Introduction To Valuation:
The Time Value Of Money
Chapter 4
Topics
1.
2.
3.
How To Determine The Future
Value Of An Investment
How To Determine The Present
Value Of Cash To Be Received At A
Future Date
How To Find The Return On An
Investment
2
Fundamental Truth in Finance:
 A dollar earned now is worth more than a dollar earned later.
 This is true because of the ability of individuals to earn interest.
Today
1 year from today
$1 received today
$1 received 1 year from today
The $1 received today is worth more than a
dollar received 1 year from now because
you can invest the dollar and earn interest.
Today
(at 10% simple rate)
$1.00
1 year from today
grows to
Present Value
(Interest going backwards)
$1.10
Future Value
(Interest going forward)
3
Fundamental Financial Concept

A dollar received today is worth
more than a dollar received later
This is because of interest 
 This is because of the discount rate 

4
Definitions:

Simple interest


Compound interest


Interest earned on both the initial principal
and the interest reinvested from prior periods
Interest on interest


Interest earned only on the original principal
amount invested
Interest earned on the reinvestment of
pervious interest payments
Compounding

The process of accumulating interest in an
investment over time to earn more interest
5
Variables for the Financial Functions Defined:
An = Annuity = Regular Payments (PMT) Made at Regular Time Intervals
Made at End of Period
LS = Lump Sum Payment = Payment Made Once
FV=Future Value (Lump Sum Value in the Future)
PV=Present Value (Lump Sum Value in the Present)
PMT = Regular Payment Made at Regular Time Intervals
i = Annual Interest Rate
n = Number of Compounding Periods per Year
x = Years
6
Future value:

The amount an investment is worth after
one or more periods
 i
FVLS = PVLS*1+ 
 n
n*x
7
Future Values: (Textbook formula)
 Textbook
FV = PV(1 + r)t
FV = future value
 PV = present value
 r = period interest rate, expressed as a decimal
 T = number of periods

 Future
value interest factor = (1 + r)t
8
100 bucks invested @ 10%
compounded yearly for 2 years
100*(1+.10)=110
110*(1+.10)=121

100*(1+.10)*(1+.10)=121
2
100*(1+.10) =121
1*2
 .10 
100* 1+

1 

 121
i

FVLS  PVLS * 1  
 n
n* x
9
10
11
How To Determine The Future Value Of
An Investment
Suppose you invest $1000 for one year at 5% per
year, compounded yearly. What is the future value
in one year?
 Interest = 1000(.05) = 50
 Value in one year = principal + interest = 1000
+ 50 = 1050
 Future Value (FV) = 1000(1 + .05) = 1050
 Suppose you leave the money in for another year.
How much will you have two years from now?
 FV = 1000(1.05)(1.05) = 1000(1.05)2 = 1102.50

12
Effects of Compounding
 Simple
interest
 Compound interest
 Consider the previous example
FV with simple interest = 1000 + 50 + 50 = 1100
 FV with compound interest = 1102.50
 The extra 2.50 comes from the interest of .05(50) =
2.50 earned on the first interest payment

13
Future Values – Example 2
 Suppose
you invest the $1000 from the
previous example for 5 years, compounded
yearly. How much would you have?

FV = 1000(1.05)5 = 1276.28
 The
effect of compounding is small for a small
number of periods, but increases as the number
of periods increases. (Simple interest would
have a future value of $1250, for a difference of
$26.28.)
14
Future Values – Example 3
 Suppose
you had a relative deposit $10 at 5.5%
interest 200 years ago , compounded yearly.
How much would the investment be worth
today?

FV = 10(1.055)200 = 447,189.84
 What
is the effect of compounding?
Simple interest = 10 + 200(10)(.055) = 210.55
 Compounding added $446,979.29 to the value of
the investment

15
Present Value


How much should you put in the bank
today in order to receive a future value
amount after one or more periods
The current value of future cash flows
discounted at the appropriate rate
PVLS =
FVLS
 i
1+ 
 n
n*x
If you know the
future amount you
would like, assume
an interest rate, and
take all the interest
that you will need to
earn out of the
future value amount
16
Present Values (Textbook formula)

How much do I have to invest today to have some
amount in the future?
Textbook FV = PV(1 + r)t
 Rearrange to solve for PV = FV / (1 + r)t

When we talk about discounting, we mean finding the
present value of some future amount.
 When we talk about the “value” of something, we are
talking about the present value unless we specifically
indicate that we want the future value.

17
Definitions:

Discount Rate


Discount


The rate used to calculate the present
value of future cash flows
Calculate the present value of some
future amounts
Discounted Cash Flow (DCF) valuation

Calculating the present value of future
cash flows to determine its value today
18
How To Determine The Present Value Of
Cash To Be Received At A Future Date
 You
want to begin saving for you daughter’s
college education and you estimate that she will
need $150,000 in 17 years. If you feel
confident that you can earn 8% per year,
compounded yearly, how much do you need to
invest today?

PV = 150,000 / (1.08)17 = 40,540.34
19
Present Values – Example 2
 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, compounded yearly,
how much did your parents invest?

PV = 19,671.51 / (1.07)10 = 10,000
20
PV – Important Relationship I
a given interest rate – the longer the time
period, the lower the present value
 For
What is the present value of $500 to be received in
5 years? 10 years? The discount rate is 10%,
compounded yearly.
 5 years: PV = 500 / (1.1)5 = 310.46
 10 years: PV = 500 / (1.1)10 = 192.77

21
PV – Important Relationship II
a given time period – the higher the interest
rate, the smaller the present value
 For

What is the present value of $500 received in 5
years if the interest rate is 10%? 15%? (both
compounded yearly).


Rate = 10%: PV = 500 / (1.1)5 = 310.46
Rate = 15%; PV = 500 / (1.15)5 = 248.58
22
Figure 4.3
23
How To Find The Return On An
Investment

If you invest $100 today in an account
that compounds interest yearly and in 8
years you have $200, what is the interest
rate?

Rule of 72 = A reasonable estimate for
the required rate to have an investment
double = 72/(i/n) = Number of periods
24
How To Find The Number Of Periods
Required For An Investment

If you want to buy an asset that
cost $100,000 and you have
$50,000 to invest now, at a rate of
12%, compounded annually, how
many years must you wait?
25
Example: Spreadsheet Strategies
 Use
the following formulas for TVM
calculations
FV(rate,nper,pmt,pv)
 PV(rate,nper,pmt,fv)
 RATE(nper,pmt,pv,fv)
 NPER(rate,pmt,pv,fv)

26
Real assets/ Financial assets
Present value and future value are
fundamental to finance
 Most instruments:


Real assets


Buildings, trucks
Financial assets

Debt, Equity, Preferred stock,
derivatives
Can be analyzed using DCF valuation
techniques
27