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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.

5. Understand the effect that compounding interest more frequently than annually has on future value and the effective annual rate of interest.

6. Describe the procedures involved in (1) determining deposits needed to accumulate to a future sum, (2) loan amortization, (3) finding interest or growth rates, and (4) finding an unknown number of periods.

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 todays 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

Computational Aids (cont.)

Figure 4.1 Time Line

Computational Aids (cont.)

Figure 4.2 Compounding andDiscounting

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:

Simple Interest

With simple interest, you dont 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

PV0 = present value or beginning amount

i = 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+i)n = PV x (FVIFi,n)

PV0 = FVn[1/(1+i)n] = FV x (PVIFi,n)

FVAn = A (1+i)n - 1 = A x (FVIFAi,n)i

PVA0 = A 1 - [1/(1+i)n] = A x (PVIFAi,n)i

Future Value of a Single Amount

Future Value techniques typically measure cash flows at the end of a projects 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 investments life and then sums these values to find the investments 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 FVIF8%,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?

FV5 = \$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

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

Figure 4.5Future 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 adollar 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 PVIF6%,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?

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. Pams opportunity cost is 8%.

Present Value of a Single Amount: Using a Financial Calculator

Present Value of a Single Amount: Using Spreadsheets

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

Figure 4.6Present ValueRelationship

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.