CHAPTER 2. Time Value of Money 6-1

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1 CHAPTER 2 Tme Value of Moey 6-

2 Tme Value of Moey (TVM) Tme Les Future value & Preset value Rates of retur Autes & Perpetutes Ueve cash Flow Streams Amortzato 6-2

3 Tme les % CF 0 CF CF 2 CF 3 Show the tmg of cash flows Tck marks occur at the ed of perods, so Tme 0 s today; Tme s the ed of the frst perod (year, moth, etc.) or the begg of the secod perod 6-3

4 Drawg tme les: $00 lump sum due 2 years; 3-year $00 ordary auty $00 lump sum due 2 years 0 2 % 3 year $00 ordary auty %

5 Drawg tme les: Ueve cash flow stream; CF 0 = -$50, CF = $00, CF 2 = $75, ad CF 3 = $50 Ueve cash flow stream %

6 What s the future value (FV) of a tal $00 after 3 years, f I/YR = 0%? Fdg the FV of a cash flow or seres of cash flows whe compoud terest s appled s called compoudg FV ca be solved by usg the arthmetc, facal calculator, ad spreadsheet methods % 00 FV =? 6-6

7 Solvg for FV: The arthmetc method After year: FV = PV ( + ) = $00 (.0) = $0.00 After 2 years: FV 2 = PV ( + ) 2 = $00 (.0) 2 =$2.00 After 3 years: FV 3 = PV ( + ) 3 = $00 (.0) 3 =$33.0 After years (geeral case): FV = PV ( + ) 6-7

8 Solvg for FV: The calculator method Solves the geeral FV equato Requres 4 puts to calculator, ad wll solve for the ffth. (Set to P/YR = ad END mode.) INPUTS OUTPUT N I/YR PV PMT FV

9 What s the preset value (PV) of $00 due 3 years, f I/YR = 0%? Fdg the PV of a cash flow or seres of cash flows whe compoud terest s appled s called dscoutg (the reverse of compoudg) The PV shows the value of cash flows terms of today s purchasg power % PV =?

10 Solvg for PV: The arthmetc method Solve the geeral FV equato for PV: PV FV PV = FV 3 / ( + ) 3 = $00 / (.0 ) 3 = $

11 Solvg for PV: The facal tables PV FV II PV 3 FV 3 II 0 PV $

12 Solvg for PV: The calculator method Solves the geeral FV equato for PV Exactly lke solvg for FV, except we have dfferet put formato ad are solvg for a dfferet varable INPUTS OUTPUT N I/YR PV PMT FV 6-2

13 Solvg for N: If sales grow at 20% per year, how log before sales double? Solves the geeral FV equato for N Same as prevous problems, but ow solvg for N INPUTS OUTPUT N I/YR PV PMT FV 6-3

14 Solvg for N: wth atural logarthm PV FV 2 20% $ l l

15 Solvg for N: wth facal tables PV, FV 2, 20% $ tables A(20% ad factor 2) betwee 3ad 4 years 6-5

16 terpolato y y 3 x x y 2 4 x years 6-6

17 Autes & Perpetutes Auty a seres of fxed, equal paymets for a specfed umber of perods Ordary Auty paymets at the ed of the perod Auty Due paymets at the begg of the perod Perpetuty a Auty forever 6-7

18 Dfferece betwee Ordary Auty ad Auty Due Ordary Auty % Auty Due PMT PMT PMT % PMT PV PMT PMT FV 6-8

19 Future Value of a Auty Solvg for FV: 3-year ordary auty of $00 at 0% $00 paymets occur at the ed of each perod, but there s o PV. FVA PMT PMT FVIFA, PMT III FVA

20 Solvg for FV: facal calculator 3-year ordary auty of $00 at 0% $00 paymets occur at the ed of each perod, but there s o PV. INPUTS OUTPUT N I/YR PV PMT FV

21 Solvg for FV: 3-year auty due of $00 at 0% Now, $00 paymets occur at the begg of each perod. Set calculator to BEGIN mode. INPUTS OUTPUT N I/YR PV PMT FV

22 Solvg for FV: 3-year auty due of $00 at 0% Now, $00 paymets occur at the begg of each perod. FVA Due PMT FVA Due PMT FVIFA, PMT III FVA Due FVA Due

23 Preset value of a Auty Solvg for PV: 3-year ordary auty of $00 at 0% $00 paymets stll occur at the ed of each perod, but ow there s o FV. INPUTS OUTPUT N I/YR PV PMT FV 6-23

24 Solvg for PV: 3-year ordary auty of $00 at 0% $00 paymets stll occur at the ed of each perod, but ow there s o FV. PVA PMT PMT PVA PMT PVIFA, PMT IV PVA 00 3 IV

25 Solvg for PV: 3 year auty due of $00 at 0% Aga, $00 paymets occur at the begg of each perod. Set calculator to BEGIN mode. INPUTS OUTPUT N I/YR PV PMT FV 6-25

26 Solvg for PV: 3 year auty due of $00 at 0% Aga, $00 paymets occur at the begg of each perod. PVA Due PMT PVIFA, PVA Due 00 3 IV

27 What s the PV of ths ueve cash flow stream? 0 0% = PV

28 Solvg for PV: Ueve cash flow stream Iput cash flows the calculator s CFLO regster: CF 0 = 0 CF = 00 CF 2 = 300 CF 3 = 300 CF 4 = -50 Eter I/YR = 0, press NPV butto to get NPV = $ (Here NPV = PV.) 6-28

29 Solvg for I: What terest rate would cause $00 to grow to $ years? Solves the geeral FV equato for I. INPUTS OUTPUT 3 N I/YR PV PMT FV 6-29

30 Solvg for I: What terest rate would cause $00 to grow to $ years? Solves the geeral FV equato for I. FV PV % 6-30

31 Classfcatos of terest rates Nomal rate ( NOM ) also called the quoted or state rate. A aual rate that gores compoudg effects NOM s stated cotracts. Perods must also be gve, e.g. 8% Quarterly or 8% Daly terest NOM = Rate/Perod X # of Perods/Year NOM = PER x m 6-3

32 Classfcatos of terest rates Perodc rate ( PER ) amout of terest charged each perod, e.g. mothly or quarterly PER = NOM / m, where m s the umber of compoudg perods per year. m = 4 for quarterly ad m = 2 for mothly compoudg 6-32

33 Classfcatos of terest rates Effectve (or equvalet) aual rate (EAR = EFF%) the aual rate of terest actually beg eared, takg to accout compoudg EFF% for 0% semaual vestmet EFF% = ( + NOM / m ) m = ( + PER ) m - = ( / 2 ) 2 = 0.25% A vestor would be dfferet betwee a vestmet offerg a 0.25% aual retur ad oe offerg a 0% aual retur, compouded semaually. 6-33

34 Why s t mportat to cosder effectve rates of retur? A vestmet wth mothly paymets s dfferet from oe wth quarterly paymets. Must put each retur o a EFF% bass to compare rates of retur. Must use EFF% for comparsos. See followg values of EFF% rates at varous compoudg levels. EAR ANNUAL = 0.00% EAR QUARTERLY = ( + 0.0/4) 4 = 0.38% EAR MONTHLY = ( + 0.0/2) 2 = 0.47% EAR DAILY (360) = ( + 0.0/360) 360 = 0.52% 6-34

35 Whe s each rate used? NOM wrtte to cotracts, quoted by baks ad brokers. Not used calculatos or show o tme les. PER Used calculatos ad show o tme les. If m =, NOM = PER = EAR. EAR Used to compare returs o vestmets wth dfferet paymets per year. Used calculatos whe auty paymets do t match compoudg perods. 6-35

36 What s the FV of $00 after 3 years uder 0% semaual compoudg? Quarterly compoudg? FV PV ( m NOM ) m PV PER umber of perods FV FV FV FV 3S 3S 3Q 3Q $00 ( ) 2 6 $00 (.05) $34.0 $ $00 (.025) $

37 Ca the effectve rate ever be equal to the omal rate? Yes, but oly f aual compoudg s used,.e., f m =. If m >, EFF% wll always be greater tha the omal rate. 6-37

38 What s the FV of a 3-year $00 auty, f the quoted terest rate s 0%, compouded semaually? 0 5% Paymets occur aually, but compoudg occurs every 6 moths. Caot use ormal auty valuato techques. 6-38

39 Method : Compoud each cash flow 0 5% FV 3 = $00(.05) 4 + $00(.05) 2 + $00 FV 3 = $

40 Method 2: Facal calculator Fd the EAR ad treat as a auty. EAR = ( / 2 ) 2 = 0.25%. INPUTS OUTPUT N I/YR PV PMT FV

41 Fd the PV of ths 3-year ordary auty. Could solve by dscoutg each cash flow, or Use the EAR ad treat as a auty to solve for PV. INPUTS OUTPUT N I/YR PV PMT FV 6-4

42 Loa amortzato Amortzato tables are wdely used for home mortgages, auto loas, busess loas, retremet plas, etc. Facal calculators ad spreadsheets are great for settg up amortzato tables. EXAMPLE: Costruct a amortzato schedule for a $,000, 0% aual rate loa wth 3 equal paymets. 6-42

43 Step : Fd the requred aual paymet All put formato s already gve, just remember that the FV = 0 because the reaso for amortzg the loa ad makg paymets s to retre the loa. INPUTS OUTPUT N I/YR PV PMT FV 6-43

44 Step : Fd the requred aual paymet 3 / yr PV 0%,000 auty factor V aual paymet A,000 3 V 0,

45 Step 2: Fd the terest pad Year The borrower wll owe terest upo the tal balace at the ed of the frst year. Iterest to be pad the frst year ca be foud by multplyg the begg balace by the terest rate. INT t = Beg bal t () INT = $,000 (0.0) = $

46 Step 3: Fd the prcpal repad Year If a paymet of $402. was made at the ed of the frst year ad $00 was pad toward terest, the remag value must represet the amout of prcpal repad. PRIN= PMT INT = $ $00 = $

47 Step 4: Fd the edg balace after Year To fd the balace at the ed of the perod, subtract the amout pad toward prcpal from the begg balace. END BAL = BEG BAL PRIN = $,000 - $302. = $

48 Costructg a amortzato table: Repeat steps 4 utl ed of loa Year BEG BAL PMT INT PRIN END BAL $,000 $402 $00 $302 $ TOTAL, ,000 - Iterest pad decles wth each paymet as the balace decles. 6-48

49 Illustratg a amortzed paymet: Where does the moey go? $ Iterest Prcpal Paymets Costat paymets. Declg terest paymets. Declg balace. 6-49

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