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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. Future Values.

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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


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.


Future Values

Example - Simple Interest

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

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


Future Values

Example - Simple Interest

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


Future Values

Example - Simple Interest

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

TodayFuture Years

12345

Interest Earned

Value100


Future Values

Example - Simple Interest

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

TodayFuture Years

12345

Interest Earned 6

Value100106


Future Values

Example - Simple Interest

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

TodayFuture Years

12345

Interest Earned 6 6

Value100106112


Future Values

Example - Simple Interest

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

TodayFuture Years

12345

Interest Earned 6 6 6

Value100106112118


Future Values

Example - Simple Interest

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

TodayFuture Years

12345

Interest Earned 6 6 6 6

Value100106112118124


Future Values

Example - Simple Interest

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

TodayFuture Years

12345

Interest Earned 6 6 6 6 6

Value100106112118124130

Value at the end of Year 5 = $130


Future Values

Example - Compound Interest

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


Future Values

Example - Compound Interest

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

Interest Earned Per Year =Prior Year Balance x .06


Future Values

Example - Compound Interest

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

TodayFuture Years

12345

Interest Earned

Value100


Future Values

Example - Compound Interest

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

TodayFuture Years

12345

Interest Earned 6.00

Value100106.00


Future Values

Example - Compound Interest

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

TodayFuture Years

12345

Interest Earned 6.00 6.36

Value100106.00112.36


Future Values

Example - Compound Interest

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

TodayFuture Years

12345

Interest Earned 6.00 6.36 6.74

Value100106.00112.36119.10


Future Values

Example - Compound Interest

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

TodayFuture Years

12345

Interest Earned 6.00 6.36 6.74 7.15

Value100106.00112.36119.10126.25


Future Values

Example - Compound Interest

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

TodayFuture Years

12345

Interest Earned 6.00 6.36 6.74 7.15 7.57

Value100106.00112.36119.10126.25133.82

Value at the end of Year 5 = $133.82


Future Values

Future Value of $100 = FV


Future Values

Future Value of any Present Value = FV

where t= number of time periods

r=is the discount rate


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


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


Future Values

Example - FV

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


Future Values

Example - FV

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


Future Values: FV with Compounding

Interest Rates


Future Value: Manhattan Island Sale

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

To answer, determine $24 is worth in the year 2008 (correct from 2006), compounded at 12.5% (long-term average annual return on S&P 500):

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


Present Values

Present Value

Value today of a future cash flow.

Discount Factor

Present value of a $1 future payment.

Discount Rate

Interest rate used to compute present values of future cash flows.


Present Values


Present Values

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


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?


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?


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)


Present Value

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


Present Value: PV of Multiple Cash Flows

Example

Your 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?


Present Value: PV of Multiple Cash Flows

  • PVs can be added together to evaluate multiple cash flows.


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.


Present Value: Perpetuities & Annuities

PV of Perpetuity Formula

C = constant cash payment

r = interest rate or rate of return


Present Value: Perpetuities & Annuities

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%?


Present Value: Perpetuities & Annuities

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?


Present Value: Perpetuities & Annuities

PV of Annuity Formula

C = cash payment

r = interest rate

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


Present Value: Perpetuities & Annuities

If PV of Annuity Formula is:

Then formula for annuity payment is:


Present Value: Perpetuities & Annuities

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:


Present Value: Perpetuities & Annuities

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


Present Value: Perpetuities & Annuities

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)?


Present Value: Perpetuities & Annuities

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)?


Present Value: Perpetuities & Annuities

Example - Annuity

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


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?


Perpetuities & Annuities

Future Value of Ordinary Annuity:


Perpetuities & Annuities

Present Value of Ordinary Annuity:


Effective Interest Rates

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

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


Effective Interest Rates

example

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


Effective Interest Rates

example

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


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 schedule

for a $1,000, 10% annual rate loan

with 3 equal payments.


402.11

Step 1: Find the required payments.

Amortization

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.

Amortization

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.

Amortization

End bal = Beg bal – Repmt

= $1,000 – $302.11 = $697.89.

Repeat these steps for Years 2 and 3

to complete the amortization table.


Amortization

BEGPRINEND

YRBALPMTINTPMTBAL

1$1,000$402$100$302$698

2 698 402 70 332 366

3 366 402 37 366 0

TOT1,206.34206.341,000

Interest declines and contribution to

principal grows. Tax implications from

lower interest paid.


$

402.11

Interest

302.11

Principal Payments

0

1

2

3

Level payments. Interest declines because outstanding balance declines. Lender earns

10% on loan outstanding, which is falling.


Amortization

  • 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.


Appendix on Time Value of Money

Future value

Present value

Rates of return


Future Value

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 lines show timing of cash flows.


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 to 3

0

1

2

3

r%

-50

100

75

50


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

0

1

2

3

10%

100

FV = ?

Finding FVs is compounding.


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


After 3 years:

FV3= PV(1 + r)3

= 100(1.10)3

= $133.10

In general,

FVn= PV(1 + r)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 + r)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

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

0

1

2

3

10%

100

PV = ?


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

Solve FVn = PV(1 + r )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 0100

N r/YR PV PMTFV

-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 + r)n;

2 = 1(1.20)n

Use calculator to solve, see next slide.


INPUTS

20 -1 0 2

N r/YR PV PMTFV

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

r%

PMT

PMT

PMT

Annuity Due

0

1

2

3

r%

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

310 0 -100

331.00

N

r/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

r/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

r/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 r = 10, then press NPV button to get NPV = 530.09. (Here NPV = PV.)


Finding the interest rate or growth rate

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

INPUTS

3-100 0 125.97

N

r/YR

PV

PMT

FV

OUTPUT

8%


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


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


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

  • Examples:

    • 8%; Quarterly

    • 8%, Daily interest (365 days)


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%.


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.


  • 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.


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

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.


rPer:

Used in calculations, shown on time lines.

If rNom has annual compounding,

then rPer = rNom/1 = rNom.


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 rNom (annual) whose

EFF% = 10.25%. In calculator,

EFF% = 10.25

P/YR = 1

NOM% = 10.25

c.

3 10.25 0 -100

INPUTS

N

r/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


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.

0

1

2

273

0.027397%

...

FV = ?

-100

273

(

)

FV

=

$100

1.00027397

273

(

)

=

$100

1.07765

=

$107.77.

Note: % in calculator, decimal in equation.


rPer=rNom/m

=10.0/365

=0.027397% per day.

INPUTS

273-100 0

107.77

N

r/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 1.75 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.


rPer = 0.027397% per day.

0

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?


rPer =0.019178% per day.

0

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:

rPer=rNom/m

=7.0/365

=0.019178% per day.

INPUTS

456-850 0

927.67

N

r/YR

PV

PMT

FV

OUTPUT

Enter rPer 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

r/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 + r)n

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

Now we must solve for r.


456-850 0 1000

0.035646%

per day

INPUTS

N

r/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.


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