Lecture # 2 time value of money

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”
1-2Slide Sets to accompany Blank & Tarquin, Engineering
Economy, 7th Edition, 2012
© 2012 by McGraw-Hill, New York, N.Y All Rights Reserved

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.

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

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

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

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!

• 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

• F = value or amount of money at some future time.
• Also F is called future worth (FW) and future value (FV);
dollars

• 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

• 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

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

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!

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

P and F
• The symbols P and F represent one-time occurrences:
• Specifically:
$P
$F
0 1 2 … … n-1 n

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

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.

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

Remember !
A = equal payments, end of each period
F = single payment, end of last period
P = single payment, beginning of first period

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.

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

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)

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.

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

Calculating factors
From formulas:

Example:
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)

Example:

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)
Slide Sets to accompany Blank & Tarquin, Engineering
Economy, 7th Edition, 2012 1-30 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved

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)

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

(P/A) and (A/P) Factor Formulas
(1 ) 1
0
(1 )
n
n
i
P A for i
i i
  
  
 
(1 )
(1 ) 1
n
n
i i
A P
i
 
  
  
(P/A,i%,n) factor
Excel: =PV(i%,n,A)
(A/P,i%,n) factor
Excel: =PMT(i%,n,P)

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 i
A 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
i
A F
i
 
    

(F/A) factor from (A/F)
 Given:
Solve for F in terms of A to yield
(1 ) 1n
i
A F
i
 
    
(1 ) 1n
i
F A
i
  
  
 
(A/F,i%,n) factor
Excel: =PMT(i%,n,,F)
(F/A,i%,n) factor
Excel: =FV(i%,n,A)

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

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

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

Graphical Presentation: MARR
0%
ROR - %
MARR - %
Safe Investment e.g.
bank
WACC - %
Acceptable range for new projects

1-40
Source: Plant design and economics for chemical engineers, 5th edition
By M.S. Peters et. al., McGraw Hill 2005.

Lecture # 2 time value of money

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