2- 1 Outline 2: Time Value of Money & Introduction to Discount Rates & Rate of Return 2.1 Future Values 2.2 Present Values 2.3 Multiple Cash Flows 2.4

2- 1

Outline 2: Time Value of Money & Introduction to Discount Rates & Rate of Return

2.1 Future Values

2.2 Present Values

2.3 Multiple Cash Flows

2.4 Perpetuities and Annuities

2.5 Effective Annual Interest Rate

2.6 Loan Amortization

Appendix on Time Value of Money

2- 2

Future Values

Future Value - Amount to which an investment will grow after earning interest.

Compound Interest - Interest earned on interest.

Simple Interest - Interest earned only on the original investment.

2- 3

Future Values

Example - Simple InterestInterest earned at a rate of 6% for five years on a principal balance of $100.

Interest Earned Per Year = 100 x .06 = $ 6

2- 4

Future Values

Example - Simple Interest

Interest earned at a rate of 6% for five years on a principal balance of $100.

2- 5

Future Values



Today Future Years

1 2 3 4 5

Interest Earned

Value 100

2- 6

Future Values



Today Future Years

1 2 3 4 5

Interest Earned 6

Value 100 106

2- 7

Future Values



Today Future Years

1 2 3 4 5

Interest Earned 6 6

Value 100 106 112

2- 8

Future Values



Today Future Years

1 2 3 4 5

Interest Earned 6 6 6

Value 100 106 112 118

2- 9

Future Values



Today Future Years

1 2 3 4 5

Interest Earned 6 6 6 6

Value 100 106 112 118 124

2- 10

Future ValuesExample - Simple Interest


Today Future Years 1 2 3 4 5

Interest Earned 6 6 6 6 6Value 100 106 112 118 124 130

Value at the end of Year 5 = $130

2- 11

Future Values

Example - Compound Interest

Interest earned at a rate of 6% for five years on the previous year’s balance.

2- 12

Future Values



Interest Earned Per Year =Prior Year Balance x .06

2- 13

Future Values



Today Future Years

1 2 3 4 5

Interest Earned

Value 100

2- 14

Future Values



Today Future Years

1 2 3 4 5

Interest Earned 6.00

Value 100 106.00

2- 15

Future Values



Today Future Years

1 2 3 4 5

Interest Earned 6.00 6.36

Value 100 106.00 112.36

2- 16

Future Values



Today Future Years

1 2 3 4 5

Interest Earned 6.00 6.36 6.74

Value 100 106.00 112.36 119.10

2- 17

Future Values



Today Future Years

1 2 3 4 5

Interest Earned 6.00 6.36 6.74 7.15

Value 100 106.00 112.36 119.10 126.25

2- 18

Future ValuesExample - Compound Interest


Today Future Years 1 2 3 4 5

Interest Earned 6.00 6.36 6.74 7.15 7.57Value 100 106.00 112.36 119.10 126.25 133.82

Value at the end of Year 5 = $133.82

2- 19

Future Values

Future Value of $100 = FV

trFV )1(100$

2- 20

Future Values

Future Value of any Present Value = FV

where t= number of time periods

r=is the discount rate

trPVFV )1(

2- 21

Future Values

if t=4:

FV = PV(1+r)(1+r) (1+r)(1+r) = PV(1+r)4

if t=10:

FV = PV(1+r)(1+r)(1+r)(1+r)(1+r)(1+r)(1+r) (1+r)(1+r)(1+r)

= PV(1+r)10

2- 22

Future Values

if t=n:

FV = PV(1+r)(1+r) (1+r)(1+r)…(1+r)

= PV(1+r)n

if t=0:

FV = PV(1+r) = PV(1+r)0 = PV

2- 23

Future Values

FV r t $100 ( )1

Example - FV

What is the future value of $100 if interest is compounded annually at a rate of 6% for five years?

2- 24

Future Values

FV r t $100 ( )1

Example - FV

What is the future value of $100 if interest is compounded annually at a rate of 6% for five years?

82.133$)06.1(100$ 5 FV

2- 25

0

1000

2000

3000

4000

5000

6000

7000

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Number of Years

FV

of

$100

0%

5%

10%

15%

Future Values: FV with Compounding

Interest Rates

2- 26

Future Value: Manhattan Island Sale

Peter Minuit bought Manhattan Island for $24 in 1626. Was this a good deal?

000,000,000,000,000,846,592$

)124.1(24$ 382

FV

To answer, determine $24 is worth in the year 2006, compounded at 12.5% (long-term average annual return on S&P 500):

FYI - The value of Manhattan Island land is FYI - The value of Manhattan Island land is a very small fraction of this number.a very small fraction of this number.

2- 27

Present Values

Present Value

Value today of a future cash

flow.

Discount Rate

Interest rate used to compute

present values of future cash flows.

Discount Factor

Present value of a $1 future payment.

2- 28

Present Values

Present Value = PV

PV = Future Value after t periods

(1+r) t

2- 29

Present Values

Since FV = PV (1+r) then solve for PV by dividing both sides by (1+r):

F V P V r t ( )1

P V

F V

rt

1

2- 30

Present Values

Example

You just bought a new computer for $3,000. The payment terms are 2 years same as cash. If you can earn 8% on your money, how much money should you set aside today in order to make the payment when due in two years?

572,2$2)08.1(3000 PV

2- 31

Present Values

Example

You are twenty years old and want to have $1 million in cash when you are 80 years old (you can expect to live to one-hundred or more). If you expect to earn the long-term average 12.4% in the stock market how much do you need to invest now?

$ 8 9 9 $ 1, ,( . )

0 0 0 0 0 01

1 1 2 4 6 0

P V F Vr n

1

1( )

2- 32

Present Values

Discount Factor = DF = PV of $1

Discount Factors can be used to compute the present value of any cash flow.

r is the discount rate (of return)

DFr t

1

1( )

2- 33

The PV formula has many applications. Given any variables in the equation, you can solve for the remaining variable.

PV FVr t

1

1( )

Present Value

2- 34

Present Value: PV of Multiple Cash Flows

ExampleYour auto dealer gives you the choice to pay $15,500 cash now, or make three payments: $8,000 now and $4,000 at the end of the following two years. If your cost of money is 8%, which do you prefer?

$15,133.06 PVTotal

36.429,3

70.703,3

8,000.00

2

1

)08.1(

000,42

)08.1(

000,41

payment Immediate

PV

PV

2- 35

Present Value: PV of Multiple Cash Flows

PVs can be added together to evaluate multiple cash flows.

P VC

rt

tt

n

( )11

P V

C

r

C

r

C

r

C

rn

n

11

22

331 1 1 1

. . .

2- 36

Present Value: Perpetuities & Annuities

Perpetuity A stream of level cash payments

that never ends.

Annuity Equally spaced level stream of cash

flows for a limited period of time.

2- 37


PV of Perpetuity Formula

C = constant cash payment r = interest rate or rate of return

P V

C

r tt

11

P VC

r

2- 38


Example - Perpetuity

In order to create an endowment, which pays $100,000 per year, forever, how much money must be set aside today in the rate of interest is 10%?

PV 100 00010 000 000,. $1, ,

2- 39


Example - continued

If the first perpetuity payment will not be received until three years from today, how much money needs to be set aside today?

PV

1 000 000

1 10 3 315, ,

( . )$751,

2- 40


PV of Annuity Formula

C = cash payment

r = interest rate

t = Number of years (periods) cash payment is received

PV C r r r t

1 11( )

2- 41


If PV of Annuity Formula is:

Then formula for annuity payment is:

PV C r r r t

1 11( )

CP V

r r r t

1 1

1

2- 42


Formula for annuity payment can be used to find loan payments. Just think of C as Payment, PV as loan amount, t as the number of months, and r must be the periodic loan r to coincide with the frequency of payments:

CP V

r r r t

1 1

1

2- 43


PV Annuity Factor (PVAF) - The present value of $1 a year for each of t years.

PVAF r r r t

1 11( )

2- 44


Example - Annuity

You are purchasing a car. You are scheduled to make 60 month installments of $500 for a $25,000 auto. Given an annual market rate of interest of 5% for a car loan, what is the price you are paying for the car (i.e. what is the PV)?

P V C

r r r t

1

1 2

1

1 2 1 1 2

10 5

1 2

1

0 51 2 1 0 5

1 2

4 9 5

6 0$ 5 0 0. . .

$ 2 6 ,

2- 45


Example - Annuity

You have just won the NJ lottery for $2 million over 25 years. How much is the “$2 million” NJ Lottery really worth at an opportunity cost rate of return of 12.4% - long-run annual stock market rate of return (ignoring income taxes)?

P V C

r r r t

1 1

10 0 0

1

1 2 4

1

1 2 4 1 1 2 4

4 4 7

2 5$ 8 0,. . .

$ 6 1 0 ,

2- 46


Example - Annuity

Now what if you took the lump-sum based on a 5% discount rate by the State of New Jersey?

P V C

r r r t

1 1

10 0 0

1

0 5

1

0 5 1 0 5

1 2 7 5 1 6

2 5$ 8 0,. . .

$ 1, ,

2- 47

Perpetuities & Annuities

Example - Future Value of annual payments

You plan to save $4,000 every year for 20 years and then retire. Given a 10% rate of interest, what will be the FV of your retirement account?

FV

FV

4 000 1 10

100

110

110 1 10

2020, ( . )

$229,

. . ( . )

2- 48


Future Value of Ordinary Annuity:

F V C r

C r C r C r

Cr

r

n t

t

n

n n

t

1

1 1 1

1 1

1

1 2 0. . .

2- 49


Present Value of Ordinary Annuity:

P V Cr

Cr

Cr

Cr

Cr r r

t

nt

n

t

1

1

1

1

1

1

1

1

1 1

1

1

1 2

. . .

2- 50

Effective Interest Rates

Effective Annual Interest Rate - Interest rate that is annualized using compound interest.

E A Rr

mnom

m

1 1

r = annual or nominal rate of interest or return

m= number of compounding periods per year

rnom/m=also known as the periodic interest rate

2- 51


example

Given a monthly rate of 1%, what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)?

2- 52


example

Given a monthly rate of 1%, what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)?

12.00%or .12=12 x .01=APR

12.68%or .1268=1-.01)+(1=EAR

r=1-.01)+(1=EAR12

12

2- 53

Amortization

Amortization is the process by which a loan is paid off. During that process, the interest and contribution amounts change every month due to the mathematics of compounding.

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

2- 54

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

Amortization

2- 55

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.

Amortization

2- 56

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 3to complete the amortization table.

Amortization

2- 57

Interest declines and contribution to principal grows. Tax implications fromlower interest paid.

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

Amortization

2- 58

$

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

2- 59

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.

Amortization

2- 60

Future valuePresent valueRates of return

Appendix on Time Value of Money

2- 61

Future Value

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 lines show timing of cash flows.

2- 62Time line for a $100 lump sum due at the end of Year 2.

100

0 1 2 Yeari%

2- 63Time line for an ordinary annuity of $100 for 3 years.

100 100100

0 1 2 3i%

2- 64

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

100 50 75

0 1 2 3r%

-50

2- 65What’s the FV of an initial $100 after 3 years if r = 10%?

FV = ?

0 1 2 310%

100

Finding FVs is compounding.

2- 66

After 1 year:

FV1 = PV + INT1 = PV + PV(r)= PV(1 + r)= $100(1.10)= $110.00

After 2 years:

FV2 = PV(1 + r)2

= $100(1.10)2

= $121.00

2- 67

After 3 years:

FV3 = PV(1 + r)3

= 100(1.10)3

= $133.10

In general,

FVn = PV(1 + r)n

2- 68

Four Ways to Find FVs

Solve the equation with a regular calculator.

Use tables.Use a financial calculator.Use a spreadsheet.

2- 69

Financial calculators solve this equation:

FVn = PV(1 + r)n

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

Financial Calculator Solution

2- 70

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 r/YR PV PMT FV

133.10

2- 71

10%

What is the PV of $100 due in 3 years if r=10%? Finding PVs is discounting, and it’s the reverse of compounding.

100

0 1 2 3

PV = ?

2- 72

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

n

nnn

r+1

1FV =

r+1

FV = PV

PV = $1001

1.10 = $100 PVIF

= $100 0.7513 = $75.13.

i,n

3

.

What interest rate would cause $100 to grow to $125.97 in 3 years?

2- 73


3 10 0 100N r/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

2- 74

Solve for n:

FVn = 1(1 + r)n; 2 = 1(1.20)n

Use calculator to solve, see next slide.

If sales grow at 20% per year, how long before sales double?

2- 75

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

3.8

Graphical Illustration:

01 2 3 4

1

2

FV

3.8

Year

INPUTS

OUTPUT

2- 76

Ordinary Annuity

PMT PMTPMT

0 1 2 3r%

PMT PMT

0 1 2 3r%

PMT

Annuity Due

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

2- 77

100 100100

0 1 2 310%

110 121FV = 331

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

2- 78

3 10 0 -100

331.00


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

INPUTS

OUTPUTr/YRN PMT FVPV

2- 79What’s the PV of this ordinary annuity?

100 100100

0 1 2 310%

90.91

82.64

75.13248.68 = PV

2- 80

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

3 10 100 0

-248.69

INPUTS

OUTPUTN r/YR PV PMT FV

2- 81

100 100

0 1 2 3

10%

100

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

2- 82

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 r/YR PV PMT FV

2- 83

0

100

1

300

2

300

310%

-50

4

90.91247.93225.39 -34.15530.08 = PV

What is the PV of this uneven cashflow stream?

2- 84

Input in “CFLO” register:CF0 = 0

CF1 = 100

CF2 = 300

CF3 = 300

CF4 = -50Enter r = 10, then press NPV button to get

NPV = 530.09. (Here NPV = PV.)

2- 85Finding the interest rate or growth rate

3 -100 0 125.97

8%

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

INPUTS

OUTPUT

N r/YR PV PMT FV

2- 86Will the FV of a lump sum be larger or smaller if we compound more often, holding interest rate

constant? Why?

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

2- 87

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.

2- 88

Rates of Return:We will deal with 3 different rates:

rNom = nominal, or stated, or quoted, rate per year.

rPer = periodic rate.

EAR = EFF% = .effective annual

rate

2- 89

rNom is stated in contracts. Periods per year (m) must also be given.

Examples: 8%; Quarterly 8%, Daily interest (365 days)

2- 90

Periodic rate = rPer = rNom/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: rPer = 8%/4 = 2%.

8% daily (365): rPer = 8%/365 = 0.021918%.

2- 91

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 + rNom/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.

2- 92

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.

2- 93Find EFF% for a nominal rate of10%, compounded semi-annually

Or use a financial calculator.

%.25.101025.0

0.105.1

0.12

10.01

11%

2

2

m

Nom

m

rEFF

2- 94

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%.

2- 95Can 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- 96

When is each rate used?

iNom: Written into contracts, quoted by banks and brokers. Not used in calculations or shownon time lines.

2- 97

rPer: Used in calculations, shown on time lines.

If rNom has annual compounding,then rPer = rNom/1 = rNom.

2- 98

(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.

2- 99

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

2- 100What’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

2- 101

Payments occur annually, but compounding occurs each 6 months.

So we can’t use normal annuity valuation techniques.

2- 1021st 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.

2- 103

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

2- 104

Or, to find EAR with a calculator:

NOM% = 10.

P/YR = 2.

EFF% = 10.25.

2- 105

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

3 10.25 0 -100 INPUTS

OUTPUT

N r/YR PV FVPMT

331.80

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

c.

2- 106

What’s the PV of this stream?

0

100

15%

2 3

100 100

90.7082.27

74.62247.59

2- 107

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.)

2- 108

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

...

2- 109

273 -100 0

107.77

INPUTS

OUTPUT

N r/YR PV FVPMT

rPer = rNom/m= 10.0/365= 0.027397% per day.

Enter i in one step.Leave data in calculator.

2- 110

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.

2- 111

rPer = 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.

......

2- 112

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?

2- 113

3 Ways to Solve:

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

rPer =0.019178% per day.

1,000

0 365 456 days

-850

......

2- 114

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.

2- 115

456 -850 0

927.67

INPUTS

OUTPUT

N r/YR PV FVPMT

Calculator Solution to FV:rPer = rNom/m

= 7.0/365= 0.019178% per day.

Enter rPer in one step.

2- 116

2. Greatest Present Wealth

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

PV = $1,000/(1.00019178)456

= $916.27.

2- 117

456 .019178 0 1000

-916.27

INPUTS

OUTPUT

N r/YR PV FV

7/365 =

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

PMT

2- 118

Find the EFF% on note and compare with 7.25% bank pays, which is your opportunity cost of capital:

FVn = PV(1 + r)n

$1,000 = $850(1 + r)456

Now we must solve for r.

3. Rate of Return

2- 119

456 -850 0 1000

0.035646% per day

INPUTS

OUTPUT

N r/YR PV FVPMT

Convert % to decimal:

Decimal = 0.035646/100 = 0.00035646.

EAR = EFF% = (1.00035646)365 – 1 = 13.89%.

2- 120

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.

Documents

2- 1 Outline 2: Time Value of Money & Introduction to Discount Rates & Rate of Return 2.1 Future Values 2.2 Present Values 2.3 Multiple Cash Flows 2.4