small fraction at the beginning of the loan repayment period, but it increases
slowly (and nonlinearly) until it becomes the dominant term toward the end.
Conversely, the interest payment term is proportionately very large at the beginning of the loan repayment period and decreases progressively until it becomes
small at the end.
This formula is predicated on the idea of making uniform payments, as is often
done with mortgages, hence the name. Naturally, there are many other ways of
repaying loans, and expressions could be derived for other situations too, if
desired.
An offshoot of the mortgage formula is the concept of an “average rate of
interest,” sometimes used in economic calculations to reflect the average cost of
long-term debt. See Appendix 11 for this concept.
Present Value of a Uniform Series
Logically, this represents the reciprocal of the loan repayment schedule.
P ¼ A
1 þ r
ð
Þ
n À 1
r 1 þ r
ð
Þ
n
(22)
The multiplier is also called the (P/A) factor or the “present worth factor for a
uniform series,” (PWFUS).
P ¼ A P=A
ð
Þ
r
n
(23)
Or
P ¼ A PWFUS
ð
Þ
r
n
(24)
The above expressions represent most of the situations encountered in economic
evaluations. There are other expressions available for other situations (e.g., when
the series of payments increases or decreases progressively at a uniform rate – the
“gradient” method), but these are far rarer and need not be discussed here.
Appendix 9: Progressions in Economic Analyses
Though trivial, the concept of arithmetic and geometric progressions is useful in
economic calculations. These concepts are outlined as a refresher.
Arithmetic Progressions
An arithmetic progression (or an arithmetic series) is a series of terms such that each
is made up of the previous term plus an additive (positive or negative) constant.
770
D.S.J. Jones and P.R. Pujado ´
slowly (and nonlinearly) until it becomes the dominant term toward the end.
Conversely, the interest payment term is proportionately very large at the beginning of the loan repayment period and decreases progressively until it becomes
small at the end.
This formula is predicated on the idea of making uniform payments, as is often
done with mortgages, hence the name. Naturally, there are many other ways of
repaying loans, and expressions could be derived for other situations too, if
desired.
An offshoot of the mortgage formula is the concept of an “average rate of
interest,” sometimes used in economic calculations to reflect the average cost of
long-term debt. See Appendix 11 for this concept.
Present Value of a Uniform Series
Logically, this represents the reciprocal of the loan repayment schedule.
P ¼ A
1 þ r
ð
Þ
n À 1
r 1 þ r
ð
Þ
n
(22)
The multiplier is also called the (P/A) factor or the “present worth factor for a
uniform series,” (PWFUS).
P ¼ A P=A
ð
Þ
r
n
(23)
Or
P ¼ A PWFUS
ð
Þ
r
n
(24)
The above expressions represent most of the situations encountered in economic
evaluations. There are other expressions available for other situations (e.g., when
the series of payments increases or decreases progressively at a uniform rate – the
“gradient” method), but these are far rarer and need not be discussed here.
Appendix 9: Progressions in Economic Analyses
Though trivial, the concept of arithmetic and geometric progressions is useful in
economic calculations. These concepts are outlined as a refresher.
Arithmetic Progressions
An arithmetic progression (or an arithmetic series) is a series of terms such that each
is made up of the previous term plus an additive (positive or negative) constant.
770
D.S.J. Jones and P.R. Pujado ´