Future value Present value Rates of return Amortization

Time Value of Money

Time lines show timing of cash flows.

CF0 CF1 CF3CF2

0 1 2 3i%

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.

100

0 1 2 Yeari%

Time line for an ordinary annuity of $100 for 3 years.

100 100100

0 1 2 3i%

Time line for uneven CFs -$50 at t = 0 and $100, $75, and $50 at the end of Years 1 through 3.

100 50 75

0 1 2 3i%

-50

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

FV = ?

0 1 2 310%

100

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 calculators solve this equation:

FVn = PV(1 + i)n.

There are 4 variables. If 3 are known, the calculator will solve for the 4th.

Financial Calculator Solution

Here’s the setup to find FV:

Clearing automatically sets everything to 0, but for safety enter PMT = 0.

Set: P/YR = 1, END

INPUTS

OUTPUT

3 10 -100 0N I/YR PV PMT FV

133.10

10%

What’s the PV of $100 due in 3 years if i = 10%?

Finding PVs is discounting, and it’s the reverse of compounding.

100

0 1 2 3

PV = ?

Solve FVn = PV(1 + i )n for PV:

n

nnn

i+11

FV = i+1

FV =PV

PV = $1001

1.10 = $100 PVIF

= $100 0.7513 = $75.13.

i,n

3

.

Financial Calculator Solution

3 10 0 100N I/YR PV PMT FV

-75.13

Either PV or FV must be negative. HerePV = -75.13. Put in $75.13 today, take out $100 after 3 years.

INPUTS

OUTPUT

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.

20 -1 0 2N I/YR PV PMT FV

3.8

Graphical Illustration:

01 2 3 4

1

2

FV

3.8

Year

INPUTS

OUTPUT

Ordinary Annuity

PMT PMTPMT

0 1 2 3i%

PMT PMT

0 1 2 3i%

PMT

Annuity Due

What’s the difference between an ordinary annuity and an annuity due?

What’s the FV of a 3-year ordinary annuity of $100 at 10%?

100 100100

0 1 2 310%

110 121FV = 331

3 10 0 -100

331.00

Financial Calculator Solution

Have payments but no lump sum PV, so enter 0 for present value.

INPUTS

OUTPUTI/YRN PMT FVPV

What’s the PV of this ordinary annuity?

100 100100

0 1 2 310%

90.91

82.64

75.13248.68 = PV

Have payments but no lump sum FV, so enter 0 for future value.

3 10 100 0

-248.69

INPUTS

OUTPUTN I/YR PV PMT FV

Find the FV and PV if theannuity were an annuity due.

100 100

0 1 2 3

10%

100

3 10 100 0

-273.55

Switch from “End” to “Begin.”Then enter variables to find PVA3 = $273.55.

Then enter PV = 0 and press FV to findFV = $364.10.

INPUTS

OUTPUTN I/YR PV PMT FV

What is the PV of this uneven cashflow stream?0

100

1

300

2

300

310%

-50

4

90.91247.93225.39 -34.15530.08 = 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?

3 -100 0 125.97

8%

$100 (1 + i )3 = $125.97.

INPUTS

OUTPUT

N I/YR PV PMT FV

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 morefrequent than once a year--for example, semiannually, quarterly,or daily--interest is earned on interestmore often.

0 1 2 310%

0 1 2 3

5%

4 5 6

134.01

100 133.10

1 2 30

100

Annually: FV3 = 100(1.10)3 = 133.10.

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.

%.25.101025.0

0.105.1

0.12

10.01

1m

i1%EFF

2

2

m

Nom

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 shownon time lines.

iPer:Used in calculations, shown on time lines.

If iNom has annual compounding,then iPer = iNom/1 = iNom.

(Used for calculations if and only ifdealing with annuities where payments don’t match interest compounding periods.)

EAR = EFF%:

Used to compare returns on investments with different payments per year.

FV of $100 after 3 years under 10% semiannual compounding? Quarterly?

= $100(1.05)6 = $134.01.FV3Q = $100(1.025)12 = $134.49.

FV = PV 1 .+ imnNom

mn

FV = $100 1 + 0.10

23S

2x3

What’s the value at the end of Year 3of the following CF stream if the quoted interest rate is 10%, compounded semiannually?

0 1

100

2 35%

4 5 6 6-mos. periods

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

100

2 35%

4 5 6

100 100.00110.25121.55331.80

FVA3 = 100(1.05)4 + 100(1.05)2 + 100= 331.80.

Could you find FV with afinancial calculator?Yes, by following these steps:

a. Find the EAR for the quoted rate:

2nd Method: Treat as an Annuity

EAR = (1 + ) – 1 = 10.25%. 0.10

22

Or, to find EAR with a calculator:

NOM% = 10.

P/YR = 2.

EFF% = 10.25.

EFF% = 10.25P/YR = 1NOM% = 10.25

3 10.25 0 -100 INPUTS

OUTPUT

N I/YR PV FVPMT

331.80

b. The cash flow stream is an annual annuity. Find kNom (annual) whose EFF% = 10.25%. In calculator,

c.

What’s the PV of this stream?

0

100

15%

2 3

100 100

90.7082.27

74.62247.59

Amortization

Construct an amortization schedulefor a $1,000, 10% annual rate loanwith 3 equal payments.

Step 1: Find the required payments.

PMT PMTPMT

0 1 2 310%

-1,000

3 10 -1000 0 INPUTS

OUTPUT

N I/YR PV FVPMT

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 afterYear 1.

End bal = Beg bal – Repmt = $1,000 – $302.11 = $697.89.

Repeat these steps for Years 2 and 3to complete the amortization table.

Interest declines. Tax implications.

BEG PRIN ENDYR 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

$

0 1 2 3

402.11Interest

302.11

Level payments. Interest declines because outstanding balance declines. Lender earns10% on loan outstanding, which is falling.

Principal Payments

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 = ?

0 1 2 273

0.027397%

-100

Note: % in calculator, decimal in equation.

FV = $100 1.00027397 = $100 1.07765 = $107.77.

273273

...

273 -100 0

107.77

INPUTS

OUTPUT

N I/YR PV FVPMT

iPer = iNom/m= 10.0/365= 0.027397% per day.

Enter i in one step.Leave data in calculator.

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 = 119.10

0 365 638 days

-100

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?

3 Ways to Solve:

1. Greatest future wealth: FV2. Greatest wealth today: PV3. Highest rate of return: Highest EFF%

iPer = 0.019178% per day.

1,000

0 365 456 days

-850

......

1. Greatest Future Wealth

Find FV of $850 left in bank for15 months and compare withnote’s FV = $1,000.

FVBank = $850(1.00019178)456

= $927.67 in bank.

Buy the note: $1,000 > $927.67.

456 -850 0

927.67

INPUTS

OUTPUT

N I/YR PV FVPMT

Calculator Solution to FV:

iPer = iNom/m= 7.0/365= 0.019178% per day.

Enter iPer in one step.

2. Greatest Present Wealth

Find PV of note, and comparewith its $850 cost:

PV = $1,000/(1.00019178)456

= $916.27.

456 .019178 0 1000

-916.27

INPUTS

OUTPUT

N I/YR PV FV

7/365 =

PV of note is greater than its $850 cost, so buy the note. Raises your wealth.

PMT

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.

3. Rate of Return

456 -850 0 1000

0.035646% per day

INPUTS

OUTPUT

N I/YR PV FVPMT

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.