Principles of Corporate Finance

Session 10

Unit II: Time Value of Money

TIME allows you the opportunity to postpone consumption and earn

INTEREST.

Why TIME?Why TIME?

Why is TIME such an important element in your decision?

Types of InterestTypes of Interest

• Compound InterestInterest paid (earned) on any previous

interest earned, as well as on the principal borrowed (lent).

• Simple Interest

Interest paid (earned) on only the original amount, or principal, borrowed (lent).

Simple Interest FormulaSimple Interest Formula

Formula SI = P0(i)(n)

SI: Simple Interest

P0: Deposit today (t=0)

i: Interest Rate per Period

n: Number of Time Periods

• SI = P0(i)(n)= $1,000(0.07)(2)= $140

Simple Interest ExampleSimple Interest Example

• Assume that you deposit $1,000 in an account earning 7% simple interest for 2 years. What is the accumulated interest at the end of the 2nd year?

FV = P0 + SI = $1,000 + $140= $1,140

• Future Value is the value at some future time of a present amount of money, or a series of payments, evaluated at a given interest rate.

Simple Interest (FV)Simple Interest (FV)

• What is the Future Value (FV) of the deposit?

The Present Value is simply the $1,000 you originally deposited. That is the value today!

• Present Value is the current value of a future amount of money, or a series of payments, evaluated at a given interest rate.

Simple Interest (PV)Simple Interest (PV)

• What is the Present Value (PV) of the previous problem?

0

5000

10000

15000

20000

1st Year 10thYear

20thYear

30thYear

Future Value of a Single $1,000 Deposit

10% SimpleInterest

7% CompoundInterest

10% CompoundInterest

Why Compound Interest?Why Compound Interest?

Fu

ture

Val

ue

(U.S

. Dol

lars

)

Simple Interest

• Year 1: 5% of $100 = $5 + $100 = $105

• Year 2: 5% of $100 = $5 + $105 = $110

• Year 3: 5% of $100 = $5 + $110 = $115

• Year 4: 5% of $100 = $5 + $115 = $120

• Year 5: 5% of $100 = $5 + $120 = $125

With simple interest, you don’t earn interest on interest.

Compound Interest

• Year 1: 5% of $100.00 = $5.00 + $100.00 = $105.00

• Year 2: 5% of $105.00 = $5.25 + $105.00 = $110.25

• Year 3: 5% of $110.25 = $5 .51+ $110.25 = $115.76

• Year 4: 5% of $115.76 = $5.79 + $115.76 = $121.55

• Year 5: 5% of $121.55 = $6.08 + $121.55 = $127.63

With compound interest, a depositor earns interest on interest!

Principles of Corporate Finance

Session 11 & 12

Unit II: Time Value of Money

The Role of Time Value in Finance

• Most financial decisions involve costs & benefits that

are spread out over time.

• Time value of money allows comparison of cash flows

from different periods.

Question?

Would it be better for a company to invest $100,000 in a product that would return a total of $200,000 in one year, or one that would return

$500,000 after two years?

Answer!

It depends on the interest rate!

The Role of Time Value in Finance

• Most financial decisions involve costs & benefits that

are spread out over time.

• Time value of money allows comparison of cash flows

from different periods.

Basic Concepts

• Future Value: compounding or growth over time

• Present Value: discounting to today’s value

• Single cash flows & series of cash flows can be

considered

• Time lines are used to illustrate these relationships

Computational Aids

• Use the Equations

• Use the Financial Tables

• Use Financial Calculators

• Use Spreadsheets

Computational Aids

Computational Aids

Computational Aids

Computational Aids

Time Value Terms

• PV0 = present value or beginning amount

• k = interest rate

• FVn = future value at end of “n” periods

• n = number of compounding periods

• A = an annuity (series of equal payments or

receipts)

Four Basic Models

• FVn = PV0(1+k)n = PV(FVIFk,n)

• PV0 = FVn[1/(1+k)n] = FV(PVIFk,n)

• FVAn = A (1+k)n - 1 = A(FVIFAk,n) k

• PVA0 = A 1 - [1/(1+k)n] = A(PVIFAk,n)

k

Future Value Example

You deposit $2,000 today at 6%

interest. How much will you have in 5

years?

$2,000 x (1.06)5 = $2,000 x FVIF6%,5

$2,000 x 1.3382 = $2,676.40

Algebraically and Using FVIF Tables

Future Value Example

You deposit $2,000 today at 6%

interest. How much will you have in 5

years?

Using Excel

PV 2,000$ k 6.00%n 5FV? $2,676

Excel Function

=FV (interest, periods, pmt, PV)

=FV (.06, 5, , 2000)

Future Value Example A Graphic View of Future Value

Compounding More Frequently than Annually

• Compounding more frequently than once a year

results in a higher effective interest rate because you

are earning on interest on interest more frequently.

• As a result, the effective interest rate is greater than

the nominal (annual) interest rate.

• Furthermore, the effective rate of interest will increase

the more frequently interest is compounded.

Compounding More Frequently than Annually

• For example, what would be the difference in future

value if I deposit $100 for 5 years and earn 12%

annual interest compounded (a) annually, (b)

semiannually, (c) quarterly, an (d) monthly?

Annually: 100 x (1 + .12)5 = $176.23

Semiannually: 100 x (1 + .06)10 = $179.09

Quarterly: 100 x (1 + .03)20 = $180.61

Monthly: 100 x (1 + .01)60 = $181.67

Compounding More Frequently than Annually

Annually SemiAnnually Quarterly Monthly

PV 100.00$ 100.00$ 100.00$ 100.00$

k 12.0% 0.06 0.03 0.01

n 5 10 20 60

FV $176.23 $179.08 $180.61 $181.67

On Excel

Continuous Compounding• With continuous compounding the number of

compounding periods per year approaches infinity.

• Through the use of calculus, the equation thus

becomes:

FVn (continuous compounding) = PV x (ekxn)

where “e” has a value of 2.7183.

• Continuing with the previous example, find the Future

value of the $100 deposit after 5 years if interest is

compounded continuously.

Continuous Compounding• With continuous compounding the number of

compounding periods per year approaches infinity.

• Through the use of calculus, the equation thus

becomes:

FVn (continuous compounding) = PV x (ekxn)

where “e” has a value of 2.7183.

FVn = 100 x (2.7183).12x5 = $182.22

Principles of Corporate Finance

Session 12 & 13

Unit II: Time Value of Money

Nominal & Effective Rates• The nominal interest rate is the stated or contractual

rate of interest charged by a lender or promised by a

borrower.

• The effective interest rate is the rate actually paid or

earned.

• In general, the effective rate > nominal rate whenever

compounding occurs more than once per year

EAR = (1 + k/m) m -1

Nominal & Effective Rates• For example, what is the effective rate of interest on

your credit card if the nominal rate is 18% per year,

compounded monthly?

EAR = (1 + .18/12) 12 -1

EAR = 19.56%

We will use the “Rule-of-72”.

Double Your Money!!!Double Your Money!!!

Quick! How long does it take to double $5,000 at a compound rate of 12%

per year (approx.)?

Approx. Years to Double = 72 / i%

72 / 12% = 6 Years[Actual Time is 6.12 Years]

The “Rule-of-72”The “Rule-of-72”

Quick! How long does it take to double $5,000 at a compound rate of 12%

per year (approx.)?

Present Value• Present value is the current dollar value of a future

amount of money.

• It is based on the idea that a dollar today is worth

more than a dollar tomorrow.

• It is the amount today that must be invested at a given

rate to reach a future amount.

• It is also known as discounting, the reverse of

compounding.

• The discount rate is often also referred to as the

opportunity cost, the discount rate, the required return,

and the cost of capital.

Present Value Example

How much must you deposit today in order to

have $2,000 in 5 years if you can earn 6%

interest on your deposit?

$2,000 x [1/(1.06)5] = $2,000 x PVIF6%,5

$2,000 x 0.74758 = $1,494.52

Algebraically and Using PVIF Tables

Present Value Example

How much must you deposit today in order to

have $2,000 in 5 years if you can earn 6%

interest on your deposit?

FV 2,000$ k 6.00%n 5PV? $1,495

Excel Function

=PV (interest, periods, pmt, FV)

=PV (.06, 5, , 2000)

Using Excel

Present Value Example A Graphic View of Present Value

Annuities• Annuities are equally-spaced cash flows of equal size.

• Annuities can be either inflows or outflows.

• An ordinary (deferred) annuity has cash flows that

occur at the end of each period.

• An annuity due has cash flows that occur at the

beginning of each period.

• An annuity due will always be greater than an

otherwise equivalent ordinary annuity because interest

will compound for an additional period.

Annuities

Future Value of an Ordinary Annuity

• Annuity = Equal Annual Series of Cash Flows

• Example: How much will your deposits grow to if you

deposit $100 at the end of each year at 5% interest for

three years.

FVA = 100(FVIFA,5%,3) = $315.25

Year 1 $100 deposited at end of year = $100.00

Year 2 $100 x .05 = $5.00 + $100 + $100 = $205.00

Year 3 $205 x .05 = $10.25 + $205 + $100 = $315.25

Using the FVIFA Tables

Future Value of an Ordinary Annuity

• Annuity = Equal Annual Series of Cash Flows

• Example: How much will your deposits grow to if you

deposit $100 at the end of each year at 5% interest for

three years.

Using Excel

PMT 100$ k 5.0%n 3FV? 315.25$

Excel Function

=FV (interest, periods, pmt, PV)

=FV (.06, 5,100, )

Future Value of an Annuity Due

• Annuity = Equal Annual Series of Cash Flows

• Example: How much will your deposits grow to if you

deposit $100 at the beginning of each year at 5%

interest for three years.

FVA = 100(FVIFA,5%,3)(1+k) = $330.96

Using the FVIFA Tables

FVA = 100(3.152)(1.05) = $330.96

Future Value of an Annuity Due

• Annuity = Equal Annual Series of Cash Flows

• Example: How much will your deposits grow to if you

deposit $100 at the beginning of each year at 5%

interest for three years.

Using Excel

Excel Function

=FV (interest, periods, pmt, PV)

=FV (.06, 5,100, )

=315.25*(1.05)

PMT 100.00$ k 5.00%n 3FV $315.25FVA? 331.01$

Present Value of an Ordinary Annuity

• Annuity = Equal Annual Series of Cash Flows

• Example: How much could you borrow if you could

afford annual payments of $2,000 (which includes

both principal and interest) at the end of each year for

three years at 10% interest?

PVA = 2,000(PVIFA,10%,3) = $4,973.70

Using PVIFA Tables

Present Value of an Ordinary Annuity

• Annuity = Equal Annual Series of Cash Flows

• Example: How much could you borrow if you could

afford annual payments of $2,000 (which includes

both principal and interest) at the end of each year for

three years at 10% interest?

Using Excel

PMT 2,000$ I 10.0%n 3PV? $4,973.70

Excel Function

=PV (interest, periods, pmt, FV)

=PV (.10, 3, 2000, )

Present Value of a Mixed Stream

• A mixed stream of cash flows reflects no particular

pattern

• Find the present value of the following mixed stream

assuming a required return of 9%.

Using Tables

Year Cash Flow PVIF9%,N PV

1 400 0.917 366.80$

2 800 0.842 673.60$

3 500 0.772 386.00$

4 400 0.708 283.20$

5 300 0.650 195.00$

PV 1,904.60$

Present Value of a Mixed Stream

• A mixed stream of cash flows reflects no particular

pattern

• Find the present value of the following mixed stream

assuming a required return of 9%.

Using EXCEL

Year Cash Flow

1 400

2 800

3 500

4 400

5 300

NPV $1,904.76

Excel Function

=NPV (interest, cells containing CFs)

=NPV (.09,B3:B7)

Present Value of a Perpetuity• A perpetuity is a special kind of annuity.

• With a perpetuity, the periodic annuity or cash flow

stream continues forever.

PV = Annuity/k

• For example, how much would I have to deposit today

in order to withdraw $1,000 each year forever if I can

earn 8% on my deposit?

PV = $1,000/.08 = $12,500