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Another NPV Example z What if the choice were trading $10 today

for $20 tomorrow.z What are the market prices in this case

y Dollars today are always worth their facevalue, that is, their price is 1

y The price of dollars tomorrow is given by theinterest rate

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Present Value z If the interest rate is 10%, what is $20

tomorrow worth today?y The amount of money you could borrow

against this payment.y Denote this amount of money as x:

x

x=

= =

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Another Example (contd) z NPV=18.18-10=8.18z You should take this opportunity.z This decision does not depend on how

you personally trade off money today vrsmoney tomorrow. It is just like copper

and aluminum.

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Review z Competitive Markets

y Both buy and sell at the same pricez Law of One Price

y Two securities with exactly the same cashflows must have exactly the same price

z NPV Rule

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Definition: NET PRESENT

VALUE z The additional value today of an investment

opportunity.z Net-Present-Value Rule (or the no-brainer rule)

y Take on any investment with a positive NPV.y Reject any investment that has a negative NPV.

z Why do peoples' preferences not affect this rule?

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Example 1z I have an offer to sell my bike for $500.

My brother also wants to buy the bike.He will pay me $545, but he can onlymake the payment in a year. If currentinterest rates are 10%, which is the better

deal?

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Present Value Formula:

r

C r

C PV

+=

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9

NET Present Value

Formula:

I

I r

C I PV NPV +==

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Example 2z Your buddy would like to start a business.

He needs $10,000. If you lend him the

money and the business is successful, hewill give you $12,000 in a year.y If interest rates are 10% what should you do?

x What is the present value of $12,000?

x What is the net present value of this investment?y Is 10% the correct rate to use?

x What would your answer be if rates were 25%?

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Multiperiodsz The key to not making a mistake is the

TIMELINE:

1

$10,000

2

$10,000

0

$0

Date:

Cash Flow:

Year 1 Year 2

End Year 1 Begin Year 2Today

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Example 3 - The Multi-

Period Casez Assume that the average college tuition

costs $20,000 dollars per annum (paid atthe end of the year). For a freshman juststarting college, what is the present valueof the cost of a four year degree when theinterest rate is 10%?

397,631.1000,20

1.1000,20

1.1000,20

1.1000,20

432

=

+++= PV

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Multiperiod PV Formula:

( ) ( )

( )nC

r

C r

C

r

C

r

C PV

n

N

n

nn

N N

1

221

= +

=+

+++

++

= K

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Net Present Valuez Just the Present Value minus the cost of

the investment:z Formula:

( ) I

r

C

V NPV N

nn

n +

=

=

=1

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Where are we?

z We understand the basic idea but ...z it is a pain to add up these series --- is

there an easier way?y YES!

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Perpetuity

z A certain (constant) cashflow forever (e.g.a consol bond).

z What is the present value of a perpetuitywith cashflow C forever?

= += 1n n

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Perpetuity Timeline

3

C

0

0

2

C

1

C

( ) ( ) ( ) ( )2 3 11 1 1 1 nnC C C C

PV r r r r

== + + + =+ + + +L

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Trick

z How much money would you have to putin the bank to get a constant cashflowstream forever?

z Formula:

r

C PV =

$105

$100

$5

30

$100

21

$105

$100

$5

$105

$100

$5

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Example

z You made your fortune in the dot-com boom(and got out in time!). As part of your legacy,

you want to endow an annual MBA graduationparty at your alma matter. You want it to be amemorable, so you budget $30,000 per year forthe party. If the university earns 8% per year on

its investments, and if the first party is in oneyears time, how much will you need to donateto endow the party?

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Solution

z PV = C / r = $30,000 / .08 = $375,000today.

3

30,000

0

0

2

30,000

1

30,000

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Example Contd

z Suppose instead the first party wasscheduled to be held 2 years from today(the current entering class). How wouldthis change the amount of the donationrequired?

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Solution

z PV = $375,000 / 1.08 = $347,222 today.

3

30,000

0

0

2

30,000

1

375,000

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Growing Perpetuityz A stream of cashflows that grows at a constant

rate forever.z

What is the present value of growingperpetuity?

where g is the growth rate

( ) ( )

( )

=

++=

+

+

++

+

++

+

=

1

1

3

2

2

1)1(

1)1(

1)1(

1

nn

n

r

g C

r

g C

r

g C r

C PV K

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Timeline

A growing perpetuity with first payment of$100 that grows at a rate of 3% has thefollowing timeline:

3

$103 1.03

= $106.09

0

0

2

$100 1.03

= $103

4

$106.09 1.03

= $109.27

1

$100

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The General Case

In general, a growing perpetuity with firstpayment C and growth rate g will havethe following series of cash flows:

3

C (1+ g )2

0

0

2

C (1+ g )

4

C (1+ g )3

1

C

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Trickz Write the growing perpetuity as a perpetuity

and apply the previous formulaz Definition:

z Now write the PV formula as

1-11

so 11

11

g r

Rr g

R ++=+++

( ) ( )

( ) ( ) +

+

+++

+

++

+=

+++

+++

++=

K

K

2

2

3

2

2

1

)1(

1

)1(

1

1

1

1)1(

1)1(

1

r

g C

r

g C

r r

C

r

g C

r

g C r

C PV

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Trick (continued)

z so the PV formulae from the previousslide becomes

( )

( )

=

=

++++=

++

+++=

1

1

11 11

1)1(

11

1

nn

nn

n

RC r r C

r

g C r r

C PV

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Trick (continued) z Now apply the perpetuity formula

z Substitute back for (1+R)( )

++

+=

+++

+=

= R

C

r r

C

R

C

r r

C PV

n

n

1

1

111

1

1 1

g r

C g r

C

r r

C PV

=

++++

+=

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Example 5

z Assuming the discount rate is 7% perannum, how much would you pay toreceive $50, growing at 5%, annually,forever?

==

= g r C

PV

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Annuity

z Pays a constant payment for a fixednumber of years (periods).

z What is the present value of an N periodannuity?

= += N

nn

1

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Timeline

N

C

0

0

2

C

1

C

( ) ( ) ( ) ( ) ( )2 3

11 1 1 1 1

N

N nn

C C C C C PV

r r r r r == + + + + =

+ + + + +L

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Trickz Write an annuity as the difference between two

perpetuities.z An N period annuity is equal to a perpetuity

minus another perpetuity whose first cashflowarrives in period N+1.

( ) ( )

48476484761 Nat beginsthat perpetuity

1

perpetuity

1

+

+=

= ++= N n nn n r

C

r

C PV

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Trick (contd)

( ) ( ) ( )

( )

( )

+

=

+

=

++

+=

=

=

N

N

nn N

nn

r r C

r C

r r C

r

C

r r

C PV

11

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Example

z You are the lucky winner of the $30Million State Lottery. You can take yourprize money either as

1. 30 payments of $1M per year (startingtoday),

2. $15M paid today. If the interest rate is 8%,which option should you take?

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Timeline

0

1M

29

1M

2

1M

1

1M

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Growing Annuity

z Pays a constantly growing cashflow for afixed number of years (N)

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Timeline

z What is the present value?

=

++= N

nn

n

1

1

N

C (1+ g ) N 1

0

0

2

C (1+ g )

1

C

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Trick

z same as before!z Express as the difference between two

growing perpetuities.z Derivation

y Let N be the number of period to maturity, C

be the first payment, and g be the growth ratex What is the first payment of the perpetuity thatneeds to to be subtracted off? N g C +

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Subtract the two perpetuities:

( ) ( )

( ) ( ) ( )

( )

++=

+

+=+

+

+

++

+=

++

++

=

=

=

+

+=

=

N

N

N

nn

n

N

N

nn

n

N nn

n

nn

n

r g

g r C

g r C

r

g g r

C r

g C

r

g

r

g C

r

g C

r

g C PV

1

1

1

1

1 Nat beginsthat perpetuity growing

1

1

perpetuity growing

1

14 4 84 4 764 4 84 4 76

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Examplez Assume that a college education means an

additional $10,000/year in starting salary, andthat this difference grows at 3% per annum.

Assume a 7% annual discount rate and a 40 yearworking life.y On graduation day, what is the value of the degree?y Assuming that college costs about $20,000/annum

(due in advance), what is the NPV of the investmentopportunity?

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Working Backwards

z Sometimes you know the PV, but you do notknow the payment

z Exampley You are considering opening a business that requires

an initial investment of $100,000. Your bank managerhas agreed to lend you this money. The terms of theloan are that you will make equal annual paymentsfor the next 10 years and will pay an interest rate of8%. What is your annual payment?

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Solution 10

-C

0

100,000

2

-C

1

-C

100,000 (annuity) 0100,000 (annuity)

PV PV

= =

101 1

100,000 1 6.710.08 1.08C C

= = 100,000

$14,9036.71

C = =

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Internal Rate of Return

z Sometime you know the monthlypayment and the PV and you would liketo know what interest rate sets them equal

z You can also think of this as the return ofthe investment

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Example

z Assume you wanted to purchase a BMWthat cost $40,000. The dealer is willing to

let you have the car with zero downpayment, so long as you are willing to payoff the car with 4 annual payments of$15,000. What interest rate is the dealercharging for this loan?

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Timeline

3

-$15,000

0

$40,000

2

-$15,000

4

-$15,000

1

-$15,000

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IRR

z The internal rate of return (IRR) is theinterest rate that sets the present value of

an investment opportunity equal to zero.

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The Discount Rate

z The discount rate is the correct rate to useto move a particular cash flow in time.

z We have not really addressed where thiscomes from.

z A deep question that we will not fully

answer in this course.

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Examplez Your cousin would like to buy your Acura.

Unfortunately, he is just a student and has verylittle money. Instead of paying for the car, he

offers to pay you $100/month forever. If theannual interest rate is 10%, how much is heoffering to pay for the car?

y What monthly interest rate would you demand onyour deposit at the bank so that you would be

indifferent between that and being paid 10%annually?

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Aside: Annual vrs MonthlyCompoundingz Say you deposit $1

y If you chose the annual interest deposit in oneyear you will have

y If you chose the monthly interest deposit inone year you will have

z So..

ar +

( )12mr +

( )1212

12

==+=+=+

am

am

r r

r r

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Example - Solution

=== r

C PV

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General Idea

z Given a discount rate of r x per x-years,the equivalent discount rate r y per y-

years is given by compounding (1+r x) fory/x periods:

1 + r y = (1 + r

x)y/ x.

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Interest Rate Quotes

z When a bank quotes an interest rate for aparticular loan, it is usually not correct to

use this quote directly as the discount ratez The discount rate often has to be

computed from the quoted rate based on

the conventions of the quote.

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Example

z If a 3 month bond has a 8% APR, howmuch interest will I earn over the life of

the bond?y Since the APR quote does not include interest

on interest and since a 3 month bond can bereinvested 4 times during the year, the bondwill earn 2% interest over its life.

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Effective Annual Rate(EAR)

y This is the amount of interest you would earnin one year assuming that you rollover the

loan and reinvest all interest payments asoften as is allowed by the terms of the loan,that is, the loan is compounded as often aspossible during the year.

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Examplez If a 3 month bond has a 8% EAR, how much

interest will I earn over the life of the bond?y Since the EAR quote does include interest on

interest and since a 3 month bond can be reinvested4 times during the year,

( ) 4

4

== =+r r

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Example 6

z Using current rates, assume that the mostyou can afford in monthly mortgage

payments is $1000. If you plan to use a 30year fixed rate mortgage with an interestrate of 7.75% and a 1% origination fee,how large a mortgage can you afford?

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Next Lecture

z Topicsy Bonds

z Readingy Chapter 5: Appendix, Sec. 5.4-5.9y BD Chapters 6 and 6X