THE TIME VALUE OF MONEY
The most important concept in finance is that of the time value of money. As we will see
in the next section on valuation, the value of a project, a bond, a company, or anything in a
financial sense is a function of the future cash flows that will be realized and the time value of
money. The easiest place to start is with future value since everyone has had a bank account
at one time or another.
Future Value
Place $100.00 in a bank earning 10% annually for three years. At the end of the first year, you
will have $110.00 in the bank account.
$100.00 * (1  .1)  $110.00
Of this amount, $100 is the principal that you put in originally, while the other $10 represents
the interest that you earned. Leaving all of the money in the bank for another year yields:
$110.00 * (1.1)  $121.00
The increase in value of $11 over the amount at the end of the first year includes another $10
(or 10%) of the original principal of $100 that you put in the bank. What is the 11th dollar? It is
interest on the interest that you earned in the first year. Leaving the money in the account for a
third year results in the bank account growing to $133.10 by the end of the third year:
$121.00 * (1.1)  $133.10
Recognizing that the $121 is just $110*(1.1) and that $110 is just $100*(1.1), we can simplify
this process by writing it as
$100.00 * (1.1)3  $133.10
The term (1.1)3 is referred to as the Future Value Interest Factor, in this case at 10% for 3
years. In general the Future Value Interest Factor can be defined as
FVIFk %, n  (1  k )n
and the relationship between Present Value and Future Value is
PV * FVIFk %, n  FV
If we look at the table at the back of the text entitled Future Value of $1 in the 10% column and
the third period row, we will find the factor 1.331 – this is how the entire table is made up.
Present Value
Generally, we’re not as much interested in what a certain amount of money today will be
worth in the future as we are interested in knowing what a future amount of money is worth
today. One reason, of course is that we know what a dollar will buy today. Thus, with the
exception of insurance companies (who need to know how much they will have to pay off
policies at a future date) and certain pension funds (who need to have enough money for
retirees in the future), we are generally more interested in Present Values. Present Value is the
amount of money that would have to be placed in a bank today earning k% in order to have the
future amount in n years. Determining the present value is called discounting. Discounting is
just the opposite of compounding, just like addition is the opposite of subtraction and
multiplication is the opposite of division. Rearranging the preceding equation to solve for the
Present Value, we obtain
PV  FV *
1
FVIFk , n
Or
PV  FV * PVIFk , n
and
PVIFk , n 
1
FVIFk , n
What is the present value of $133.10 three years from now if you can earn 10% interest in the
bank? The answer, of course, is $100.00 as we just saw with future value. Mathematically, it
would be determined as
PV  FV *
1
FVIF19%, 3
 FV * PVIF10%, 3
 $133.10 * .7513
 $100.00
If you look at the table in the back of the text entitled Present Value of $1 in the 10% column
and the third period row, you will see the factor 0.7513 which is just the reciprocal of the
FVIF10%,3 of 1.331 and is how the entire table is calculated.
Annuities
An annuity is a series of equal payments that occurs every period. Suppose we wanted
to determine the present value of an annuity of $100 per year for three years beginning in one
year. Using the present value interest factors we could calculate the present value of each
$100 and then add all of the present values together:
0
1
2
3
100
100
100
.9091
90.91
.8264
82.64
.7513
75.13
--------248.68
The value of an annuity of $100 per year for three years when discounted at 10% is $248.68 in
today’s dollar terms. Looking at our calculation mathematically, we could factor out the
common element of $100 and simplify the calculation
100 (.9091) + 100 (.8264) + 100 (.7513)
= 100 (.9091 + .8264 + .7513)
= 100 (2.4868)
= 100 * PVIFA10%,3
As indicated, the sum of the individual present value factors is what we call the Present Value
Interest Factor of an Annuity. The table in the back of the text entitled Present Value of an
Annuity of $1 is constructed simply by adding successive Present Value factors for successive
years of an annuity. Notice that the PVIFA for a one-year annuity is the same as the PVIF for
one year since they are the same. You will also notice in the annuity table that the factor for
10% and 3 periods is 2.4869 which is just slightly different than what we got since we have a
little rounding error involved.
How would you explain to someone what you mean by saying that the present value of an
annuity of $100 per year for three years at 10% is $248.68? What do you mean by present
value? Remember that it is just the amount that you would have to put in the bank today in
order to have those future cash flows. Suppose we put $248.68 in a bank today that pays 10%
interest:
$248.68 Initial deposit
24.87 Interest for the first year
$273.55 Amount at end of first year
- 100.00 Withdrawal at end of first year
$ 173.55 Remaining balance at end of first year
17.36 Interest for the second year
$ 190.91 Amount at end of second year
- 100.00 Withdrawal at end of second year
$ 90.91 Remaining balance at end of second year
9.09 Interest for the third year
$100.00 Amount at end of third year
- 100.00 Withdrawal at end of third year
-0- Account is empty
Notice that the value of the three-year annuity of $100 is only $248.68 today. It is NOT
$300 because we would be ignoring the time value of money. Think about the lottery. It is paid
in 25 annual payments with the first one being paid immediately and equal payments in each of
the following 24 years. The winner of a $25 million jackpot would, technically, receive $25
million but it would not be worth that much. In fact, if you were the winner and chose the cash
option for payment, you would only receive about $13 million today. This is equivalent to
discounting the 25 payments of $1 million to a present value using about a 6% rate of interest,
probably about right given the cost of debt to the state when it has to borrow. The same thing
is true with these athletic contracts where $50 million is paid for a five-year contract. The fact
is, the player may receive $5 million per year during the five years in which they are playing,
while the balance is paid out over the following ten years. It only adds up to $50 million if you
ignore the time value of money (I’d still like 10% as the agent).
Would you rather have $100 at the end of each of the next three years or $100 at the
beginning of each of the next three years? At the beginning, of course. Why? Because you
can start earning interest sooner.
0
100
1
2
100
100
3
.9091
90.91
.8264
82.64
--------273.55
273.55 - 248.68 = 24.87
The difference in value of $24.87 is the present value of the additional interest ($10 per year)
that can be earned. Since we have essentially moved the $100 in year 3 to today, we can earn
an extra $10 per year of interest on that amount. What is the present value of $10 of additional
interest each year for three years?
$10*(2.4868) = $24.87
When an annuity is paid at the beginning of each period, it is referred to as an annuity due.
Feeling pretty comfortable with annuities? What is the present value of an annuity of
$100 per year beginning in three years?
0
1
2
3
4
100
100
5
100
Solution #1
0
0'
2
1
1'
3
2'
4
3'
5
100
100
100
2.4869
$ 248.69
.8264
$ 205.52
Would you rather have $205.52 today or $248.69 in two years? Or $100 at the end of years 3,
4 and 5? The answer is “they are all the same thing”. But don’t call it $300 because you are
ignoring the time value of money and comparing apples with oranges!
Solution #2
0
1
2
3
100
4
5
100
100
3.7908 - 1.7355 = 2.0553
$ 205.53
PVIFA5 - PVIFA2
This approach is taking advantage of the fact that we know the annuity factors are just the
summation of the individual present value factors. Thus, by subtracting the 2-year factor from
the 5-year factor we are isolating the years 3 – 5.
Periods shorter than 1 year
Would you rather have $100 at the end of each of the next three years or $50 every six
months for the next three years? Take the $50 every six months. Why? Because you can
start earning interest sooner.
0
1
50
50
$ 253.79
PVIFA5%,6 = 5.0757
2
50
50
3
50
50
Note now, however, that our periods are no longer years but half-years. We therefore
had to convert our interest rate to the same basis. Unless otherwise stated, an interest rate is
virtually always expressed in annual terms. Thus, for half-year periods we need to divide our
10% annual rate of interest in half to match the number of half-year periods in one year.
Similarly, quarterly payments would require we divide the annual interest rate by four while
monthly payments would have us divide the per annum rate by twelve.
Compound Interest vs. Simple Interest
1½% * 12 = 18% APR (simple interest)
(1.015)12 = 1.196 or 19.6% (compound interest)
Finding Growth Rates and Lengths of Time
How long will it take to double your money if you can earn 8% interest?
0
?
1,000
2,000
Utilizing the Present Value Factors,
PV = FV * PVIFk,n
1,000 = 2,000 * PVIF8%,n
.5 = PVIF8%,n
Looking at the Present Value of $1 table, we can see that there are several elements that are
equal to .5000; the question is “Which one is the right one?” What we know, however, is that
we should be looking in the 8% column since we are earning an 8% rate of interest. Looking
down the column we find the factor .5000 in only one row which we can now identify as
belonging to period 9.
n=9
Thus, it will take 9 years to double our money if we are earning an 8% rate of interest.
The tables can similarly be used to determine interest rates and rates of growth.
Moreover, the compound average rate of growth that is determined by such an approach is the
only correct method of doing so.
Arithmetic Average vs. Geometric Average
To illustrate the proper means of determining an average rate of return or average
growth rate, consider putting $1,000 into a mutual fund that had the following year-end values:
Year 0
1,000
Year 1
2,000
Year 2
1,000
+100%
- 50%
50%/2 = 25% average per year
In the first year, we doubled our money, or earned a +100% rate of interest. In the
second year, we lost half of our money, or earned a –50% rate of return. When individual
growth rates are added up and divided by the number of years, we obtain what is called an
arithmetic average. In this example, it implies that we earned, on average, a 25% rate of
return. It is obvious that we did not. In fact, we averaged a zero rate of return since we ended
up with the same amount of money that we started with. This does not occur when Present
Value/Future Value factors are employed.
PV * FVIF
= FV
1,000 * (1 + k)2 = 1,000
1 + k = 1.00 or 0%
This type of average is referred to as a geometric average and indicates the true rate of return
that was earned. Although other methods of determining the geometric average exist, the use
of Present Value/Future Value factors is, by far, the simplest means of doing so.