CE 314 Engineering Economy
Interest Formulas
METHODS OF COMPUTING INTEREST
1) SIMPLE INTEREST - Interest is computed using the principal only. Only
applicable to bonds and savings accounts.
2) COMPOUND INTEREST - Interest is calculated on the principal plus the total
amount of interest accrued in previous periods.
"Interest on top of Interest"
Example:
An individual borrows $18,000 at an interest rate of 7% per year to be paid back in a
lump sum payment at the end of 4 years. Compute the total amount of interest charged
over the 4-year period using the simple interest and compound interest formulas.
Compute the total amount owed after 4 years using simple and compound interest.
Using simple interest:
Interest = Principal (number of periods) (interest rate)
I = P(n)(i)
I = 18,000 (4)(0.07) = $5,040
And the
Amount owed = Principal + Interest accrued
F=P+I
F = 18,000 + 5,040 = $23,040
Using compound interest:
Year
Interest Charge
1
2
3
4
18,000 (0.07)
19,260 (0.07)
20,608.20(0.07)
22,050.77(0.07)
Accrued Amount
= $1,260
= $1,348.20
= $1,442.57
= $1,543.55
18,000 + 1,260
19,260 + 1,348.20
20,608.20 + 1,442.57
22,050.77 + 1,543.55
=$19,260
= $20,608.20
= $22,050.77
= $23,594.32
Total Interest charged = $23,594.32 - $18,000 = $5,594.32
(11% increase)
Interest Formulas:
Symbols
PF A GgA1 n i t-
Present value, value of money at the present (time = 0); $'s
Future value, value of money at some time in the future; $'s
Uniform Series, a series of consecutive, equal, end of time period amounts of
money; $'s/ month, $'s/ year, etc.
Constant arithmetic-gradient, period-by-period linear increase or decrease in cash
flow; $’s/month, $’s/year, etc.
Geometric gradient, period-by-period constant increase or decrease in cash flow;
$’s/month, $’s/year, etc.
First payment in a geometric gradient (time =1), $’s
Number of interest periods; months, years, etc.
Interest rate or rate of return per period; percent per month, percent per year, etc.
time, stated in periods; months, years, etc.
Derivation of the relationship between a future amount and a present amount: (Single
Payment Formulas)
Previous example:
P = $18,000
i = 7% per year
n=4
F4 = ?
F1 = 18,000 + 18,000 (0.07) = $19,260
F1 = P + P(i) = P (1 + i)
F2 = F1 + F1(i) = P (1 + i) + [P (1 + i)](i) = P (1 + i) [1 + i] = P (1 + i)2
F3 = P (1 + i)2 + P (1 + i)2 (i) = P (1 + i)2 (1 + i) = P (1 + i)3
In general, F = P (1 + i)n
The term (1 + i)n is called the single payment compound amount factor.
To compute a present amount from a future amount, solve for P:
F = P (1 + i)n
P = F / (1 + i)n
The term 1 / (1 + i)n is called the single payment present worth factor.
Derivation of the relationship between a uniform series and a future worth and a uniform
series and a present worth:
F = A1 (1 + i)4 + A2 (1 + i)3 + A3 (1 + i)2 + A4 (1 + i) + A5
But: A1=A2=A3=A4=A5=A
Equation 1:
F= A [(1 + i)4 + (1 + i)3 + (1 + i)2 + (1 + i) + 1]
A [(1 + i)4 + (1 + i)3 + (1 + i)2 + (1 + i) + 1] - F = 0
Now multiply each side by (1 + i):
Equation 2:
F(1 + i) = A [ (1 + i)5 + (1 + i)4 + (1 + i)3 + (1 + i)2+ (1 + i)]
A [ (1 + i)5 + (1 + i)4 + (1 + i)3 + (1 + i)2+ (1 + i)] - F( 1 + i) = 0
A [ (1 + i)5 + (1 + i)4 + (1 + i)3 + (1 + i)2+ (1 + i)] - F - Fi = 0
Equation 2 - Equation 1:
A [ (1 + i)5 + (1 + i)4 + (1 + i)3 + (1 + i)2+ (1 + i)]
- F - Fi
=0
A [(1 + i)4 + (1 + i)3 + (1 + i)2 + (1 + i) + 1] - F
A [ (1 + i)5
=0
- 1]
- Fi = 0
A [ (1 + i)5 - 1] - Fi = 0
Fi = A [ (1 + i)5 - 1]
F = A{ [ (1 + i)5 - 1] / i}
In general,
F = A{ [ (1 + i)n - 1] / i}
The term { [ (1 + i)n - 1] / i} is called the uniform series compound amount factor.
A = F{ i / [ (1 + i)n - 1]}
The term { i / [ (1 + i)n - 1]} is called the sinking fund factor.
Sinking fund is the annual amount invested by a company to finance a proposed
expenditure.
Derivation of the relationship between a uniform series and a present amount:
A
= F{ i / [ (1 + i)n - 1]}
and
F = P (1 + i)n
Substitute P (1 + i)n for F in equation 1:
A
= P (1 + i)n{ i / [ (1 + i)n - 1]} = P [i( + i)n/ (1 + i)n - 1]
The term [i( + i)n/ (1 + i)n - 1] is called the capital recovery factor. Capital recovery
refers to the amount of money required each year to offset an initial investment.
To compute a present amount from a uniform series. Solve for P:
A = P [i(1 + i)n/ (1 + i)n - 1]
P = A {[(1 + i)n - 1] / i( 1 + i)n}
An arithmetic gradient is a cash flow series that either increases or decreases by a
constant amount:
To compute a present amount from a linear gradient series use:
The term in the brackets is called the arithmetic-gradient series present worth factor.
To compute an equivalent annual series from a linear gradient use:
The term in the brackets is called the arithmetic-gradient uniform-series factor.
To compute a future amount from a linear gradient series use:
The term in the brackets is called the arithmetic-gradient series future worth factor.
The general equations for calculating total present worth are PT = PA + PG and
PT = PA - PG.
The general equations for calculating the equivalent total annual series are AT = AA + AG
and AT = AA - AG.
It is common for cash flow series, such as operating costs, construction costs, and
revenues to increase or decrease from period to period by a constant percentage. The
uniform rate of change defines a geometric gradient series of cash flows:
To compute a present amount from a geometric gradient series use:
Use only if g does not equal i.
The term in the brackets is called the geometric-gradient-series present worth factor.
Use if i = g.
To compute a future worth from a geometric gradient series use:
F = A1[((1 + i)n - (1 + g)n)/(i - g)]
use only if i does not equal g.
The term [(1-(1 + g)n(1 + i)-n)/(i - g)] is called the geometric-gradient-series future worth
factor.
F = nA1(1 + i)n-1
use if i = g.
Standard Notation:
To compute a future amount given a present amount:
F = P (F/P, i%, n)
“Looking for a F given a P”
To compute a present amount given a future amount:
P = F (P/F, i%, n)
“Looking for a P given a F”
To compute a present amount given a geometric-gradient-series:
P = A1(P/A1,g,i,n)
Tables are available on pages 727-755 in your textbook, which have factors computed for
all of the formulas (excluding the geometric-gradient-series) for different values of i and
n.
Convention:
The present value of a series cash flow is computed one period prior to the first series
payment.
The future value of a series cash flow is computed at the same time period as the last
series payment.
The present value of a linear gradient series is computed by breaking the linear gradient
into two parts: a uniform series cash flow and a conventional linear gradient series. The
present value of a conventional linear gradient series is computed two periods prior to the
first payment in the conventional linear gradient.
The future value of a conventional linear gradient is computed at the same time period as
the last payment in the conventional linear gradient.