Chapter 4: Time Value of Money

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1 FIN 301 Class Notes Chapter 4: Time Value of Moey The cocept of Time Value of Moey: A amout of moey received today is worth more tha the same dollar value received a year from ow. Why? Do you prefer a $100 today or a $100 oe year from ow? why? - Cosumptio forgoe has value - Ivestmet lost has opportuity cost - Iflatio may icrease ad purchasig power decrease Now, Do you prefer a $100 today or $110 oe year from ow? Why? You will ask yourself oe questio: - Do I have ay thig better to do with that $100 tha ledig it for $10 extra? - What if I take $100 ow ad ivest it, would I make more or less tha $110 i oe year? Note: Two elemets are importat i valuatio of cash flows: - What iterest rate (opportuity rate, discout rate, required rate of retur) do you wat to evaluate the cash flow based o? - At what time do these the cash flows occur ad at what time do you eed to evaluate them? 1

2 Time Lies: Show the timig of cash flows. Tick marks occur at the ed of periods, so Time 0 is today; Time 1 is the ed of the first period (year, moth, etc.) or the begiig of the secod period i% CF0 CF1 CF2 CF3 Example 1 : $100 lump sum due i 2 years i 100 Today Ed of Ed of Period 1 Period 2 (1 period (2 periods form ow) form ow) Example 2 : $10 repeated at the ed of ext three years (ordiary auity ) i

3 Calculatios of the value of moey problems: The value of moey problems may be solved usig 1- Formulas. 2- Iterest Factor Tables. (see p.684) 3- Fiacial Calculators (Basic keys: N, I/Y, PV, PMT, FV). I use BAII Plus calculator 4- Spreadsheet Software (Basic fuctios: PV, FV, PMT, NPER,RATE). I use Microsoft Excel. 3

4 FUTUR VALUE OF A SINGLE CASH FLOW Examples: You deposited $1000 today i a savig accout at BacFirst that pays you 3% iterest per year. How much moey you will get at the ed of the first year? i=3% FV1 0 1 $1000 You led your fried $500 at 5% iterest provided that she pays you back the $500 dollars plus iterest after 2 years. How much she should pay you? i=5% FV $500 You borrowed $10,000 from a bak ad you agree to pay off the loa after 5 years from ow ad durig that period you pay 13% iterest o loa. $10, Preset Value of Moey i=13% Ivestmet Compoudig 4 FV5 Future Value of Moey

5 Detailed calculatio: Simple example: Ivest $100 ow at 5%. How much will you have after a year? FV 1 FV 1 Or = PV + INT = PV + (PV i) = PV (1 + i) = $100 + INT = $100 + ($100.05) = $100 + $5 = $105 FV 1 = $100 (1+0.05) = $100 (1.05) = $105 5

6 Aother example: Ivest $100 at 5% (per year) for 4 years PV = $100 FV 1 = $105 FV 2 = $ FV 3 = $ FV 4 = $ Iterest added: + $ $ $ $5.79 FV 1 = 100 (1.05) = $105 FV 2 = 105 (1.05) = $ = 100 (1.05) (1.05) = $ = 100 (1.05) 2 = $ FV 3 = (1.05) = $ = 100 (1.05) (1.05) (1.05)= $ = 100 (1.05) 3 = $ FV 4 = $100 (1.05) (1.05) (1.05) (1.05) = PV (1+i) (1+i) (1+i) (1+i) = PV (1+i) 4 I geeral, the future value of a iitial lump sum is: FV = PV (1+i) 6

7 To solve for FV, You eed 1- Preset Value (PV) 2- Iterest rate per period (i) 3- Number of periods () Remarks: As PV, FV. As i, FV. As, FV. 1- By Formula FV = PV (1 ) 0 + i 2- By Table I 0, FV IF, = (1 + i ) i 3- By calculator (BAII Plus) FV = PV ( FV IF ) i Clea the memory: CLR TVM CE/C 2d FV INPUTS OUTPUT N I/Y PV PMT CPT FV Notes: - To eter (i) i the calculator, you have to eter it i % form. - Use +/- To chage the sig of a umber. For example, to eter -100: 100 +/- - To solve the problems i the calculator or excel, PV ad FV caot have the same sig. If PV is positive the FV has to be egative. 7

8 Example: Jack deposited $1000 i savig accout earig 6% iterest rate. How much will jack moey be worth at the ed of 3 years? Time lie ? 6% 1000 Before solvig the problem, List all iputs: I = 6% or 0.06 N= 3 PV= 1000 PMT= 0 Solutio: By formula: FV = PV (1+i) FV 3 = $1000 (1+0.06) 3 = $1000 (1.06) 3 = $ = $ 1,191 By Table: FV = PV FVIF i, FV 3 = $1000 FVIF 6%,3 = $ = $ 1,191 8

9 By calculator: Clea the memory: CLR TVM CE/C 2d FV INPUTS OUTPUT By Excel: N I/Y PV PMT CPT FV =FV (0.06, 3, 0,-1000, 0) 1,

10 PRESENT VALUE OF A SINGLE CASH FLOW Examples: You eed $10,000 for your tuitio expeses i 5 years how much should you deposit today i a savig accout that pays 3% per year? $10,000 PV0 i=3% FV5 Oe year from ow, you agree to receive $1000 for your car that you sold today. How much that $1000 worth today if you use 5% iterest rate? PV0 $ i=5% 1 FV1 Preset Value of Moey Discoutig Future Value of Moey 10

11 Detailed calculatio FV = PV (1 + i) FV PV 0 = (1 + i ) 1 = (1 + i ) PV 0 FV Example: $100 $105 $ $ = $ PV 4 = FV 4 = $ PV 3 = FV 4 [1/(1+i)] = $ [1/(1.05)] = $ PV 2 = FV 4 [1/(1+i)(1+i)] = $ [1/(1.05)(1.05)] = $ [1/(1.05) 2 ] = $

12 Or PV 2 = FV 3 [1/ (1+i)] = $ [1/ (1.05)] = $ PV 1 = FV 4 [1/(1+i)(1+i) (1+i)] = $ [1/(1.05)(1.05) (1.05)] = $ [1/(1.05) 3 ] = $105 Or PV 1 = FV 2 [1/ (1+i)] = $ [1/ (1.05)] = $105 PV 0 = FV 4 [1/ (1+i) (1+i) (1+i) (1+i)] = FV 4 [1/(1+i) 4 ] = $ [1/(1.05)(1.05) (1.05) (1.05)] = $ [1/(1.05) 4 ] = $100 I geeral, the preset value of a iitial lump sum is: PV 0 = FV [1/(1+i) ] 12

13 To solve for PV, You eed 4- Future Value (FV) 5- Iterest rate per period (i) 6- Number of periods () Remarks: As As As FV, PV i, PV, PV 1- By Formula 0 1 PV = FV (1 + i ) PV = FV ( PVIF ) 2- By Table II 0, 1 = (1 + i ) PV IF i, 3- By calculator (BAII Plus) i Clea the memory: CLR TVM CE/C 2d FV INPUTS OUTPUT N I/Y PV FV PMT CPT PV

14 Example: Jack eeded a $1191 i 3 years to be off some debt. How much should jack put i a savig accout that ears 6% today? Time lie % 3 $1191? Before solvig the problem, List all iputs: I = 6% or 0.06 N= 3 FV= $1191 PMT= 0 Solutio: By formula: PV 0 = FV 3 [1/(1+i) ] PV 0 = $1,191 [1/(1+0.06) 3 ] = $1,191 [1/(1.06) 3 ] = $1,191 (1/1.191) = $1, = $1000 By Table: = FV PVIF i, PV 0 = $1,191 PVIF 6%,3 = $1, = $

15 By calculator: Clea the memory: CLR TVM CE/C 2d FV INPUTS OUTPUT By Excel: N I/Y PV FV PMT CPT PV =PV (0.06, 3, 0, 1191, 0)

16 Solvig for the iterest rate i You ca buy a security ow for $1000 ad it will pay you $1,191 three years from ow. What aual rate of retur are you earig? FV PV 1 By Formula: i = 1 i = = 0.06 FV = PV ( FVIF ) By Table: 0, i FV IF = i, FV PV FV IF i,3 = = From the Table I at =3 we fid that the iterest rate that yield FVIF is 6% PV = FV ( PV IF ) Or 0 i, PV IF = i, PV FV 1000 PV IF i,3 = = From the Table II at =3 we fid that the iterest rate that yield PVIF is 6% 16

17 By calculator: Clea the memory: CLR TVM CE/C 2d FV INPUTS OUTPUT N PV PV FV PMT CPT I/Y

18 Solvig for : Your fried deposits $100,000 ito a accout payig 8% per year. She wats to kow how log it will take before the iterest makes her a millioaire. ( L FV ) ( l PV) By Formula: = L 1+ i ( ) FV = $1,000,000 PV = $100, i = 1.08 = ( ) ( ) l 1,000,000 l 100,000 l(1.08) By Table: 0, = = FV = PV ( FVIF ) i 30 years FV IF = i, FV PV 0 1,000,000 FV IF 8, = = ,000 From the Table I at i=8 we fid that the umber of periods that yield 10 FVIF is 30 PV = FV ( PVIF ) Or 0 i, PV PV IFi, = FV 100,000 PV IF 8, = = 0.1 1,000,000 0 From the Table II at i=8 we fid that the umber of periods that yield 0.1 PVIF is 30 18

19 By calculator: Clea the memory: CLR TVM CE/C 2d FV INPUTS OUTPUT 8-100,000 1,000,000 0 I/Y PV FV PMT CPT N FUTURE VALUE OF ANNUTIES A auity is a series of equal paymets at fixed itervals for a specified umber of periods. PMT = the amout of periodic paymet Ordiary (deferred) auity: Paymets occur at the ed of each period. Auity due: Paymets occur at the begiig of each period. 19

20 Ordiary Due i PM PM PM i PM PM PM Example: Suppose you deposit $100 at the ed of each year ito a savigs accout payig 5% iterest for 3 years. How much will you have i the accout after 3 years? % $ Time ( ) ( ) 1 2 FVAN = PMT 1+ i + PMT 1 + i PMT (Hard to use this formula) PMT PMT PMT PMT PMT PMT 20

21 ( i ) 1+ 1 FV AN = PMT i = PMT ( FV IFA ) i, Future Value Iterest Factor for a Auity Note: For a auity due, simply multiply the aswer above by (1+i). So FVAND (auity due) = PMT ( FVIFA )(1 + i). ( i ) 1+ 1 = PMT 1 + i Auity: ( i ) i, 21

22 Auity Due: 22

23 Remark: FVIFA = FVIF + FVIF + FVIF i,3 i,2 i,1 i,0 To solve for the future value of Auities, You eed: 1-Payemt or auity amout (PMT) 2-Iterest rate per period (i) 3-Number of periods () 1-BY Formula: ( 1+ i ) 1 FV AN = PMT i ( 1+ i ) 1 FV AND = PMT 1 + i i ( 1 ) FVAND = FVAN + i == Ordiary Auity ( ) == Auity Due 2- BY Table III: FV A N = PMT ( FV IFA ) == Ordiary Auity i, i, ( ) FV AND = PMT ( FV IFA ) 1+ i == Auity Due 23

24 3- BY calculator: Ordiary Auity: 1- Clea the memory: CLR TVM CE/C 2d FV 2- Set paymet mode to END of period: BGN SET 2d 2d PMT ENTER 3- Make sure you ca see END writte o the scree the press CE/C NOTE: If you do ot see BGN writte o the upper right side of the scree, you ca skip Step 2 ad 3. INPUTS OUTPUT N I/Y PV PMT CPT FV

25 Auity Due: Clea the memory: CLR TVM CE/C 2d FV Set paymet mode to BGN of period: BGN SET 2d 2d PMT ENTER Make sure you ca see BGN writte o the scree the press CE/C INPUTS OUTPUT N I/Y PV PMT CPT FV

26 Example: You agree to deposit $500 at the ed of every year for 3 years i a ivestmet fud that ears 6%. Time lie 0 1 6% $500 2 $500 3 $500 FV=? Before solvig the problem, List all iputs: I = 6% or 0.06 N= 3 PMT=500 PV= 0 FV=? Solutio: By formula: FV AN ( i ) 1+ 1 = PMT i 3 ( ) 1 = = 500 = 1, FV AN = PMT ( FV IFA ) By Table:, FV A N 3 6,3 i = 500( FV IFA ) = 500(3.184) = 1,592 26

27 By calculator: Clea the memory: CLR TVM CE/C 2d FV Make sure you do ot see BGN writte o the upper right side of the scree. INPUTS OUTPUT N I/Y PV -500 PMT CPT FV 1, By Excel: =FV (0.06, 3, -500, 0, 0) 27

28 Now assume that you deposit the $500 at the begiig of the year ot at the ed of the year. Time lie 0 $500 6% 1 $500 2 $500 3 FV=? Before solvig the problem, List all iputs: I = 6% or 0.06 N= 3 PMT=500 (beg) PV= 0 FV=? Solutio: By formula: ( i ) 1+ 1 FV AND = PMT 1 + i i ( ) 1 FV A ND 3 = 500 ( ) = 500 (1.06) = 1, ( ) By Table: FV AND = PMT ( FVIFA )( 1+ i ) FV AND 3 6,3 i, ( ) = 500( FV IFA ) = 500(3.184)(1.06) = 1,

29 By calculator: Clea the memory: CLR TVM CE/C 2d FV Set paymet mode to BGN of period: BGN SET 2d 2d PMT ENTER Make sure you ca see BGN writte o the scree the press CE/C INPUTS OUTPUT N I/Y PV -500 PMT CPT FV 1, By Excel: =FV (0.06, 3, -500, 0, 1) 29

30 PRESENT VALUE OF ANNUTIES Problem: You have a choice a) $100 paid to you at the ed of each of the ext 3 years or b) a lump sum today. i = 5%, sice you would ivest the moey at this rate if you had it. How big does the lump sum have to be to make the choices equally good? Time PVAN 3 = Formula: PVA = PMT ( 1+ i) ( 1+ i) ( 1+ i) 1 = PMT = PMT PMT ( 1+ i) i 1 ( PVIFA ) i, PMT Preset Value Iterest Factor 30

31 PVA = $ = $100 3 ( ) = $ Note: For auities due, simply multiply the aswer above by (1+i) PVAND (auity due) = PMT (PVIFA i, ) (1+i) To solve for the preset value of Auities, You eed: 1-Payemt or auity amout (PMT) 2-Iterest rate per period (i) 3-Number of periods () 1- BY Formula: 1 1 ( 1+ i ) PVAN = PMT i == Ordiary Auity 1 1 ( 1+ i ) PVAND = PMT ( 1+ i) i == Auity Due ( 1 ) PVAND = PVAN + i 31

32 2- BY Table IV: PVAN = PMT ( PVIFAi, ) == Ordiary Auity i, ( ) PVAND = PMT ( PVIFA ) 1+ i == Auity Due 3- BY calculator: Ordiary Auity: Clea the memory: CLR TVM CE/C 2d FV Make sure you do ot see BGN writte o the upper right side of the scree. INPUTS OUTPUT N I/Y FV PMT CPT PV

33 Auity Due: Clea the memory: CLR TVM CE/C 2d FV Set paymet mode to BGN of period: BGN SET 2d 2d PMT ENTER Make sure you ca see BGN writte o the scree the press CE/C INPUTS OUTPUT N I/Y FV PMT CPT PV

34 Example: You agree to receive $500 at the ed of every year for 3 years i a ivestmet fud that ears 6%. Time lie 0 PV=? 6% 1 $500 2 $500 3 $500 Before solvig the problem, List all iputs: I = 6% or 0.06 N= 3 PMT=500 FV= 0 PV=? Solutio: By formula: PVAN = PMT 1 1 ( 1+ i) i PVAN 1 = ( ) = 0.06 = $1,

35 PVAN = PMT ( PVIFA ) By Table:, PVAN i = 500( PVIFA ) 3 6,3 = 500(2.673) = 1, By calculator: Clea the memory: CLR TVM CE/C 2d Make sure you do ot see BGN writte o the upper right side of the scree. FV INPUTS OUTPUT N I/Y FV PMT CPT PV 1, By Excel: =PV (0.06, 3, -500, 0, 0) 35

36 Now assume that you receive the $500 at the begiig of the year ot at the ed of the year. Time lie 0 $500 6% 1 $500 2 $500 3 PV=? Before solvig the problem, List all iputs: I = 6% or 0.06 N= 3 PMT=500 (beg) FV= 0 PV=? Solutio By formula: 1 1 ( 1+ i ) PVAND = PMT ( 1+ i) i 1 1 ( ) PVAND = ( ) = 500 (1.06) = 1,

37 By Table: PVAND = PMT ( PVIFAi, ) ( 1+ i) PVAND = 500( PVIFA )( ) By calculator: 3 6,3 = 500(2.673)(1.06) = 1, Clea the memory: CLR TVM CE/C 2d FV Set paymet mode to BGN of period: BGN SET 2d 2d PMT ENTER Make sure you ca see BGN writte o the scree the press CE/C INPUTS OUTPUT N I/Y FV PMT CPT PV 1, By Excel: =PV (0.06, 3, -500, 0, 1) 37

38 Perpetuities A perpetuity is a auity that cotiues forever. 1 1 (1 ) + i PVAN = PMT i 1 As gets very large, 0 ( 1+ i) 1 0 PVPER0 ( perpetuity) = PMT i 1 PMT = PMT = i i Formula: PVPER 0 = PMT i 38

39 UNEVEN CASH FLOWS How do we get PV ad FV whe the periodic paymets are uequal? Preset Value $ PV = CF 0 CF i CF CF 2 ( 1+ i) ( 1+ i) Future Value % $ ( 1+ i) + CF ( 1+ i) CF ( 1 ) 0 FV = CF i 39

40 Example: Preset Value of Ueve Cash Flows 40

41 By Calculator: Clea the memory: CF 2d CE/C Iput cash flows i the calculator s CF register: CF0 = 0 0 ENTER CF1 = 100 C ENTER F01 1 ENTER CF2 = 200 C ENTER F02 1 ENTER CF3 = 300 C ENTER F03 1 ENTER Press NPV, the the it will ask you to eter the Iterest rate (I) Eter I = ENTER Use to get to the NPV o the scree Whe you read NPV o the scree, press CPT You will get NPV = $ (Here NPV = PV.) NOTE: To calculate the future value of ueve cash flows, it is much easier to start by calculatig the Preset value of the cash flows usig NPV fuctio the calculate the future value usig the future value of a sigle cash flow rules. The sigle cash flow i this case will be the preset value. 41

42 Simple ad Compoud Iterest Example: Simple Iterest Iterest paid o the pricipal sum oly Compoud Iterest Iterest paid o the pricipal ad o iterest Calculate the future value of $1000 deposited i a savig accout for 3 years earig 6%. Also, calculate the simple iterest, the iterest o iterest, ad the compoud iterest. FV3 = 1000 (1.06) 3 = $1, Pricipal = PV = $1000 Compoud iterest = FV PV = = Simple Iterest = PV * i * =1000 * 0.06 * 3 = $180 Iterest o iterest = Compoud iterest - Simple Iterest = =

43 Effect of Compoudig over Time Other Compoudig Periods So far, our problems have used aual compoudig. I practice, iterest is usually compouded more frequetly. 43

44 Example: You ivest $100 today at 5% iterest for 3 years. Uder aual compoudig, the future value is: FV 3 = PV 1 ( + i) = $100(1.05) = $100(1.1576) = $ What if iterest is compouded semi-aually (twice a year)? The the periods o the time lie are o loger years, but half-years! Time: 5% i = Periodic iterest rate = = 2.5% 2 = No.of periods = 3 2 = 6 FV FV % PV=100 = PV (1 + i) = $100(1.025) 6 = $100(1.1597) = $ moths FV 6 =? Note: the fial value is slightly higher due to more frequet compoudig. 44

45 Will the FV of a lump sum be larger or smaller if compouded more ofte, holdig the stated I% costat? LARGER, as the more frequetly compoudig occurs, iterest is eared o iterest more ofte % Aually: FV 3 = $100(1.10) 3 = $ % 100 Importat: Whe workig ay time value problem, make sure you keep straight what the relevat periods are! = the umber of periods i = the periodic iterest rate From ow o: = m* i = i/m Semiaually: FV 6 = $100(1.05) = $ Where m = 1 m = 2 m = 4 m = 12 m = 52 m = 365 for aual compoudig for semiaual compoudig for quarterly compoudig for mothly compoudig for weekly compoudig for daily compoudig For cotiuously compoudig: (1+i) = e i = FV = PV (e) i = PV = FV (e) - 45

46 EFFECTIVE INTREST RATE You have two choices: 1-11% aual compouded rate of retur o CD 2-10% mothly compouded rate of retur o CD How ca you compare these two omial rates? A omial iterest rate is just a stated (quoted) rate. A APR (aual percetage rate) is a omial rate. For every omial iterest rate, there is a effective rate. The effective aual rate is the iterest rate actually beig eared per year. To compare amog differet omial rates or to kow what is the actual rate that you re gettig o ay ivestmet you have to use the Effective aual iterest rate. i i 1 1 Effective Aual Rate: eff = + m To compare the two rates i the example, m i eff = 1+ 1 = compoudig) 1 or 11% (Nomial ad Effective rates are equal i aual 2- i eff = 1+ 1 = or % You should choose the first ivestmet. 46

47 To compute effective rate usig calculator: ICONV 2d 2 Eter Nomial Rate NOM 10 ENTER Eter compoudig frequecy per year (m) C/Y 12 ENTER Compute the Effective rate EFF CPT Nomial Versus Real Iterest Rate Nomial rate rf is a fuctio of: Iflatio premium i :compesatio for iflatio ad lower purchasig power. Real risk-free rate r f : compesatio for postpoig cosumptio. (1 + r ) = (1 + r )(1 + i ) f f r = r + i + r i f f f r r + i f f 47

48 Amortized Loas A amortized loa is repaid i equal paymets over its life. Example: You borrow $10,000 today ad will repay the loa i equal istallmets at the ed of the ext 4 years. How much is your aual paymet if the iterest rate is 9%? Time 0 9% PVA N= $10,000 PMT PMT PMT 4 PMT Iputs: The periods are years. (m=1) = 4 i = 9% PVAN 4 = $10,000 FV =0 PMT =? PV AN = PMT ( PV IFA ) 4 9%,4 $10,000 = PMT (3.240) $10,000 PMT = = $3,

Iterest amout = Begiig balace * i Pricipal reductio = aual paymet - Iterest amout Edig balace = Begiig balace - Pricipal reductio Begiig balace: Start with pricipal amout ad the equal to previous

49 Iterest amout = Begiig balace * i Pricipal reductio = aual paymet - Iterest amout Edig balace = Begiig balace - Pricipal reductio Begiig balace: Start with pricipal amout ad the equal to previous year s edig balace. As a loa is paid off: at the begiig, much of each paymet is for iterest. later o, less of each paymet is used for iterest, ad more of it is applied to payig off the pricipal. 49

Chapter 5 Time Value of Money

Chapter 5 Time Value of Money Chapter 5 Time Value of Moey 1. Suppose you deposit $100 i a bak that pays 10% iterest per year. How much will you have i the bak oe year later? 2. Suppose you deposit $100 i a bak that pays 10% per year.