Chapter 6 The Time Value of Money
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1 Chapter 6 The Time Value of Moey Which would you rather have? $100 today OR $100 oe year from today Sooer is better! Iterest rate 4% Time Lies are VERY IMPORTANT!!! $100? 0 1 Time 0 is today; Time 1 is oe period from today How much will you have accumulated at Time 2? Time 3? $100 $104?? Future Value PV = preset value or startig amout, say $100 i = iterest rate, say 4% per year (4% = 0.04) INT = dollars of iterest you ear durig the year; for Year 1, $100 x 0.04 = $4 FV = future value after periods or $100 + $4 = $104 after 1 year Future Value--year 1 FV = $100 ( ) = $100(1.04) = $104 Future Value--years 2 & 3 FV 2 = $104 (1.04) = $100 (1.04) (1.04) = $100 (1.04) 2 = FV 3 = 1
2 Future Value -- Compoud Iterest Iterest eared o iterest = PV(1 i) FV + Alterative method Fiacial Calculator Preset Value If you are preseted with the decisio betwee $100 today or $ at the ed of five years, your choice will deped o your opportuity cost Opportuity cost: the rate of retur o the best available alterative ivestmet of equal risk 4 5 PV =? FV = Start with future value formula: FV = PV(1 + i) rearrage: FV PV = (1 + i) = FV 1 ( 1 + i) = FV (1 + i) 2
3 Example Fid the preset value of $ i five years whe the applicable iterest rate (opportuity cost) is 5% 5 PV = FV (1 + i) = $121.67(1.05) = $95.33 Solvig for Time ad Iterest Rates FV = PV(1 + i) Four variables: PV, FV, i ad If you kow ay three, you ca solve for the fourth Solvig for i For $82.19 you ca buy a security that will pay you $100 after five years We kow PV, FV, ad, but we do ot kow i i =? 4 5 $82.19 $100 FV = PV(1 + i) $100 = $82.19(1 + i) 5 Solve for i: 3
4 Solvig for Suppose you kow that the security will provide a retur of 10 percet per year, that it will cost $68.30, ad that you will receive $100 at maturity, but you do ot kow whe the security matures. You kow PV, FV, ad i, but you do ot kow - the umber of periods. 10% =? $100 FV = PV(1 + i) $100 = $68.30(1.10) = (1.10) solve for Auity Auity is a series of paymets of a equal amout at fixed itervals for a specified umber of periods Ordiary (deferred) auity has paymets at the ed of each period Auity due has paymets at the begiig of each period FVA is the future value of a auity over periods Future Value of a Auity The future value of a auity is the amout received over time plus the iterest eared o the paymets from the time the paymet is received util the (future) date beig valued The future value of each paymet may be calculated separately ad the the total summed If you deposit $100 at the ed of each year for three years i a savigs accout that pays 4% iterest per year, how much will you have at the ed of three years? 4% = 100 (1.04) = 100 (1.04) = 100 (1.04)
5 Future Value of a Auity FVA = PMT ( 1 + i) i ( 1.04) FVA3 = $100 = $100(3.1216) = $ Auities Due If the three $100 paymets had bee made at the begiig of each year, the auity would have bee a auity due. Each paymet would shift to the left oe year ad each paymet would ear iterest for a additioal year (period). Future Value of a Auity $100 at the ed of each year 4% = 100 (1.04) = 100 (1.04) = 100 (1.04) Future Value of a Auity Due $100 at the start of each year 4% = 100 (1.04) = 100 (1.04) = 100 (1.04)
6 Future Value of a Auity Due (1 + i) 1 FVAD = PMT (1 + i) i Numerical solutio: 3 (1.04) 1 FVAD = $100 (1.04).04 = $ [( ) ] = $ Preset Value of a Auity If you were offered a three-year auity with paymets of $100 at the ed of each year Or a lump sum paymet today that you could put i a savigs accout payig 4% iterest per year How large must the lump sum paymet be to make it equivalet to the auity? 100(1.04) 100(1.04) 1 = 2 = (1.04) 3 = % ( 1 i) 1 (1 i) 3 1 (1.04) PVA PMT + + = = PMT = $100 = $100(2.7751) = $ i i
7 Preset Value of a Auity Due Paymets at the begiig of each year Paymets all come oe year sooer Each paymet would be discouted for oe less year Preset value of auity due will exceed the value of the ordiary auity by oe year s iterest o the preset value of the ordiary auity 4% (1.04) 1 = 100(1.04) 2 = Numerical solutio: PVAD 1 (1 + i) = PMT i 1 (1.04) = ( 1 + i) = $100 (1.04) = $100(2.7751)(1.04) $ Perpetuities Perpetuity - a stream of equal paymets expected to cotiue forever Paymet PVP = = Iterest Rate PMT i 7
8 Compoudig Periods Aual compoudig iterest is added oce a year Semiaual compoudig iterest is added twice a year 4% aual iterest compouded semiaually would pay 2% every six moths adjust the periodic rate ad umber of periods before calculatig Iterest may also be compouded quarterly, mothly, daily, ad cotiuously. Iterest rate 4% compouded semi-aually Example Semi- $100 $102 $ $ $ aual 4 periods Usig aual compoudig: Aual $100 $ $ Periods 1 year 2 years Iterest Rates Simple (Quoted) Iterest Rate rate used to compute the iterest paymet paid per period Effective Aual Rate (EAR) aual rate of iterest actually beig eared, cosiderig the compoudig of iterest i EAR = 1 + m simple 1.0 Aual Percetage Rate (APR) the periodic rate multiplied by the umber of periods per year ot adjusted for compoudig m 8
9 Amortized Loas Loas that are repaid i equal paymets the life of the loa Example: Borrow $15,000 to repay i three equal paymets at the ed of the ext three years, with 8% iterest due o the outstadig loa balace at the begiig of each year 8% 15,000 PMT PMT PMT 1 - (1.08) $15,000 = PMT = PMT(2.5771) PMT = $5, A amortizatio schedule shows how a loa will be repaid with a breakdow of iterest ad pricipal o each paymet date Year Begiig Amout (1) Paymet (2) Iterest a (3) Repaymet of Pricipal b (2)-(3)=(4) Remaiig Balace (1)-(4)=(5) 1 $ 15, $ 5, , , a Iterest is calculated by multiplyig the loa balace at the begiig of the year by the iterest rate. Therefor, iterest i Year 1 is $15,000(0.08) = $1,200; i Year 2, it is $10,379.50(0.08)=$830.36; ad i Year 3, it is $5,389.36(0.08) = $ (rouded). b Repaymet of pricipal is equal to the paymet of $5, mius the iterest charge for each year. 9
10 Other examples: What is the PV of the followig cash flows? $100 $200 -$ $ $ Some problems take multiple steps to solve. For istace, 1. You are ow 25 years old ad plaig for retiremet. Additioally, you predict that you will live for 20 years after retiremet. You pla to make aual deposits ito your retiremet accout for the ext 40 years. If the iterest rate is a costat 12% durig this 6-year period, how much must you deposit aually ito your accout? 2. Your brother-i-law borrowed $1,000 from you 10 years ago ad the disappeared. Yesterday he retured ad expressed a desire to pay back the loa, icludig the iterest that accrued sice he borrowed the moey. Assumig that you had agreed to charge him 7% ad assumig that he wishes to make five equal aual paymets begiig i oe year, how much would your brother-i-law have to pay you aually i order to extiguish the debt? 10
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