Lecture Essentials of corporate finance (2/e) – Chapter 4: Introduction to valuation: the time value of money

After completing this unit, you should be able to compute the future value of an investment made today, be able to compute the present value of cash to be received at some future date, be able to compute the return on an investment.

Introduction to valuation:

The time value of money

Chapter 4

Key concepts and skills

Be able to compute the following:

• The future value of an investment made today

• The present value of cash to be received at some future date

• The return on an investment

Be able to predict how long it will take for an investment

to

reach a desired value

Be able to solve time value of money problems using:

• formulas

• a financial calculator

Chapter outline

• Future value and compounding

• Present value and discounting

• More on present and future values

Basic definitions

• Present value (PV)

– The current value of future cash flows

discounted at the appropriate discount

rate.

– Value at t=0 on a time line

• Future value (FV)

– The amount an investment is worth after

one or more periods.

– ‘Later’ money on a time line

Basic definitions (cont.)

Future values: General

• FVIF(r,t)=Future value interest factor for

$1 invested at r% for t periods

• Future value interest factor = (1 + r) t

• Calculator note: y x key

Time line of cash flows

•Tick marks at ends of periods

Time line for a $100 lump sum due

at the end of year 2

100

r%

Future values: Example 1

• Investing for a single period

– Suppose you invest $100 for one year at 10% (r) per year What is the future value in one year?

• Interest = 100(.1) = $10

• Value in one year = principal + interest = 100 + 10 = $110

• Future value (FV) = 100(1 + 1) = $110

• Investing for more than one period

– Suppose you leave the money in for another year How much will you have two years from now?

• FV = 100(1.1)(1.1) = 1000(1.1)2 = $121

– ‘Interest on interest’—interest earned

on reinvestment of previous interest payments

Future values of $100 at 10%

Future values:

Simple interest and compound

Interest

Future value of $1 for different

interests and rates

Calculating future value

• Using a calculator

– Calculator key: y x

• Using the Future Value Factor Table

– Table 4.2

Future value: Calculator

hints

• Texas Instruments BA-II Plus

– FV = Future value

– PV = Present value

– I/Y = Period interest rate

• PV must equal 1 for the I/Y to be the period rate

• Interest is entered as a percentage, not a decimal

– N = Number of periods

– Remember to clear the registers (CLR

TVM) after each problem.

– Other calculators are similar in format.

Future value: Example 2

• Suppose you invest the $100 from the previous example for 5 years How much would you

have?

– Formula solution:

• FV=PV(1+r)t=100(1.10)5=161.05

• The effect of compounding is small for a small

number of periods, but increases as the number

Future value: Example 3

• Suppose you had a relative deposit $5

at 6% interest 200 years ago How

much would the investment be worth

Quick quiz: Part 1

• What is the difference between simple interest and compound interest?

• Suppose you have $500 to invest and

you believe that you can earn 8% per

year over the next 15 years.

– How much would you have at the end of

15 years using compound interest?

– How much would you have using simple

interest?

Present value and

discounting

• The current value of future cash flows

discounted at the appropriate discount rate

• Value at t=0 on a time line

• Answers the questions:

– How much do I have to invest today to

have a particular amount in the future?

– What is the current value of a particular

amount to be received in the future?

– Risk and uncertainty

– Discount rate = ƒ (time, risk)

Present value (cont.)

Present value Example 1—Single period

• Suppose you need $400 to buy textbooks next year You can earn 7% on your

money How much do you have to put up today?

Present value Example 2—Multiperiod

• You would like to buy a new car You have $50

000, but the car costs $68 500 If you can earn

9%, how much do you have to invest today to buy the car in two years? Do you have enough?

Assume the price will stay the same.

Present value Example 3—Multiperiod

• Your parents set up a trust fund for you

10 years ago that is now worth $19

671.51 If the fund earned 7% per year, how much did your parents invest?

Present value:

Important relationship I

For a given interest rate:

– The longer the time period,

the lower the present value.

t ) r 1

(

FV PV

For a given r, as t increases, PV decreases.

Present value:

Important relationship I (cont.)

• What is the present value of $500 to be received in 5 years? 10 years? The

Present value:

Important relationship II

For a given time period:

– The higher the interest rate,

the smaller the present value.

t

) r 1

(

FV PV

For a given t, as r increases, PV decreases.

Present value:

Important relationship II (cont.)

• What is the present value of $500

received in 5 years if the interest rate is 10%? 15%?

– Rate = 10%: PV = 500 / (1.1)5 = $310.46

– Rate = 15%: PV = 500 / (1.15)5 = $248.58

• Calculator:

– 10%: N = 5; I/Y = 10; FV = 500; CPT PV = -310.46

– 15%: N = 5; I/Y = 15; FV = 500; CPT PV =

Present value of $1 for different

periods and rates

Quick quiz: Part 2

• What is the relationship between

present value and future value?

• Suppose you need $15 000 in 3 years

If you can earn 6% annually, how much

do you need to invest today?

• If you could invest the money at 8%,

would you have to invest more or less

than at 6%? How much more or less?

The basic PV equation:

Refresher

There are 4 parts to this equation:

– PV, FV, r and t

– Know any 3, solve for the 4th

• Be sure to remember the sign

convention

+CF = Cash INFLOW -CF = Cash OUTFLOW

Present vs future value—

Evaluating investments

• Your company proposes to buy an asset

for $335 This investment is very safe You will sell the asset in 3 years for $400 You know you could invest the $335 elsewhere

at 10% with very little risk What do you

think of the proposed investment?

– Not a good investment

– FV of 335= 335(1.1) 3 = 445.89

– Future value of $335 is more than the value of asset in three years.

Discount rate

• To find the implied interest rate,

rearrange the basic PV equation

and solve for r:

FV = PV(1 + r) t

r = (FV / PV) 1/t – 1

• If using formulas with a calculator,

make use of both the y x and the 1/x

keys.

Discount rate: Example 1

• You are looking at an investment that will pay

$1200 in 5 years if you invest $1000 today What is the implied rate of interest?

– r = (1200 / 1000) 1/5 – 1 = 03714 = 3.714% – Calculator – the sign convention matters!

• N = 5

• PV = -1000 (you pay $1000 today)

• FV = 1200 (you receive $1200 in 5 years)

• CPT I/Y = 3.714%

Discount rate: Example 2

• Suppose you are offered an investment

that will allow you to double your money in

6 years You have $10 000 to invest What

is the implied rate of interest?

Discount rate: Example 3

• Suppose you have a 1-year-old son and you

want to provide $75 000 in 17 years towards

his university education You currently have

$5000 to invest What interest rate must you

earn to have the $75 000 when you need

Quick quiz: Part 3

• What are some situations in which you

might want to compute the implied interest rate?

• Suppose you are offered the following

investment choices:

– You can invest $500 today and receive $600

in 5 years The investment is considered low risk.

– You can invest the $500 in a bank account

paying 4%.

• What is the implied interest rate for the

first choice and which investment should Copyright 2011 McGraw-Hill Australia Pty Ltd

4-38

Finding the number of

ln(

ln

r PV

FV t

Calculator: CPT N

Number of periods:

Example 1

• You want to purchase a new car and you are willing to pay $20 000 If you can

invest at 10% per year and you currently

have $15 000, how long will it be before

you have enough money to pay cash for

the type of house you want costs about

$150 000 and you can earn 7.5% per

year, how long will it be before you

have enough money for the deposit

and legal fees?

Number of periods: Example 2

Example: Spreadsheet

strategies

• The formula icon is very useful when

you can’t remember the exact formula.

• Double-click on the Excel icon to open

a spreadsheet containing four different examples.

Example: Work the Web

• Many financial calculators are available online.

• Click on the web surfer icon to go to the present value portion of the

moneychimp website and work the

following example:

– You need $40 000 in 15 years If you can earn 9.8% interest, how much do you

need to invest today?

– You should get $9 841.

Summary of TVM calculations

Quick quiz: Part 4

• When might you want to compute the

number of periods?

• Suppose you want to buy some new

furniture for your family room You

currently have $500 and the furniture

you want costs $1600 If you can earn 6%, how long will you have to wait if

you don’t add any additional money?

Chapter 4

END