The Time Value of

Money

Prepared by

Professor Wei Wang

Queen’s University

© 2011 McGraw–Hill Ryerson Limited

5-1

Chapter Outline

5.1 The One-Period Case

5.2 The Multiperiod Case

5.3 Compounding Periods

5.4 Simplifications

5.5 What Is a Firm Worth?

5.6 Summary and Conclusions

© 2011 McGraw–Hill Ryerson Limited

5-2

The One-Period Case: Future Value

LO5.1

• If you were to invest $10,000 at 5-percent interest for one year, your investment would grow to $10,500

$500 would be interest ($10,000 × .05)

$10,000 is the principal repayment ($10,000 × 1)

$10,500 is the total due. It can be calculated as:

$10,500 = $10,000×(1.05).

The total amount due at the end of the investment is called the Future Value (FV).

© 2011 McGraw–Hill Ryerson Limited

5-3

The One-Period Case: Future Value

LO5.1

• In the one-period case, the formula for FV can be written as:

FV = C0×(1 + r)

Where C0 is cash flow at date 0 and r is the appropriate interest rate.

C0×(1 + r)

C0 = $10,000

Year

0

$10,000 1.05

FV = $10,500

1

© 2011 McGraw–Hill Ryerson Limited

5-4

The One-Period Case: Present Value

LO5.1

• If you were to be promised $10,000 due in one year when interest rates are at 5-percent, your investment would be worth $9,523.81 in today’s dollars.

$10,000

$9,523.81

1.05

The amount that a borrower would need to set aside today to be able to meet the promised payment of

$10,000 in one year is call the Present Value (PV) of

$10,000.

Note that $10,000 = $9,523.81×(1.05).

© 2011 McGraw–Hill Ryerson Limited

5-5

The One-Period Case: Present Value

LO5.1

• In the one-period case, the formula for PV can be written as:

C1

PV

1 r

Where C1 is cash flow at date 1 and r is the appropriate interest rate.

PV = $9,523.81

C1/(1 + r)

$10,000/1.05

Year

0

C1 = $10,000

1

© 2011 McGraw–Hill Ryerson Limited

5-6

The One-Period Case: Net Present Value LO5.1

• The Net Present Value (NPV) of an investment is the present value of the expected cash flows, less the cost of the investment.

• Suppose an investment that promises to pay

$10,000 in one year is offered for sale for

$9,500. Your interest rate is 5%. Should you buy? $10,000

NPV $9,500

1.05

NPV $9,500 $9,523.81

NPV $23.81

Yes!

© 2011 McGraw–Hill Ryerson Limited

5-7

The One-Period Case: Net Present Value LO5.1

In the one-period case, the formula for NPV can be written as:

NPV Cost PV

If we had not undertaken the positive NPV project considered on the last slide, and instead invested our

$9,500 elsewhere at 5-percent, our FV would be less than the $10,000 the investment promised and we would be unambiguously worse off in FV terms as well: $9,500×(1.05) = $9,975 < $10,000.

© 2011 McGraw–Hill Ryerson Limited

5-8

The Multiperiod Case: Future Value LO5.2

• The general formula for the future value of an investment over many periods can be written as: FV = C0×(1 + r)T

Where

C0 is cash flow at date 0, r is the appropriate interest rate, and

T is the number of periods over which the cash is invested. © 2011 McGraw–Hill Ryerson Limited

5-9

The Multiperiod Case: Future Value LO5.2

• Suppose that Jay Ritter invested in the initial public offering of the Modigliani company.

Modigliani pays a current dividend of $1.10, which is expected to grow at 40-percent per year for the next five years.

• What will the dividend be in five years?

FV = C0×(1 + r)T

$5.92 = $1.10×(1.40)5

© 2011 McGraw–Hill Ryerson Limited

5-10

Future Value and Compounding

LO5.2

• Notice that the dividend in year five, $5.92, is considerably higher than the sum of the original dividend plus five increases of 40percent on the original $1.10 dividend:

$5.92 > $1.10 + 5×[$1.10×.40] = $3.30

This is due to compounding.

© 2011 McGraw–Hill Ryerson Limited

5-11

Future Value and Compounding

$1.10 (1.40)