Secton 5.3 Annutes, Future Value, and Snkng Funds Ordnary Annutes A sequence of equal payments made at equal perods of tme s called an annuty. The tme between payments s the payment perod, and the tme from the begnnng of the frst payment perod to the end of the last perod s called the term of the annuty. Annutes can be used to accumulate funds for example, when you make regular deposts n a savngs account. Or they can be used to pay out funds as when you receve regular payments from a penson plan after you retre. Annutes that pay out funds are consdered n the next secton. Ths secton deals wth annutes n whch funds are accumulated by regular payments nto an account or nvestment that earns compound nterest. The future value of such an annuty s the fnal sum on depost that s, the total amount of all deposts and all nterest earned by them. We begn wth ordnary annutes ones where the payments are made at the end of each perod and the frequency of payments s the same as the frequency of compoundng the nterest. EXAMPLE: $1500 s deposted at the end of each year for the next 6 years n an account payng 8% nterest compounded annually. Fnd the future value of ths annuty. Soluton: The Fgure below shows the stuaton schematcally. To fnd the future value of ths annuty, look separately at each of the $1500 payments. The frst $1500 s deposted at the end of perod 1 and earns nterest for the remanng 5 perods. The compound amount produced by ths payment s A = P(1+) n = 1500(1+.08) 5 = 1500(1.08) 5 The second $1500 payment s deposted at the end of perod 2 and earns nterest for the remanng 4 perods. So the compound amount produced by the second payment s 1500(1+.08) 4 = 1500(1.08) 4 Contnue to compute the compound amount for each subsequent payment, as shown n the Fgure below. Note that the last payment earns no nterest. 1
The last column of the Fgure above shows that the total amount after 6 years s the sum 1500+1500 1.08+1500 1.08 2 +1500 1.08 3 +1500 1.08 4 +1500 1.08 5 = 1500(1+1.08+1.08 2 +1.08 3 +1.08 4 +1.08 5 ) (1) It s known that Applyng ths algebrac fact to the expresson n parentheses (wth x = 1.08 and n = 6). It shows that the sum (the future value of the annuty) s 1500 1.086 1 1.08 1 = $11,003.89 Ths Example s the model for fndng a formula for the future value of any annuty. Suppose that a payment of R dollars s deposted at the end of each perod for n perods, at an nterest rate of per perod. Then the future value of ths annuty can be found by usng the procedure n the Example, wth these replacements: The future value S n the Example call t S s the sum (1), whch now becomes S = R [ 1+(1+)+(1+) 2 +...+(1+) n 2 +(1+) n 1] Apply the algebrac fact n the box above to the expresson n brackets (wth x = 1+). Then we have (1+) n 1 (1+) n 1 S = R = R (1+) 1 The quantty n brackets n the rght-hand part of the precedng equaton s sometmes wrtten s n (read s-angle-n at ). So we can summarze as follows. 2
EXAMPLE: A rooke player n the Natonal Football League just sgned hs frst 7-year contract. To prepare for hs future, he deposts $150,000 at the end of each year for 7 years n an account payng 4.1% compounded annually. How much wll he have on depost after 7 years? Soluton: Hs payments form an ordnary annuty wth R = 150,000, n = 7, and =.041. The future value of ths annuty s (1+) n 1 (1+.041) 7 1 S = R = 150,000.041 [ ] (1.041) 7 1 = 150,000 = $1,188,346.11.041 EXAMPLE: Allyson, a college professor, contrbuted $950 a month to the CREF stock fund (an nvestment vehcle avalable to many college and unversty employees). For the past 10 years ths fund has returned 4.25%, compounded monthly. (a) How much dd Allyson earn over the course of the last 10 years? Soluton: Allyson s payments form an ordnary annuty, wth monthly payment R = 950. The nterest per month s =.0425, and the number of months n 10 years s n = 10 12 = 120. 12 The future value of ths annuty s (1+) n 1 (1+.0425/12) 120 1 S = R = 950 = $141, 746.90.0425/12 (b) As of Aprl 14, 2013, the year to date return was 9.38%, compounded monthly. If ths rate were to contnue, and Allyson contnues to contrbute $950 a month, how much would the account be worth at the end of the next 15 years? Soluton: Deal separately wth the two parts of her account(the $950 contrbutons n the future and the $141,746.90 already n the account). The contrbutons form an ordnary annuty as n part (a). Now we have R = 950, =.0938/12, and n = 12 15 = 180. So the future value s (1+) n 1 (1+.0938/12) 180 1 S = R = 950 = $372, 068.65.0938/12 Meanwhle, the $141,746.90 from the frst 10 years s also earnng nterest at 9.38%, compounded monthly. By the compound amount formula, the future value of ths money s A = P(1+) n = 141,746.90(1+.0938/12) 180 = $575,691.85 So the total amount n Allyson s account after 25 years s the sum $372,068.65+$575,691.85 = $947,760.50 3
Snkng Funds A snkng fund s a fund set up to receve perodc payments. Corporatons and muncpaltes use snkng funds to repay bond ssues, to retre preferred stock, to provde for replacement of fxed assets, and for other purposes. If the payments are equal and are made at the end of regular perods, they form an ordnary annuty. EXAMPLE: A busness sets up a snkng fund so that t wll be able to pay off bonds t has ssued when they mature. If t deposts $12,000 at the end of each quarter n an account that earns 5.2% nterest, compounded quarterly, how much wll be n the snkng fund after 10 years? Soluton: The snkng fund s an annuty, wth R = 12,000, =.052/4, and n = 4(10) = 40. The future value s (1+) n 1 (1+.052/4) 40 1 S = R = 12,000 = $624, 369.81.052/4 So there wll be about $624,370 n the snkng fund. EXAMPLE: A frm borrows $6 mllon to buld a small factory. The bank requres t to set up a $200,000 snkng fund to replace the roof after 15 years. If the frm s deposts earn 6% nterest, compounded annually, fnd the payment t should make at the end of each year nto the snkng fund. Soluton: Ths stuaton s an annuty wth future value S = 200,000, nterest rate =.06, and n = 15. Solve the future-value formula for R: (1+) n 1 (1+.06) 15 1 S = R = 200,000 = R.06 hence R = 200, 000 [ ] (1+.06) 15 = 200,000 1 23.27597.06 = $8592.55. So the annual payment s about $8593. EXAMPLE: As an ncentve for a valued employee to reman on the job, a company plans to offer her a $100,000 bonus, payable when she retres n 20 years. If the company deposts $200 a month n a snkng fund, what nterest rate must t earn, wth monthly compoundng, n order to guarantee that the fund wll be worth $100,000 n 20 years? Soluton: The snkng fund s an annuty wth R = 200, n = 12(20) = 240, and future value S = 100,000. We must fnd the nterest rate. If x s the annual nterest rate n decmal form, then the nterest rate per month s = x/12. Insertng these values nto the future-value formula, we have (1+) n 1 (1+x/12) 240 1 R = S = 200 = 100,000 x/12 Ths equaton s hard to solve algebracally. You can get a rough approxmaton by usng a calculator and tryng dfferent values for x. Wth a graphng calculator, you can get an accurate soluton by graphng [ ] (1+x/12) 240 1 y 1 = 200 and y 2 = 100,000 x/12 and fndng the x-coordnate of the pont where the graphs ntersect. The Fgure on the rght shows that the company needs an nterest rate of about 6.661%. 4
Annutes Due The formula developed prevously s for ordnary annutes annutes wth payments at the end of each perod. The results can be modfed slghtly to apply to annutes due annutes where payments are made at the begnnng of each perod. An example wll llustrate how ths s done. Consder an annuty due n whch payments of $100 are made for 3 years, and an ordnary annuty n whch payments of $100 are madefor4years,bothwth5%nterest,compounded annually. The Fgure on the rght computes the growth of each payment separately. The Fgure shows that the future values are thesame, exceptforone$100paymentonthe ordnary annuty (shown n red). So we can use the Future Value of an Ordnary Annuty formula [ ] (1+) n 1 S = R tofndthefuturevalueofthe4-yearordnary annuty and then subtract one $100 payment to get the future value of the 3-year annuty due: Essentally the same argument works n the general case. 5
EXAMPLE: Payments of $500 are made at the begnnng of each quarter for 7 years n an account payng 8% nterest, compounded quarterly. Fnd the future value of ths annuty due. Soluton: In 7 years, there are n = 28 quarterly perods. For an annuty due, add one perod to get n+1 = 29, and use the formula wth =.08/4 =.02: (1+) n+1 1 (1+.02) 29 1 S = R R = 500 500 = $18,896.12.02 After 7 years, the account balance wll be $18,896.12. EXAMPLE: Jay Rechten plans to have a fxed amount from hs paycheck drectly deposted nto an account that pays 5.5% nterest, compounded monthly. If he gets pad on the frst day of the month and wants to accumulate $13,000 n the next three-and-a-half years, how much should he depost each month? Soluton: Jay s deposts form an annuty due whose future value s S = 13, 000. The nterest rate s =. There are 3 12+6 = 42 months n three-and-a-half years. Snce ths s an annuty due, add one perod, so that n+1 = 43. Then solve the future-value formula for the payment R: [ ] (1+) n+1 1 R R = S [ ] (1+) 43 1 R R = 13,000 [ ] (1+) 43 1 R R 1 = 13,000 ([ ] (1+) 43 1 R ) 1 = 13,000 therefore R = 13, 000 ([ ] (1+) 43 1 ) = 13,000 46.4103 = 280.110 1 Jay should have $280.11 deposted from each paycheck. 6