A Master Tme Value of Money Formula Floyd Vest For Fnancal Functons on a calculator or computer, Master Tme Value of Money (TVM) Formulas are usually used for the Compound Interest Formula and for Annutes. (See Formula 7 below. See the Appendx n the TI83 (p. A55) or TI84 manuals. The manuals are almost dentcal for fnance. You can download parts of a manual at www.t.com/calc.) The followng s a dervaton of the TVM Formula for Future Value (FV) wth examples and exercses. You Try It #1 Check the Appendx, Fnancal Functons n the TI83/84 manual and fnd the Tme Value of Money Formula for FV. Compare t to Formula 7 n ths artcle. Compound Interest Formula. Frst we need to derve the Compound Interest Formula, whch s (1) FV = PV(1 + ), where FV s the future value, PV s the present value, s the nterest rate per compoundng perod, and s the number of compoundng perods. We are usng the termnology n the TI83 and TI84 manuals. Example 1 For an example we calculate the FV for PV = $100 nvested for ten years at the annual nomnal rate of 4%, compounded quarterly. The rate per compoundng perod s = 0.04 = (annual nomnal rate) (number of compoundng perods per year). s the 4 number of compoundng perods, whch s 10 4 = 40 quarterly perods. So FV = 100 1+ 0.04 40! $ # 4 % & = $148.89. The nvestment of $100 grew to $148.89 n ten years. You Try It #2 For Example 1 above, what s the Future Value f 4% s compounded daly? Money Market Funds usually compound daly. A Master Tme Value of Money Formula Sprng, 2011 1
For the dervaton, consder that compound nterest earns nterest on nterest as recorded n the followng table: Begnnng PV Interest FV at end of perod Perod 1 PV (PV) PV(1 + ) Perod 2 PV(1 + ) [PV(1 + )] PV(1 + ) PV(1 + ) = PV(1 + ) 2 Perod 3 PV(1 + ) 2 [PV(1 + ) 2 ] PV(1 + ) 3 Perod PV(1 + ) 1 [PV(1 + ) 1 ] PV(1 + ) We conclude, from the last lne of the table, the Compound Interest Formula for FV to be FV = PV(1 + ) as descrbed above. Future Value of an Ordnary Annuty. To contnue the dervaton of the TVM Formula, we need the formula for the Future Value of an Ordnary Annuty of payments (PMT), each drawng compound nterest at the rate per perod as llustrated on the followng tmelne. 0 1 2 3... 1 PMT PMT PMT PMT PMT By the Compound Interest Formula, the FV of the Ordnary Annuty s the accumulaton of PMTs and the nterest on each: FV = PMT(1 + ) 1 + PMT(1 + ) 2 + PMT(1 + ) 3 +... + PMT(1 + ) 1 + PMT. Multplyng through by (1 + ) we get (1 + )FV = PMT(1 + ) + PMT(1 + ) 1 + PMT(1 + ) 2 + + PMY(1 + ) 2 + PMT(1 + ). ext we subtract to get (1 + )FV FV = PMT(1 + ) PMT, wth ntermedate terms subtractng out. Solvng for FV we have (2) FV = PMT (1 ) + 1 as the formula for the Future Value of an Ordnary Annuty where PMT s the payments over perods, each drawng compound nterest at the rate per perod. For an Ordnary Annuty, the PMTs occur at the end of perods. A Master Tme Value of Money Formula Sprng, 2011 2
Example 2 Consder payments of $500 = PMT nvested at the end of each year for = 30 years at the rate = 6% = 0.06 per year. The Future Value of the savngs s FV = 500 (1+ 0.06)30!1% $ ' = $39,529.09. # 0.06 & The amount nvested was 30 500 = $15,000 n PMTs to accumulate to $39,529.09. You Try It #3 For Example 2 above, what s the Future Value of $500 per month for thrty years n a retrement nvestment program? If there s also a PV nvested at tme 0 on the tme lne, then (3) FV = PMT (1 ) + 1 + PV(1 + ). See the Exercses for an example. Future Value of an Annuty Due. Consder payments PMT occurrng at the begnnng of each perod as llustrated on the followng tmelne: 0 1 2 3... 1 PMT PMT PMT PMT PMT In ths case each PMT draws compound nterest for an addtonal perod and accumulates to the FV so (1 + ) 1 (4) FV = (1 + ) PMT + PV (1 + ) for an Annuty Due. For the Master TVM Formula, the followng formula s used for two cases: k = 0 f end of perod payments, k = 1 f begnnng of perod payments, wth G = 1 + (k), where s the nterest rate per perod, to gve (5) (1 + ) 1 FV = G PMT + PV (1 + ) ote that for end of perod payments, G = 1 + (0) = 1 gvng Formula 3; and for begnnng of perod payments G = 1 + 1() = 1 + gvng Formula 4. A Master Tme Value of Money Formula Sprng, 2011 3
Algebrac manpulaton s conducted on Formula 5 to get the TVM Formula for FV to be (6) FV = PMT G PMT G (1 ) PV + + +. The algebra s left as an exercse. Cash Flow Sgn Conventons. One property of the fnal Master TVM Formula s that sgn changes are made to accommodate Cash Flow Sgn Conventons. For example consder a Cash Flow tmelne wth PV and PMTs negatve and FV postve. 0 1 2 3... 1 FV PV PMT PMT PMT PMT PMT In an nvestment of PV and PMTs, money gong out s consdered negatve, and ndcated under the tmelne, and FV comng n at the end s postve, and ndcated wth an upward arrow on the tmelne. Master TVM Formula. To accommodate ths sgn conventon, we wll n Formula 6 change PMT to PMT and PV to PV to get Formula 7. PMT (7) FV = G PMT G (1 ) PV + +, where we have an annuty wth payments PMT, nterest rate per perod of, perods, and a PV at tme zero. G s descrbed above. Ths Master TVM Formula for FV s n the manuals for the TI83 and TI84. Fnancal functons on other calculators may have dfferent sgn conventons. (See the TIBA35, TIBAII, and HP19B.) Example 3 We wll do an example for Formula 7. Consderng the above tmelne, we nvest PV = $100,000, PMT of $10,000 at the end of each year for = 25 years n a retrement program averagng 6% per year. otce the sgn conventons. What s the FV of the savngs program? Substtutng we have G = 1 snce we have end of perod PMTs. FV =!10,000 0.06! (1+ 0.06) 25!100,000 +!10,000 % $ # 0.06 '. FV = $977,832.19. & You Try It #4 For Example 3 above, nvestments earn 6% compounded monthly and the famly has $200,000 n nvestments and are savng $500 at the end of each month for the next 25 years. How much s n ther retrement fund? A Master Tme Value of Money Formula Sprng, 2011 4
To get the TVM Formulas for PV and PMT from Formula 7, you smply use algebra to solve. To solve for, you can use logarthms. These solutons are left as exercses. To solve for for an Annuty requres an teratve program. See the references for some teratve calculator programs. To get the Compound Interest Formula from Formula 7, smply let PMT = 0, and remember the sgn changes. For the Compound Interest Formula, you can solve algebracally for. The concept of dscountng and PVOA. From the Compound Interest Formula, FV = PV(1 + ) we get PV = FV(1 + I). Ths s referred to as dscountng FV to get PV. For an Ordnary Annuty, we can dscount each of the PMTs to get a PV of an Ordnary Annuty (PVOA) as the sum, PVOA = 1 PMT (1 + ) 2 + PMT (1 + ) 3 + PMT (1 + ) ( 1) + + PMT (1 + ) + PMT (1 + ). It left as an exercse to show that the PVOA of an Ordnary Annuty s (8) PVOA = PMT 1 (1 ) +, and to derve a formula for the Present Value of an Annuty Due (PVAD). The concept of dscountng s mportant because t comes up n loans, mortgages, long term fnancal plannng, nflaton, bond prcng, dscountng Cash Flows to Intal Equty or et Present Value n busness plannng, Yeld to Maturty (YTM), Internal Rate of Return (IRR), other applcatons. ote here that the term PV has taken on two meanngs. Examples for Exercses The followng s an example of the format of some of the applcaton problems n the Exercses requestng formulas, answers, commentary, tmelnes, knowns, unknowns, varables, formulas, Fnancal Functons, code, and summary. Instructons: Solve the followng problems wth formulas, a scentfc calculator pad, lst and label knowns, unknowns, draw a tme lne. Use any fnancal formulas that are convenent. Then, solve wth Fnancal Functons on a calculator or computer. See the calculator manual for the requred code. The problems are taken from Chapter 14 of the TI83 and TI84 manuals. Wrte code and commentary. Identfy the output, and put on unts. Summarze the answer to the problem. Problem: Consder a problem on p. 143 of the TI83 manual. A car costs $9000. You can afford monthly payments of $250 a month for four years. What (APR) annual percentage rate wll make ths possble? A Master Tme Value of Money Formula Sprng, 2011 5
Answer: From the problem, = 4 12 = 48 months. PV = $9000. PMT = $250 a month. FV = 0. P/Y = 12. Pmt:End snce we have an ordnary annuty. The queston s, what s I%? Usng Formula 8 for the PV of an Ordnary Annuty, we have 48 1 (1 + ) 9000 = 250, where s the monthly rate as a decmal and APR = 12. There s no closed form formula for solvng for for an annuty. TI83 code and commentary for the problem: 2 nd Fnance Enter (To select TVM Solver.) ()250 Enter 0 Enter 12 Enter Enter (To select Pmt:End for payments.) Press (To hghlght the I% entry.) Press Alpha Solve (To solve for I%.) (You see a cursor on I%.) You read I% = 14.90%. Summary: Wth an APR of I% 14.90%, the purchaser can afford the car. (I% = 100() (the number of compoundng perods per year).) A Master Tme Value of Money Formula Sprng, 2011 6
Exercses For some of the followng exercses, solve wth mathematcs of fnance formulas and wth fnancal functons on a calculator. Use any mathematcs of fnance formula that s convenent. For all problems, show all your work, label all nputs, show formulas, label all answers, and summarze. For the basc mathematcs of fnance formulas and ther dervatons, see Luttman or Kastng n Unt 1 of ths course. 1. At what annual nterest rate, compounded monthly, wll a depost of $1250 accumulate to $2000 n seven years? See p. 143 of the TI83 Manual. Consder Formula 9 below.! (9) FV = PV 1+ r $ # k % &, where r s the annual nomnal rate compounded k tmes per year. s the number of compoundng perods. I% s a percent. r = I % 100 2. For a mortgage of $100,000 at 18% per year for 30 years, what s the monthly payment? See p. 144 n the TI 83 Manual. 3. For a mortgage of $100,000 at 8.5% per year for 30 years, what s the monthly payment? See p. 146 of the TI 83 Manual. 4. For a 30year, 11% mortgage wth $1000 a month payments, what s the amount of the mortgage? See p. 147 of the TI 83 Manual. 5. On p. 143 of the TI 83 Manual, one problem s I% = 6%, PV = 9000, PMT = 350, FV = 0, P/Y = 3, what s? Do ths problem. 6. Set up the Cash Flow equaton for the second tmelne on p. 148 and verfy the answers. The npv (et Present Value) at 6% = $2920.65 means the sum of the cash flows dscounted at 6% gves $2920.65. The (Internal Rate of Return) rr = 27.88% makes the npv equal to zero. The Internal Rate of Return s the rate that dscounts the remanng cash flows to the ntal equty CF0. To do ths calculaton wth a scentfc calculator pad, store the 1 + n STO and use RCL. 7. By algebra, derve Formula 6 from Formula 5. 8. Derve from Formula 3, the TVM Formula n the Appendx of the TI 83 Manual for, # PMT! FV & ln % PMT + PV ( whch s = $ ' for G = 1. ln(1+ ). A Master Tme Value of Money Formula Sprng, 2011 7
9. Derve from Formula 3, the TVM Formula n the Appendx of the TI 83 Manual for PMT 1 PMT PV wth G = 1, whch s PV = FV. Indcate the requred (1 + ) sgn changes to satsfy the TVM sgn conventons for cash flows. 10. Derve from Formula 3, the TVM Formula n the Appendx of the TI 83 Manual for PV + FV PMT wth G = 1, whch s PMT = PV + (1 + ) 1 and ndcate the requred sgn changes to satsfy the TVM sgn conventons for cash flows. You may want to use the PV (1 + ) PV dentty = PV + (1 + ) 1 (1 + ) 1. 11. What do you change n Formula 7 to get the Compound Interest Formula? Make changes and show the formula. 12. Derve the Formula for the Sum of an Ordnary Annuty from Formula 7. What strange results do you get and what sgn change s requred? 13. Derve a formula for the Present Value of an Annuty Due. 14. Wrte a paragraph wth cash flow tmelnes dscussng some of the entres and sgn changes requred n the TVM Solver and requred by the TVM formulas. 15. Consder a fund of PV dollars that provdes annual wthdrawals of PMTs for years. Draw a cash flow tmelne wth sgn conventons and gve the sgns for PV and PMTs. Gve a formula for ths fund and wthdrawals. 16. (a) Consder a loan of PV dollars that s repad wth PMTs. Draw a cash flow tmelne wth sgn conventons and gve the sgn changes for the varables. Gve a formula for ths loan and payments. (b) otce that fnancal varables have multple meanngs. It s ths multvalency and abstractness that gves the power and generalty to mathematcs. In some publcatons, even more generc fnancal terms such as Rent (R), Prncple (P), and Sum (S) are used. Sometmes appled problems n fnance requre formulas that are not among the common ones. Then you need to derve your own formula. For an example of such a dervaton, derve a formula for the Sum S n of the frst n terms of a geometrc sequence wth frst term a, common rato r, second term ar and nth term ar n 1. Use the technque used to derve Formula 2 above. 17. A smplfcaton of a formula for n the Appendx to the TI83 Manual s ln(1 ) = e + 1. Prove ths formula. Use the defnton of ln(1 + ). 18. Some problems requre two tmelnes. A person makes a debt of $2000 that must be pad back n ten years at 5% nterest compounded semannually. To accumulate money A Master Tme Value of Money Formula Sprng, 2011 8
to pay the debt, after one year they start savng each year wth the last depost made when the debt s due. The nvestment pays 4% compounded annually. How much should they save each year? 19. From Money magazne, June 2011, p. 65: You don t start savng untl age 45, but then dutfully nvest $10,000 a year untl 65, assumng annualzed 8% returns. At age 65 you ll have $460,000. Your brother starts at 35, saves $10,000 a year but quts at age 45. At age 65 he has $675,000. Make the mstake of startng to save late, and you ll have to work hard to make up the error. Problem: Do the calculatons to get ther numbers. Keep tryng and you ll get them. Explan Money magazne s calculatons. 20. As ths course progresses, accumulate at least twenty reasons for knowng the fnancal mathematcs and calculator sklls to complement the Fnancal Functons on a calculator. Items such as The Fnancal Functons on a calculator are referred to as, black boxes. 21. From Forbes, May 9, 2011: In comparng ETFs (Exchange Traded Funds), t s mportant to calculate the effect on earnngs of expense ratos charged for managng the nvestment. Vanguard MSCI Emergng Market (VWO, 49) has an expense rato of 22 bass ponts. BlackRock s Shares charges 69 bass ponts. If your $10,000 earns 9% before expenses, the dfference n expense ratos wll balloon nto a dfference n fnal amount of $16,000 n your pocket n 2041. Over 30 years that dfference of half a percentage pont compounds to 13%. Problem: See f you can get these numbers. If you keep tryng, you wll fgure t out. What lesson dd you learn about nvestng? Dd you compound wth (1 + 0.09 0.0022) or (1 + 0.09)(1 0.0022)? Whch s correct? How much dfference does t make? A Master Tme Value of Money Formula Sprng, 2011 9
Sde Bar otes Load Funds and oload Funds. From USA Today, May 27, 2011: Let s look at the Amercan Funds Growth Fund of Amerca. The A shares have ganed 4.17% a year for the last decade. To put ths n perspectve: A $10,000 nvestment n the fund would have ganed $5,046 n 10 years. If you pad the fund s maxmum 5.75% sales charge, however, your total return was 3.55%,. Your gan has now shrunk to $4,174. You owe taxes on the dstrbutons. your after tax return would be 3.21%. And f you had sold the fund after 10 years and pad taxes on you gans, you d be left wth a 2.96% average annual gan. So you nvested $10,000 n a taxable account, pad the sales charge, pad taxes on dstrbutons and gans. Your $10,000 s now $13,387. At that rate, you d double your money n about 24 years. But you can reduce your cost. One easy way of course, s to buy a noload fund. You pay no commsson. If you had the money nvested n a Roth IRA or 401k, t would be aftertax money and there would be no ncome taxes. See IRS Publcaton 590. For a noload fund, see www.vanguard.com for the Vanguard Wellngton Fund, whch s the naton s oldest balanced fund. 6.20% n the last 10 years. Expense rato 0.34%. Some salesmen of load funds wll tell you that you wll pay ndrectly a commsson on a noload fund. How s that wth a 0.34% expense rato on the noload fund, and a 0.69% expense rato on the load fund. Problem: See f your can reproduce the above calculatons. You can do most of them. A Master Tme Value of Money Formula Sprng, 2011 10
References Most of the followng references are cted n the annotated bblography for ths course, and provde dervatons, exercses, and calculator programs for practce wth the fnancal formulas, calculator Fnancal Functons, and graphs. Vest, Floyd Tme, Money, & Polynomals, HMAP PullOut, Consortum 37. Vest, Floyd, What Can a Fnancal Calculator Do for a Mathematcs Teacher or Student? Journal of Computers n Mathematcs and Scence Teachng, pp. 6778, Fall 1989. Vest, Floyd, Computerzed Busness Calculus Usng Calculators, Journal of Computers n Mathematcs and Scence Teachng, Summer,1991. Vest, Floyd, Tm and Tom s Fnancal Adventure, HMAP PullOut, Consortum 39. Vest, Floyd, Makng Money Wth Algebra, HMAP PullOut, Consortum 33. A Master Tme Value of Money Formula Sprng, 2011 11
Answers to You Try Its! You Try It #2 FV = 100 1+ 0.04 $ # 365 % & 3650 = $149.18. (! # You Try It #3 FV = 500 ) $ % & 0.06 12 1+ 0.06 12 360 + '1 = $502,257.49,! You Try It #4 FV = 200,000 1+ 0.06 $ # 12 % & 892,993.93 + 346,496.97 = $1,239,490.90 300 (! # + 500 ) $ % & 0.06 12 1+ 0.06 12 300 + '1 =, A Master Tme Value of Money Formula Sprng, 2011 12
Answers to Exercses 1. To solve the problem wth a formula, use the Compound Interest Formula 9 gven above: Substtutng n Formula 9, we have 2000 = 1250 1+ r 84! $ # 12% & snce there are 7 12 = 84 compoundng perods. Usng algebra and a scentfc calculator pad to solve for r, we have r = 0.06733 = 6.733% as the annual nomnal rate. For the fnancal calculator code and explanaton for the TVM Solver, see page 143 of the TI 83 Manual, or see the TI 84 Manual. 2. We wll use Formula 8 above. Substtutng we get 1! 1+ 0.18!360 ( % + # $ 12 & ' 100,000 = PMT 0.18. 12 ), Solvng and usng a scentfc calculator pad gves PMT = $1507.09 as the monthly payment. We could have used Formula 3 above wth FV = 0 and PMT would have come out negatve. For a fnancal calculator code, see p. 144 of the TI 83 Manual, or see the TI 84 Manual, for the TVM Solver. 3. For ths problem, the TI83 Manual presents the tvm_pmt fnancal functon, whch s menu tem 2 under 2 nd Fnance. Code and comments: Put the knowns n the TVM Solver. Put 0 n PMT. Code: 2 nd Fnance (to tvm_pmt ) Enter Enter (On the home screen you read 768.91. The monthly payment s $768.91.) 4. Substtutng nto Formula 6 and calculatng wth a scentfc calculator pad gves PV = $105,006.35, as the amount of the mortgage. The TI 83 Manual presents the tvm_pv [(,I%,PMT,FV,P/Y,C/Y)] functon for ths problem. On the home screen, you store wth the STO> key, 360 n, 11 n I%, ()1000 n PMT, 0 n FV, 12 n P/Y and calculate PV. Code and comments: (On the home screen.) 360 2 nd Fnace > Enter (To Sto 360 n.) Alpha : 11 2 nd Fnace > (To I%) Enter Alpha : ()1000 2 nd Fnace > (To PMT.) Enter Alpha : 12 2 nd Fnance > (To P/Y.) Enter Enter 2 nd Fnace (To tvm_pv.) Enter Enter (You read 105006.35.) The amount of the mortgage s $105,006.36. (It s more convenent to enter values drectly nto the TVM Solver when possble.) The above code ddn t store values to the TVM Solver. 5. Substtutng nto Formula 8 above and usng logs to solve, you get = 36.47 payments. The TVM Solver gves the same answer. A Master Tme Value of Money Formula Sprng, 2011 13
11. To get the Compound Interest Formula FV = PV(1 + ) from Formula 7, set PMT = 0, G = 1 and make a sgn change. 12. To get the Formula for the Sum of an Ordnary Annuty from Formula 7, set PV = 0 and G = 1, and make a sgn change to get Formula 2. 13. To get a Formula for the Present Value of an Annuty Due, use Formula 8 and snce n 1 (1 + ) PMTs are at the begnnngs of perods, use PVAD = PMT (1 + ). 15. For the fund, PV s negatve and PMT s money comng n and s postve. 16. For the loan, PV s money comng n and s postve, PMTs are money gong out and are negatve. 18. They should save $272.96 at the end of each year for ten years. A Master Tme Value of Money Formula Sprng, 2011 14
Teachers otes The TVM Formulas. Ths artcle presents dervatons of the basc mathematcs of fnance formulas found n Luttman and Kastng n Unt 1 of ths course, but n the form of the TVM Formulas n the appendx to the TI83/84. These formulas utlzed tme lne sgn conventons for varables. For example for a loan, PV s postve snce t s thought of as money comng n, and PMT s negatve snce t s thought of as money gong out. All of ths s explaned n the artcle. TI 83/84 Fnancal Functons are ndcated n some the exercses. Students can use other calculators f they wsh. Instructons for use of the TI83/84 are nclude. A lfetme fle on fnancal mathematcs. Have students set up and mantan a lfetme fle on personal fnance and fnancal mathematcs. In the future, ths artcle wll help students use ther TI83/84 n personal fnance. Students should be encouraged to buy ther own calculators. They wll last them a lfetme. Addtonal examples and exercses. The artcles n the references provde addtonal examples and exercses whch students can do wth ther graphng calculators and fnancal calculators. Some gve programs for solvng for n annuty formulas, as well as assocated graphs. A Master Tme Value of Money Formula Sprng, 2011 15