# Lecture 3: Annuity. Study annuities whose payments form a geometric progression or a arithmetic progression.

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1 Lecture 3: Annuty Goals: Learn contnuous annuty and perpetuty. Study annutes whose payments form a geometrc progresson or a arthmetc progresson. Dscuss yeld rates. Introduce Amortzaton Suggested Textbook Readngs: Chapter 2: 2.2, 2.3, and Practce Problems: Secton 2.2: 1-5, 7-10, Secton 2.3: 1-5, 7-8, 10 Secton 2.4: 1(a) part (), (b) part (), 2, practce examples 2.24, 2.25 n the text book.

2 Lecture 3: More Annutes 2 Dfferng nterest and payment perods It may happen that the quoted nterest rate has a compoundng perod that doesn t concde wth the annuty payment perod. For the purpose of evaluaton we can fnd the nterest rate per payment perod that s equvalent to the quoted nterest rate, or fnd the equvalent payment n each quoted nterest perod. Example 1: (Example 2.12 (a)) On the last day of March, June, September and December, Smth makes a depost of \$1000 nto a savng account that earns nomnal rate (12) = 9%. The frst depost s on Mar 31, 1995 and the last s December 31, What s the balance on January 1, 2011? m-thly payable annutes Example 2: (Example 2.12 (b)) In the above example, f the nterest rate s quoted at an effectve annual rate of 10%, what s the balance n Smth s account on January 1, 2011? m thly payable annuty-mmedate If the effectve annual nterest rate s, and m payments of X are made each year, then the accumulated value over n years s Ks (m) n = K (1 + )n 1 (m) = Ks n (m) where K = mx. The present value of ths seres of payments s Ka (m) n = Ka n (m)

3 Lecture 3: More Annutes 3 Perpetutes If an annuty has no end pont, t s called a perpetuty. We cannot fnd the future value of a perpetuty, but we can always calculate the present value. Annuty-mmedate: a = lm n a n = 1 Smlarly, for annuty-due: ä = 1 d and for m thly payable annuty-mmedate: a (m) = 1 (m) Contnuous Annuty If payments are made more frequently, t s more convenent to approxmate the calculaton by assumng the payments are made contnuously. The accumulated value of the contnuous annuty, pad at 1 per perod for n perods, denoted by s n, s s n = δ s n Smlarly a n = δ a n. If accumulaton s based on force of nterest δ r, then and s n δr = n 0 a n δr = n 0 e R t 0 δr dr dt e R n t δr dr dt. Also s n δr = a n δr e R t 0 δr dr.

4 Lecture 3: More Annutes 4 Geometrc Progresson Sometmes the annuty payment s adjusted perodcally for nflaton. Such an annuty would have payments that ncrease geometrcally. Example 3: (Example 2.17, page 110) Smth wshes to purchase a 20-year annuty at an effectve annual rate of 11% wth annual payments begnnng one year from now. Smth antcpates an effectve annual nflaton rate over the next 20 years s 4%, so he would lke each payment after the frst to be 4% larger than the prevous one. If Smth s frst payment s 26000, what s the present value of the annuty? Suppose a seres of n perodc payments has a frst payment of amount 1 and all subsequent payments are (1 + r) tmes the prevous payment. At a rate per payment perod, one perod before the frst payment, what s the present value, and what s the accumulated value at the tme of the fnal payment? The present value of the seres s 1 ( ) 1+r n 1+ r and the accumulated value at the tme of the last payment s 1 ( 1+r 1+ r ) n (1 + ) n = (1 + )n (1 + r) n r Queston: What s the present value of the above seres f r =?

5 Lecture 3: More Annutes 5 Dvdend Dscount Model The value of a share of stock s the present value of the future dvdends that wll be pad on the stock. In general we assume a constant rate of ncrease n the amount of the dvdend pad, so that the future stream of dvdends forms a geometrc payment perpetuty. Example 4: Common Stock X pays a dvdend of 50 at the end of the frst year, wth each subsequent annual dvdend beng 5% greater than the preceddng one. Suppose the effectve annual nterest rate s 10%. What s the theoretcal prce of the stock? If the next dvdend payable one year from now s of amount K, the annual compound growth rate of the dvdend s r, and the nterest rate used for calculatng present value s, the present value one payment perod before the frst dvdend payment s ( 1+r 1+ K [ v + (1 + r)v 2 + (1 + r) 2 v 3 + ] 1 = K lm n r ) n = K r under the assumpton that > r. Ths s usually referred to as the theoretcal prce of the stock.

6 Lecture 3: More Annutes 6 Arthmetc Progresson Example 5: (Increasng annutes) Consder an annuty whose frst payment s 1. If each subsequent payment ncreases by 1 for n perods, wth an nterest rate per perod and equally spaced payments, what s the present value of the seres of payments one perod before the frst payment? The present value s (Ia) n = än nv n and the accumulated value at the fnal payment s (Is) n = s n n Increasng perpetuty mmedate If the payments n an ncreasng annuty mmedate are allowed to contnue forever, the present value s (Ia) = lm n ä n nv n = Example 6: (Decreasng annutes) Consder an annuty whose frst payment s n. If each subsequent payment s of amount 1 less than the prevous payment, wth an nterest rate per perod, what s the present value of the seres of payments one perod before the frst payment? The present value s (Da) n = n a n and the accumulated value at the fnal payment s (Ds) n = n(1 + )n s n = (Da) n (1 + ) n

7 Lecture 3: More Annutes 7 Example 7: Mary make deposts of 1000 nto an account wth an effectve annual rate of 10% at the end of each year for 5 years. Each year just after the nterest s credted, Mary wthdraw only the nterest and redepost t nto a second account wth an effectve annual nterest rate of 8%. What s the value of Mary s nvestment at the end of 5 years? Yeld Rates The nterest rate earned by the lender s referred to yeld rate earned on the nvestment. The expresson yeld rate s used n dfferent nvestment contexts wth dfferent meanngs. In each case t wll be mportant to relate the meanng of the yeld rate to the context n whch t s beng used. Example 8: Consder a 10-year loan of at = 5%. Fnd the yeld rate n each of the followng cases. 1. The loan s repad by a sngle payment at the end of 10 years. 2. The loan s repad by 10 equal annual payments wth the frst payment one year from now. Then the payments are renvested at 3% as they are receved. 3. The loan receve 10 level nterest payments of 500 per year at the end of each year, plus a return of the entre prncpal at the end of ten years. The nterest payments are also renvested at 3%.

8 Lecture 3: More Annutes 8 Amortzaton Method Defnton: An amortzed loan of amount L made at tme 0 at perodc nterest rate and to be repad by n payments of amounts K 1, K 2,, K n at tmes 1, 2,, n (where the payment perod corresponds to the nterest perod) s based on the equaton L = K 1 v + K 2 v K n v n 1 The amortzaton method of loan repayment apples payments frst to nterest, wth excess payment appled to outstandng prncpal. Amortzaton Schedule t Payment Interest Due Prncpal Repad Outstandng Prncpal 0 L 1 2. t t+1. n 0 At tme t, Interest due n the payment s I t Outstandng balance just after the payment s OB t Prncpal repad n the payment s P R t Example 9: Smth takes out a \$100, 000 mortgage to buy a house. The mortgage s repad wth monthly payments of \$716.43, startng one month after the mortgage begns, for 20 years at a nomnal rate of (12) = 6%. How much of the second payment s used to repay nterest?

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