) s n. (2.6) Equaton (2.5) can also be derved from (2.1) and (2.3). Lkewse, (2.6) can be derved from (2.2) and (2.4).

7 Annutes 45 As an annuty-due of n payments conssts of a payment at tme 0 and an annuty-mmedate of n 1 payments, the frst payment of whch s to be made attme1,wehave ä n =1+a n 1. (2.7) Smlarly, f we consder an annuty-mmedate wth n +1payments at tme 1, 2,, n+1as an annuty-due of n payments startng at tme 1 plus a fnal payment at tme n +1, we can conclude s n+1 = s n +1. (2.8) Equatons (2.7) and (2.8) can also be proved algebracally usng (2.1) through (2.4). Readers are nvted to do ths as an exercse. Example 2.6: A company wants to provde a retrement plan for an employee who s aged 55 now. The plan wll provde her wth an annuty-mmedate of $7,000 every year for 15 years upon her retrement at the age of 65. The company s fundng ths plan wth an annuty-due of 10 years. If the rate of nterest s 5%, what s the amount of nstallment the company should pay? Soluton: equal to We frst calculate the present value of the retrement annuty. Ths s [ ] 1 (1.05) 15 7,000 a 15 = 7,000 =$72, Ths amount should be equal to the future value of the company s nstallments P, whch s P s 10. Now from (2.4), we have so that s 10 = (1.05)10 1 = , 1 (1.05) 1 P = 72, =$5, Perpetuty, Deferred Annuty and Annuty Values at Other Tmes A perpetuty s an annuty wth no termnaton date,.e., there s an nfnte number of payments, wth n. An example that resembles a perpetuty s the dvdends of a preferred stock. As a stock has no maturty date, f the dvdend amounts are fxed, the payments are lke a perpetuty. To calculate the present value of a

8 46 CHAPTER 2 perpetuty, we note that, as v<1, v n 0 when n. Thus, from (2.1), we conclude that the present value of a perpetuty of payments of 1 unt when the frst payment s made one perod later, s a = 1. (2.9) For the case when the frst payment s made mmedately, we have, from (2.3), ä = 1 d. (2.10) A deferred annuty s one for whch the frst payment starts some tme n the future. Consder an annuty wth n unt payments for whch the frst payment s due at tme m +1. Ths can be regarded as an n-perod annuty-mmedate to start at tme m, and ts present value s denoted by m a n (or m a n for short). Thus, we have m a n = v m a n [ ] 1 v = v m n = vm v m+n = (1 vm+n ) (1 v m ) = a m+n a m. (2.11) To understand the above equaton, note that the deferred annuty can be regarded as a (m + n)-perod annuty-mmedate wth the frst m payments removed. It can be seen that the rght-hand sde of the last lne of (2.11) s the present value of a (m + n)-perod annuty-mmedate mnus the present value of a m-perod annuty-mmedate. Fgure 2.3 llustrates (2.11). From (2.11), we have the followng results (note that the roles of m and n can be nterchanged) a m+n = a m + v m a n = a n + v n a m. (2.12) Multplyng the above equatons throughout by 1+,we have ä m+n = ä m + v m ä n = ä n + v n ä m. (2.13)

9 Annutes 47 Fgure 2.3: Illustraton of equaton (2.11) Cash flow Tme m m +1 m + n Present value of annuty a m a m+n v m a n We also denote v m ä n as m ä n, whch s the present value of a n-payment annuty of unt amounts due at tme m, m + 1,, m + n 1. If we multply the equatons n (2.12) throughout by (1 + ) m+n, we obtan a n s m+n = (1+) n s m + s n = (1+) m s n + s m. (2.14) Note that a (m + n)-perod annuty-mmedate can be regarded as the sum of a m-perod annuty-mmedate to start at tme 0 and a n-perod annuty-mmedate to start at tme m. The frst lne n (2.14) s the sum of the future values of the two annuty-mmedates at tme m + n. Fnally, note that the roles of m and n can be nterchanged so that the second lne of (2.14) s an alternatve way to wrte the future value of the (m + n)-perod annuty-mmedate. Fgure 2.4 llustrates the frst lne of (2.14). It s also straghtforward to see that We now return to (2.2) and wrte t as s m+n = (1+) n s m + s n = (1+) m s n + s m. (2.15) s m+n =(1+) m+n a m+n, from whch we multply both sdes of the equaton by v m to obtan v m s m+n =(1+) n a m+n, (2.16)

10 48 CHAPTER 2 Fgure 2.4: Illustraton of equaton (2.14) Cash flow Tme m m +1 m + n Future value of annuty s m s m+n s n (1 + ) n s m for arbtrary postve ntegers m and n. Ths equaton expresses the value at tme n of a (m+n)-perod annuty-mmedate startng at tme 0 n two dfferent ways. The left-hand sde dscounts the future value at tme m + n backwards by m perods, whle the rght-hand sde brngs the present value at tme 0 forward by n perods. Fgure 2.5 llustrates (2.16). Fgure 2.5: Illustraton of equaton (2.16) Cash flow Tme n n +1 m + n Annuty value at tme n a m+n (1 + ) n a m+n s m+n v m s m+n

11 Annutes 49 Also, t can be checked that the followng result holds: v m s m+n =(1+) n ä m+n. 2.4 Annutes under Other Accumulaton Methods We have so far dscussed the calculatons of the present and future values of annutes assumng compound nterest. In ths secton we shall extend our dscusson to other nterest-accumulaton methods. We shall consder a general accumulaton functon a( ) and assume that the functon apples to any cash-flow transactons n the future. Thus, as stated n Secton 1.7, any payment at tme t>0starts to accumulate nterest accordng to a( ) as a payment made at tme 0. Gven the accumulaton functon a( ), the present value of a unt payment due 1 at tme t s a(t). Thus, the present value of a n-perod annuty-mmedate of unt payments s n 1 a n = a(t). (2.17) t=1 Lkewse, the future value at tme n of a unt payment at tme t<ns a(n t). Thus, the future value of a n-perod annuty-mmedate of unt payments s s n = n a(n t). (2.18) t=1 If (1.35) s satsfed so that a(n t) = a(n) a(t) s n = n t=1 for n>t>0,then a(n) n a(t) = a(n) 1 a(t) = a(n) a n. (2.19) Ths result s satsfed for the compound-nterest method, but not the smple-nterest method or other accumulaton schemes for whch equaton (1.35) does not hold. Example 2.7: Suppose δ(t) =0.02t for 0 t 5, fnda 5 and s 5. Soluton: t=1 We frst calculate a(t), whch, from (1.26), s ( t ) a(t) = exp 0.02s ds 0 = exp(0.01t 2 ). Hence, from (2.17), a 5 = 1 e e e e =4.4957, e0.25

12 50 CHAPTER 2 and, from (2.18), s 5 =1+e e e e 0.16 = Note that a(5) = e 0.25 =1.2840, sothat a(5) a 5 = = s 5. Note that n the above example, a(n t) =exp[0.01(n t) 2 ] and a(n) a(t) =exp[0.01(n2 t 2 )], so that a(n t) a(n) a(t) and (2.19) does not hold. Example 2.8: Calculate a 3 and s 3 f the nomnal rate of nterest s 5% per annum, assumng (a) compound nterest, and (b) smple nterest. Soluton: and (a) Assumng compound nterest, we have a 3 = 1 (1.05) =2.7232, s 3 =(1.05) = (b) For smple nterest, the present value s a 3 = 3 t=1 1 a(t) = 3 t=1 and the future value at tme 3 s s 3 = 3 a(3 t) = t=1 1 1+rt = =2.7310, 3 (1 + r(3 t)) = =3.15. t=1 At the same nomnal rate of nterest, the compound-nterest method generates hgher nterest than the smple-nterest method. Therefore, the future value under the compound-nterest method s hgher, whle ts present value s lower. Also, note that for the smple-nterest method, a(3) a 3 = = , whchs dfferent from s 3 =3.15.

13 Annutes Payment Perods, Compoundng Perods and Contnuous Annutes We have so far assumed that the payment perod of an annuty concdes wth the nterest-converson perod. Now we consder the case where the payment perod dffers from the nterest-converson perod. In general, the formulas derved n the prevous sectons can be appled wth beng the effectve rate of nterest for the payment perod (not the nterest-converson perod). The examples below llustrate ths pont. Example 2.9: Fnd the present value of an annuty-due of $200 per quarter for 2 years, f nterest s compounded monthly at the nomnal rate of 8%. Soluton: Ths s the stuaton where the payments are made less frequently than nterest s converted. We frst calculate the effectve rate of nterest per quarter, whch s [ ] 3 1=2.01%. 12 As there are n =8payments, the requred present value s [ ] 1 (1.0201) ä = (1.0201) 1 =$1, Example 2.10: Fnd the present value of an annuty-mmedate of $100 per quarter for 4 years, f nterest s compounded semannually at the nomnal rate of 6%. Soluton: Ths s the stuaton where payments are made more frequently than nterest s converted. We frst calculate the effectve rate of nterest per quarter, whch s [ ] 1 2 1=1.49%. 2 Thus, the requred present value s [ ] 1 (1.0149) a = 100 =$1,

14 52 CHAPTER 2 It s possble to derve algebrac formulas to compute the present and future values of annutes for whch the perod of nstallment s dfferent from the perod of compoundng. We frst consder the case where payments are made less frequently than nterest converson, whch occurs at tme 1, 2,, etc. Let denote the effectve rate of nterest per nterest-converson perod. Suppose an m-payment annuty-mmedate conssts of unt payments at tme k, 2k,, mk. We denote n = mk, whch s the number of nterest-converson perods for the annuty. Fgure 2.6 llustrates the cash flows for the case of k =2. Fgure 2.6: Payments less frequent than nterest converson (k =2) Cash flow Interest-converson perod Payment perod (k =2) n 2 n 1 n k 2k (m 1)k mk The present value of the above annuty-mmedate s (we let w = v k ) v k + v 2k + + v mk = w + w [ w m 1 w m ] = w 1 w [ ] 1 v = v k n 1 v k 1 v n = (1 + ) k 1 = a n, (2.20) s k and the future value of the annuty s (1 + ) n a n s k = s n s k. (2.21)

15 Annutes 53 Note that (2.20) has a convenent nterpretaton. Consder an annuty-mmedate wth nstallments of 1/s k each at tme 1,, k. The future value of these nstallments at tme k s 1. Thus, the m-payment annuty-mmedate of unt payments at tme k, 2k,, mk has the same present value as an mk-payment of annutymmedate wth nstallments of 1/s k at tme 1, 2,, n,whchsa n /s k. We now consder the case where the payments are made more frequently than nterest converson. Let there be an mn-payment annuty-mmedate due at tme 1 m, 2 m,, 1, 1+ 1 m,, 2,, n, andlet be the effectve rate of nterest per nterest-converson perod. Thus, there are mnpayments over n nterest-converson 1 perods. Suppose each payment s of the amount m, so that there s a nomnal amount of unt payment n each nterest-converson perod. Fgure 2.7 llustrates the cash flows for the case of m =4. We denote the present value of ths annuty at tme 0 by a (m), whch can be n computed as follows (we let w = v 1 m ) where a (m) n = 1 m (v 1 m + v 2 m + + v + v 1+ 1 m + + v n) = 1 m (w + w2 + + w mn ) = 1 [w 1 ] wmn m 1 w = 1 [v 1m 1 v n ] m 1 v 1 m [ ] = 1 1 v n m (1 + ) 1 m 1 = 1 vn r (m), (2.22) [ ] r (m) = m (1 + ) 1 m 1 (2.23) s the equvalent nomnal rate of nterest compounded m tmes per nterestconverson perod (see (1.19)). The future value of the annuty-mmedate s s (m) n = (1+) n a (m) n = (1 + )n 1 = r (m) r (m) s n. (2.24)

16 54 CHAPTER 2 Fgure 2.7: Payments more frequent than nterest converson (m =4) Cash flow Interest-converson perod n 2 n 1 n Payment perod (k =2) k 2k (m 1)k mk The above equaton parallels (2.22), whch can also be wrtten as a (m) n = r (m) a n. If the mn-payment annuty s due at tme 0, 1 m, 2 m,,n 1 m, we denote ts present value at tme 0 by ä (m), whch s gven by n ä (m) n =(1+) 1 m a (m) n Thus, from (1.22) we conclude ä (m) n The future value of ths annuty at tme n s =(1+) 1 m [ 1 v n r (m) ]. (2.25) = 1 vn d (m) = d d (m) än. (2.26) s (m) n For deferred annutes, the followng results apply =(1+) n ä (m) n = d d (m) s n. (2.27) and q a (m) n q ä (m) n = v q a (m) n, (2.28) = v q ä (m) n. (2.29)

17 Annutes 55 Example 2.11: Solve the problem n Example 2.9 usng (2.20). Soluton: We frst note that = = Nowk =3and n =24so that from (2.20), the present value of the annuty-mmedate s 200 a s [ ] 1 (1.0067) 24 = 200 (1.0067) 3 1 = $1, Fnally, the present value of the annuty-due s (1.0067) 3 1, =$1, Example 2.12: Solve the problem n Example 2.10 usng (2.22) and (2.23). Soluton: Note that m =2and n =8. Wth =0.03, we have, from (2.23) Therefore, from (2.22), we have r (2) =2 [ ] = a (2) = 1 (1.03) = As the total payment n each nterest-converson perod s $200, the requred present value s = $1, Now we consder (2.22) agan. Suppose the annutes are pad contnuously at the rate of 1 unt per nterest-converson perod over n perods. Thus, m and we denote the present value of ths contnuous annuty by ā n.aslm m r (m) = δ, we have, from (2.22), ā n = 1 vn δ = 1 vn ln(1 + ) = δ a n. (2.30) For a contnuous annuty there s no dstncton between annuty-due and annutymmedate. Whle theoretcally t may be dffcult to vsualze annutes that are

18 56 CHAPTER 2 pad contnuously, (2.30) may be a good approxmaton for annutes that are pad daly. Note that (2.30) does not depend on the frequency of nterest converson. The present value of an n-perod contnuous annuty of unt payment per perod wth a deferred perod of q s gven by q ā n = v q ā n =ā q+n ā q. (2.31) To compute the future value of a contnuous annuty of unt payment per perod over n perods, we use the followng formula s n =(1+) n ā n = (1 + )n 1 ln(1 + ) = δ s n. (2.32) We now generalze the above results to the case of a general accumulaton functon a( ), whch s not necessarly based on the compound-nterest method. The present value of a contnuous annuty of unt payment per perod over n perods s ā n = n 0 v(t) dt = n 0 ( exp t 0 ) δ(s) ds dt. (2.33) To compute the future value of the annuty at tme n, we assume that, as n Secton 1.7, a unt payment at tme t accumulates to a(n t) at tme n, forn> t 0. 1 Thus, the future value of the annuty at tme n s n n ( n t ) s n = a(n t) dt = exp δ(s) ds dt. (2.34) Varyng Annutes We have so far consdered annutes wth level payments. Certan fnancal nstruments may provde cash flows that vary over tme. For example, a bond may pay coupons wth a step-up coupon rate, makng lower coupon payments (or even deferred payments) ntally wth ncreased payments later. We llustrate how varyng annutes may be evaluated. In partcular, we consder annutes the payments of whch vary accordng to an arthmetc progresson. Thus, we consder an annutymmedate and assume the ntal payment s P, wth subsequent payments P + D, P +2D,, etc., so that the jth payment s P +(j 1)D. Note that we allow D to be negatve so that the annuty can be ether steppng up or steppng down. However, for a n-payment annuty, P +(n 1)D must be postve so that negatve cash flow s ruled out. Fgure 2.8 presents the tme dagram of ths annuty. 1 Note that ths s not the only assumpton we can adopt, nor s t the most reasonable. In Chapter 3 we shall see another possble assumpton.

19 Annutes 57 Fgure 2.8: Increasng annuty-mmedate Cash flow P P + D P +2D P +(n 1)D Tme n We can see that the annuty can be regarded as the sum of the followng annutes: (a) an n-perod annuty-mmedate wth constant amount P, and(b) n 1 deferred annutes, where the jth deferred annuty s a (n j)-perod annutymmedate wth level amount D to start at tme j, forj =1,,n 1. Thus, the present value of the varyng annuty s n 1 n 1 Pa n + D v j a n j = Pa n + D v j (1 vn j ) j=1 j=1 ( n 1 ) j=1 = Pa n + D vj (n 1)v n ( n ) j=1 = Pa n + D vj nv n [ an nv n ] = Pa n + D. (2.35) For an n-perod ncreasng annuty wth P = D = 1, we denote ts present and future values by (Ia) n and (Is) n, respectvely. Readers are nvted to show that (Ia) n = än nv n (2.36) and (Is) n = s n+1 (n +1) = s n n. (2.37) For an ncreasng n-payment annuty-due wth payments of 1, 2,, nat tme 0, 1,, n 1, the present value of the annuty s (Iä) n =(1+)(Ia) n. (2.38)

20 58 CHAPTER 2 Ths s the sum of an n-perod level annuty-due of unt payments and a (n 1)- payment ncreasng annuty-mmedate wth startng and ncremental payments of 1. Thus, we have (Iä) n =ä n +(Ia) n 1. (2.39) For the case of a n-perod decreasng annuty wth P = n and D = 1, we denote ts present and future values by (Da) n and (Ds) n, respectvely. Fgure 2.9 presents the tme dagram of ths annuty. Fgure 2.9: Cash flow Tme Decreasng annuty-mmedate wth P = n and D = 1 n n 1 n n Readers are nvted to show that (Da) n = n a n (2.40) and (Ds) n = n(1 + )n s n. (2.41) We consder two types of ncreasng contnuous annutes. Frst, we consder the case of a contnuous n-perod annuty wth level payment (.e., at a constant rate) of τ unts from tme τ 1 through tme τ. We denote the present value of ths annuty by (Iā) n, whch s gven by n (Iā) n = τ τ=1 τ τ 1 v s ds = n τ τ=1 τ τ 1 e δs ds. (2.42) The above equaton can be smplfed to (see Exercse 2.42) (Iā) n = än nv n. (2.43) δ

21 Annutes 59 Second, we may consder a contnuous n-perod annuty for whch the payment n the nterval t to t + t s t t,.e., the nstantaneous rate of payment at tme t s t. We denote the present value of ths annuty by (Īā) n, whch s gven by (see Exercse 2.42) n n (Īā) n = tv t dt = te δt dt = ān nv n. (2.44) 0 0 δ We now consder an annuty-mmedate wth payments followng a geometrc progresson. Let the frst payment be 1, wth subsequent payments beng 1+k tmes the prevous one. Thus, the present value of an annuty wth n payments s (for k ) n 1 v + v 2 (1 + k)+ + v n (1 + k) n 1 = v [v(1 + k)] t t=0 n 1 ] t [ 1+k = v 1+ t=0 ( ) 1+k n 1 = v k 1+ ( ) 1+k n 1 1+ =. (2.45) k Note that n the above equaton, f k = the present value of the annuty s nv, as can be concluded from the frst lne of (2.45). Example 2.13: An annuty-mmedate conssts of a frst payment of $100, wth subsequent payments ncreased by 10% over the prevous one untl the 10th payment, after whch subsequent payments decreases by 5% over the prevous one. If the effectve rate of nterest s 10% per payment perod, what s the present value of ths annuty wth 20 payments? Soluton: The present value of the frst 10 payments s (note that k = ) (1.1) 1 = $ For the next 10 payments, k = 0.05 and ther present value at tme 10 s (note that the payment at tme 11 s 100(1.10) 9 (0.95)) ( ) (1.10) (0.95) =1,

22 60 CHAPTER 2 Hence, the present value of the 20 payments s ,148.64(1.10) 10 =$1, Example 2.14: An nvestor wshes to accumulate $1,000 at the end of year 5. He makes level deposts at the begnnng of each year for 5 years. The deposts earn a 6% annual effectve rate of nterest, whch s credted at the end of each year. The nterests on the deposts earn 5% effectve nterest rate annually. How much does he have to depost each year? Soluton: Let the level annual depost be A. The nterest receved at the end of year 1 s 0.06A, whch ncreases by 0.06A annually to A at the end of year 5. Thus, the nterests from the deposts form a 5-payment ncreasng annuty wth P = D =0.06A, earnng annual nterest of 5%. Hence, we have the equaton of value 1,000 = 5A +0.06A(Is) From (2.37) we obtan (Is) = , sothat A = 2.7 Term of Annuty 1, = $ Gven the rate of nterest, the term of the annuty and the szes of the annuty payments, the formulas derved above enable us to calculate the present and future values of the annuty. The problem, however, may sometmes be reversed. For example, we may consder the followng: (a) gven an ntal nvestment that generates level annuty payments at a gven rate of nterest, how long wll the annuty payments last? (b) gven an ntal nvestment that generates a fxed number of level annuty payments, what s the rate of return for ths nvestment? Obvously the frst queston requres us to solve for the value of n gven other nformaton, whle the second requres the soluton of. We now consder the case where the annuty perod may not be an nteger. Ths wll throw lght on the frst queston above. For example, we consder a n+k,where

23 Annutes 61 n s an nteger and 0 <k<1. We note that a n+k = 1 vn+k = (1 vn )+ ( v n v n+k) [ (1 + ) = a n + v n+k k ] 1 = a n + v n+k s k. (2.46) Thus, a n+k s the sum of the present value of a n-perod annuty-mmedate wth unt amount and the present value of an amount s k pad at tme n + k. Note that s k should not be taken as the future value of an annuty-mmedate as k s less than 1, the tme of the frst payment. When the equaton of value does not solve for an nteger n, care must be taken to specfy how the last payment s to be calculated. The example below llustrates ths problem. Example 2.15: A prncpal of $5,000 generates ncome of $500 at the end of every year at an effectve rate of nterest of 4.5% for as long as possble. Calculate the term of the annuty and dscuss the possbltes of settlng the last payment. Soluton: The equaton of value 500 a n = 5,000 mples a n =10, whch can be solved to obtan n = Thus, the nvestment may be pad off wth 13 payments of $500, plus an addtonal amount A at the end of year 13. Computng the future value at tme 13, we have the equaton of value 500 s A = 5,000 (1.045)13, whch mples A = $ We conclude that the last payment s = $ Alternatvely, f A s not pad at tme 13 the last payment B may be made at the end of year 14, whch s gven by B = = $ Fnally, f we adopt the approach n (2.46), we let n =13and k =0.58 so that the last payment C to be pad at tme years s gven by [ (1.045) 0.58 ] 1 C = 500s k = 500 = $

24 62 CHAPTER 2 Note that A < C < B, whch s as expected, as ths follows the order of the occurrence of the payments. In prncple, all three approaches are justfed. Generally, the effectve rate of nterest cannot be solved analytcally from the equaton of value. Numercal methods must be used for ths purpose. Example 2.16: A prncpal of $5,000 generates ncome of $500 at the end of every year for 15 years. What s the effectve rate of nterest? Soluton: The equaton of value s so that a 15 = 5, =10, 1 (1 + ) 15 a 15 = =10. A smple grd search provdes the followng results a A fner search provdes the answer 5.556%. The soluton of Example 2.16 requres the computaton of the root of a nonlnear equaton. Ths can be done numercally usng the Excel Solver, whch s llustrated n Exhbt 2.1. We enter a guessed value of 0.05 n Cell A1 n the Excel worksheet. The followng expresson s then entered n Cell A2: (1 (1 + A1)ˆ( 15))/A1, whch computes a 15 wth set to the value at Cell A1. The Solver s then called up (from Tools, followed by Solver). In the Solver Parameters wndow we set the target cell to A2, and the target value to 10 by changng the value n Cell A1. The answer s found to be Note that the exact procedure depends on the verson of the Excel Solver.

25 Annutes 63 Exhbt 2.1: Use of Excel Solver for Example 2.16 Alternatvely, we can use the Excel functon RATE to calculate the rate of nterest that equates the present value of an annuty-mmedate to a gven value. Specfcally, consder the equatons a n A =0 and ä n A =0. (2.47) Gven n and A, wewshtosolvefor, whch s the rate of nterest per payment perod of the annuty-mmedate or annuty-due. The use of the Excel functon RATE to compute s descrbed as follows: 3 Excel functon: RATE(np,1,pv,type,guess) np = n, pv = A, type = 0 (or omtted) for annuty-mmedate, 1 for annuty-due guess = startng value, set to 0.1 f omtted Output =, rate of nterest per payment perod of the annuty To use RATE to solve for Example 2.16 we key n =RATE(15,1,-10). 3 We use bold face fonts to denote Excel functons or the nputs of an Excel functon that are mandatory. Input varables that are optonal are denoted n normal fonts. For brevty we have adapted the specfcaton of the functons for our usage. Readers should refer to the Excel Help Menu for the complete specfcaton of the functons.

26 64 CHAPTER Summary 1. Annutes are seres of payments at equal ntervals. An annuty-mmedate makes payments at the end of the payment perods, and an annuty-due makes payments at the begnnng of the perods. 2. Algebrac formulas for the present and future values of annuty-mmedate and annuty-due can be derved usng geometrc progresson. These formulas facltate the calculaton of nstallments to pay off a loan, or nstallments requred to accumulate to a targeted amount n the future. 3. Perpetuty has no maturty and makes payments ndefntely. Deferred annuty makes the frst payment some tme n the future. 4. Present- and future-value formulas can be derved for a general accumulaton functon. 5. When the payment perod and the nterest-converson perod are not the same, a smple approach s to compute the effectve rate of nterest for the payment perod and do the computatons usng ths rate. Algebrac approach can be used for both the cases of payments more frequent than nterest converson and less frequent than nterest converson. 6. Computatons for contnuous annuty can be used as approxmatons to frequent payments such as daly annutes. 7. Annutes nvolvng smple step-up or step-down of payments n an arthmetc progresson can be computed usng basc annuty formulas. 8. The equaton of value can be adopted to solve for n or gven other nformaton. The calculaton of from the equaton of value generally admts no analytc soluton. Numercal methods such as grd search may be used. Exercses 2.1 Let the effectve rate of nterest be 5%. Evaluate (a) s 10, (b) ä 5, (c) (Is) 7, (d) (Ia) 7, (e) a.

27 Annutes Let d (2) =2%. Evaluate each of the followng and draw a tme dagram for the cash flows nvolved. (a) a (2) 2, (b) s (4) 4, (c) ä ( ) Descrbe the meanng of a 20.3 evaluated at annual rate of nterest 2%. 2.4 (a) Explan why ä n a n. (b) Explan why a (12) a n n. 2.5 Let the effectve rate of nterest be 4%. Fnd the accumulated value of an annuty whch pays $750 (a) annually for 8 years, (b) bennally for 8 years, (c) semannually for 4 years, assumng the payments are due at the end of every payment perod. 2.6 A loan of $10,000 s repad by 20 level nstallments at the end of every 6 months. The nomnal rate of nterest s 8%, convertble half yearly. Fnd the total amount of nterest payment made over the 10-year perod. 2.7 The present value of a seres of payments of $5 at the end of every 3 years s $10. Fnd the effectve rate of nterest per year. 2.8 Fnd the present value of a 20-year annuty-due of $50 per year f the effectve rate of nterest s 6% for the frst 12 years and 5% thereafter. 2.9 Assumng smple nterest at 4%, evaluate (a) (Ia) 3, (b) (Is) 3. Does the formula a(3)(ia) 3 =(Is) 3 hold? 2.10 Fnd the present value (evaluated at the begnnng of year 2006) of the perpetuty whch pays $1,000 at the begnnng of every year startng from year 2006, but provdes no payment at the begnnng of every leap year. Express your answer n terms of the effectve rate of nterest.

28 66 CHAPTER Nova deposts $1,000 nto her bank account at the begnnng of every year. The bank credts annual nterest of 6%. At the end of every year, she takes out the nterest earned from the bank account and puts t nto an nvestment fund whch earns 3% every half year. Fnd the nterest earned n the fund durng the frst half of the nnth year The prce of a watch s $25,000 and you fnance t wth a down payment and 18 nstallments of $1,000 payable at the end of every month. The nomnal nterest rate s 2.5% convertble monthly. Fnd the down payment The prce of a LCD montor s $1,649. Fnd the monthly payments commencng at tme 0 and contnung for 1.5 years for the purchase f the nterest charged s 2% convertble monthly Let δ(t) = 0.01(t +1), t 0. Fnd the current value at tme 3 for $1 payable at tme 1 and 5, $2 payable at tme 2 and 4, and $3 payable at tme 3. You may adopt the assumpton n Secton 1.7 for the future values of future payments Prove (2.7) and (2.8) algebracally A person plans to accumulate a sum of $50,000 at the end of 48 months by equal nstallments at the begnnng of each month. If the effectve rate of nterest s 1% per month, what s the amount of the nstallment? 2.17 A sum of $20,000 s used to buy a deferred perpetuty-due that pays $2,100 every year for the frst 5 years and $1,000 per year thereafter. If the annual effectve rate s 6%, fnd the deferred perod Fnd the present value of a 10-year annuty that pays $500 at the end of each month for the frst 4 years and $2,000 at the end of each quarter for the last 6 years. The annual effectve rate of nterest s 7% A sum of $20,000 s used to buy a 4-year deferred perpetuty-mmedate that pays M every quarter. If the annual effectve rate of nterest s 8%, fnd M Prove (2.36) and (2.37) Prove (2.40) and (2.41) Consder an annuty-due that pays 1 unt at the begnnng of every k other nterest-converson perods, for a total of m payments. Derve the present value of ths annuty. What s the future value of ths annuty at tme n = mk?

29 Annutes Express s(52) n s (12) n n terms of the nomnal rates of nterest and dscount A couple want to save $100,000 to pay for ther daughter s educaton. They put $2,000 nto a fund at the begnnng of every month. Interest s compounded monthly at a nomnal rate of 7%. How long does t take for the fund to reach $100,000? 2.25 Denote the present value and future value (at tme n)for1 at tme 0; 2 at tme 1;, n at tme n 1, by(iä) n and (I s) n, respectvely. Show that (Iä) n = än d nvn 1 and (I s) n = s n n d wth the help of formulas (2.36) and (2.37) John deposts $500 nto the bank at the end of every month. The bank credts monthly nterest of 1.5%. Fnd the amount of nterest earned n hs account durng the 13th month Consder lendng $1 at the begnnng of each year for n years. The borrower s requred to repay nterest at the end of every year. After n years, the prncpal s returned. (a) Draw a tme dagram and fnd the present value of all nterests. (b) How much prncpal s returned after n years? (c) Formula (2.36) can be wrtten as ä n = (Ia) n + nv n. Explan verbally the meanng of the expresson above. (d) Usng (Ia) n +(Da) n =(n+1)a n and (2.36), derve formula (2.40) Payments of $500 per quarter are made over a 5-year perod commencng at the end of the frst month. Show that the present value of all payments 2 years pror to the frst payment s $2,000(ä (4) 7 ä(4) 2 ), where the annuty symbols are based on an effectve rate of nterest. What s the accumulated value of all payments 7 years after the frst payment? Express your answers n a form smlar to the expresson above.

30 68 CHAPTER Fnd the accumulated value at the end of 12 years of a fund where $1,000 s deposted at the begnnng of each year for the frst 4 years, $2,000 s deposted at the begnnng of each year for the subsequent 4 years, and $3,000 s deposted at the begnnng of each year for the fnal 4 years. Express your answer n terms of a sum of accumulated values s n Let δ(t) =(2+t) 1, t 0. (a) Fnd v(t). (b) Fnd a Deposts of $1,000 per year are placed nto a fund contnuously for 20 years at 4% force of nterest. After 20 years, annual wthdrawals commence and contnue for 30 years at the same effectve rate of nterest, wth the frst payment to be made 1 year after the last depost. Fnd the amount of each wthdrawal Assumng smple rate of dscount d, fnd an expresson for ä n for n<1/d Descrbe the meanng of the followng wth the help of a tme dagram: (a) a 18 a 2, (b) 1 s 3, (c) s 9 s Express, n terms of a n and s n, (a) the present value at tme 0 of $1 payable at tme 4, 8, 12, 16, 20,, 60, (b) the accumulated value at tme 25 of $1 payable at tme 5, 8, 11, 14, 17 and Show that the current value at tme 16 of $1 payable at tme 3, 7, 11,,31 s s 16 + a 16 s 3 + a Fnd the present value of monthly payments of $2,400, $2,300,, $1,400, $1,300 commencng at tme 0 at d (12) = 12%.

31 Annutes A paused ranbow s a cash-flow stream whch pays $1 at tme 1 and 2n; $2 at tme 2 and 2n 1; ; $(n 1) at tme n 1 and n +2;and$n at tme n and n +1. Fnd the present value of the paused ranbow (a) Fnd the present value of a perpetuty of $1 at tme 0, $2 at tme 3, $3 at tme 6, $4 at tme 9, at an effectve rate of nterest of 3%. [Hnt: Use the result n Exercse 2.25.] (b) Hence, fnd the present value of a perpetuty whch pays $1 at the end of the 3rd, 4th and 5th year, $2 at the end of the 6th, 7th and 8th year, $3 at the end of the 9th, 10th, 11th year and so on, at an effect rate of nterest of 3% Express (D s) n n terms of and n under smple nterest. [Hnt: The symbol (D s) n s defned smlarly as n Exercse 2.25.] 2.40 Erc makes deposts nto a retrement fund earnng an annual effectve rate of 7%. The frst depost of $1,000 s made on hs 38th brthday and the last depost s made on hs 64th brthday. Every year hs depost ncreases by 3%. When he attans age 65, he wll wthdraw all the money n the retrement fund to purchase an annuty-mmedate whch provdes hm wth monthly payment for 25 years. The nomnal rate of nterest on the annuty s convertble monthly at 6%. Fnd the amount of each monthly payment Albert purchased a house for $1,500,000 ffteen years ago. He put 30% down and fnanced the balance by a 20-year real estate mortgage at 6%, convertble monthly. Albert decdes to pay the remanng loan balance n full by a sngle payment together wth the nstallment just due. Fnd the prepayment penalty, whch s one-thrd of the lender s nterest loss Prove (2.43) and (2.44) An nvestor wshes to accumulate $1,000 at the end of year 5. He makes level deposts at the end of each year. The deposts earn a 6% annual effectve rate of nterest, whch s credted at the end of each year. The nterests on the deposts earn 5% effectve nterest rate annually. How much does he have to depost each year? 2.44 A prncpal of $20,500 generates ncome of $3,000 at the end of every 2 years at an effectve rate of nterest of 3% per annum for as long as possble. Calculate the term of the annuty and dscuss the possbltes of settlng the last payment.

32 70 CHAPTER You wsh to accumulate $50,000 by semannual deposts of $800 commencng at tme 0 and for as long as necessary. The nomnal rate of nterest s 6% convertble monthly. Fnd how many regular deposts are necessary An nvestment of $1,000 s to be used to make payments of $15 at the end of the frst year, $30 at the end of the second year, $45 at the end of the thrd year, etc., every year for as long as possble. A drop payment s pad 1 year after the last regular payment. Calculate the tme and the amount of the drop payment at an annual rate of nterest of 4%. [Hnt: Use a tral-and-error approach, but start wth a reasonable number of payments.] 2.47 Use the Excel RATE functon to solve for the effectve nterest rate of the followng: (a) a 18 =11, (b) ä 18 =11, (c) s 18 =28, (d) s 18 =28. [Hnt: Refer to the Excel Help Menu for the full specfcaton of the RATE functon.] Advanced Problems 2.48 If ā t = t 1+t for t 0, fndδ(t). [Hnt: Dfferentate ā t wth respect to t.] 2.49 Draw tme dagrams and gve nterpretatons to the followng two formulas: (a) n t=1 s t =(Is) n, (b) n t=1 a t =(Da) n. Hence, prove formulas (2.37) and (2.40) Phlp borrowed a loan of $12,000 for 29 years and fnanced t by 15 level payments payable at the end of every alternate year, wth the frst payment due at the end of the frst year. The effectve annual nterest rate s 4%. Just before the ffth payment was due, Phlp lost hs job and the lender reduced the level payment by extendng the loan for 6 more years. (a) Fnd the amount of the orgnal nstallments. (b) Fnd the outstandng loan balance just before the ffth payment s due.

33 Annutes 71 (c) Fnd the amount of extra nterest that Phlp has to pay because of the extenson of the loan Ada, Betty and Chara receve an nhertance n the form of a seres of level payments of $4,000 at the end of every year. They decde to let Ada start recevng the payments frst, followed by Betty and fnally Chara such that they receve the same present value at an annual nterest of 4%. (a) How many regular payments should Ada receve and how large s the fnal rregular payment? (b) Descrbe the payment pattern of Betty A ranbow s a cash flow stream whch pays $1 at tme 1 and 2n 1, $2at tme 2 and 2n 2,, $(n 1) at tme n 1 and n +1,and$n at tme n for n>1. (a) Wth the use of a tme dagram, argue that the current value at tme n of a ranbow s s n ä n. (b) By usng the formula x + y 2 xy for x, y 0, show that v k +(1+ ) k 2 for any postve nteger k. (c) Hence or otherwse, show that s n ä n n 2. When does the equalty hold? What s the sgnfcance of ths result? 2.53 It s gven that s n = x and s 2n = y. Express s kn n terms of x and y for k beng a postve nteger.