# Interest and Principal Formulas

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```					                             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                        Accrued Amount

1      18,000 (0.07)          = \$1,260        18,000 + 1,260         =\$19,260
2      19,260 (0.07)          = \$1,348.20     19,260 + 1,348.20      = \$20,608.20
3      20,608.20(0.07)        = \$1,442.57     20,608.20 + 1,442.57   = \$22,050.77
4      22,050.77(0.07)        = \$1,543.55     22,050.77 + 1,543.55   = \$23,594.32

Total Interest charged = \$23,594.32 - \$18,000 = \$5,594.32
(11% increase)
Interest Formulas:

Symbols

P-      Present value, value of money at the present (time = 0); \$'s
F -     Future value, value of money at some time in the future; \$'s
A -     Uniform Series, a series of consecutive, equal, end of time period amounts of
money; \$'s/ month, \$'s/ year, etc.
G-      Constant arithmetic-gradient, period-by-period linear increase or decrease in cash
flow; \$’s/month, \$’s/year, etc.
g-      Geometric gradient, period-by-period constant increase or decrease in cash flow;
\$’s/month, \$’s/year, etc.
A1 -    First payment in a geometric gradient (time =1), \$’s
n -     Number of interest periods; months, years, etc.
i -     Interest rate or rate of return per period; percent per month, percent per year, etc.
t-      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                  =0

A [ (1 + i)5                                                          - 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.

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