TVM Lecture

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.

TVM Lecture

Recommended

Recommended

More Related Content

What's hot

What's hot (20)

Similar to TVM Lecture

Similar to TVM Lecture (20)

More from Bich Lien Pham

More from Bich Lien Pham (20)

TVM Lecture