Basic Time Value of Money Principles
The fundamental idea supporting
the concept of time value of money is actually very simple. Simply put, a
dollar received today is worth more than a dollar received a year from now.
It is worth more because the dollar can be invested and allowed to earn interest
for that year. At the end of the year, the dollar received today is worth
the original dollar plus the interest it has earned during the year.
Solving time value of money
problems involve either compounding or discounting
either lump sums (or single payments) or annuities (a series of equal payments).
Special forms of these four
In financial matters, the word compounding
means the deposit of a dollar sum into a bank account or other
interest-bearing investment, the drawing of interest periodically on the
invested amount, and the further drawing of interest on the interest received in
prior periods. The compounding period is that period of time when
the interest is paid. That interest paid the first period now earns
interest in subsequent periods. The shorter the compounding period, the
greater the accumulation over time.
For example, if $10,000 is
deposited into a bank account that pays 6% interest compounded annualy
will accumulate to $10, 600 by the end of the year ($10,000 + $600
interest). If instead, the bank pays that same interest rate compounded monthly,
the account will accumulate $10, 616.78. Compounding monthly earned $16.78
more interest. This is because the $50 interest paid from the very first
month (6%/12*$10,000=$50) then earns interest the remaining 11 months.
The second month the account earns $50.25 ($10,050*6%/12). Each month the
interest earned is added to the account and earns interest in the subsequent
THE FUTURE VALUE OF A LUMP
SUM (SINGLE PAYMENTS)
If you deposit $1,000 in an
investment yielding 8 percent, during the first year you will make $80 in
interest. Thus, the total value of your investment after the passage of a year
will be $1,080.
If you leave the $1,080 in the
investment, how much will the value of your investment be at the end of two
years? You will, of course, have your $1,080 from the first year, plus 8 percent
on the $1,080 during the second year. For the second year you would receive
$86.40 in interest to add to your $1,080. The total value of your investment at
the end of two years would therefore be $1,166.40.
Similarly, for the third year,
your interest would be $93.31, and the total value of your investment at the end
of the third year would be $1,000 plus $86.40 plus $93.31 for a total of
As can be seen, the process of
compounding involves investment of a given sum, drawing of interest on that sum
for a period, and then drawing of interest on the original sum plus the interest
accumulated in prior periods. The general formula for compounding is:
THE FUTURE VALUE OF AN
ANNUITY (SERIES OF EQUAL PAYMENTS)
Suppose that, instead of
depositing $1,000 at the beginning of a period, you deposit $1,000 at the end of
each year for three years. What would be the total value of your investment at
the end of the three years, drawing interest at 8 percent per year? The process
of compounding in this case is similar to that of the example above, except that
the payments are usually made at the end of each year, and the values thereof
are added together. Thus, a $1,000 payment would be made at the end of year one,
another $1,000 payment would be made at the end of year two, and another payment
would be made at the end of year three.
The payment at the end of year
one would be worth $1,000; it would draw interest of $80 during the second year,
and $1,000 would be added to it, making a total value at the end of year two of
$2,080. $166.40 would accumulate in interest during the third year, and $1,000
would be added at the end of the third year, making a total value of $3,246.40.
This amount, then, is the future value of a series of three payments of $1,000
each, accumulating interest at 8 percent annually. Such a series of payments is
called a level annuity.
The general formula for the
accumulation of a given amount each period is easily derived using Equation 1.
The derivation of the general formula may be seen in Equation 2.
SINKING FUND ACCUMULATION
A sinking fund is an account
into which is deposited money every period so that the account will grow to the
desired amount in the future. The future value of a series of equal
payments was the unknown that had to be calculated in the previous compounding
situation. Now suppose that you wish to invest a certain amount which over a
three-year period will accumulate to $1,000. How much would you have to invest
each year for three years at 8 percent interest to reach the desired figure?
This question is literally and
mathematically the reciprocal of the prior question that asks, "How much is
the future value of a series of known payments?" To find how much
must be invested or put into a sinking fund drawing interest at 8 percent, one
need only take the reciprocal of the future value. Thus, if $1000 were deposited
at the end of the year for three years into an account paying 8% per year, the
account would accumulate to $3,246.40. The reciprocal in this example
would be $1,000 divided by $3,246.40, or one divided by 3.2464 which would yield
.30803. The quotient must then be multiplied by 1,000, since it is the sum
which we wish to accumulate. The resulting answer is approximately
$308.03, which means that if this amount were invested at the end of each three
years, at the end of the three year period the value of the accumulated total
would be $1,000. The formula for the Sinking fund (SF) is easily found by
solving Equation 2 for the payment (PMT) as seen in Equation 3.
Discounting may be regarded as
the reciprocal process of compounding. In compounding we seek to find the future
value of a payment or series of payments, or the payments necessary to
accumulate to a desired future value. In discounting we know the future
payments, or the total amount to be paid off, and we must calculate the present
value of those future payments or of the individual payments necessary to
amortize (pay off) a present value.
THE PRESENT VALUE OF A
LUMP SUM (SINGLE PAYMENT)
The present value of a lump sum
payment due at some time in the future is the reciprocal of the future value of
a lump sum payment invested today. Thus, its formula is shown in Equation 4
The present value of $1,000 due
at the end of the fifth year is $680.58. Or to state it slightly differently, if
an investor pays $680.58 today for the right to obtain $1,000 in five years, (s)he
will have made an 8 percent return on the investment.
THE PRESENT VALUE OF AN
ANNUITY (SERIES OF EQUAL PAYMENTS)
A series of equal payments due
in the future is also known as a level annuity. The formula for the present
value of an annuity is shown in Equation 5. While not illustrated, this formula
is also easily derived using the same algebraic technique used in Equation 2.
The algebraic solution to Equation 5 is left as a student exercise.
This calculation of an
annuity's present value is extremely important because it demonstrates that the
value of any income stream, whether the payments are equal or whether they vary,
can be obtained by summing the present values of the individual payments. Thus,
the present value of an annuity of $1.00 per year for 5 years is the sum of the
present values of annual payments. These amounts are shown below and summed:
If the amount to be invested
annually is $1,000 the present value of the annuity is therefore 3.992710 x
1,000 = $3,992.71.
INSTALLMENT TO AMORTIZE A
The periodic payments necessary
to repay a present value, such as an amount of borrowed money, constitute the
amortization or repayment of the loan. The principal amount borrowed is nothing
more than the present value of the series of periodic payments discounted at the
interest rate. The amortization installment formula shown in Equation 6 is the
reciprocal of the present value of an annuity formula.
PAYMENTS IN ADVANCE
Some income streams provide
payments at the beginning of the periods. For example, apartment rentals are
often paid at the beginning of the rental period. Such streams can be compounded
or discounted easily by recognizing that the beginning of any period is the end
of the prior period. For example, an ordinary annuity of $1,000.00 for five
years, whose value discounted at 8 percent is $3,992.71 can be converted to
payments in advance by shifting all payments backward one year. The resulting
annuity requires a payment at time zero (now), which is worth 100 percent of the
amount paid, plus payments at the end of each year (beginning of the next year)
for four years. The value of the annuity with payments in advance is therefore
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