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©2005 Binghamton UniversityState University of New YorkState University of New York

ISE 211 Engineering Economy

More Interest Formulas

(Chapter 4)

Uniform Series Compound Interest Formulas Uniform Series Compound Interest Formulas

Many times we will find situations where there are a uniform

series of receipts or disbursement. Examples: Automobile loan, house payments, etc. The series A is defined as: An end-of-period cash receipt or

disbursement in a uniform series, continuing for n periods – the

entire series equivalent to P or F at interest rate i.

Uniform Series Compound Interest Formulas (cont’d) Uniform Series Compound Interest Formulas (cont’d)

The uniform series is equivalent to a future worth value, F,

at interest rate i, and can be expressed as follows:

In functional notation:

The term ___________________ is called the uniform series

compound amount factor.

Or we can say that the term _________________ is called

the uniform series sinking fund factor.

Example 1Example 1

A man deposits $500 in a credit union at the

end of each year for five years. The credit

union pays 5% interest, compounded

annually. At the end of five years,

immediately following his fifth deposit, how

much will he have in his account?

Solution:

Example 2 Example 2

Jim Hayes read that in western United States, a ten-

acre parcel of land could be purchased for $1000

cash. Jim decided to save a uniform amount at the

end of each month so that he would have the

required $1000 at the end of one year. The local

credit union pays 6% interest, compounded monthly.

How much would Jim have to deposit each month?

Solution:

How can we find P if we know A?How can we find P if we know A?

The factor __________________ is called the uniform

series capital recovery factor.

The factor __________________ is called the uniform

series present worth factor.

Example 1Example 1

On January 1st a man deposits $5000 in a

credit union that pays 8% interest,

compounded annually. He wishes to withdraw

all the money in five equal end-of-year sums,

beginning December 31st of the first year. How

much should he withdraw each year?

Solution:

Example 2Example 2

An investor holds a time payment purchase contract

on some machine tools. The contract calls for the

payment of $140 at the end of each month for a five

year period. The first payment is due in one month.

He offers to sell you the contract for $6800 cash

today. If you otherwise can make 1% per month on

your money, would you accept or reject the

investor’s offer?

Solution:

Example 3Example 3

Suppose we decided to pay the $6,800 for

the time purchase contract in Example 2.

What monthly rate of return would we

obtain on our investment?

Solution:

Example 4Example 4

Using a 15% interest rate, compute the value of F

in the following cash flow:

Year Cashflow

1 +100

2 +100

3 +100

4 0

5 F

Solution:

Example 5Example 5

Every month starting today, you save $50

for 12 months. If your account earns 3%

interest, compounded monthly, how much

will you have just after your last deposit?

Solution:

Example 6Example 6

Consider the following situation, and find

the value of P.

Solution:

P=?

2030

200

i = 15%

Relationships Between Compound Interest FactorsRelationships Between Compound Interest Factors

Single Payment:

Compound amount factor = 1 / Present worth

factor

(F/P, i, n) = 1 / (P/F, i, n)

Uniform Series:

Capital recovery factor = 1 / Present worth factor

(A/P, i,n) = 1 / (P/A, i, n)

Compound amount factor = 1 / Sinking fund factor

(F/A, i, n) = 1 / (A/F, i, n)

Miscellaneous RelationshipsMiscellaneous Relationships

N

J

JiFPniAP1

),,/(),,/(

1

1

),,/(1),,/(N

J

JiPFniAF

1),,/(),,/( niFAniPA i

Arithmetic Gradient Arithmetic Gradient

In an arithmetic series, a constant amount is added each period.

Suppose the amount added = G.

Examples: 2, 4, 6, 8, …. (G=2). : 225, 250, 275, 300, … (G=25)

The cash flow diagram for such a situation is shown as:

Arithmetic Gradient (cont’d) Arithmetic Gradient (cont’d)

In an arithmetic series, a constant amount is added each period.

This can be expressed as follows:

Arithmetic Gradient (cont’d) Arithmetic Gradient (cont’d)

The arithmetic gradient present

worth factor is given as follows:

The arithmetic gradient uniform

series factor:

Example 1 Example 1 A man purchased a new automobile. He wishes to set aside enough

money in a bank account to pay the maintenance on the car for the

first five years. It has been estimated that the maintenance cost of an

automobile is as follows:

Year Maintenance Cost

1 $120

2 150

3 180

4 210

5 240

Assume the maintenance costs occur at the end of each year and that

the bank pays i=5%. How much should he deposit in the bank now?

Example 2Example 2On a certain piece of machinery, it is estimated that the

maintenance expense will be as follows:

Year Maintenance Cost

1 $100

2 200

3 300

4 400

What is the equivalent uniform annual maintenance

cost for the machinery if 6% interest is used?

Example 3Example 3

Consider the following cash flows

Year Amount

1 $24,000

2 18,000

3 12,000

4 6,000

Assuming 10% interest, what is the equivalent

uniform annual series over 4 years?

Example 4Example 4

Compute the value of P in the diagram below, using

10% interest rate.

P

0 0050

150100

Geometric Gradient Geometric Gradient

This illustrates the situation when the period-by-period change

is a uniform rate, g.

Example: The maintenance costs for an automobile are $100

the first year and increasing at a uniform rate, g, of 10% per year.

Illustration:

Geometric Gradient (cont’d)Geometric Gradient (cont’d)

From the previous table, we can calculate the maintenance in

any given year as follows: $100(1+g)n-1

Which can be written in general as follows:

An = A1 (1+g)n-1 Where g = geometric gradient -- uniform rate of cash flow

increase/decrease from period to period

A1 = value of cash flow at year 1

An = value of cash flow at any year n.

Geometric Gradient (cont’d)Geometric Gradient (cont’d) What if we are interested in obtaining the present worth of the

geometric series shown below?P= ??

This can be done using the following equation:

Geometric Gradient (cont’d)Geometric Gradient (cont’d)

This can be expressed as follows in functional

notation: P = A1(P/A, g, i, n) i ≠ g

The term A1(P/A, g, i, n) is the geometric series

present worth factor where i g.

In the special case where i = g, the equation

becomes: P = A1 n (1 + i)-1

Example 1 Example 1

The first year maintenance for a new

automobile is estimated to be $100, and it

increases at a uniform rate of 10% per

year. What is the present worth of cost of

the first five years of maintenance in this

situation, using an 8% interest rate?

Solution:

Nominal and Effective Interest Nominal and Effective Interest

Example: Consider a situation of a person depositing

$100 into a bank account that pays 5% interest

compounded semi-annually. How much would be in

the savings account at the end of one year?

Solution:

What interest rate compounded annually, results in

this value of F = ______?

Solution:

Nominal and Effective Interest (cont’d)Nominal and Effective Interest (cont’d)

Definitions:

r = nominal interest rate per interest period

(usually one year) – interest rate per interest period

without considering the compounding effect.

i = effective interest rate per interest period (e.g.,

year) – the interest rate taking into account the effecting

of compounding per interest period.

ia = effective interest rate per year (annum)

m= # of compounding sub-periods per time period

The effective interest rate per year is given as follows:

ia = (1 + r/m)m – 1

ia = (1 + i)m – 1 Example: If a savings bank pays 1.5% interest

every three months, what are the nominal and

effective interest rates per year?

Nominal and Effective Interest (cont’d)Nominal and Effective Interest (cont’d)

Nominal and Effective Interest (cont’d)Nominal and Effective Interest (cont’d)

ExampleExample

You go to a check cashing store to get a $50 advance on your

pay check. They are willing to do so provided you write a check

for $60 dated for 1 week from now.

a) What nominal interest rate per year are you paying?

b) What effective interest rate per year are you paying?

c) How much would you have to write the check for if

your pay back period is 1 year instead of 1 week?

Solution:

What if compounding interval What if compounding interval cash flow period? cash flow period?

Example: A bank pays 8% nominal annual interest per

year, compounded quarterly. A person deposits $5000

now (at time 0). After the end of each of 5 years, she

wants to withdraw an equal amount of money. What is

this amount?

Solution # 1: