Understanding the Time Value of Money
Congratulations!!! You have won a cash prize! You have
two payment options:
A. Receive $10,000 now
OR
B. Receive $10,000 in three years.
Okay, the above offer is hypothetical, but play along
with me here... Which option would you choose?
What Is Time Value?
If you're like most people, you would choose to receive
the $10,000 now. After all, three years is a long time
to wait. Why would any rational person defer payment into
the future when he or she could have the same amount of
money now? For most of us, taking the money in the present
is plain instinctual. So at the most basic level, the
time value of money demonstrates that, all things being
equal, it is better to have money now rather than later.
But why is this? A $100 bill has the same value as a
$100 bill one year from now, doesn't it? Actually, although
the bill is the same, you can do much more with the money
if you have it now: over time you can earn more interest
on your money.
Back to our example: by receiving $10,000 today, you
are poised to increase the future value of your money
by investing and gaining interest over a period of time.
For option B, you don't have time on your side, and the
payment received in three years would be your future value.
To illustrate, we have provided a timeline:
If you are choosing option A, your future value will
be $10,000 plus any interest acquired over the three years.
The future value for option B, on the other hand, would
only be $10,000. But stay tuned to find out how to calculate
exactly how much more option A is worth, compared to option
B.
Future Value Basics
If you choose option A and invest the total amount at
a simple annual rate of 4.5%, the future value of your
investment at the end of the first year is $10,450, which
is calculated by multiplying the principal amount of $10,000
by the interest rate of 4.5% and then adding the interest
gained to the principal amount:
Future value of investment at end of first year:
= ($10,000 x 0.045) + $10,000
= $10,450
You can also calculate the total amount of a one-year
investment with a simple manipulation of the above equation:
Original equation: ($10,000 x 0.045) + $10,000 = $10,450
Manipulation: $10,000 x [(1 x 0.045) + 1] = $10,450
Final equation: $10,000 x (0.045 + 1) = $10,450
The manipulated equation above is simply a removal of
the like-variable $10,000 (the principal amount) by dividing
the entire original equation by $10,000.
If the $10,450 left in your investment account at the
end of the first year is left untouched and you invested
it at 4.5% for another year, how much would you have?
To calculate this, you would take the $10,450 and multiply
it again by 1.045 (0.045 +1). At the end of two years,
you would have $10,920:
Future value of investment at end of second year:
= $10,450 x (1+0.045)
= $10,920.25
The above calculation, then, is equivalent to the following
equation:
Future Value = $10,000 x (1+0.045) x (1+0.045)
Think back to math class in junior high, where you learned
the rule of exponents, which says that the multiplication
of like terms is equivalent to adding their exponents.
In the above equation, the two like terms are (1+0.045),
and the exponent on each is equal to 1. Therefore, the
equation can be represented as the following:
We can see that the exponent is equal to the number of
years for which the money is earning interest in an investment.
So, the equation for calculating the three-year future
value of the investment would look like this:
This calculation shows us that we don't need to calculate
the future value after the first year, then the second
year, then the third year, and so on. If you know how
many years you would like to hold a present amount of
money in an investment, the future value of that amount
is calculated by the following equation:
Present Value Basics
If you received $10,000 today, the present value would
of course be $10,000 because present value is what your
investment gives you now if you were to spend it today.
If $10,000 were to be received in a year, the present
value of the amount would not be $10,000 because you do
not have it in your hand now, in the present. To find
the present value of the $10,000 you will receive in the
future, you need to pretend that the $10,000 is the total
future value of an amount that you invested today. In
other words, to find the present value of the future $10,000,
we need to find out how much we would have to invest today
in order to receive that $10,000 in the future.
To calculate present value, or the amount that we would
have to invest today, you must subtract the (hypothetical)
accumulated interest from the $10,000. To achieve this,
we can discount the future payment amount ($10,000) by
the interest rate for the period. In essence, all you
are doing is rearranging the future value equation above
so that you may solve for P. The above future value equation
can be rewritten by replacing the P variable with present
value (PV) and manipulated as follows:
Let's walk backwards from the $10,000 offered in option
B. Remember, the $10,000 to be received in three years
is really the same as the future value of an investment.
If today we were at the two-year mark, we would discount
the payment back one year. At the two-year mark, the present
value of the $10,000 to be received in one year is represented
as the following:
Present value of future payment of $10,000 at end of
year two
Note that if today we were at the one-year mark, the
above $9,569.38 would be considered the future value of
our investment one year from now.
Continuing on, at the end of the first year we would
be expecting to receive the payment of $10,000 in two
years. At an interest rate of 4.5%, the calculation for
the present value of a $10,000 payment expected in two
years would be the following:
Present value of $10,000 in one year
Because of the rule of exponents, we don't have to calculate
the future value of the investment every year counting
back from the $10,000 investment at the third year. We
could put the equation more concisely and use the $10,000
as FV. So, here is how you can calculate today's present
value of the $10,000 expected from a three-year investment
earning 4.5%:
So the present value of a future payment of $10,000 is
worth $8,762.97 today if interest rates are 4.5% per year.
In other words, choosing option B is like taking $8,762.97
now and then investing it for three years. The equations
above illustrate that option A is better not only because
it offers you money right now but because it offers you
$1,237.03 ($10,000 - $8,762.97) more in cash. Furthermore,
if you invest the $10,000 that you receive from option
A, your choice gives you a future value that is $1,411.66
($11,411.66 - $10,000) greater than the future value of
option B.
Present Value of a Future Payment
Let's add a little spice to our investment knowledge.
What if the payment in three years is more than the amount
you'd receive today? Say you could receive either $15,000
today or $18,000 in four years. Which would you choose?
The decision is now more difficult. If you choose to receive
$15,000 today and invest the entire amount, you may actually
end up with an amount of cash in four years that is less
than $18,000. You could find the future value of $15,000,
but since we are always living in the present, let's find
the present value of $18,000 if interest rates are currently
4%. Remember that the equation for present value is the
following:
In the equation above, all we are doing is discounting
the future value of an investment. Using the numbers above,
the present value of an $18,000 payment in four years
would be calculated as the following:
Present Value
From the above calculation we now know our choice is
between receiving $15,000 or $15,386.48 today. Of course
we should choose to postpone payment for four years.
Conclusion
These calculations demonstrate that time literally is
money--the value of the money you have now is not the
same as it will be in the future and vice versa. So, it
is important to know how to calculate the time value of
money so that you can distinguish between the worth of
investments that offer you returns at different times.
Shauna Croome (Investopedia.com)
August 27th, 2003
