Chapter 5 Notes
Learning Goals
Discuss the role of time value in finance and the use of computational aids to
simplify its application.
Understand the concept of future value and its calculation for a single amount;
understand the effects on future value and the true rate of interest of
compounding more frequently than annually.
Understand the concept of present value, its calculation for a single
amount, and the relationship of present to future cash flow.
Find the future value and present value of an ordinary annuity, the future value
of an annuity due, and the present value of a perpetuity.
Calculate the present value of a mixed stream of cash flows, describe the
procedures involved in:
 Determining deposits to accumulate to a future sum
 Loan amortization
 Finding interest or growth rates
The Role of Time Value in Finance

Most financial decisions involve costs and 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!
Basic Concepts

Future Value
 Compounding or growth over time

Present Value
 Discounting to today’s value

Single cash flows and 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
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
PV0
k
FVn
n
A
= present value or beginning amount
= interest rate
= future value at end of “n” periods
= number of compounding periods
= an annuity (series of equal payments or receipts)
Four Basic Models
Future Value Example
 Algebraically and Using FVIF Tables
 You deposit $2,000 today at 6% interest. How much will you have in 5
years?
 Using Microsoft® Excel
 You deposit $2,000 today at 6% interest. How much will you have in 5
years?
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.

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, and (d) monthly?
Continuous Compounding
 With continuous compounding the number of compounding periods per year approaches
infinity.
 Through the use of calculus, the equation thus becomes:
Nominal and 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 is greater than the nominal rate
whenever compounding occurs more than once per year.
 For example, what is the effective rate of interest on your credit
card if the nominal rate is 18% per year, compounded monthly?
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
 Algebraically and Using PVIF Tables
 How much must you deposit today in order to have $2,000 in 5 years if you
can earn 6% interest on your deposit?
 Using Microsoft® Excel
 How much must you deposit today in order to have $2,000 in 5 years if you
can earn 6% interest on your deposit?
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.
Future Value of an Ordinary Annuity
 Using the FVIFA Tables
 An annuity is an 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 Microsoft® Excel
 An annuity is an 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?
Future Value of an Annuity Due
 Using the FVIFA Table
 An annuity is an 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 Microsoft® Excel
 An annuity is an 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.
Present Value of an Ordinary Annuity
 Using PVIFA Tables
 An annuity is an 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 Microsoft® Excel
 An annuity is an 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?
Present Value of a Mixed Stream

Using Microsoft® Excel
 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%.
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.
Loan Amortization
Determining Interest or Growth Rates
 At times, it may be desirable to determine the compound interest rate or growth rate implied by
a series of cash flows.
 For example, you invested $1,000 in a mutual fund in 1994 which grew as shown in the table
below?
 At times, it may be desirable to determine the compound interest rate or growth rate implied by
a series of cash flows.
 For example, you invested $1,000 in a mutual fund in 1994 which grew as shown in the table
below?