Transcript The Time Value of Money

The Time Value of Money
Timothy R. Mayes, Ph.D.
FIN 3300: Chapter 5
1
What is Time Value?



We say that money has a time value because that
money can be invested with the expectation of
earning a positive rate of return
In other words, “a dollar received today is worth
more than a dollar to be received tomorrow”
That is because today’s dollar can be invested so that
we have more than one dollar tomorrow
2
The Terminology of Time Value




Present Value - An amount of money today, or the
current value of a future cash flow
Future Value - An amount of money at some future
time period
Period - A length of time (often a year, but can be a
month, week, day, hour, etc.)
Interest Rate - The compensation paid to a lender (or
saver) for the use of funds expressed as a percentage
for a period (normally expressed as an annual rate)
3
Abbreviations
PV - Present value
 FV - Future value
 Pmt - Per period payment amount
 N - Either the total number of cash flows or
the number of a specific period
 i - The interest rate per period

4
Timelines
A timeline is a graphical device used to clarify the
timing of the cash flows for an investment
Each tick represents one time period
PV
0
Today
FV
1
2
3
4
5
5
Calculating the Future Value

Suppose that you have an extra $100 today that you
wish to invest for one year. If you can earn 10% per
year on your investment, how much will you have in
one year?
-100
?
0
1
2
3
4
5
FV1  1001  010
.   110
6
Calculating the Future Value (cont.)

Suppose that at the end of year 1 you decide to
extend the investment for a second year. How much
will you have accumulated at the end of year 2?
0
-110
?
1
2
3
4
5
FV2  1001  010
. 1  010
.   121
or
2
FV2  1001  010
.   121
7
Generalizing the Future Value

Recognizing the pattern that is developing,
we can generalize the future value
calculations as follows:
FVN  PV1  i

N
If you extended the investment for a third
year, you would have:
FV3  1001  010
.   13310
.
3
8
Compound Interest


Note from the example that the future value is
increasing at an increasing rate
In other words, the amount of interest earned each
year is increasing
• Year 1: $10
• Year 2: $11
• Year 3: $12.10

The reason for the increase is that each year you are
earning interest on the interest that was earned in
previous years in addition to the interest on the
original principle amount
9
Compound Interest Graphically
4000
3833.76
3500
5%
Future Value
3000
10%
15%
2500
20%
2000
1636.65
1500
1000
672.75
500
265.33
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Years
10
The Magic of Compounding


On Nov. 25, 1626 Peter Minuit, a Dutchman, reportedly
purchased Manhattan from the Indians for $24 worth of beads
and other trinkets. Was this a good deal for the Indians?
This happened about 371 years ago, so if they could earn 5% per
year they would now (in 1997) have:
$1,743,577,261.65 = 24(1.05) 371

If they could have earned 10% per year, they would now have:
$54,562,898,811,973,500.00 = 24(1.10) 371
That’s about 54,563 Trillion dollars!
11
The Magic of Compounding (cont.)





The Wall Street Journal (17 Jan. 92) says that all of New York
city real estate is worth about $324 billion. Of this amount,
Manhattan is about 30%, which is $97.2 billion
At 10%, this is $54,562 trillion! Our U.S. GNP is only around $6
trillion per year. So this amount represents about 9,094 years
worth of the total economic output of the USA!
At 5% it seems the Indians got a bad deal, but if they earned
10% per year, it was the Dutch that got the raw deal
Not only that, but it turns out that the Indians really had no
claim on Manhattan (then called Manahatta). They lived on
Long Island!
As a final insult, the British arrived in the 1660’s and
unceremoniously tossed out the Dutch settlers.
12
Calculating the Present Value
So far, we have seen how to calculate the
future value of an investment
 But we can turn this around to find the
amount that needs to be invested to achieve
some desired future value:

PV 
FVN
1  i
N
13
Present Value: An Example

Suppose that your five-year old daughter has just
announced her desire to attend college. After some
research, you determine that you will need about
$100,000 on her 18th birthday to pay for four years of
college. If you can earn 8% per year on your
investments, how much do you need to invest today
to achieve your goal?
PV 
100,000
. 
108
13
 $36,769.79
14
Annuities


An annuity is a series of nominally equal payments
equally spaced in time
Annuities are very common:
•
•
•
•

Rent
Mortgage payments
Car payment
Pension income
The timeline shows an example of a 5-year, $100
annuity
0
100
100
100
100
100
1
2
3
4
5
15
The Principle of Value Additivity
How do we find the value (PV or FV) of an
annuity?
 First, you must understand the principle of
value additivity:

• The value of any stream of cash flows is equal to
the sum of the values of the components

In other words, if we can move the cash flows
to the same time period we can simply add
them all together to get the total value
16
Present Value of an Annuity

We can use the principle of value additivity to find
the present value of an annuity, by simply summing
the present values of each of the components:
N
PVA 
 1  i
t 1
Pmt t
t

Pmt 1
1  i
1

Pmt 2
1  i
2
  
Pmt N
1  i
N
17
Present Value of an Annuity (cont.)

Using the example, and assuming a discount rate of
10% per year, we find that the present value is:
PVA 
100
. 
110
1

100
. 
110
2

100
. 
110
3

100
. 
110
4

100
. 
110
5
 379.08
62.09
68.30
75.13
82.64
90.91
379.08
0
100
100
100
100
100
1
2
3
4
5
18
Present Value of an Annuity (cont.)
Actually, there is no need to take the present
value of each cash flow separately
 We can use a closed-form of the PVA equation
instead:

N
PVA 
 1  i
t 1
Pmt t
t
1  1

N

1  i 

 Pmt 

i




19
Present Value of an Annuity (cont.)

We can use this equation to find the present
value of our example annuity as follows:
1  1

5

110
.  

PVA  Pmt 
  379.08
010
.





This equation works for all regular annuities,
regardless of the number of payments
20
The Future Value of an Annuity

We can also use the principle of value additivity to
find the future value of an annuity, by simply
summing the future values of each of the
components:
N
FVA 

Pmt t 1  i
Nt
 Pmt 1 1  i
N 1
 Pmt 2 1  i
N 2
     Pmt N
t 1
21
The Future Value of an Annuity (cont.)

Using the example, and assuming a discount rate of
10% per year, we find that the future value is:
FVA  100110
.   100110
.   100110
.   100110
.   100  610.51
4
3
2
1
146.41
133.10
121.00
110.00
0
100
100
100
100
100
1
2
3
4
5
}
= 610.51
at year 5
22
The Future Value of an Annuity (cont.)

Just as we did for the PVA equation, we could
instead use a closed-form of the FVA
equation:
N
FVA 
 Pmt 1  i
t
t 1

Nt
 1  i N  1 

 Pmt 
i


This equation works for all regular annuities,
regardless of the number of payments
23
The Future Value of an Annuity (cont.)

We can use this equation to find the future
value of the example annuity:
5
 110
.   1
  610.51
FVA  100
.
 010

24
Annuities Due


Thus far, the annuities that we have looked at begin
their payments at the end of period 1; these are
referred to as regular annuities
A annuity due is the same as a regular annuity,
except that its cash flows occur at the beginning of
the period rather than at the end
100
5-period Annuity Due
5-period Regular Annuity
0
100
100
100
100
100
100
100
100
100
1
2
3
4
5
25
Present Value of an Annuity Due



We can find the present value of an annuity due in
the same way as we did for a regular annuity, with
one exception
Note from the timeline that, if we ignore the first cash
flow, the annuity due looks just like a four-period
regular annuity
Therefore, we can value an annuity due with:
PVAD
1  1

 N 1


1  i

 Pmt 
  Pmt
i




26
Present Value of an Annuity Due (cont.)

Therefore, the present value of our example
annuity due is:
PVAD

1  1

 51

110
.  

 100
  100  416.98
010
.




Note that this is higher than the PV of the,
otherwise equivalent, regular annuity
27
Future Value of an Annuity Due

To calculate the FV of an annuity due, we can
treat it as regular annuity, and then take it
one more period forward:
FVAD
 1  i N  1 
1  i
 Pmt 
i


Pmt
Pmt
Pmt
Pmt
Pmt
0
1
2
3
4
5
28
Future Value of an Annuity Due (cont.)

The future value of our example annuity is:
FVAD

5
 110
.   1
110
 100
.   67156
.
.
 010

Note that this is higher than the future value
of the, otherwise equivalent, regular annuity
29
Deferred Annuities
A deferred annuity is the same as any other
annuity, except that its payments do not
begin until some later period
 The timeline shows a five-period deferred
annuity

0
1
2
100
100
100
100
100
3
4
5
6
7
30
PV of a Deferred Annuity


We can find the present value of a deferred annuity
in the same way as any other annuity, with an extra
step required
Before we can do this however, there is an important
rule to understand:
When using the PVA equation, the resulting PV is
always one period before the first payment occurs
31
PV of a Deferred Annuity (cont.)
To find the PV of a deferred annuity, we first
find use the PVA equation, and then discount
that result back to period 0
 Here we are using a 10% discount rate

PV2 = 379.08
PV0 = 313.29
0
0
0
100
100
100
100
100
1
2
3
4
5
6
7
32
PV of a Deferred Annuity (cont.)
Step 1:
Step 2:
1  1

5

110
.  

PV2  100
  379.08
010
.




PV0 
379.08
. 
110
2
 313.29
33
FV of a Deferred Annuity
The future value of a deferred annuity is
calculated in exactly the same way as any
other annuity
 There are no extra steps at all

34
Uneven Cash Flows
Very often an investment offers a stream of
cash flows which are not either a lump sum
or an annuity
 We can find the present or future value of
such a stream by using the principle of value
additivity

35
Uneven Cash Flows: An Example (1)

Assume that an investment offers the following cash
flows. If your required return is 7%, what is the
maximum price that you would pay for this
investment?
100
0
PV 
200
1
300
2
100
. 
107
1

3
200
. 
107
2

4
300
. 
107
3
5
 513.04
36
Uneven Cash Flows: An Example (2)

Suppose that you were to deposit the following
amounts in an account paying 5% per year. What
would the balance of the account be at the end of the
third year?
300
0
500
1
2
700
3
4
5
FV 300105
.   500105
.   700  1,555.75
2
1
37
Non-annual Compounding




So far we have assumed that the time period is equal
to a year
However, there is no reason that a time period can’t
be any other length of time
We could assume that interest is earned semiannually, quarterly, monthly, daily, or any other
length of time
The only change that must be made is to make sure
that the rate of interest is adjusted to the period
length
38
Non-annual Compounding (cont.)

Suppose that you have $1,000 available for
investment. After investigating the local banks, you
have compiled the following table for comparison. In
which bank should you deposit your funds?
Bank
First National
Second National
Third National
Interest Rate
10%
10%
10%
Compounding
Annual
Monthly
Daily
39
Non-annual Compounding (cont.)



To solve this problem, you need to determine which
bank will pay you the most interest
In other words, at which bank will you have the
highest future value?
To find out, let’s change our basic FV equation
slightly:
Nm
i

FV  PV 1  

m
In this version of the equation ‘m’ is the number of
compounding periods per year
40
Non-annual Compounding (cont.)

We can find the FV for each bank as follows:
First National Bank:
Second National Bank:
Third National Bank:
FV  1,000110
.   1100
,
1
010
. 

FV  1,000 1 


12 
12
010
. 

FV  1,000 1 


365 
365
 1104
, .71
 110516
, .
Obviously, you should choose the Third National Bank
41
Continuous Compounding



There is no reason why we need to stop increasing
the compounding frequency at daily
We could compound every hour, minute, or second
We can also compound every instant (i.e.,
continuously):
F  Pe

rt
Here, F is the future value, P is the present value, r is
the annual rate of interest, t is the total number of
years, and e is a constant equal to about 2.718
42
Continuous Compounding (cont.)

Suppose that the Fourth National Bank is offering to
pay 10% per year compounded continuously. What
is the future value of your $1,000 investment?
F  1,000e


0 .10 1
 110517
, .
This is even better than daily compounding
The basic rule of compounding is: The more frequently
interest is compounded, the higher the future value
43
Continuous Compounding (cont.)

Suppose that the Fourth National Bank is offering to
pay 10% per year compounded continuously. If you
plan to leave the money in the account for 5 years,
what is the future value of your $1,000 investment?
F  1,000e
0.10 5
 1,648.72
44