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CHAPTER 7 Time Value of Money. Future value Present value Rates of return Amortization. Time lines show timing of cash flows. 0. 1. 2. 3. i%. CF 0. CF 1. CF 2. CF 3. Tick marks at ends of periods, so Time 0 is today; Time 1 is the end of Period 1; or the beginning of Period 2.
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CHAPTER 7Time Value of Money • Future value • Present value • Rates of return • Amortization
Time lines show timing of cash flows. 0 1 2 3 i% CF0 CF1 CF2 CF3 Tick marksat ends of periods, so Time 0 is today; Time 1 is the end of Period 1; or the beginning of Period 2.
Time line for a $100 lump sum due at the end of Year 2. 0 1 2 Year i% 100
Time line for an ordinary annuity of $100 for 3 years. 0 1 2 3 i% 100 100 100
Time line for uneven CFs -$50 at t = 0 and $100, $75, and $50 at the end of Years 1 through 3. 0 1 2 3 i% -50 100 75 50
What’s the FV of an initial $100 after 3 years if i = 10%? 0 1 2 3 10% 100 FV = ? Finding FVs is compounding.
After 1 year: FV1 = PV + INT1 = PV + PV(i) = PV(1 + i) = $100(1.10) = $110.00. After 2 years: FV2 = PV(1 + i)2 = $100(1.10)2 = $121.00.
After 3 years: FV3 = PV(1 + i)3 = 100(1.10)3 = $133.10. In general, FVn = PV(1 + i)n.
Four Ways to Find FVs • Solve the equation with a regular calculator. • Use tables. • Use a financial calculator. • Use a spreadsheet.
Financial Calculator Solution Financial calculators solve this equation: FVn = PV(1 + i)n. There are 4 variables. If 3 are known, the calculator will solve for the 4th.
Here’s the setup to find FV: INPUTS 3 10 -100 0 N I/YR PV PMT FV 133.10 OUTPUT Clearing automatically sets everything to 0, but for safety enter PMT = 0. Set: P/YR = 1, END
What’s the PV of $100 due in 3 years if i = 10%? Finding PVs is discounting, and it’s the reverse of compounding. 0 1 2 3 10% 100 PV = ?
Solve FVn = PV(1 + i )n for PV: . 3 1 ö æ ( ) ÷ PV = $100 = $100 PVIF ç ø è i, n 1.10 ( ) = $100 0.7513 = $75.13.
Financial Calculator Solution INPUTS 3 10 0 100 N I/YR PV PMT FV -75.13 OUTPUT Either PV or FV must be negative. Here PV = -75.13. Put in $75.13 today, take out $100 after 3 years.
If sales grow at 20% per year, how long before sales double? Solve for n: FVn = 1(1 + i)n; 2 = 1(1.20)n Use calculator to solve, see next slide.
INPUTS 20 -1 0 2 N I/YR PV PMT FV 3.8 OUTPUT Graphical Illustration: FV 2 3.8 1 Year 0 1 2 3 4
What’s the difference between an ordinaryannuity and an annuitydue? Ordinary Annuity 0 1 2 3 i% PMT PMT PMT Annuity Due 0 1 2 3 i% PMT PMT PMT
What’s the FV of a 3-year ordinary annuity of $100 at 10%? 0 1 2 3 10% 100 100 100 110 121 FV = 331
Financial Calculator Solution INPUTS 3 10 0 -100 331.00 N I/YR PV PMT FV OUTPUT Have payments but no lump sum PV, so enter 0 for present value.
What’s the PV of this ordinary annuity? 0 1 2 3 10% 100 100 100 90.91 82.64 75.13 248.68 = PV
INPUTS 3 10 100 0 N I/YR PV PMT FV OUTPUT -248.69 Have payments but no lump sum FV, so enter 0 for future value.
Find the FV and PV if theannuity were an annuity due. 0 1 2 3 10% 100 100 100
Switch from “End” to “Begin.” Then enter variables to find PVA3 = $273.55. INPUTS 3 10 100 0 -273.55 N I/YR PV PMT FV OUTPUT Then enter PV = 0 and press FV to find FV = $364.10.
What is the PV of this uneven cashflow stream? 4 0 1 2 3 10% 100 300 300 -50 90.91 247.93 225.39 -34.15 530.08 = PV
Input in “CFLO” register: CF0 = 0 CF1 = 100 CF2 = 300 CF3 = 300 CF4 = -50 • Enter I = 10, then press NPV button to get NPV = 530.09. (Here NPV = PV.)
What interest rate would cause $100 to grow to $125.97 in 3 years? $100 (1 + i )3 = $125.97. INPUTS 3 -100 0 125.97 N I/YR PV PMT FV OUTPUT 8%
Will the FV of a lump sum be larger or smaller if we compound more often, holding the stated I% constant? Why? LARGER! If compounding is more frequent than once a year--for example, semiannually, quarterly, or daily--interest is earned on interest more often.
0 1 2 3 10% 100 133.10 Annually: FV3 = 100(1.10)3 = 133.10. 0 1 2 3 4 5 6 0 1 2 3 5% 100 134.01 Semiannually: FV6 = 100(1.05)6 = 134.01.
We will deal with 3 different rates: iNom = nominal, or stated, or quoted, rate per year. iPer = periodic rate. EAR = EFF% = . effective annual rate
iNom is stated in contracts. Periods per year (m) must also be given. • Examples: • 8%; Quarterly • 8%, Daily interest (365 days)
Periodic rate = iPer = iNom/m, where m is number of compounding periods per year. m = 4 for quarterly, 12 for monthly, and 360 or 365 for daily compounding. • Examples: 8% quarterly: iPer = 8%/4 = 2%. 8% daily (365): iPer = 8%/365 = 0.021918%.
Effective Annual Rate (EAR = EFF%): The annual rate that causes PV to grow to the same FV as under multi-period compounding. Example: EFF% for 10%, semiannual: FV = (1 + iNom/m)m = (1.05)2 = 1.1025. EFF%= 10.25% because (1.1025)1 = 1.1025. Any PV would grow to same FV at 10.25% annually or 10% semiannually.
An investment with monthly payments is different from one with quarterly payments. Must put on EFF% basis to compare rates of return. Use EFF% only for comparisons. • Banks say “interest paid daily.” Same as compounded daily.
How do we find EFF% for a nominal rate of 10%, compounded semiannually? Or use a financial calculator.
EAR = EFF% of 10% EARAnnual = 10%. EARQ = (1 + 0.10/4)4 – 1 = 10.38%. EARM = (1 + 0.10/12)12 – 1 = 10.47%. EARD(360) = (1 + 0.10/360)360 – 1 = 10.52%.
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.
When is each rate used? iNom: Written into contracts, quoted by banks and brokers. Not used in calculations or shown on time lines.
iPer: Used in calculations, shown on time lines. If iNom has annual compounding, then iPer = iNom/1 = iNom.
EAR = EFF%: Used to compare returns on investments with different payments per year. (Used for calculations if and only if dealing with annuities where payments don’t match interest compounding periods.)
FV of $100 after 3 years under 10% semiannual compounding? Quarterly? mn i æ ö Nom FV = PV 1 . + ç ÷ è ø n m 2x3 0.10 æ ö FV = $100 1 + ç ÷ è ø 3S 2 = $100(1.05)6 = $134.01. FV3Q = $100(1.025)12 = $134.49.
What’s the value at the end of Year 3of the following CF stream if the quoted interest rate is 10%, compounded semiannually? 4 5 0 1 2 3 6 6-mos. periods 5% 100 100 100
Payments occur annually, but compounding occurs each 6 months. • So we can’t use normal annuity valuation techniques.
1st Method: Compound Each CF 0 1 2 3 4 5 6 5% 100 100.00 100 110.25 121.55 331.80 FVA3 = 100(1.05)4 + 100(1.05)2 + 100 = 331.80.
Could you find FV with afinancial calculator? 2nd Method: Treat as an Annuity Yes, by following these steps: a. Find the EAR for the quoted rate: EAR = (1 + ) – 1 = 10.25%. 2 0.10 2
Or, to find EAR with a calculator: NOM% = 10. P/YR = 2. EFF% = 10.25.
b. The cash flow stream is an annual annuity. Find kNom (annual) whose EFF% = 10.25%. In calculator, EFF% = 10.25 P/YR = 1 NOM% = 10.25 c. 3 10.25 0 -100 INPUTS N I/YR PV PMT FV OUTPUT 331.80
What’s the PV of this stream? 0 1 2 3 5% 100 100 100 90.70 82.27 74.62 247.59
Amortization Construct an amortization schedule for a $1,000, 10% annual rate loan with 3 equal payments.
402.11 Step 1: Find the required payments. 0 1 2 3 10% -1,000 PMT PMT PMT 3 10 -1000 0 INPUTS N I/YR PV PMT FV OUTPUT
Step 2: Find interest charge for Year 1. INTt = Beg balt (i) INT1 = $1,000(0.10) = $100. Step 3: Find repayment of principal in Year 1. Repmt = PMT – INT = $402.11 – $100 = $302.11.