introduction to valuation: the time value of money chapter 4 4-1

Introduction to valuation: The time value of money

Chapter 4

4-1

Key concepts and skillsBe 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 • a spreadsheet

Copyright 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4-2

Chapter outline

• Future value and compounding• Present value and discounting• More on present and future values

Copyright © 2011 McGraw-Hill Australia Pty LtdPPTs t/a Essentials of Corporate Finance 2e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh

4-3

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

Copyright 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh

4-4

Basic definitions (cont.)• Interest rate (r)– Discount rate– Cost of capital–Opportunity cost of capital– Required return– Terminology depends on usage


4-5

Future values: General formula• FV = PV(1 + r)t

– FV = Future value– PV = Present value– r = Period interest rate, expressed as a decimal– T = Number of periods

• FVIF(r,t)=Future value interest factor for $1 invested at r% for t periods

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

• Calculator note: yx keyCopyright 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh

4-6

Time line of cash flows•Tick marks at ends of periods

• Time 0 is today

• Time 1 is the end of Period 1

CF0 CF1 CF2

0 1 2 3r%

+CF = Cash INFLOW -CF = Cash OUTFLOW PMT = Constant CF


4-7

Time line for a $100 lump sum due at the end of year 2

100

0 1 2 Yearr%


4-8

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


4-9

Future values: Effects of compounding

• Simple interest – Interest earned only on the original

principal• Compound interest– Interest earned on principal and on

interest received– ‘Interest on interest’—interest earned

on reinvestment of previous interest payments


4-10

Future values: Effects of compounding

• Consider the previous example:– FV w/simple interest

= 100 + 10 + 10 = 120– FV w/compound interest

=100(1.10)2 = 121.00 – The extra 1.00 comes from the interest

of .10(10) = 1.00 earned on the first interest payment.


4-11

Future values of $100 at 10%


4-12

Future values: Simple interest and compound Interest


4-13

Future value of $1 for different interests and rates


4-14

Calculating future value• Using a calculator– Calculator key: yx

• Using the Future Value Factor Table– Table 4.2


4-15

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.


4-16

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 of periods increases. – Key strokes :

• N=5,• I/Y=10• PV=-100• CPT-FV• FV=161.05


4-17

Future value: Example 3• Suppose you had a relative deposit $5 at

6% interest 200 years ago. How much would the investment be worth today?– FV = 5(1.06)200 = $575 629.50

• What is the effect of compounding?– Simple interest = 5 + 200(5)(.06) = 5+60=65– Compounding added $575564.5 to the value

of the investment.


4-18

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?Copyright 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh

4-19

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?Copyright 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh

4-20

Present value• Present value = the current value of an

amount to be received in the future.• Why is it worth less than face value?– Opportunity cost– Risk and uncertainty– Discount rate = ƒ (time, risk)


4-21

Present value (cont.)FV = PV(1 + r)t

• Rearrange to solve for basic present value equation

PV = FV / (1+r)t

PV = FV(1+r)-t

• ‘Discounting’ = finding the present value of one or more future amounts.

• (1+r)-t is the present value factor.Copyright 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh

4-22

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?– Formula solution:

• PV=FV(1+r)-t=400/(1+0.07)• PV= 373.83

– Calculator keys:• 1 N• 7 I/Y• 400 FV• CPT PV = -373.83


4-23

Present valueExample 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.– PV=FV/(1+r) t

– =68500/(1.09)2

– 57655.08, so still short of the required amount by $7655• Calculator keys:

• 2 N• 9 I/Y• 68500 FV• CPT PV = -57655.08

4-24Copyright 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh

Present valueExample 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?– PV = 19 671.51 / (1.07)10 = $10 000

• Calculator keys:• 10 N• 7 I/Y• 19671.51 FV• CPT PV = -10000


4-25

Present value:Important relationship I

For a given interest rate:– The longer the time period, the lower the present value.

t)r1(

FVPV

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


4-26

Present value:Important relationship I (cont.)

• What is the present value of $500 to be received in 5 years? 10 years? The discount rate is 10%.– 5 years: PV = 500 / (1.1)5 = $310.46– 10 years: PV = 500 / (1.1)10 = $192.77

• Calculator: – 5 years: N = 5; I/Y = 10; FV = 500; CPT PV = -

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

192.77


4-27

Present value:Important relationship II

For a given time period: –The higher the interest rate, the smaller the present value.

t)r1(

FVPV

For a given t, as r increases, PV decreases.Copyright 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh

4-28

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 = -

248.58Copyright 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh

4-29

Present value of $1 for different periods and rates


4-30

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?


4-31

The basic PV equation: Refresher

PV = FV / (1 + r)t

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


4-32

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.


4-33

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 yx and the 1/x keys.Copyright 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh

4-34

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%


4-35

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?

• Formula:– r = (20 000 / 10 000)1/6 – 1 = .122462 = 12.25%

• Calculator: – N = 6 – FV = 20 000– PV = 10 000– CPT I/Y = 12.25%


4-36

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 it?– Formula:

• r = (75 000 / 5 000)1/17 – 1 = .172688 = 17.27%– Calculator: – N = 17– FV = 75 000 – PV = 5 000 – CPT I/Y = 17.27%


4-37

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 you choose?Copyright 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh

4-38

Finding the number of periods• Start with basic equation and solve for t:

FV = PV(1 + r)t

)1ln(

ln

rPVFV

t

Calculator: CPT N


4-39

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 car?– Formula:

• t = ln(20 000 / 15 000)/ln(1.1) = 3.02 years– Calculator:

• I/Y = 10• FV = 20 000• PV = 15 000• CPT N = 3.02 years


4-40

Number of periods: Example 2• Suppose you want to buy a new house. You

currently have $15 000 and you figure you need to have a 10% deposit plus an additional 5% in legal fees. If 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?


4-41

Number of periods: Example 2 (cont.)

• How much do you need to have in the future?– Deposit = .1(150 000) = $15 000– Legal fees = .05(150 000 – 15 000) = $6 750– Total needed = 15 000 + 6 750 = $21 750

• Compute the number of periods:– PV = -15 000– FV = 21 750– I/Y = 7.5– CPT N = 5.14 years

• Using the formula:– t = ln(21 750 / 15 000) / ln(1.075) = 5.14 years


4-42

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.


4-43

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.


4-44

http://www.moneychimp.com/articles/finworks/fmpresval.htm

Summary of TVM calculations


4-45

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?


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Chapter 4

END

4-47

introduction to valuation: the time value of money chapter 4 4-1

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