Chapter 4. Time Value of Money. Learning Goals. Learning Goals (cont.)

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Chapter 4

Time Value of Money

Learning Goals

1. Discuss the role of time value in finance, the use of computational aids, and the basic patterns of cash flow.

2. Understand the concept of future value and present value, their calculation for single amounts, and the relationship between them.

3. Find the future value and the present value of both an ordinary annuity and an annuity due, and the present value of a perpetuity.

Learning Goals (cont.)

4. Calculate both the future value and the present value

of a mixed stream of cash flows.

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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: Your father has offered to give you some money and asks that you choose one of the following two alternatives:

– $1,000 today, or – $1,100 one year from now.

• What do you do?

The Role of Time Value in Finance (cont.)

• The answer depends on what rate of interest you could earn on any money you receive today.

• For example, if you could deposit the $1,000 today at 12% per year, you would prefer to be paid today.

• Alternatively, if you could only earn 5% on deposited funds, you would be better off if you chose the $1,100 in one year.

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

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Computational Aids

• Use the Equations

• Use the Financial Tables

• Use Financial Calculators

• Use Electronic Spreadsheets

Computational Aids (cont.)

Figure 4.1 Time Line

Computational Aids (cont.)

Figure 4.2 Compounding and

Discounting

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Computational Aids (cont.) Figure 4.3 Calculator Keys

Computational Aids (cont.) Figure 4.4 Financial Tables

Basic Patterns of Cash Flow

• The cash inflows and outflows of a firm can be described by its general pattern.

• The three basic patterns include a single amount, an annuity, or a mixed

stream:

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Simple Interest

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

Compound Interest

• With compound interest, a depositor earns interest on 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

Time Value Terms

• PV

₀

= present value or beginning amount

• i = interest rate

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Four Basic Models

• FV

_n

= PV

₀

(1+i)

ⁿ

= PV x (FVIF

_i,n

)

• PV

₀

= FV

_n

[1/(1+i)

ⁿ

] = FV x (PVIF

_i,n

)

• FVA

_n

= A (1+i)

ⁿ

- 1 = A x (FVIFA

_i,n

) i

• PVA

₀

= A 1 - [1/(1+i)

ⁿ

] = A x (PVIFA

_i,n

) i

Future Value of a Single Amount

• Future Value techniques typically measure cash flows at the end of a project’s life.

• Future value is cash you will receive at a given future date.

• The future value technique uses compounding to find the future value of each cash flow at the end of an investment’s life and then sums these values to find the investment’s future value.

• We speak of compound interest to indicate that the amount of interest earned on a given deposit has become part of the principal at the end of the period.

$100 x (1.08)

¹

= $100 x FVIF

_8%,1

$100 x 1.08 = $108

Future Value of a Single Amount: Using FVIF Tables

• If Fred Moreno places $100 in a savings account

paying 8% interest compounded annually, how

much will he have in the account at the end of

one year?

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FV

₅

= $800 X (1 + 0.06)

⁵

= $800 X (1.338) = $1,070.40

Future Value of a Single Amount: The Equation for Future Value

• Jane Farber places $800 in a savings account paying 6% interest compounded annually. She wants to know how much money will be in the account at the end of five years.

Future Value of a Single Amount:

Using a Financial Calculator

Future Value of a Single Amount:

Using Spreadsheets

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Future Value of a Single Amount:

A Graphical View of Future Value

Figure 4.5 Future Value Relationship

Present Value of a Single Amount

• 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.

• Calculating present value is also known as discounting.

• The discount rate is often also referred to as the opportunity cost, the discount rate, the required return, or the cost of capital.

$300 x [1/(1.06)

¹

] = $300 x PVIF

_6%,1

$300 x 0.9434 = $283.02

Present Value of a Single Amount: Using PVIF Tables

• Paul Shorter has an opportunity to receive $300

one year from now. If he can earn 6% on his

investments, what is the most he should pay now

for this opportunity?

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PV = $1,700/(1 + 0.08)

⁸

= $1,700/1.851 = $918.42

Present Value of a Single Amount: The Equation for Future Value

• Pam Valenti wishes to find the present value of

$1,700 that will be received 8 years from now.

Pam’s opportunity cost is 8%.

Present Value of a Single Amount: Using a Financial Calculator

Present Value of a Single Amount: Using

Spreadsheets

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Present Value of a Single Amount: A Graphical View of Present Value

Figure 4.6 Present Value Relationship

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.

Types of Annuities

Note that the amount of both annuities total $5,000.

• Fran Abrams is choosing which of two annuities to receive. Both are 5-year $1,000 annuities; annuity A is an ordinary annuity, and annuity B is an annuity due.

Fran has listed the cash flows for both annuities as

shown in Table 4.1 on the following slide.

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Table 4.1 Comparison of Ordinary Annuity and Annuity Due Cash Flows ($1,000, 5 Years)

Finding the Future Value of an Ordinary Annuity

• Fran Abrams wishes to determine how much money she will have at the end of 5 years if he chooses annuity A, the ordinary annuity and it earns 7% annually.

Annuity a is depicted graphically below:

Future Value of an Ordinary Annuity:

Using the FVIFA Tables

FVA = $1,000 (FVIFA,7%,5)

= $1,000 (5.751)

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Future Value of an Ordinary Annuity:

Using a Financial Calculator

Future Value of an Ordinary Annuity:

Using Spreadsheets

Present Value of an Ordinary Annuity

• Braden Company, a small producer of plastic toys, wants to

determine the most it should pay to purchase a particular

annuity. The annuity consists of cash flows of $700 at the end

of each year for 5 years. The required return is 8%.

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Present Value of an Ordinary Annuity:

The Long Method

Table 4.2 Long Method for Finding the Present Value of an Ordinary Annuity

Present Value of an Ordinary Annuity:

Using PVIFA Tables

PVA = $700 (PVIFA,8%,5)

= $700 (3.993)

= $2,795.10

Present Value of an Ordinary Annuity:

Using a Financial Calculator

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Present Value of an Ordinary Annuity:

Using Spreadsheets

FVA = $1,000(FVIFA,7%,5)(1+.07)

= $1,000 (5.751) (1.07)

= $6,154

Future Value of an Annuity Due:

Using the FVIFA Tables

• Fran Abrams now wishes to calculate the future value of an annuity due for annuity B in Table 4.1. Recall that annuity B was a 5 period annuity with the first annuity beginning immediately.

Future Value of an Annuity Due:

Using a Financial Calculator

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Future Value of an Annuity Due:

Using Spreadsheets

PVA = $700 (PVIFA,8%,5) (1.08)

= $700 (3.993) (1.08)

= $3,018.40

Present Value of an Annuity Due:

Using PVIFA Tables

• In the earlier example, we found that the value of Braden Company’s $700, 5 year ordinary annuity discounted at 8% to be about $2,795. If we now assume that the cash flows occur at the beginning of the year, we can find the PV of the annuity due.

Present Value of an Annuity Due:

Using a Financial Calculator

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Present Value of an Annuity Due:

Using Spreadsheets

PV = Annuity/Interest Rate

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

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.

• 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?

Future Value of a Mixed Stream

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Future Value of a Mixed Stream:

Using Excel

Future Value of a Mixed Stream (cont.)

Table 4.3 Future Value of a Mixed Stream of Cash Flows

Present Value of a Mixed Stream

• Frey Company, a shoe manufacturer, has been

offered an opportunity to receive the following

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Present Value of a Mixed Stream

• If the firm must earn at least 9% on its investments, what is the most it should pay for this opportunity?

• This situation is depicted on the following time line.

Present Value of a Mixed Stream:

Using Excel

Present Value of a Mixed Stream

Table 4.4 Present Value of a Mixed Stream of Cash

Flows

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Compounding Interest 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 Interest More Frequently Than Annually (cont.)

• Fred Moreno has found an institution that will pay him 8%

annual interest, compounded quarterly. If he leaves the money in the account for 24 months (2 years), he will be paid 2% interest compounded over eight periods.

Table 4.5 Future Value from Investing $100 at 8% Interest

Compounded Semiannually over 24 Months (2 Years)

Compounding Interest More Frequently Than Annually (cont.)

Table 4.6 Future Value from Investing $100 at 8%

Interest Compounded Quarterly over 24 Months (2 Years)

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Compounding Interest More Frequently Than Annually (cont.)

Table 4.7 Future Value at the End of Years 1 and 2 from Investing $100 at 8% Interest, Given Various Compounding Periods

Compounding Interest More Frequently Than Annually (cont.)

• A General Equation for Compounding More Frequently than Annually

Compounding Interest More Frequently Than Annually (cont.)

• A General Equation for Compounding More Frequently than Annually

– Recalculate the example for the Fred Moreno example

assuming (1) semiannual compounding and (2) quarterly

compounding.

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Compounding Interest More Frequently Than Annually: Using a Financial Calculator

Compounding Interest More Frequently Than Annually: Using a Spreadsheet

Continuous Compounding

• With continuous compounding the number of compounding periods per year approaches infinity.

• Through the use of calculus, the equation thus becomes:

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FV

_n

(continuous compounding) = PV x (e

^kxn

) where “e” has a value of 2.7183.

FVn = 100 x (2.7183)

^.08x2

= $117.35

Continuous Compounding (cont.)

• With continuous compounding the number of compounding periods per year approaches infinity.

• Through the use of calculus, the equation thus becomes:

Continuous Compounding:

Using a Financial Calculator

Continuous Compounding:

Using a Spreadsheet

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Nominal & Effective Annual Rates of Interest

• 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

Nominal & Effective Annual Rates of Interest (cont.)

• Fred Moreno wishes to find the effective annual rate associated with an 8% nominal annual rate (I = .08) when interest is compounded (1) annually (m=1); (2) semiannually (m=2); and (3) quarterly (m=4).

Special Applications of Time Value:

Deposits Needed to Accumulate to a Future Sum

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PMT = $30,000/5.637 = $5,321.98 Special Applications of Time Value: Deposits Needed to Accumulate to a Future Sum (cont.)

• Suppose you want to buy a house 5 years from now and you estimate that the down payment needed will be

$30,000. How much would you need to deposit at the end of each year for the next 5 years to accumulate

$30,000 if you can earn 6% on your deposits?

Special Applications of Time Value: Deposits Needed to Accumulate to a Future Sum (cont.)

Special Applications of Time Value: Deposits

Needed to Accumulate to a Future Sum (cont.)

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Special Applications of Time Value:

Loan Amortization

Table 4.8 Loan Amortization Schedule

($6,000 Principal, 10% Interest, 4-Year Repayment Period)

Special Applications of Time Value:

Loan Amortization (cont.)

Special Applications of Time Value:

Interest or Growth Rates

• At times, it may be desirable to determine the

compound interest rate or growth rate implied

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PVIF

_i,5yrs

= PV/FV = ($1,250/$1,520) = 0.822

PVIF

_i,5yrs

= approximately 5%

Special Applications of Time Value:

Interest or Growth Rates (cont.)

Special Applications of Time Value:

Interest or Growth Rates (cont.)

Special Applications of Time Value:

Interest or Growth Rates (cont.)

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Ann Bates wishes to determine the number of years it will take for her initial $1,000 deposit, earning 8% annual interest, to grow to equal $2,500. Simply stated, at an 8% annual rate of interest, how many years, n, will it take for Ann’s $1,000 (PV

_n

) to grow to $2,500 (FV

_n

)?

Special Applications of Time Value:

Finding an Unknown Number of Periods

• At times, it may be desirable to determine the number of time periods needed to generate a given amount of cash flow from an initial amount.

PVIF

_8%,n

= PV/FV = ($1,000/$2,500) = .400

PVIF

_8%,n

= approximately 12 years Special Applications of Time Value:

Finding an Unknown Number of Periods (cont.)

Special Applications of Time Value:

Finding an Unknown Number of Periods (cont.)

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Special Applications of Time Value:

Finding an Unknown Number of Periods (cont.)

Table 4.9 Summary of Key Definitions, Formulas, and Equations for Time Value of Money (cont.)

Table 4.9 Summary of Key Definitions,

Formulas, and Equations for Time Value

of Money