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Lecture # 2 Time Value of Money – Interest theory 1-1 Dr. A. Alim

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Lecture # 2

Time Value of Money – Interest theory

1-1

Dr. A. Alim

Time Value of Money

The “time value” of money is the most

important concept in engineering economy

All firms make use of investment of funds

Investments are expected to earn a return

Money possesses a “time value”

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Equivalence

Different sums of money at different times may be equal in economic value

0 1

$100 now

$106 one

year from now

Interest rate = 6% per year

$100 now is said to be equivalent to $106 one year from now, if

the $100 is invested at the interest rate of 6% per year.

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Interest Rate and Rate of Return

Interest – the manifestation of the time value of money

Rental fee that one pays to use someone else’s

money

Difference between an ending amount of money and a

beginning amount of money

Interest rate (%) = interest accrued per time unit

x 100%original amount

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Rate of Return

Interest earned over a period of time is expressed as

a percentage of the original amount, specifically;

interest accrued per time unit

Rate of return (%) = x 100%original amount

Borrower’s perspective – interest rate paid

Lender’s perspective – interest rate earned

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Simple and Compound Interest

Simple Interest:

Interest = (principal)(number of periods)(interest rate)

Future value (F) = principal(P) + P x n x i

Compound Interest:

Interest earns interest on interest

Compounds over time

Interest = (principal + all accrued interest) (interest rate)

Future value (F) = principal(P) x (1 + i)n

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Terminology and Symbols

• Specific symbols and their respective definitions have

been developed for use in engineering economy

• Symbols tend to be standard in most engineering

economy texts world-wide

• Mastery of the symbols and their respective meanings

is most important in understanding of the subsequent

material!

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Terminology and Symbols

• P = value or amount of money at a time designated as the

present or time 0.

• Also P is referred to as present worth (PW), present value

(PV), net present value (NPV), discounted cash flow (DCF),

and capitalized cost (CC); dollars

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Terminology and Symbols

• F = value or amount of money at some future time.

• Also F is called future worth (FW) and future value (FV);

dollars

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Terminology and Symbols

• A = series of consecutive, equal, end-of-period

amounts of money.

• Also A is called the annual worth (AW) , equivalent

uniform annual worth (EUAW); dollars per year,

dollars per month

• n = number of interest periods; years, months, days

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Terminology and Symbols

• i = interest rate or rate of return per time period;

percent per year, percent per month

• t = time, stated in periods; years, months, days, etc

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Cash Flows: Their Estimation and

Diagramming

Definition of terms Cash Inflows - amount of funds flowing into the

firm

Cash Outflows – amount of funds flowing out of the firm

Net Cash Flow equals cash inflows – cash outflows

Assumption for analysis – end of period Funds flow at the end of a given (interest) period

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Cash Flow Diagrams

A typical cash flow diagram might look like:

0 1 2 … … … n-1 n

1. Draw a time line

One time period

0 1 2 … … … n-1 n

2. Show the cash flows

Cash flows are shown as directed arrows (+ for up or – for down) ---

(+) inflow; (-) outflow

Always assume end-of-period

cash flows!

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IMPORTANT

• Cash flow diagrams are not limited to P, F, and/or A.

For example, this could be a very real diagram:

P

F

0 1 2 … … n-1 n

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P and F

• The symbols P and F represent one-time occurrences:

• Specifically:

$P

$F

0 1 2 … … n-1 n

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P and F

• It should be clear that a present value P represents

a single sum of money at some time prior to a future

value F

• This is an important basic point to remember

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Annual Amounts

• It is important to note that the symbol A always

represents a uniform mount (i.e., the same amount

each period) that extends through consecutive

interest periods.

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Annual Amounts

• Cash Flow diagram for annual amounts might look like

the following:

…………

$A $A $A $A $A

A = equal, end of period cash flow amounts

$P

0 1 2 3 .. N-1 N

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Remember !

A = equal payments, end of each period

F = single payment, end of last period

P = single payment, beginning of first period

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Interest Rate – i% per period

• The interest rate i is assumed to be a compound rate,

unless specifically stated

as “simple interest”

• The rate i is expressed in percent per interest period,

for example, 12% per year.

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Standard Notation for

Interest Factors

Standard notation has been adopted to

represent the various interest factors

Consists of two cash flow symbols, the

interest rate, and the number of time periods

General form: (X/Y,i%,n) X represents what is unknown

Y represents what is known

i and n represent input parameters; can be known or

unknown depending upon the problem

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Notation - continued

Example: (F/P,6%,20) is read as:

To find F, given P when the interest rate is 6% and

the number of time periods equals 20.

In problem formulation, the standard notation

is often used.

Example : F = P (F/P,6%,20)

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Notation - continued

There are 3 sources to determine the values of

these factors:

Tables at the back of many books provide

tabulations of common values for i% and n.

Using EXCEL functions.

Calculating the factors from formulas.

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Introduction To Solution By Computer

Application of Microsoft’s Excel© spreadsheet program

Good review in Appendix “A”, Blank’s Book.

Excel financial functions Present Value P: =PV(i%,n,A,F) e.g.

=PV(i%,n,A) or PV(i%,n,,F)

Future Value F: =FV(i%,n,A,P)

Equal, periodic value A: =PMT(i%,n,P,F)

No. of periods: =NPER((i%,A,P,F)

Compound interest rate: =RATE(n,A,P,F)

Compound interest rate: =IRR(first_cell:last_cell)

Present value of a series: =NPV(i%,second_cell:last_cell) + first_cell

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Calculating factors

From formulas:

Example:

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P

F

5 years

i = 8%

Cash flow diagram from the point of view

of the graduate.

Example:

P = 10000 dollars

i = 8%

n = 5 years

1) Using tables:

From Blank's book table 12, for 8% interest rate and 5 years, we find:

F/P = 1.4693 then F= $10000 x 1.4693 = $14,693

2) Using formula:

F/P = (1+i)^5 = 1.46932808 then F = $14,693.28

3) Using EXCEL function FV

EXCEL assigns a negative value to cash outflow and a positive value to cash inflow

P in this case is a positive cash flow, hence F calculated will be negative.

F = ($14,693.28)

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Example:

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Summary: Compounding Factors

1. Single-Payment compound amount factor (F from P)

2. Single-Payment present worth factor (P from F)

3. Uniform-series present worth factor (P from A)

4. Capital recovery factor (A from P)

5. Uniform-series compound amount factor (F from A)

6. Sinking fund factor (A from F)

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Single-Payment Factors(F/P and P/F)

Objective:

Derive factors to determine the present or future

worth of a cash flow

Cash Flow Diagram – basic format

0 1 2 3 n-1 n

P0

Fn

i% / period

P0 = Fn1/(1+i)n →(P/F,i%,n) factor: Excel: =PV(i%,n,,F)

Fn = P0(1+i)n →(F/P,i%,n) factor: Excel: =FV(i%,n,,P)

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Uniform-Series: Present Worth Factor (P/A)

and

Capital Recovery Factor(A/P)

Cash flow profile for P/A factor

. . . .

0 1 2 3 n-2 n-1 n $A per interest period

i% per interest period

Required: To find P given A

Cash flows are equal, uninterrupted and flow at the end of

each interest period

Find P

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(P/A) and (A/P) Factor Formulas

(1 ) 1 0

(1 )

n

n

iP A for i

i i

(1 )

(1 ) 1

n

n

i iA P

i

(P/A,i%,n) factor

Excel: =PV(i%,n,A)

(A/P,i%,n) factor

Excel: =PMT(i%,n,P)

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Sinking Fund Factor and Uniform Series

Compound Amount Factor

(A/F and F/A)

Cash flow diagram for (A/F) factor

Start with what has already been developed

1 (1 )

(1 ) (1 ) 1

n

n n

i iA F

i i

. . . .

0 1 2 3 n-2 n-1 n A=? per interest period

i% per interest period

F = given

Find A, given F

(1 ) 1n

iA F

i

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(F/A) factor from (A/F)

Given:

Solve for F in terms of A to yield

(1 ) 1n

iA F

i

(1 ) 1niF A

i

(A/F,i%,n) factor

Excel: =PMT(i%,n,,F)

(F/A,i%,n) factor

Excel: =FV(i%,n,A)

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Minimum Attractive Rate of Return

Investors expect to earn a return on their

investment (commitment of funds) over time

A profitable investment should earn (return)

funds in excess of the investment amounts

Economic projects should earn a reasonable

return, which is termed:

MARR – Minimum attractive rate of return

Also termed the “hurdle” rate for an investment

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The MARR

The MARR is established by the financial managers of the firm

The MARR is expressed as a percent value

Most, if not all, projects should earn at a rate equal to or greater than the established MARR

MARR is set based upon: The cost of all types of capital

Allowance for risk

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Types of Financing

Equity Financing – the firm uses funds either from

retained earnings, new stock issues, or owner’s

infusion of money

Debt Financing – the firm borrows funds from outside

sources

The cost of debt financing = the interest rate charged on the

debt (loan) amounts

Weighted average cost of capital (WACC) = Σ Xi (int. rate)i

The MARR is approximated from the weighted

average cost of all sources of capital to the firm

A firm’s ROR > MARR > WACC

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Graphical Presentation: MARR

0%

ROR - %

MARR - %

Safe Investment e.g.

bank WACC - %

Acceptable range for new projects

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1-40

Source: Plant design and economics for chemical engineers, 5th edition

By M.S. Peters et. al., McGraw Hill 2005.

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