# CHAPTER 2. Time Value of Money 2-1

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1 CHAPTER 2 Time Value of Money 2-1

2 Time Value of Money (TVM) Time Lines Future value & Present value Rates of return Annuities & Perpetuities Uneven cash Flow Streams Amortization 2-2

3 Time lines i% CF 0 CF 1 CF 2 CF 3 Show the timing of cash flows Tick marks occur at the end of periods, so Time 0 is today; Time 1 is the end of the first period (year, month, etc.) or the beginning of the second period 2-3

4 Drawing time lines: \$100 lump sum due in 2 years; 3-year \$100 ordinary annuity \$100 lump sum due in 2 years i% 3 year \$100 ordinary annuity i%

5 Drawing time lines: Uneven cash flow stream; CF 0 = -\$50, CF 1 = \$100, CF 2 = \$75, and CF 3 = \$50 Uneven cash flow stream i%

6 What is the future value (FV) of an initial \$100 after 3 years, if I/YR = 10%? Finding the FV of a cash flow or series of cash flows when compound interest is applied is called compounding FV can be solved by using the arithmetic, financial calculator, and spreadsheet methods % 100 FV =? 2-6

7 Solving for FV: The arithmetic method After 1 year: FV 1 = PV ( 1 + i ) = \$100 (1.10) = \$ After 2 years: FV 2 = PV ( 1 + i ) 2 = \$100 (1.10) 2 =\$ After 3 years: FV 3 = PV ( 1 + i ) 3 = \$100 (1.10) 3 =\$ After n years (general case): FV n = PV ( 1 + i ) n 2-7

8 Solving for FV: The calculator method Solves the general FV equation Requires 4 inputs into calculator, and will solve for the fifth. (Set to P/YR = 1 and END mode.) INPUTS OUTPUT N I/YR PV PMT FV

9 What is the present value (PV) of \$100 due in 3 years, if I/YR = 10%? Finding the PV of a cash flow or series of cash flows when compound interest is applied is called discounting (the reverse of compounding) The PV shows the value of cash flows in terms of today s purchasing power % PV =?

10 Solving for PV: The arithmetic method Solve the general FV equation for PV: PV FV = 1+ n ( i) n PV = FV 3 / ( 1 + i ) 3 = \$100 / ( 1.10 ) 3 = \$

11 Solving for PV: The financial tables PV = FV n II i PV = 3 FV 3 II 10 PV = \$ =

12 Solving for PV: The calculator method Solves the general FV equation for PV Exactly like solving for FV, except we have different input information and are solving for a different variable INPUTS OUTPUT N I/YR PV PMT FV 2-12

13 Solving for N: If sales grow at 20% per year, how long before sales double? Solves the general FV equation for N Same as previous problems, but now solving for N INPUTS OUTPUT N I/YR PV PMT FV 2-13

14 Solving for N: with natural logarithm PV = 1 FV = 2 i = 20% ( ) n \$ 1 = ( ) n = 2 ( 1.20) ln 2 nln = n0.18 = 0.69 n =

15 Solving for N: with financial tables PV = 1, FV = 2, i = 20% ( ) n \$ 1 = ( ) n = 2 ( ) n 1.20 = 2 tables A1(20% and factor 2) between 3 and 4 years 2-15

16 interpolation y = 3 x 1 y = n y 2 = 4 x 1 x 2 = = 2 = n = 3 + = ( ) 3.79 years 2-16

17 Annuities & Perpetuities Annuity a series of fixed, equal payments for a specified number of periods Ordinary Annuity payments at the end of the period Annuity Due payments at the beginning of the period Perpetuity an Annuity forever 2-17

18 Difference between Ordinary Annuity and Annuity Due Ordinary Annuity i% Annuity Due PMT PMT PMT i% PMT PV PMT PMT FV 2-18

19 Future Value of an Annuity Solving for FV: 3-year ordinary annuity of \$100 at 10% \$100 payments occur at the end of each period, but there is no PV. FVA ( 1+ i) n 1 = PMT = PMT i, n i ( ) n FVIFA = PMT III i FVA = 100 ( ) = 100 ( 3.310) =

20 Solving for FV: financial calculator 3-year ordinary annuity of \$100 at 10% \$100 payments occur at the end of each period, but there is no PV. INPUTS OUTPUT N I/YR PV PMT FV

21 Solving for FV: 3-year annuity due of \$100 at 10% Now, \$100 payments occur at the beginning of each period. Set calculator to BEGIN mode. INPUTS OUTPUT N I/YR PV PMT FV

22 Solving for FV: 3-year annuity due of \$100 at 10% Now, \$100 payments occur at the beginning of each period. FVA FVA ( Due) = PMT ( 1+ i) i n 1 ( 1+ i) ( ) ( )( ) n Due = PMT FVIFA 1+ i = PMT III ( 1+ i) i, n i FVA FVA ( Due) = 100 ( ) ( ) ( Due) = 100( 3.310)( 1.10) =

23 Present value of an Annuity Solving for PV: 3-year ordinary annuity of \$100 at 10% \$100 payments still occur at the end of each period, but now there is no FV. INPUTS OUTPUT N I/YR PV PMT FV 2-23

24 Solving for PV: 3-year ordinary annuity of \$100 at 10% \$100 payments still occur at the end of each period, but now there is no FV. PVA PVA = = 1 PMT PMT 1 ( ) n ( ) n 1+ i 1+ i = PMT i ( 1+ i) ( ) n PVIFA = PMT IV i, n i 1 i n 3 PVA = 100 IV10 = =

25 Solving for PV: 3 year annuity due of \$100 at 10% Again, \$100 payments occur at the beginning of each period. Set calculator to BEGIN mode. INPUTS OUTPUT N I/YR PV PMT FV 2-25

26 Solving for PV: 3 year annuity due of \$100 at 10% Again, \$100 payments occur at the beginning of each period. PVA ( Due) = PMT( PVIFA )( 1 i) n i, n + PVA ( ) 3 Due = 100 IV ( ) 10 = =

27 What is the PV of this uneven cash flow stream? 0 10% = PV

28 Solving for PV: Uneven cash flow stream Input cash flows in the calculator s CFLO register: CF 0 = 0 CF 1 = 100 CF 2 = 300 CF 3 = 300 CF 4 = -50 Enter I/YR = 10, press NPV button to get NPV = \$ (Here NPV = PV.) 2-28

29 Solving for I: What interest rate would cause \$100 to grow to \$ in 3 years? Solves the general FV equation for I. INPUTS OUTPUT 3 N I/YR PV PMT FV 2-29

30 Solving for I: What interest rate would cause \$100 to grow to \$ in 3 years? Solves the general FV equation for I. ( i) n FV = PV = ( + i) = ( 1+ i) 3 i = i = = 0.08 = 8% 2-30

31 Classifications of interest rates Nominal rate (i NOM ) also called the quoted or state rate. An annual rate that ignores compounding effects i NOM is stated in contracts. Periods must also be given, e.g. 8% Quarterly or 8% Daily interest i NOM = Rate/Period X # of Periods/Year i NOM = i PER xm 2-31

32 Classifications of interest rates Periodic rate (i PER ) amount of interest charged each period, e.g. monthly or quarterly i PER = i NOM / m, where m is the number of compounding periods per year. m = 4 for quarterly and m = 12 for monthly compounding 2-32

33 Classifications of interest rates Effective (or equivalent) annual rate (EAR = EFF%) the annual rate of interest actually being earned, taking into account compounding EFF% for 10% semiannual investment EFF% = ( 1 + i NOM / m ) m 1 = ( 1 + i PER ) m -1 = ( / 2 ) 2 1 = 10.25% An investor would be indifferent between an investment offering a 10.25% annual return and one offering a 10% annual return, compounded semiannually. 2-33

34 Why is it important to consider effective rates of return? An investment with monthly payments is different from one with quarterly payments. Must put each return on an EFF% basis to compare rates of return. Must use EFF% for comparisons. See following values of EFF% rates at various compounding levels. EAR ANNUAL = 10.00% EAR QUARTERLY = ( /4) 4 1 = 10.38% EAR MONTHLY = ( /12) 12 1 = 10.47% EAR DAILY (360) = ( /360) = 10.52% 2-34

35 When is each rate used? i NOM written into contracts, quoted by banks and brokers. Not used in calculations or shown on time lines. i PER Used in calculations and shown on time lines. If m = 1, i NOM = i PER = EAR. EAR Used to compare returns on investments with different payments per year. Used in calculations when annuity payments don t match compounding periods. 2-35

36 What is the FV of \$100 after 3 years under 10% semiannual compounding? Quarterly compounding? FV n = PV ( 1 + i m NOM ) m n = PV 1 ( + i ) PER number of periods FV FV FV FV 3S 3S 3Q 3Q = \$100 ( 1 + ) 2 6 = \$100 (1.05) = \$ = \$ = \$100 (1.025) = \$

37 Can the effective rate ever be equal to the nominal rate? Yes, but only if annual compounding is used, i.e., if m = 1. If m > 1, EFF% will always be greater than the nominal rate. 2-37

38 What s the FV of a 3-year \$100 annuity, if the quoted interest rate is 10%, compounded semiannually? 0 1 5% Payments occur annually, but compounding occurs every 6 months. Cannot use normal annuity valuation techniques. 2-38

39 Method 1: Compound each cash flow 0 1 5% FV 3 = \$100(1.05) 4 + \$100(1.05) 2 + \$100 FV 3 = \$

40 Method 2: Financial calculator Find the EAR and treat as an annuity. EAR = ( / 2 ) 2 1 = 10.25%. INPUTS OUTPUT N I/YR PV PMT FV

41 Find the PV of this 3-year ordinary annuity. Could solve by discounting each cash flow, or Use the EAR and treat as an annuity to solve for PV. INPUTS OUTPUT N I/YR PV PMT FV 2-41

42 Loan amortization Amortization tables are widely used for home mortgages, auto loans, business loans, retirement plans, etc. Financial calculators and spreadsheets are great for setting up amortization tables. EXAMPLE: Construct an amortization schedule for a \$1,000, 10% annual rate loan with 3 equal payments. 2-42

43 Step 1: Find the required annual payment All input information is already given, just remember that the FV = 0 because the reason for amortizing the loan and making payments is to retire the loan. INPUTS OUTPUT N I/YR PV PMT FV 2-43

44 Step 1: Find the required annual payment n = 3 i / yr = 10% PV = 1,000 annuity factor V n i = ( ) n 1+ i i ( 1+ ) n i 1 annual payment A n 3 = 1,000 V10 = 1, =

45 Step 2: Find the interest paid in Year 1 The borrower will owe interest upon the initial balance at the end of the first year. Interest to be paid in the first year can be found by multiplying the beginning balance by the interest rate. INT t = Beg bal t (i) INT 1 = \$1,000 (0.10) = \$

46 Step 3: Find the principal repaid in Year 1 If a payment of \$ was made at the end of the first year and \$100 was paid toward interest, the remaining value must represent the amount of principal repaid. PRIN= PMT INT = \$ \$100 = \$

47 Step 4: Find the ending balance after Year 1 To find the balance at the end of the period, subtract the amount paid toward principal from the beginning balance. END BAL = BEG BAL PRIN = \$1,000 - \$ = \$

48 Constructing an amortization table: Repeat steps 1 4 until end of loan Year BEG BAL PMT INT PRIN END BAL 1 \$1,000 \$402 \$100 \$302 \$ TOTAL 1, ,000 - Interest paid declines with each payment as the balance declines. 2-48

49 Illustrating an amortized payment: Where does the money go? \$ Interest Principal Payments Constant payments. Declining interest payments. Declining balance. 2-49

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