One-Time Payment Factor
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ScribdTranslationsSingle payment factor
The fundamental factor in economic engineering is the one that determines the amount of
money F that accumulates after n years (or periods), from a single present value P with
compound interest once per year (or per period). Remember that compound interest refers to
interest paid on interest. Therefore, if an amount P is invested at some time t = 0, the amount of
money F1 that will have accumulated in one year from the time of investment at an interest rate
of i percent per year will be:
Reverse the situation to calculate the P-value for a given quantity F occurring n periods in the
future. Just solve the above equation for P.
Below are the meanings of the symbols to be used in single payment financial formulas:
P: Present value of something that is received or paid at time zero.
F: Future value of something that will be received or paid at the end of the evaluated period.
n: Number of periods (months, quarters, years, among others) elapsed between what is
received and what is paid, or the opposite; that is, the period of time necessary to carry out a
transaction. It should be noted that n may or may not be presented continuously depending on
the situation being evaluated.
i : Interest rate recognized per period, either on the investment or the financing obtained; the
interest considered in single payment relationships is compound.
F/P: Find F when P is given. Example: (F/P, 6%, 20) means obtaining the value that when
multiplied by a given P allows finding the future amount of money F, which will be accumulated
in 20 periods, if the interest rate is 6% per period.
Example:
An industrial engineer received a $12,000 bonus that he wants to invest now. He wants to
calculate the equivalent value after 24 years, when he plans to use all the resulting money as a
down payment on a vacation home on an island. Assume a rate of return of 8% per year for
each of the 24 years. a) Determine the amount you can initially pay.
Solution
The symbols and their values are:
P = $12,000 F =? i = 8% per year n = 24 years
The cash flow diagram is the same as in Figure
F = P (1 + i)n = 12 000(1 + 0.08)24
= 12 000(6.341181)
= $76 094.17
The slight difference in the responses is due to rounding error introduced by the tabulated factor
values. An equivalent interpretation of this result is that today's $12,000 will equal $76,094 after
24 years of growing at 8% per year, compounded annually.
Present value factor
The equivalent present value P of a uniform series A of cash flows at the end of the period is
shown in Figure -14a. An expression for the present value can be determined by considering
each value of A as a future value F, calculating its present value with the P/F factor and then
adding the results:
Cash flow diagrams to determine a) P of a uniform series and b) A for a present value.
Present value (PV) is the present value of a future sum of money or cash flows, given a specific
rate of return as of the valuation date. It will always be less than or equal to the future value,
because money has the potential to earn interest, a characteristic known as the time value of
money.
The concept of present value is one of the most fundamental and widespread in the world of
finance. It is the basis for stock and bond prices. Also financial models for banking and
insurance, and valuation of pension funds.
Present value is used in reference to future value. Comparing present value to future value best
illustrates the principle of the time value of money and the need to charge or pay additional
interest rates based on risk.
The most commonly applied present value model uses compound interest. The standard
formula is:
Present value (PV)= FV / (1 + i) ^ n, where
FV is the future amount of money to be discounted.
N is the number of compound periods between the current date and the future date.
i is the interest rate for one compounding period. Interest is applied at the end of a compounding
period, for example, annually, monthly, daily.
The interest rate i is given as a percentage, but is expressed as a number in the formula.
For example, if $1,000 is to be received in five years, and the effective annual interest rate
during this period is 10%, then the present value of this amount is:
PV= $1,000 / (1+0.10) ^ 5= $620.92.
The interpretation is that for an effective annual interest rate of 10%, a person would be
indifferent between receiving $1,000 in five years or $620.92 today.
Example
How much money should be allocated to pay now for $600 guaranteed each year for 9 years, starting
next year, at a rate of return of 16% per year?
Solution
The cash flow diagram. The present value is:
P = $2 763.90
Future value factor
In some cases, the amount of money deposited and the amount of money received after a
specified number of years are known but the interest rate or rate of return is unknown. When a
single payment and a single receipt, a uniform series of payments received, or a conventional
uniform gradient of payments received are involved, the unknown rate can be determined for “i”
by a direct solution of the time value of money equation. However, when there are non-uniform
payments, or many factors, the problem must be solved by a trial and error method, or
numerically.
Example:
Formasa Plastics has large manufacturing plants in Texas and Hong Kong. Your president
wants to know the equivalent future value of a capital investment of $1 million each year for 8
years, starting one year from now. Formasa's capital earns at a rate of 14% per year.
Solution:
The cash flow diagram shows annual payments starting at the end of year 1 and ending in the
year for which you want to calculate the future value. Cash flows are stated in units of $1,000.
The F value at 8 years is
F = 1,000 = 1,000(13.2328) = $13,232.80
Literature
Blank, Leland T. Tarquin Anthony J. Economic Engineering, Mc Graw Hill
Publishing House.
Baca Urbina, Gabriel, Project Evaluation, Mc Graw Hill Publishing House.
Newnan, Donald G. Economic Analysis in Engineering, Mc Graw Hill Publishing
House.