corporate finance : time value of money

7/24/2019 Corporate Finance : Time Value of Money

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Corporate Finance 1

Part 2: Time Value of MoneyProgramme: MSc in Finance

ACADEMIC YEAR 2015-2016

Professor: Francesco Reggiani


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Outline: Part 2

Time Value of Money

I. Basic Conceptsa. Present value

b. Future value

c. Comparing present value and future value

II. When There Are Multiple Cash Flowsa. Perpetuities

b. Present value of an ordinary annuity

c. Present value of an annuity due

d. Future value of an ordinary annuitye. Future value of an annuity due

f. Present value of an uneven cash flow series

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Outline: Part 2

Time Value of Money(continued)

III. Determining Interest Rates

a. Individual cash flows

b. Annuities

c. Uneven cash flow series

IV. Determining Time Periods

a. Individual cash flows

b. Annuities

V. Applications in Financial Management

a. Investment decision making

b. Effective interest rates

c. Annual percentage rate

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I. Basics Concepts

4

The Italian firm, Montana SpA, is attempting to choose betweentwo proposals for new products. Both proposals will provide

additional cash flows over a four-year period and will initially

cost 10,000. The cash flows from the proposals are as follows:

Year NewProduct A

NewProduct B

1 0 4,000

2 0 4,000

3 0 4,000

4 20,000 4,000

Total 20,000 16,000

Which is Best?


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Basic Concepts(continued)

Equal cash flows are not of equal value if received in

different time periods

In finance, the timing of all cash flows must be considered

so that they correspond to the same point in time

We will examine cash flows in terms of

The present moment in time

Some future moment in time

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Basic ConceptsPresent Value

0 21 . . . . n3

PV0

CFn

yearsorperiodsofNumbern

nat timeflowcashtheofAmountCF

rateinterestAnnualk

CFofzeroat timeluePresent vaPV

Where

k)(1

CFPV

n

n0

n

n0

=

==

=

+=

6


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Basic ConceptsPresent Value

(continued)

An exampleWhat is the present value of $110 received one year from now

if k = 10%?

Received four years from now?

Ten years from now?

$1000.10)1/(PV 10 =+=CF

$75.130.10)/(1PV 440 =+= CF

$42.410.10)(1/PV 10100 =+= CF

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Basic ConceptsFuture Value

0 21 . . . . n3

CF0

FVn

n

0n k)(1FV += CF

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Basic Conceptsb. Future Value

(concluded)

An exampleWhat is the future value in one year of $100 if k = 10%?

What would be the future value in two years?

In three years?

$110$100(1.10)k)(1FV 01 ==+= CF

$121$100(1.10)k)(1FV 2202 ==+= CF

$133.10$100(1.10)k)(1FV 3303 ==+= CF

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Basic Conceptsc. Comparing Present and Future Values

The PV and FV are reciprocals of one another

These relationships hold for all comparable rates and timeperiods

Present values are always less than the future value

n

0n

nn

0

k)(1PVFV

k)(1FV

PV

+=

+=

10


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Basic ConceptsComparing Present to Future Values

(concluded)

FV, PV, interest rates and time

$3.00

$0.50

$1.00

$1.50

$2.00

$2.50

2 4 6 8 100

$3.00

$0.50

$1.00

$1.50

$2.00

$2.50

2 4 6 8 100

5%

10%

0%

Period

Future value of $1

FUTURE VALUE PRESENT VALUE

0%

5%

10%

Period

Present value of $1

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II. When There Are Multiple Cash Flows

a. PerpetuitiesA perpetuity is a series of cash flows of a constant

amount that continues indefinitely

Present value of a perpetuity starting in one year isthe annual cash flow (CF) divided by the discountrate

An example

Suppose you are about to receive a $100 per yearperpetuity discounted at 9%. What is its value?

Answer:

$1,111.110.09

$100

k

CFPV0 ===

12

if n is big enough (infinite), then:


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When There Are Multiple Cash Flows(continued)

Growing perpetuitiesThe size of the future cash flows is growing at a constant

rate of "g" percent per year

As long as k is greater than g, the present value of a

growing perpetuity is

g-k

CFPV0=

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When There Are Multiple Cash Flows(continued)

An exampleIf the $100 perpetuity in the previous example grows by

4% per year, what is the value of this growing

perpetuity?

Answer:

000,2$04.009.0

100$

g-k

CFPV0 =

==

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When There Are Multiple Cash Flowsb. Present Value of an Ordinary Annuity

An annuity is a limited-life perpetuityAn ordinary annuity is when the cash flows occur

at the endof each period

The present value of an ordinary annuity is equal tothe present value of a typical perpetuity minus the

present value of a delayed perpetuity that starts at

the end of period n, and can be calculated by the

following equation

( )

=

+

k

k111

0

n

PV CF

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When There Are Multiple Cash FlowsPresent Value of an Ordinary Annuity

(continued)

An exampleHow much would you be willing to pay for an investment

that provides $50 per year for each of the next ten years

if you expect a 9% return on your money?

Answer:

( )88.320$$50PV

0.09

09.0111

0

10

=

=

+

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When There Are Multiple Cash FlowsPresent Value of an Ordinary Annuity

(concluded)

How much if you receive $500 per year for each of 3years at 8.75%.?

Answer:

?

2 31

$500 $500 $500

( )

31.271,1$$500PV 0.08750875.01

11

0

3

=

=

+

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When There Are Multiple Cash Flowsc. Present Value of an Annuity Due

An annuity due is when cash inflows occur at thebeginningof each period, not the end

Each of the payments is shifted back one period, or

year, on the time line

In the previous example they now occur at t = 0, t =

1, and t = 2

$500 $500 $500

?

21

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When There Are Multiple Cash FlowsPresent Value of an Annuity Due

(continued)

General equationTo calculate the present value of an annuity due, we

multiply the present value of an ordinary annuity by the

term (1 + k)

( ) ( )k1PVk

k111

0

n

+

=

+

CF

( ) ( ) 55.382,1$0.08751$500PV

examplepreviousFor the

0.0875

0875.0111

0

3

=+

=

+

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When There Are Multiple Cash Flowsd. Future Value of an Ordinary Annuity

General equation

An example

Suppose you are to receive $500 at the end of each year

for three years. You immediately invest it at 8%interest. What is the value at the end of three years?

( )

+=

k

1k1FV

n

n CF

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When There Are Multiple Cash FlowsFuture Value of an Ordinary Annuity

(concluded)

$500

?

21

$500$500

03

( )

20.623,1$

0.08

108.01$500FV

3

n

=

+=

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When There Are Multiple Cash Flowse. Future Value of an Annuity Due

General equation

An example

What if you had received the $500 annuity from the

previous example at the beginning of each period ?

Answer:

( )( )k1

k

1k1FV

n

n +

+= CF

( )( ) 06.753,1$08.01

0.08

10.081$500FV

3

n =+

+=

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When There Are Multiple Cash Flowsf. Present Value of an Uneven Cash Flow Series

General equation

An example

What is the present value of an uneven series of cash

flows of $500 at year 1, $600 at year 2, and $700 at

year 3, if the discount rate is 12.5%?

( )= +

=n

1tt

t0

k1

CFPV

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When There Are Multiple Cash FlowsPresent Value of an Uneven Cash Flow Series

(concluded)

Answer:

Make use of any embedded annuities that are present

in the series

( ) ( )15.410,1$

125.1700$

)125.1(600$

125.1500$PV 320

=

++=

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III. Determining Interest Rates

Individual Cash flows

An exampleYou borrow $1,000 today and agree to repay principal and

interest of $1,610.31 in five years. What is k?

Answer:

( )

10%k

9.99726%1$1,000

$1,610.31k

1

PV

CFk

k1CFPV

51

n1

0

n

nn

0

=

=

=

+=

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Determining Interest Ratesb. Annuities

An exampleYou borrow $2,294.90 and agree to repay $708.30 at the

end of each of the next four years. What is k?

Answer:

9%B.2Tablefrom

2400.330.708$/90.294,2$)(PVA

CF

PV)(PVA)(PVAPV

?%,4yr

0nk,nk,0

==

== CF

%00.9

8.9959022%kgetwecalculatorfinancialaUsing

=

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Determining Interest Rates

c. Uneven Cash Flow Series

The unknown interest rate can also be determined if the cashflow series is uneven

An example

Suppose you invest $352.31 today, and the series of paymentspromised is $80 at t = 1, $125 at t = 2, and $225 at t = 3.What is your expected annual compound percentage return?

Answer:

Find the discount rate that equates the present value of the cashinflows with the present value, $352.31

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Uneven Cash Flow Series(continued)

Step 1: Try 12 percentYear Cash Inflow X = PV (Cash inflows)

1 $ 80 0.893 $ 71.44

2 125 0.797 99.62

3 225 0.712 160.20$331.26

Because the PV of the inflows of $331.26 is lessthan $352.31, the

discount rate must be lowered

n12%,PV

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Uneven Cash Flow Series(continued)

Step 2: Try 8 percent

Year Cash Inflow X = PV (Cash inflows)

1 $ 80 0.926 $ 74.08

2 125 0.857 107.12

3 225 0.794 178.65

$359.85

Because the PV of the inflows of $359.85 is greaterthan $352.31, the discount

rate must be increased

n8%,PV

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Uneven Cash Flow Series(concluded)

Step 3: Try 9 percent

Year Cash Inflow X = PV (Cash inflows)

1 $ 80 0.917 $ 73.36

2 125 0.842 105.25

3 225 0.772 173.70

$352.31

Because the PV of the inflows of $352.31 exactly equals $352.31, the discount

rate is 9%

n9%,PV

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IV. Determining Time Periodsa. Individual Cash Flows

An exampleSuppose that you just won $20,000. You intend to use the

money to buy a car that cost $30,000. If you can earn 13

percent a year on the $20,000, how long will it be before

you can purchase the car?Answer:

( )

( )

years3.32n

og(1.13)log(3/2)/ln

nlog(1.13)3/2log

1.13$20,000

$30,000

13)$20,000(1.$30,000

n

n

=

=

=

=

=

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Determining Time Periodsb. Annuities

An example

Suppose that you want to save $5,000 and you can deposit

$100 a month into an account earning 0.5 percent a

month, how long will it take you to save the $5,000?

( )

( ) ( )

( )

( )

months44.74n

log(1.005)log(1.25)/n

)nlog(1.0051.25log

005.125.1

1.0051005.0$100

$5,000

$100$5,000

n

005.0

11.005

=

=

=

=

=+

=

n

n

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V. Applications in Financial Managementa. Investment Decision Making

Net present value (NPV) of the investment

An example

A project requires an initial investment at t = 0 of $77,800and it is expected to generate cash inflows of $26,250 att = 1, $40,950 at t = 2, and $73,500 at t = 3. If there isno risk associated with the project, and the return on 3-year government securities is 8%, what is the project

NPV?

investmentInitial-inflowscashfutureofluePresent vaNPV=

( )=

+

=n

1t

0t

t

k1

CFNPV CF

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Applications in Financial Management

Investment Decision Making(continued)

Answer:

Internal rate of returnIRR is determined by solving for the unknown discount

rate in

The IRR can be found using a financial calculator

The IRR for the previous example is 30.06%

$39,960

$77,800.08)$73,500/(1.08)$40,950/(108$26,250/1.NPV 32

=++=

( )

=

=+

n

1t

0t

t PV

IRR1

CF

34

A li ti i Fi i l M t


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Investment Decision Making(continued)

Decision criteriaAccept a proposed project if its NPV > 0

Accept a proposed project if its IRR > k

What about risk?

Until now we have assumed the project was a sure thing.That assumption is unrealistic

In practice, not many investments (except short-term

government securities) provide a risk-free return

Risk is one of the major items that has to beconsidered in every financial decision

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Investment Decision Making(concluded)

The opportunity cost of capitalIt is the rate k employed as the discount rate when

calculating NPV, or the rate that IRR is compared to

It is considered the required rateof returnbecause it is

what investors require to make an investment now andreceive cash inflows at some time in the future

It is a hurdle rate when it is employed as the standard

against which the IRR is compared

It is an opportunity costbecause it is the return forgone byinvesting in a specific asset rather than investing in

some equally risky investment

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Applications in Financial Managementb. Effective Interest Rates

More frequent intervalsA nominal interest rates is the quoted rate per year

An effective interest rate is the true rate per time period

myear,perperiodsgcompoundinofNumberkrate,interestNominalperiodperrateEffective =

( )[ ]

( )nm0n

nm

n0

k/m1PVFV

k/m1

FVPV

+=

+

=

37

EXAMEN


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Applications in Financial ManagementEffective Interest Rates

(continued)

Using continuous discounting and compounding

=

=

=

==

k

1)(eFV

k

)e(1PV

ePVFV

eFVe

FVPV

kn

n

kn

0

kn

0n

kn

nknn

0

CF

CF

38


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(continued)

Effective annual interest rates

An example

A bank quotes you a nominal rate of 12% compounded

quarterly. A friend quotes you a nominal rate of 11.9%

compounded daily. Is your friend a friend?

[ ] [ ] 12.5509%14/12.011/mk1k 4mnominalannualeffective

=+=+=

Bank

[ ] [ ] 12.6348%1365/119.011/mk1k 365mnominalannualeffective

=+=+=

Hence, your friendsrate is actually higher than the banks !!

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(concluded)

An exampleIf a lender wants to earn an effective annual rate of 9% on

a loan with quarterly compounding, what nominal rate

will be quoted?

Answer:

( )

8.71%k

4k11.09

14

k10.09

nominal

nominal41

4

nominal

=

+=

+=

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Annual, Semi-Annual and Continuous

Compounding

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Applications in Financial Managementc. Annual Percentage Rate (APR)

Many loans have front or back end fees relating to

management costs, administration, etc

In the EU, all loans must state the effective interest rate that

includes all costs, not just the interest payments

This is known as the Annual Percentage Rate (APR)


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An example

The sale price of a car is 30,000.

The quoted rate is a simple annual interest rate of 12 percent on

the original borrowed amount over three years, payable in 36

monthly installments.

The finance company also charges an administration fee of 250.

What does this mean?

The lender will charge 12 percent interest on the original loan of

30,000 every year for three years.

Each year, the interest charge will be (12% of 30,000) 3,600

making a total interest payment of 10,800 over three years.

Applications in Financial ManagementAnnual Percentage Rate (APR)


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OriginalAmount

The car costs 30,000

Interest andFees

Total Interest is 10,800 Admin Fee is 250

MonthlyPayment

(30,000 + 10,800)/36 = 1,133.33



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What is the APR of this loan?

This gives an Annual Percentage Rate (APR) of 24.13%!

The lender must also state the total amount paid at the end of the

loan, which, in this case, is 41,049.88 and the total charge forcredit is 11,049.88 (41,049.88 - 30,000).

1 2 36

12 12 12

1,133.33 1,133.33 1,133.3330,000 250

(1 APR) (1+APR) (1+ APR)

= + + + +

+

L


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