CHAPTER 7 Time Value of Money Future value Present value Rates of return Amortization

Time lines show timing of cash flows. 1 2 3 i% CF0 CF1 CF2 CF3 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.

Time line for a $100 lump sum due at the end of Year 2. 1 2 Year i% 100

Time line for an ordinary annuity of $100 for 3 years. 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. 1 2 3 i% -50 100 75 50

What’s the FV of an initial $100 after 3 years if i = 10%? 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. 1 2 3 10% PV = ? 100

( ) ( ) Solve FVn = PV(1 + i )n for PV: . 1 æ ö PV = $100 ç ÷ = $100 3 1 æ ö ( ) PV = $100 ç ÷ = $100 PVIF è ø i, n 1.10 = $100 ( 0.7513 ) = $75.13.

Financial Calculator Solution 3 10 0 100 N I/YR PV PMT FV -75.13 INPUTS 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.

Graphical Illustration: INPUTS 20 -1 0 2 N I/YR PV PMT FV 3.8 OUTPUT Graphical Illustration: FV 2 3.8 1 Year 1 2 3 4

What’s the difference between an ordinary annuity and an annuity due? 1 2 3 i% PMT PMT PMT Annuity Due 1 2 3 i% PMT PMT PMT

What’s the FV of a 3-year ordinary annuity of $100 at 10%? 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? 1 2 3 10% 100 100 100 90.91 82.64 75.13 248.68 = PV

Have payments but no lump sum FV, so enter 0 for future value. 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 the annuity were an annuity due. 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 cash flow stream? 1 2 3 4 10% 100 300 300 -50 90.91 247.93 225.39 -34.15 530.08 = PV

Input in “CFLO” register: 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? 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.

1 2 3 10% 100 133.10 Annually: FV3 = 100(1.10)3 = 133.10. 1 2 3 1 2 3 4 5 6 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 FV = PV ç 1 . + Nom ÷ n è ø m 2x3 0.10 æ ö FV = $100 ç 1 + ÷ 3S è ø 2 = $100(1.05)6 = $134.01. FV3Q = $100(1.025)12 = $134.49.

What’s the value at the end of Year 3 of the following CF stream if the quoted interest rate is 10%, compounded semiannually? 1 2 3 4 5 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 1 2 3 4 5 6 5% 100 100 100.00 110.25 121.55 331.80 FVA3 = 100(1.05)4 + 100(1.05)2 + 100 = 331.80.

Could you find FV with a financial calculator? 2nd Method: Treat as an Annuity Could you find FV with a financial calculator? Yes, by following these steps: a. Find the EAR for the quoted rate: EAR = (1 + ) – 1 = 10.25%. 0.10 2 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? 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.

Step 1: Find the required payments. 1 2 3 10% -1,000 PMT PMT PMT 3 10 -1000 0 INPUTS N I/YR PV PMT FV OUTPUT 402.11

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.

Step 4: Find ending balance after Year 1. End bal = Beg bal – Repmt = $1,000 – $302.11 = $697.89. Repeat these steps for Years 2 and 3 to complete the amortization table.

Interest declines. Tax implications. BEG PRIN END YR BAL PMT INT PMT BAL 1 $1,000 $402 $100 $302 $698 2 698 402 70 332 366 3 366 402 37 366 0 TOT 1,206.34 206.34 1,000 Interest declines. Tax implications.

10% on loan outstanding, which is falling. $ 402.11 Interest 302.11 Principal Payments 1 2 3 Level payments. Interest declines because outstanding balance declines. Lender earns 10% on loan outstanding, which is falling.

Amortization tables are widely used--for home mortgages, auto loans, business loans, retirement plans, etc. They are very important! Financial calculators (and spreadsheets) are great for setting up amortization tables.

On January 1 you deposit $100 in an account that pays a nominal interest rate of 10%, with daily compounding (365 days). How much will you have on October 1, or after 9 months (273 days)? (Days given.)

( ) ( ) iPer = 10.0% / 365 = 0.027397% per day. ... FV = $100 1 2 273 0.027397% ... -100 FV = ? FV = $100 ( 1.00027397 ) 273 273 = $100 ( 1.07765 ) = $107.77. Note: % in calculator, decimal in equation.

Leave data in calculator. iPer = iNom/m = 10.0/365 = 0.027397% per day. INPUTS 273 -100 0 107.77 N I/YR PV PMT FV OUTPUT Enter i in one step. Leave data in calculator.

Now suppose you leave your money in the bank for 21 months, which is 1 Now suppose you leave your money in the bank for 21 months, which is 1.75 years or 273 + 365 = 638 days. How much will be in your account at maturity? Answer: Override N = 273 with N = 638. FV = $119.10.

iPer = 0.027397% per day. ... ... FV = $100(1 + .10/365)638 365 638 days ... ... -100 FV = 119.10 FV = $100(1 + .10/365)638 = $100(1.00027397)638 = $100(1.1910) = $119.10.

You are offered a note that pays $1,000 in 15 months (or 456 days) for $850. You have $850 in a bank that pays a 7.0% nominal rate, with 365 daily compounding, which is a daily rate of 0.019178% and an EAR of 7.25%. You plan to leave the money in the bank if you don’t buy the note. The note is riskless. Should you buy it?

1. Greatest future wealth: FV 2. Greatest wealth today: PV iPer = 0.019178% per day. 365 456 days ... ... -850 1,000 3 Ways to Solve: 1. Greatest future wealth: FV 2. Greatest wealth today: PV 3. Highest rate of return: Highest EFF%

1. Greatest Future Wealth Find FV of $850 left in bank for 15 months and compare with note’s FV = $1,000. FVBank = $850(1.00019178)456 = $927.67 in bank. Buy the note: $1,000 > $927.67.

Calculator Solution to FV: iPer = iNom/m = 7.0/365 = 0.019178% per day. INPUTS 456 -850 0 927.67 N I/YR PV PMT FV OUTPUT Enter iPer in one step.

2. Greatest Present Wealth Find PV of note, and compare with its $850 cost: PV = $1,000/(1.00019178)456 = $916.27.

7/365 = INPUTS 456 .019178 0 1000 -916.27 N I/YR PV PMT FV OUTPUT PV of note is greater than its $850 cost, so buy the note. Raises your wealth.

3. Rate of Return Find the EFF% on note and compare with 7.25% bank pays, which is your opportunity cost of capital: FVn = PV(1 + i)n $1,000 = $850(1 + i)456 Now we must solve for i.

Convert % to decimal: Decimal = 0.035646/100 = 0.00035646. INPUTS 456 -850 0 1000 0.035646% per day N I/YR PV PMT FV OUTPUT Convert % to decimal: Decimal = 0.035646/100 = 0.00035646. EAR = EFF% = (1.00035646)365 – 1 = 13.89%.

Using interest conversion: P/YR = 365. NOM% = 0.035646(365) = 13.01. EFF% = 13.89. Since 13.89% > 7.25% opportunity cost, buy the note.