978-1259277160 Chapter 9 Lecture Note

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Time Value of Money

Author’s Overview

This is one of the most important chapters in the book as far as student comprehension is

concerned. The instructor should first determine how much prior knowledge of time value of

money the students have acquired from accounting or lower mathematics. While most

students are generally familiar with the concepts of future value and present value, they often

lack the ability to identify and categorize the nature of the problem before them.

The material in this chapter will serve as a springboard to the remaining chapters in this

section on valuation, cost of capital and capital budgeting related topics. A good background

in time value of money will ease the transition. The authors suggest a liberal use of

homework problems and a quiz to reinforce the importance of this material.

This chapter uses color-coordinated figures to explain the relationship between present value

and future value and present value to the present value of annuities, and future value to future

value of annuities. These color coded figures are helpful to those more visually oriented

students who may not understand the mathematical relationships between these time value

calculations.

For faculty who want to emphasize Excel spreadsheets and calculator keystrokes, this chapter

provides an excellent opportunity to develop both skills. Using spreadsheets with time-value

exercises can be especially instructive in understanding the concept of how higher discount

rates generate lower cash flows and visa versa.

Chapter Concepts

LO1. Money has a time value associated with it and therefore a dollar received today is worth

more than a dollar received in the future.

LO2. The future value is based on the number of periods over which the funds are to be

compounded at a given interest rate.

LO3. The present value is based on the current value of funds to be received.

LO4. Not only can future value and present value be computed, but other factors such as yield

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or rate of return can be determined as well.

LO5. Compounding or discounting may take place on a less than annual basis such as

semiannually or monthly.

Authors’ Note:

In previous editions of Foundations of Financial Management, the authors have emphasized

the use of time value tables in the text. This has presented several problems. With the advent

of computer generated problem sets, the tables were not precise and caused students who used

calculators and spreadsheets much frustration because of rounding errors created by

truncating the table values. This also caused difficulty for instructors because students could

get different answers using the tables than with calculators or spreadsheets. In this edition,

we have revamped this chapter to introduce calculator and Excel spreadsheets as the primary

way to calculate time value problems. Because many of our users like to use tables, we have

kept that pedagogy, but now it is found at the end of the chapter.

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Annotated Outline and Strategy

I. Relationship to the Capital Outlay Decision: Capital budgeting involves the analysis

of whether or not funds invested today will be more than offset by the value of the funds

received from the investment in the future.

II. Future Value — Single Amount

A. In determining future value, we measure the value of an amount that is invested

now and allowed to grow at a given interest rate over a specified period of time.

B. The relationship may be expressed by the following formula:

The FVIF term is found using the calculator keystrokes. We use a $1,000 example

rather than $1 (also see Appendix A).

C. Using a financial calculator to derive the future value of a single payment, enter

the known values for the following:

N enter the number of interest periods

I/Y enter the annual interest rate

PV enter the present value amount

PMT enter zero

FV leave blank

Then press the CPT key and then the FV key

(The resulting value will appear as a negative number.)

Perspective 9-1: One or two numerical examples using this spreadsheet are helpful.

III. Present Value — Single Amount

A. The present value of a future sum is the investment required today, at a given

interest rate that will equal the future sum at a specified point in time.

PPT Relationship of Present Value and Future Value (Figure 9-1)

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FV = PV(1 )n

FV = PV FVIF

Perspective 9-2: Use Figure 9-1 to demonstrate the relationship between present and future

value.

B. The relationship may be expressed in the following formula:

C. The formula may be restated as:

The PVIF term is found using calculator keystrokes. We use $1,000 rather than

$1 in our example (also see Appendix B).

D. Using a financial calculator to derive the present value of a single future

payment, enter the known values for the following:

N enter the number of interest periods

I/Y enter the annual interest rate

PV leave blank

PMT enter zero

FV enter the future value amount

Then press the CPT key and then the PV key

(The resulting value will appear as a negative number.)

Perspective 9-3: We would suggest using the Excel spreadsheet example on page 261 to

show the relationship of the present value to the discount rate and time period.

IV. Interest Rate – Single Amount

A. When we know the future value, the present value, and the time period, we can

solve for the interest rate or the rate of return on an investment.

B. Using Formula 9-3 we find that:

C. We can use calculator keystrokes or the Excel spreadsheet to solve for the

interest rate.

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PV = FV (1 )n

PV = FV PVIF

FV 1

iæ ö

= –

ç ÷

è ø

V. Number of Periods – Single Amount

A. Following up on the interest rate example, we can use Formula 9-4 to solve for

the number of periods it takes to grow (compound) $1,000 to $1,464.10 if we

know the rate of return is 10%.

B. Once again, using the spreadsheet example on page 263 or the calculator

keystrokes, you can demonstrate how many years it takes to compound $1,000 to

$1,464.10.

VI. Future Value – Annuity

A. An annuity represents consecutive payments or receipts of equal amounts over

equal time intervals.

B. The annuity value is normally assumed to take place at the end of each period.

C. The future value of an annuity represents the sum of the future value of the

individual flows.

PPT Compounding Process for Annuity (Figure 9-2)

D. The formula for the future value for an annuity is given in Formula 9-5:

FVA = A  FVIFA

The FVIFA term is found using the calculator keystrokes or the Excel spreadsheet

on page 265 (also see Appendix C).

E. Using a financial calculator to derive the future value of an annuity, enter the

known values for the following:

N enter the number of interest periods

I/Y enter the annual interest rate

PV enter zero

PMT enter the periodic payment

FV leave blank

Then press the CPT key and then the FV key

(The resulting value will appear as a negative number.)

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Finance in Action: Powerball Jackpot Decisions

The odds of winning are now 1 in 292 million after Powerball changed the number of choices

for the white and red numbers and many students will be quite familiar with the $1.5 billion

Powerball prize won on January 13, 2016. However the purpose of the box is to demonstrate

the impact that present value has on an annuity prize taken as a lump sum payout. As a side bar

you might want to bring in what taxes do the lump sum payout.

VII. Present Value — Annuity

A. The present value of an annuity represents the sum of the present value of the

individual cash flows.

B. The formula for the present value of an annuity is presented in Formula 9-6 (also

see Appendix D for PVIFA)

PVA = A  PVIFA

C. Using a financial calculator to derive the present value of an annuity, enter the

known values for the following:

N enter the number of interest periods

I/Y enter the annual interest rate

PV leave blank

PMT enter the periodic payment

FV enter zero

Then press the CPT key and then the PV key

(The resulting value will appear as a negative number.)

D. The annuity value associated with a present value is often associated with

withdrawal of funds from an initial deposit or the repayment of a loan.

Perspective 9-4: An example that might ring true with the students is to have them use the

Excel spreadsheet on page 267 to calculate their annual or monthly student loan repayment

once they graduate from college and have a job. It might be an eye-opening exercise.

VIII. Graphical Presentation of Time Value Relationships

A. Use Figures 9-3 and 9-4 to show that present value and future value are inversely

related.

B. Use Figures 9-5 and 9-6 to demonstrate how annuities are the sums of single

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period values.

IX. Determining the Annuity Value (The amount of the periodic payment that would grow

to the requisite future value or be discounted to the requisite present value)

A. Annuity Equaling a Future Value

1. The formula for the future value for an annuity is found in Formula 9-7:

PPT Relationship of Present Value to Annuity (Table 9-1)

Perspective 9-3: Demonstrate how annuities work in everyday situations.

PPT Payoff Table for Loan (Amortization Table) (Table 9-2)

2. Using a financial calculator to derive the annuity value required to

achieve a future amount, enter the known values for the following:

N enter the number of interest periods

I/Y the periodic interest rate

PV enter zero

PMT leave blank

FV enter the future value amount

Then press the CPT key and then the PMT key

(The resulting annuity value will appear as a negative value.)

B. Annuity Equaling a Present Value

1. The formula for the present value of an annuity is presented in Formula 9-8:

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(1 ) 1

=é ù

+ –

ê ú

ë û

2. Using a financial calculator to derive the annuity value required to achieve a

present amount, enter the known values for the following:

N enter the number of interest periods

I/Y the periodic interest rate

PV enter the present value amount

PMT leave blank

FV enter zero

Then press the CPT key and then the PMT key

(The resulting annuity value will appear as a negative value.)

3. Finding annuity payments using Excel’s PMT function: The PMT

function assumes the each payment is at the end of a period. Either the

PV or FV argument must be entered. See the example on page 276.

X. Finding Interest Rates and the Number of Payments

A. Finding Annuity Interest Rates Using Calculator Keystrokes or Excel: See

discussion on page 274.

B. Finding the Number of Annuity Payments Using Calculator Keystrokes or Excel:

See discussion on page 275.

XI. Compounding over Additional Periods

A. Compounding Semiannually requires us to cut the annual interest rate in half and

to multiply the number of periods by two. Consequently quarterly compounding

divides the interest rate by four and multiplies the periods by four. See case 1

and case 2 on page 275.

B. Patterns of Payment with a Deferred Annuity: Sometimes an annuity does not

start being received until several years in the future, and under these conditions,

we need to calculate the present value of each payment. There are several ways

to do the calculation. See examples on pages 276 and 277

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1(1 )

=é ù

–

ê ú

ë û

D. Annuities Due: Annuities that start at the beginning of the period rather than at

the end of the period. Use the Excel spreadsheet on page 278 to demonstrate both

present value and future values of annuities due.

XI. Alternative Calculations Using TVM Tables – APPENDICES 9A & B Page 288-294

A. Future Value — Single Amount

1. In determining future value, we measure the value of an amount that is

invested now and allowed to grow at a given interest rate over a specified

period of time.

2. The relationship may be expressed by the following formula:

The FVIF term is found in Table 9-3 (also see Appendix A).

B. Present Value — Single Amount

1. The present value of a future sum is the investment required today at a

given interest rate that will equal the future sum at a specified point in time.

2. The relationship may be expressed in the following formula:

PV = FV (1 )n

The PVIF term is found in Table 9-4 (also see Appendix B).

C. Future Value — Annuity

1. An annuity represents consecutive payments or receipts of equal amounts

over equal time intervals.

2. The annuity value is normally assumed to take place at the end of each

period.

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FV = PV(1 )n

FV = PV FVIF

3. The future value of an annuity represents the sum of the future value of

the individual flows.

4. The formula for the future value for an annuity is:

FVA = A  FVIFA

The FVIFA term is found in Table 9-5 (also see Appendix C).

D. Present Value — Annuity

1. The present value of an annuity represents the sum of the present value of

the individual cash flows.

2. The formula for the present value of an annuity is:

PVA = A  PVIFA

The PVIFA term is found in Table 9-6 (also see Appendix D).

Other Chapter Supplements

Cases for Use with Foundations of Financial Management

Case 10, Allison Boone, M.D. (time value of money)

Case 11, Billy Wilson, All American (time value of money)

Case 12, Sarah Gilbert, Retiree (time value of money)

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