chapters 4 and 5.fp

Lecture # 1 EFI-FIE1B

Chapters 4, 5, 8, 9, & 10;

Book: Fundamentals of Corporate Finance

(European edition) by David Hillier

Chapters 4 & 5

Lesson 1

Source Power Point Slides:

Fundamentals of corporate finance

(European edition) by David Hillier

Future value and compounding

Future value (FV) = the amount an investment is worth after

one or more periods ( page 75)

Compounding = the process of accumulating interest on an

investment over time to earn more interest ( page 76 )

Interest on interest = interest is earned on the reinvestment

of previous interest payments ( example 4.1 )

Compounded interest = interest earned on both the initial

principal and the interest reinvested from prior periods (

example 4.2 )

Simple interest= interest earned only on the original

principal amount invested.

Future value = €1 * (1+r)t

Example

Example (cont’d)

Three Rules of Time Travel

Financial decisions often require combining cash

flows or comparing values. Three rules govern these

processes.

The Three Rules of Time Travel

The 1st Rule of Time Travel

A dollar today and a dollar in one year are

not equivalent.

It is only possible to compare or combine

values at the same point in time.

– Which would you prefer: A gift of $1,000

today or $1,210 at a later date?

– To answer this, you will have to compare the

alternatives to decide which is worth more.

One factor to consider: How long is “later?”

The 2nd Rule of Time Travel

To move a cash flow forward in time, you

must compound it.

– Suppose you have a choice between

receiving $1,000 today or $1,210 in two

years. You believe you can earn 10% on the

$1,000 today, but want to know what the

$1,000 will be worth in two years. The time

line looks like this:

The 2nd Rule of Time Travel (cont’d)

Future Value of a Cash Flow

Example

Problem

– Suppose you have a choice between

receiving $5,000 today or $10,000 in five

years. You believe you can earn 10% on the

$5,000 today, but want to know what the

$5,000 will be worth in five years.

Example (cont’d)

0 3 4 521

$5,000 $5, 500 $6,050 $6,655 $7,321 $8,053x 1.10 x 1.10 x 1.10 x 1.10 x 1.10

Solution

– The time line looks like this:

– In five years, the $5,000 will grow to:

$5,000 × (1.10)5 = $8,053

– The future value of $5,000 at 10% for five years

is $8,053.

– You would be better off forgoing the gift of $5,000 today and

taking the $10,000 in five years.

Present value and discounting

The 3rd Rule of Time Travel

To move a cash flow backward in time, we

must discount it.

Present Value of a Cash Flow

(1 ) (1 )

n

n

CPV C r

r

Example

Example

Alternative Example

Problem

– Suppose you are offered an investment that

pays $10,000 in five years. If you expect to

earn a 10% return, what is the value of this

investment today?

Alternative Example

Solution

– The $10,000 is worth:

$10,000 ÷ (1.10)5 = $6,209

Applying the Rules of Time Travel

Recall the 1st rule: It is only possible to

compare or combine values at the same

point in time. So far we’ve only looked at

comparing.

– Suppose we plan to save $1000 today, and

$1000 at the end of each of the next two

years. If we can earn a fixed 10% interest

rate on our savings, how much will we have

three years from today?

Applying the Rules of Time Travel

The time line would look like this:

Applying the Rules of Time Travel

Applying the Rules of Time Travel

Applying the Rules of Time Travel

The Three Rules of Time Travel

Example

Example

Alternative Example

0 3 4 521

$10,000$5,000

Problem

– Assume that an investment will pay you $5,000

now and $10,000 in five years.

– The time line would like this:

Alternative Example

0 3 4 521

$6,209 $10,000$5,000

$11,209 ÷ 1.105

Solution

– You can calculate the present value of the combined

cash flows by adding their values today.

– The present value of both cash flows is $11,209.

Alternative Example

Solution

– You can calculate the future value of the

combined cash flows by adding their values

in Year 5.

– The future value of both cash flows is

$18,053.

0 3 4 521

$5,000 $8,053x 1.105

$10,000

$18,053

Alternative Example

0 3 4 521

$11,209 $18,053÷ 1.10

5

0 3 4 521

$11,209 $18,053x 1.10

5

Present

Value

Future

Value

Valuing a Stream of Cash Flows

Based on the first rule of time travel we

can derive a general formula for valuing a

stream of cash flows: if we want to find

the present value of a stream of cash

flows, we simply add up the present

values of each.

Valuing a Stream of Cash Flows

Present Value of a Cash Flow Stream

0 0

( ) (1 )

N N

nn n

n n

CPV PV C

r

Example

Example

Example

0 321

$2,000 $2,000 $2,000

Problem

– What is the future value in three years of the

following cash flows if the compounding rate

is 5%?

Example

Solution

Or

0 321

$2,000

$2,000

x 1.05 x 1.05

$2,315x 1.05

$2,205

$2,000x 1.05 x 1.05

$2,100

$6,620x 1.05

0 321

$2,000

x 1.05

$4,100$2,100

$4,305

$2,000 $2,000

x 1.05

$6,305

x 1.05$6,620

(1 ) n

nFV PV r

Future Value of Cash Flow Stream

Future Value of a Cash Flow Stream with

a Present Value of PV

Calculating the Net Present Value

Calculating the NPV of future cash flows

allows us to evaluate an investment

decision.

Net Present Value compares the present

value of cash inflows (benefits) to the

present value of cash outflows (costs).

Example

Example

Example

0 321

$1,000$3,000 $2,000

Problem

– Would you be willing to pay $5,000 for the

following stream of cash flows if the discount rate

is 7%?

Example

Solution

– The present value of the benefits is:

3000 / (1.05) + 2000 / (1.05)2 + 1000 / (1.05)3 =

5366.91

– The present value of the cost is $5,000,

because it occurs now.

– The NPV = PV(benefits) – PV(cost)

= 5366.91 – 5000 = 366.91

Perpetuities and Annuities

Perpetuities

– When a constant cash flow will occur at

regular intervals forever it is called a

perpetuity.

Perpetuities and Annuities

The value of a perpetuity is simply the

cash flow divided by the interest rate.

Present Value of a Perpetuity

( in perpetuity) C

PV Cr

Example

Example

Example

Problem

– You want to endow a chair for a female

professor of finance at your alma mater.

You’d like to attract a prestigious faculty

member, so you’d like the endowment to add

$100,000 per year to the faculty member’s

resources (salary, travel, databases, etc.) If

you expect to earn a rate of return of 4%

annually on the endowment, how much will

you need to donate to fund the chair?

Example

Solution

– The timeline of the cash flows looks like this:

– This is a perpetuity of $100,000 per year. The funding

you would need to give is the present value of that

perpetuity. From the formula:

– You would need to donate $2.5 million to endow the

chair.C $100,000

PV $2,500,000r .04

Perpetuities and Annuities

Annuities

– When a constant cash flow will occur at regular

intervals for a finite number of N periods, it is

called an annuity.

– Present Value of an Annuity

N

1nnN32 )r1(

C

)r1(

C...

)r1(

C

)r1(

C

)r1(

CPV

Present Value of an Annuity

To find a simpler formula, suppose you invest $100 in

a bank account paying 5% interest. As with the

perpetuity, suppose you withdraw the interest each

year. Instead of leaving the $100 in forever, you close

the account and withdraw the principal in 20 years.

Present Value of an Annuity

You have created a 20-year annuity of $5

per year, plus you will receive your $100

back in 20 years. So:

Re-arranging terms:

)years20in100($PV)yearper5$ofannuityyear20(PV100$

31.62$)05.1(

100100

)years20in100($PV100$)yearper5$ofannuityyear20(PV

20

Present Value of an Annuity

For the general formula, substitute P for

the principal value and:

N N

PV(annuityof Cfor N periods)

P PV(Pin period N)

P 1P P 1

(1 r) (1 r)

Example

Example

Future Value of an Annuity

Future Value of an Annuity

(annuity) V (1 )

1 1 (1 )

(1 )

1 (1 ) 1

N

N

N

N

FV P r

Cr

r r

C rr

Example

Example

Growing Cash Flows

Growing Perpetuity

– Assume you expect the amount of your

perpetual payment to increase at a constant

rate, g.

Present Value of a Growing Perpetuity

(growing perpetuity)

CPV

r g

Textbook Example 4.10

Example

Alternative Example

Problem

– In Alternative Example 4.7, you planned to

donate money to endow a chair at your alma

mater to supplement the salary of a qualified

individual by $100,000 per year. Given an

interest rate of 4% per year, the required

donation was $2.5 million. The University

has asked you to increase the donation to

account for the effect of inflation, which is

expected to be 2% per year. How much will

you need to donate to satisfy that request?

Alternative Example

The timeline of the cash flows looks like

this:

The cost of the endowment will start at $100,000, and

increase by 2% each year. This is a growing

perpetuity. From the formula:

C $100,000PV $5,000,000

r .04 .02

You would need to donate $5.0 million to endow the

chair.

Growing Cash Flows

Growing Annuity

– The present value of a growing annuity with

the initial cash flow c, growth rate g, and

interest rate r is defined as:

– Present Value of a Growing Annuity

1 1 1

( ) (1 )

N

gPV C

r g r

Example

Example

Alternative Example

Problem

– You want to begin saving for your

retirement. You plan to contribute $12,000

to the account at the end of this year. You

anticipate you will be able to increase your

annual contributions by 3% each year for the

next 45 years. If your expected annual

return is 8%, how much do you expect to

have in your retirement account when you

retire in 45 years?

Alternative Example

Example

Example

Example

Example

Non-Annual Cash Flows

The same time value of money concepts

apply if the cash flows occur at intervals

other than annually.

The interest and number of periods must

be adjusted to reflect the new time period.

Example

Example

Solving for the Cash Payments

Sometimes we know the present value or

future value, but do not know one of the

variables we have previously been given

as an input.

Solving for the Cash Payments

For example, when you take out a loan

you may know the amount you would like

to borrow, but may not know the loan

payments that will be required to repay it.

Example

Example

The Internal Rate of Return

In some situations, you know the present

value and cash flows of an investment

opportunity but you do not know the

internal rate of return (IRR), the interest

rate that sets the net present value of the

cash flows equal to zero.

Example

Example

Example

Example

Solving for the Number of Periods

In addition to solving for cash flows or

the interest rate, we can solve for the

amount of time it will take a sum of

money to grow to a known value.

Example

Example

Chapters Quiz

1. Can you compare or combine cash flows at

different times?

2. How do you calculate the present value of a

cash flow stream?

3. What benefit does a firm receive when it

accepts a project with a positive NPV?

4. How do you calculate the present value of a

a. Perpetuity?

b. Annuity?

c. Growing perpetuity?

d. Growing annuity?