Section 5.4 Annuities, Present Value, and Amortization


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1 Secton 5.4 Annutes, Present Value, and Amortzaton Present Value In Secton 5.2, we saw that the present value of A dollars at nterest rate per perod for n perods s the amount that must be deposted today (at the same nterest rate) n order to produce A dollars n n perods. Smlarly, the present value of an annuty s the amount that must be deposted today (at the same compound nterest rate as the annuty) to provde all the payments for the term of the annuty. It does not matter whether the payments are nvested to accumulate funds or are pad out to dsperse funds; the amount needed to provde the payments s the same n ether case. We begn wth ordnary annutes. EXAMPLE: Your rch aunt has funded an annuty that wll pay you $1500 at the end of each year for sx years. If the nterest rate s 8%, compounded annually, fnd the present value of ths annuty. Soluton: Look separately at each payment you wll receve. Then fnd the present value of each payment the amount needed now n order to make the payment n the future. The sum of these present values wll be the present value of the annuty, snce t wll provde all of the payments. To fnd the frst $1500 payment (due n one year), the present value of $1500 at 8% annual nterest s needed now. Accordng to the presentvalue formula for compound nterest wth A = 1500, =.08, and n = 1, ths present value s A P = (1+) = 1500 n (1+.08) = = 1500( ) $ Ths amount wll grow to $1500 n one year. For the second $1500 payment (due n two years), we need the present value of $1500 at 8% nterest, compounded annually for two years. The presentvalue formula for compound nterest (wth A = 1500, =.08, and n = 2) shows that ths present value s A P = (1+) = 1500 n (1+.08) = = ( ) $ Less money s needed for the second payment because t wll grow over two years nstead of one. A smlar calculaton shows that the thrd payment (due n three years) has present value $1500( ). Contnue n ths manner to fnd the present value of each of the remanng payments, as summarzed n the Fgure below. 1
2 The lefthand column of the Fgure above shows that the present value s = 1500( ) Now apply the algebrac fact that (1) to the expresson n parentheses (wth x = 1.08 and n = 6). It shows that the sum (the present value of the annuty) s [ ] [ ] = 1500 = $ Ths amount wll provde for all sx payments and leave a zero balance at the end of sx years (gve or take a few cents due to roundng to the nearest penny at each step). The Example above s the model for fndng a formula for the future value of any ordnary annuty. Suppose that a payment of R dollars s made at the end of each perod for n perods, at nterest rate per perod. Then the present value of ths annuty can be found by usng the procedure n that Example, wth these replacements: The future value n the Example above s the sum n equaton (1), whch now becomes P = R[(1+) 1 +(1+) 2 +(1+) (1+) ] Apply the algebrac fact n the box above to the expresson n brackets (wth x = 1+). Then we have [ ] [ ] 1 (1+) 1 (1+) P = R = R (1+) 1 The quantty n brackets n the rghthand part of the precedng equaton s sometmes wrtten a n (read aanglen at ). So we can summarze as follows. 2
3 CAUTION: Do not confuse the formula for the present value of an annuty wth the one for the future value of an annuty. Notce the dfference: The numerator of the fracton n the presentvalue formula s 1 (1+), but n the futurevalue formula, t s (1+) n 1. EXAMPLE: Jm Rles was n an auto accdent. He sued the person at fault and was awarded a structured settlement n whch an nsurance company wll pay hm $600 at the end of each month for the next seven years. How much money should the nsurance company nvest now at 4.7%, compounded monthly, to guarantee that all the payments can be made? Soluton: The payments form an ordnary annuty. The amount needed to fund all the payments s the present value of the annuty. Apply the presentvalue formula wth R = 600, n = 7 12 = 84, and =.047/12 (the nterest rate per month). The nsurance company should nvest [ ] [ ] 1 (1+) 1 (1+.047/12) 84 P = R = 600 = $42, /12 EXAMPLE: To supplement hs penson n the early years of hs retrement, Ralph Taylor plans to use $124,500 of hs savngs as an ordnary annuty that wll make monthly payments to hm for 20 years. If the nterest rate s 5.2%, how much wll each payment be? Soluton: The present value of the annuty s P = $124, 500, the monthly nterest rate s =.052/12, and n = = 240 (the number of months n 20 years). Solve the presentvalue formula for the monthly payment R: [ ] 1 (1+) P = R [ ] 1 (1+.052/12) ,500 = R.052/12 R = 124, 500 [ ] 1 (1+.052/12) 240 = $ /12 Taylor wll receve $ a month (about $10,026 per year) for 20 years. EXAMPLE: Surnder Snah and Mara Gonzalez are graduates of Kenyon College. They both agree to contrbute to an endowment fund at the college. Snah says he wll gve $500 at the end of each year for 9 years. Gonzalez prefers to gve a sngle donaton today. How much should she gve to equal the value of Snahs gft, assumng that the endowment fund earns 7.5% nterest, compounded annually? Soluton: Snah s gft s an ordnary annuty wth annual payments of $500 for 9 years. Its future value at 7.5% annual compound nterest s [ ] [ ] [ ] (1+) n 1 (1+.075) S = R = 500 = 500 = $ We clam that for Gonzalez to equal ths contrbuton, she should today contrbute an amount equal to the present value of ths annuty, namely, [ ] [ ] [ ] 1 (1+) 1 (1+.075) P = R = 500 = 500 = $
4 To confrm ths clam, suppose the present value P = $ s deposted today at 7.5% nterest, compounded annually for 9 years. Accordng to the compound nterest formula, P wll grow to A = P(1+) n = (1+.075) 9 = $ the future value of Snah s annuty. So at the end of 9 years, Gonzalez and Snah wll have made dentcal gfts. The Example above llustrates the followng alternatve descrpton of the present value of an accumulaton annuty. Corporate bonds, whch were ntroduced n Secton 5.1, are routnely bought and sold n fnancal markets. In most cases, nterest rates when a bond s sold dffer from the nterest rate pad by the bond (known as the coupon rate). In such cases, the prce of a bond wll not be ts face value, but wll nstead be based on current nterest rates. The next example shows how ths s done. EXAMPLE:A15year$10,000 bondwtha5%couponratewasssuedfveyearsagoandsnow beng sold. If the current nterest rate for smlar bonds s 7%, what prce should a purchaser be wllng to pay for ths bond? Soluton: Accordng to the smple nterest formula, the nterest pad by the bond each halfyear s I = Prt = 10, = $250 Thnk of the bond as a twopart nvestment: The frst s an annuty that pays $250 every sx months for the next 10 years; the second s the $10,000 face value of the bond, whch wll be pad when the bond matures, 10 years from now. The purchaser should be wllng to pay the present value of each part of the nvestment, assumng 7% nterest, compounded semannually. The nterest rate per perod s =.07/2, and the number of sxmonth perods n 10 years s n = 20. So we have: Present value of annuty [ ] 1 (1+) P = R [ ] 1 (1+.07/2) 20 = /2 = $ Present value of $10,000 n 10 years P = A(1+) = 10,000(1+.07/2) 20 = $ So the purchaser should be wllng to pay the sum of these two present values: $ $ = $
5 Loans and Amortzaton If you take out a car loan or a home mortgage, you repay t by makng regular payments to the bank. From the bank s pont of vew, your payments are an annuty that s payng t a fxed amount each month. The present value of ths annuty s the amount you borrowed. EXAMPLE: Chase Bank n Aprl 2013 advertsed a new car auto loan rate of 2.23% for a 48month loan. Shelley Fasulko wll buy a new car for $25,000 wth a down payment of $4500. Fnd the amount of each payment. (Data from: Soluton: After a $4500 down payment, the loan amount s $20,500. Use the present value formula for an annuty, wth P = 20,500, n = 48, and =.0223/12 (the monthly nterest rate). Then solve for payment R. [ ] 1 (1+) P = R [ ] 1 ( /12) 48 20,500 = R.0223/12 R = 20, 500 [ ] 1 ( /12) 48 = $ /12 A loan s amortzed f both the prncpal and nterest are pad by a sequence of equal perodc payments. The perodc payment needed to amortze a loan may be found, as n the Example above, by solvng the presentvalue formula for R. EXAMPLE: In Aprl 2013, the average rate for a 30year fxed mortgage was 3.43%. Assume a down payment of 20% on a home purchase of $272,900. (Data from: Fredde Mac.) (a) Fnd the monthly payment needed to amortze ths loan. Soluton: Thedownpayments.20(272,900) = $54,580. Thus,theloanamountP s$272,900 $54,580 = $218,320. We can now apply the formula n the precedng box, wth n = 12(30) = 360 (the number of monthly payments n 30 years), and monthly nterest rate =.0343/12. R = P (218,320)(.0343/12) = = $ (1+) 1 ( /12) 360 Monthly payments of $ are requred to amortze the loan. 5
6 (b) After 10 years, approxmately how much s owed on the mortgage? Soluton: You may be tempted to say that after 10 years of payments on a 30year mortgage, the balance wll be reduced by a thrd. However, a sgnfcant porton of each payment goes to pay nterest. So, much less than a thrd of the mortgage s pad off n the frst 10 years, as we now see. After 10 years (120 payments), the 240 remanng payments can be thought of as an annuty. The present value for ths annuty s the (approxmate) remanng balance on the mortgage. Hence, we use the presentvalue formula wth R = , =.0343/12, and n = 240: [ ] 1 ( /12) 240 P = = $168, (.0343/12) So the remanng balance s about $168, The actual balance probably dffers slghtly from ths fgure because payments and nterest amounts are rounded to the nearest penny. The Example above, part (b), llustrates an mportant fact: Even though equal payments are made to amortze a loan, the loan balance does not decrease n equal steps. The method used to estmate the remanng balance n the Example works n the general case. If n payments are needed to amortze a loan and x payments have been made, then the remanng payments form an annuty of n x payments. So we apply the presentvalue formula wth n x n place of n to obtan ths result. Amortzaton Schedules The remanngbalance formula s a quck and convenent way to get a reasonable estmate of the remanng balance on a loan, but t s not accurate enough for a bank or busness, whch must keep ts books exactly. To determne the exact remanng balance after each loan payment, fnancal nsttutons normally use an amortzaton schedule, whch lsts how much of each payment s nterest, how much goes to reduce the balance, and how much s stll owed after each payment. EXAMPLE: Beth Hll borrows $1000 for one year at 12% annual nterest, compounded monthly. (a) Fnd her monthly payment. Soluton: Apply the amortzaton payment formula wth P = 1000, n = 12, and monthly nterest rate =.12/12 =.01. Her payment s R = P 1 (1+) = 1000(.01) = $ (1+.01) 12 (b) After makng fve payments, Hll decdes to pay off the remanng balance. Approxmately how much must she pay? 6
7 (b) After makng fve payments, Hll decdes to pay off the remanng balance. Approxmately how much must she pay? Soluton: Apply the remanngbalance formula just gven, wth R = 88.85, =.01, and n x = 12 5 = 7. Her approxmate remanng balance s [ ] [ ] 1 (1+) (n x) 1 (1+.01) 7 B = R = = $ (c) Construct an amortzaton schedule for Hll s loan. Soluton: An amortzaton schedule for the loan s shown n the table below. It was obtaned as follows: The annual nterest rate s 12% compounded monthly, so the nterest rate per month s 12%/12 = 1% =.01. When the frst payment s made, one month s nterest, namely,.01(1000) = $10, s owed. Subtractng ths from the $88.85 payment leaves $78.85 to be appled to repayment. Hence, the prncpal at the end of the frst payment perod s = $921.15, as shown n the payment 1 lne of the table. When payment 2 s made, one month s nterest on the new balance of $ s owed, namely,.01(921.15) = $9.21. Contnue as n the precedng paragraph to compute the entres n ths lne of the table. The remanng lnes of the table are found n a smlar fashon. Note that Hll s remanng balance after fve payments dffers slghtly from the estmate made n part (b). The fnal payment n the amortzaton schedule n the last Example, part (c), dffers from the other payments. It often happens that the last payment needed to amortze a loan must be adjusted to account for roundng earler and to ensure that the fnal balance wll be exactly 0. 7
8 Annutes Due We want to fnd the present value of an annuty due n whch 6 payments of R dollars are made at the begnnng of each perod, wth nterest rate per perod, as shown schematcally n the Fgure on the rght. The present value s the amount needed to fund all 6 payments. Snce the frst payment earns no nterest, R dollars are needed to fund t. Now look at the last 5 payments by themselves n the Fgure on the rght. If you thnk of these 5 payments as beng made at the end of each perod, you see that they form an ordnary annuty. The money needed to fund them s the present value of ths ordnary annuty. So the present value of the annuty due s gven by [ ] Present value of the ordnary 1 (1+) 5 R+ = R+R annuty of 5 payments Replacng 6 by n and 5 by n 1, and usng the argument just gven, produces the general result that follows. EXAMPLE: The Illnos Lottery Wnner s Handbook dscusses the optons of how to receve the wnnngs for a $12 mllon Lotto jackpot. One opton s to take 26 annual payments of approxmately $461,538.46, whch s $12 mllon dvded nto 26 equal payments. The other opton s to take a lumpsum payment (whch s often called the cash value ). If the Illnos lottery commsson can earn 4.88% annual nterest, how much s the cash value? Soluton: The yearly payments form a 26payment annuty due. An equvalent amount now s the present value of ths annuty. Apply the presentvalue formula wth R = 461,538.46, =.0488, and n = 26: [ ] 1 (1+) (n 1) P = R+R [ ] 1 ( ) 25 = 461, , = $7,045, The cash value s $7,045,
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