Present Value On-Line Notes
Accounting and the Time Value of Money
In accounting, the term time value of money is used to indicate a
relationship between time and money—that a dollar received today is worth
more than a dollar promised at some time in the future. A dollar available
today can be invested and interest received on the investment. The concept
of the time value of money is used frequently in financial accounting as well
as business generally.
The foundation of the concept of the time value of money is compound
interest. John Maynard Keynes, the legendary English economist, supposedly
referred to compound interest as magic. Compound interest is computed on
principal and on any interest earned that has not been paid or withdrawn. It
is the return (or growth of) the principal for two or more time periods.
Compounding computes interest not only on the principal but also on the
interest earned to date on that principal, assuming the interest is left on
deposit.
To illustrate these concepts, let us assume that Pacific Corp deposited $250
million in an escrow account with Wells Fargo Bank at the beginning of the
year 2000 as a commitment toward a power plant to be completed December
31, 2003. How much will be on deposit at the end of 4 years if interest is
10% compounded semiannually?
The $250 million placed in escrow will earn interest for eight compounding
periods (4 years with compounding occurring twice per year, hence eight
times) at an interest rate of 5% per compounding period (10% per year, 5%
every six months). The answer may be obtained in several different ways
such as the use of interest tables or an electronic calculator. You should
obtain and learn to use a financial calculator. Nevertheless, we will focus in
these notes on the formulae programmed in financial calculators to allow us
to obtain the results we desire quickly and relatively painlessly.
Understanding the formulae will help you to become a wise user of the
computing power your calculator possesses.
The formula used to determine the future amount to which the deposit at
the beginning of 2000 will accumulate is as follows:
Present amount x (1 + i)n = Future amount
In the above formula, i = the interest rate and n = the number of periods.
The solution (rounding the value for the future amount to five places)would
therefore be:
$250,000,000 x (1 + .05)8 = $369, 365,000.
The deposit of $250,000,000 will accumulate to $369,365,000 by December
31, 2003.
Suppose we reverse the problem. Suppose Pacific Corp wants to know the
amount that must be deposited at the beginning of year 2000 in order to
accumulate the required $369,365,000 by December 31, 2003. We solve
this problem by taking the present value of the amount to be accumulated by
the end of 2003 using the following formula:
Future amount x
1
= Present amount
(1 + i)n
Inserting the numbers into the formula we have:
$369,365,000 x
1
= $250,000,000
8
(1 + .05)
Pacific Corp may decide to deposit $75,000 at the end of each 6-month
period for the next 3 years to accumulate money for the purchase of
equipment. The annual interest rate is 10%. The term annuity can be used
to describe the $75,000 semiannual deposits. An annuity is an equal periodic
payment. An annuity requires that the periodic payment (or receipt) always
be the same amount, the interval between payments (receipts) always be the
same, and the interest be compounded once each interval.
How much would Pacific Corp accumulate by depositing $75,000 every six
months for the next 3 years? The answer can be obtained using the
following formula:
Periodic payment x [(1 + i)n –1]/i = Future value
Inserting the numbers into the formula we have:
$75,000 x [(1 + .05)6 – 1] /.05 = $510,143.25.
Six 6-month deposits of $75,000 earning 5% per period will grow to
$510,143.46.
Now suppose that Pacific Corp wants to deposit an amount today so that a
series of withdrawals of $75,000 at the end of every six months could be
made for three years, again assuming the interest rate to be 10%. This
represents the present value of an ordinary annuity. The present value of an
ordinary annuity is the single sum that, if invested at compound interest now,
would provide for an annuity (a series of withdrawals) for a certain number
of future periods (or the present value of a series of equal rents to be
withdrawn at equal intervals).
The solution to this problem is obtained using the following formula:
Periodic withdrawal x [((1 – 1/(1 + i)n))]/i = Amount to be deposited
Inserting the numbers into the formula we have:
$75,000 x [((1 – 1/(1 + .05)6))]/.05 = $380,676.75
Is this really the correct amount? Try it! Do the following:
The $380,679.75 is deposited on January 1, 2000. At the end of the first
six months of the year 2000, the $380,679.75 will have earned
$380,679.75 x .05 = $19,034 of interest. The total principal and interest is
now $380,679.75 + $19,034 = $399,713.75. Now subtract the first
$75,000 withdrawal from the $399,679.75 available, leaving $324,713.75.
The $324,713.75 earns interest for the last six months of 2000 in the
amount of $324,713.75 x .05 = $16,235.69. The amount available is
$324,713.75 + $16,235.69 = $340,949.44. Subtract the second $75,000
payment from $340,949.44 and $265,949.44 is left. All of this can be
shown in a table called an amortization table as follows:
DATE
PAYMENTS
INTEREST
PRINCIPAL
REDUCTION
Jan. 1, 2001
June 30, 2001
Dec. 31, 2001
June 30, 2002
Dec. 31, 2002
June 30, 2003
Dec. 31, 2003
$75,000
$75,000
$75,000
$75,000
$75,000
$75,000
$19,034.00
$16,235.69
$13,297.47
$10,212.35
$ 6,972.96
$ 3,567.78
$55,966.00
$58,764.31
$61,702.53
$64,787.65
$68,027.04
$71,432.22
PRINCIPAL
BALANCE
$380,679.75
$324,713.75
$265,949.44
$204,246.91
$139,459.26
$ 71,432.22
-
Note: Rounding errors necessitate rounding the interest for the period
ended 12/31/03 from $3,571.61 to $3,567.78, a difference of $3.83.
The amortization table above also could be applied to another situation with
which you might be more familiar. Suppose you bought a house the total
cost of which is $380,679.90. You would need to pay $75,000 every six
months in order to pay off the loan at the end of three years with 10%
annual interest. Although the time and the payment amounts are not
realistic for the purchase of a home, the process would be similar in
determining the amount of your monthly payments required to pay off the
loan over fifteen years or thirty years at a given interest rate.
Notes Payable and Capital Leases
Another application of present value relates to notes payable and capital
leases. Notes payable are written promises to pay a certain sum of money on
a specified future date and may arise from purchases, financing, or other
transactions. A lease is a contractual agreement between an lessor and a
lessee that gives the lessee the right to use specific property, owned by the
lessor, for a specified period of time in return for stipulated, and generally
periodic, cash payments (rents).
Plant assets frequently are purchased on long-term credit contracts through
the use of notes payable. Assets purchased on long-term credit contracts
should be accounted for at the present value of the consideration exchanged
between the contracting parties at the date of the transaction. For
example, a company may purchase on January 2, 2001, a piece of equipment
by paying $5,000 down and issuing a 10% note payable in four equal annual
installments of $2,755 beginning December 31, 2001. Assume that the
market value of the equipment is not readily determinable. The annual
payment amount includes both principal and interest. How would this
transaction be recorded?
The equipment should be valued at the amount of the cash payment plus the
present value of the payments on the note. The historical cost principle
requires that assets be valued at their current cash equivalent price
irrespective of the method of financing. Interest charges should not be
included as part of the recorded cost of the asset except under special
circumstances beyond the scope of the course. Accordingly, the equipment
would be valued as follows:
Cash down payment
Present value of note payments:
$2,755 x [((1 – (1 + i)n))/i]
=
$2,755 x [((1 – (1 + .10)4))/.10
Total cost of equipment
$ 5,000
8,733
$13,733
The entry to record the purchase would be:
Equipment ................................................................13,733
Cash ...............................................................
Note Payable ...............................................
5,000
8.733
The entry to record the first payment on December 31, 2001, would be:
Interest Expense ($8,733 x .10) ....................... 873
Note Payable ........................................................... 1,882
Cash ...............................................................
2,755
The amortization table for this note would be as follows:
DATE
PAYMENTS
INTEREST
PRINCIPAL
REDUCTION
Jan. 1, 2001
Dec. 31, 2001
Dec. 31, 2002
June 30, 2003
Dec. 31, 2004
$2,755
$2,755
$2,755
$2,755
$873
$685
$478
$251
$1,882
$2,070
$2,277
$2,504
PRINCIPAL
BALANCE
$8,733
$6,851
$4,781
$2,504
-0-
A capital lease is really a long-term purchase agreement. The lessor
transfers substantially all the benefits and risks of ownership to the lessee.
The lessor recognizes a sale by removing the asset from the balance sheet
and replacing it with a receivable. The lessee records an asset and a liability
generally equal to the present value of the rental payments. The accounting
by the lessee for the acquisition of the equipment in the previous example
under the same terms under a lease agreement would result in the same
dollar amounts. The account title “notes payable” would be replaced by the
title “obligations under capital leases”.