Date post: | 14-Jun-2015 |
Category: |
Documents |
Upload: | bich-lien-pham |
View: | 254 times |
Download: | 0 times |
Lecture # 3
1. Compounding Factors
2. Effect of Inflation
1-1 Dr. A. Alim
Determination of Unknown Interest Rate
Class of problems where the interest rate, i%,
is the unknown value
For simple, single payment problems (i.e., P
and F only), solving for i% given the other
parameters is not difficult
For annuity and gradient type problems,
solving for i% can be tedious
Trial and error method
Use EXCEL
Slide Sets to accompany Blank & Tarquin, Engineering
Economy, 7th Edition, 2012 1-2 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
The IRR and RATE Spreadsheet Functions
Define the total cash flow as a column of
values within Excel
Apply the IRR function:
=IRR(first_cell:last_cell, guess value)
If the cash flow series is an A value then apply
the RATE function:
=RATE(number_years, A,P,F)
Slide Sets to accompany Blank & Tarquin, Engineering
Economy, 7th Edition, 2012 1-3 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
The IRR and RATE Spreadsheet Functions
Slide Sets to accompany Blank & Tarquin, Engineering
Economy, 7th Edition, 2012 1-4 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
Example 1
End of Year Cash Flow
0 (1,200.00)$
1 354.00$ IRR = 26.3%
2 700.00$
3 216.00$
4 953.00$
Example 2
End of Year Cash Flow
0 (1,200.00)$
1 400.00$ Rate = 12.6%
2 400.00$
3 400.00$ IRR = 12.6%
4 400.00$
Determination of Unknown Number of Years
Class of problems where the number of time periods (years) is the unknown
In single payment type problems, solving for n is straight forward
In other types of cash flow profiles, solving for n requires trial and error.
In Excel, given A, P, or F, and i% values apply: =NPER(i%,A,P,F) to return the value of n
Slide Sets to accompany Blank & Tarquin, Engineering
Economy, 7th Edition, 2012 1-5 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
Determination of Unknown Number of Years
Slide Sets to accompany Blank & Tarquin, Engineering
Economy, 7th Edition, 2012 1-6 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
Example:
P -1200
A 400
i 12.60%
Number of periods = 4
In practice – interest rates do not stay the same
over time unless by contractual obligation.
There can exist “variation” of interest rates over
time – quite normal!
If required, how do we handle that situation?
Interest rates that vary over time
Slide Sets to accompany Blank & Tarquin, Engineering
Economy, 7th Edition, 2012 1-7 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
Interest Rates that vary over time
Best illustrated by an example.
Assume for now that interest rate is constant 7% for the
whole 4 years period:
0 1 2 3 4
$70,000 $70,000
$35,000 $25,000
7% 7% 7% 7%
Slide Sets to accompany Blank & Tarquin, Engineering
Economy, 7th Edition, 2012 1-8 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
Interest Rates that vary over time
Best illustrated by an example.
Assume for now that interest rate is constant 7% for the
whole 4 years period:
0 1 2 3 4
$70,000 $70,000
$35,000 $25,000
7% 7% 7% 7%
P =
Slide Sets to accompany Blank & Tarquin, Engineering
Economy, 7th Edition, 2012 1-9 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
Interest Rates that vary over time
Best illustrated by an example.
Assume for now that interest rate is constant 7% for the
whole 4 years period:
0 1 2 3 4
$70,000 $70,000
$35,000 $25,000
7% 7% 7% 7%
P = 70,000(P/A,7%,2) + 35,000(P/F,7%,3) + 25,000(P/F,7%,4)
Slide Sets to accompany Blank & Tarquin, Engineering
Economy, 7th Edition, 2012 1-10 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
Interest Rates that vary over time
Best illustrated by an example.
Now, assume the following future profits:
0 1 2 3 4
$70,000 $70,000
$35,000 $25,000
7% 7% 9% 10%
(P/F,7%,1)
(P/F,7%,1)
(P/F,9%,1)
(P/F,10%,1)
(P/F,7%,1)
(P/F,7%,1) (P/F,7%,1)
(P/F,7%,1) (P/F,7%,1) (P/F,9%,1)
Slide Sets to accompany Blank & Tarquin, Engineering
Economy, 7th Edition, 2012 1-11 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
Varying Rates: Present Worth
To find the Present Worth:
Bring each cash flow amount back to the
appropriate point in time at the interest rate
according to:
P = F1(P/F,i1,1) + F2(P/F,i1,1)(P/F,i2,1) + …
+ Fn(P/F,i1,1)(P/F,i2,1)(P/F,i3,1)…(P/F,in,1)
This Process can get computationally involved!
Slide Sets to accompany Blank & Tarquin, Engineering
Economy, 7th Edition, 2012 1-12 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
Period-by-Period Analysis
P0 =
$7000(P/F,7%,1) +
$7000(P/F,7%,1)(P/F,7%,1) +
$35000(P/F,9%,1)(P/F,7%,1)2 +
$25000(P/F,10%,1)(P/F,9%,1)(P/F,7%,1)2
Equals: $172,816 at t = 0…
Work backwards one period at a
time until you get to “0”.
Slide Sets to accompany Blank & Tarquin, Engineering
Economy, 7th Edition, 2012 1-13 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
Period-by-Period Analysis
To obtain the equivalent uniform series A
over all n years, substitute the symbol A for
each Fi, then solve for A:
Slide Sets to accompany Blank & Tarquin, Engineering
Economy, 7th Edition, 2012 1-14 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
Period-by-Period Analysis
P0 = $172,816 =
$A (P/F,7%,1) +
$A (P/F,7%,1)(P/F,7%,1) +
$A (P/F,9%,1)(P/F,7%,1)2 +
$A (P/F,10%,1)(P/F,9%,1)(P/F,7%,1)2
Solve for A = $51,777 per year
Slide Sets to accompany Blank & Tarquin, Engineering
Economy, 7th Edition, 2012 1-15 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
Varying Rates: Approximation
An alternative approach that approximates the present value:
Average the interest rates over the appropriate number of time
periods. In the previous example {7% + 7% + 9% + 10%} / 4 =
8.25%
This approach is only an approximation. Students MUST NOT use
this method in solving problems in quizzes or examinations.
Slide Sets to accompany Blank & Tarquin, Engineering
Economy, 7th Edition, 2012 1-16 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
Inflation - Definition
The increase in the amount of money
necessary to obtain the same amount of product or service before the inflated price was present;
Social Phenomena where too much money chases too few goods/services;
Harmful impact because the purchasing power of the currency changes downward in value.
Slide Sets to accompany Blank & Tarquin, Engineering
Economy, 7th Edition, 2012 1-17 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
Deflation
Where the value , i.e. the purchasing power of the currency increases over time.
Less amounts of the currency can purchase more goods and services than before.
Not very commonly seen……2009 was the first year with deflation (or zero inflation) in a very long time!
Slide Sets to accompany Blank & Tarquin, Engineering
Economy, 7th Edition, 2012 1-18 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
Inflation rate - f
The measure of the annual rate of change in the value of a currency.
Similar to an interest rate but should not be viewed as an interest rate.
f is a percentage value similar to the interest rate.
Let n represent the period of time between now and a future date, then:
Future cost = Current cost (1 +f )n
Note: this does not involve time value of money (compounding)
Slide Sets to accompany Blank & Tarquin, Engineering
Economy, 7th Edition, 2012 1-19 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
Example to Consider
Assume a firm desires to purchase a productive asset that costs $209,000 in today’s dollars.
Assume an inflation rate of say, 4% per year;
In 10 years, that same piece of equipment would cost:
$209,000(1.04)10 = $309,371!
Does not include an interest rate or rate of return consideration.
The $309,371 are called “future dollars”.
Slide Sets to accompany Blank & Tarquin, Engineering
Economy, 7th Edition, 2012 1-20 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
Inflation can be Significant
From the previous example we see that
even at a modest 4% rate of inflation, the future impact on cost can be and often is significant!
The previous example does not consider the time value of money.
A proper engineering economy analysis should consider both inflation and the time value of money.
Slide Sets to accompany Blank & Tarquin, Engineering
Economy, 7th Edition, 2012 1-21 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
How is inflation measured*?
Consumer Price Index (CPI) - a government measure of the price change of a “market basket” of goods and services (national inflation rate)
Producer Price Indices – a government measure of price changes for specific industries Chemical and Petrochemical Process Plants Farm Products Consulting Engineering Services Price Indexes Residential Building Construction Input Price Indexes Industrial product price indexes Raw materials price indexes Energy consumer price indices
* Ref. : J.C. Paradi, Centre for Management of Technology and Entrepreneurship ( 1996-2004)
1-22 Slide Sets to accompany Blank & Tarquin, Engineering
Economy, 7th Edition, 2012
© 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
CPI
The CPI for a given period relates the average price of a fixed “basket” of goods in the given period to the average price of the same basket of goods in a base period.
Current CPI base year is 1982 -84.
Base year index is set at 100.
The index for any other year indicates the number of dollars needed in that year to buy the basket of goods that cost $100 in 1982-84.
1-23 Slide Sets to accompany Blank & Tarquin, Engineering
Economy, 7th Edition, 2012
© 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
Historical CPI
Consumer Price Index historical summary – 1982-84 = 100%
Year CPI % Change Year CPI % Change Year CPI % Change
1972 41.8 3.2 1983 99.6 3.2 1994 148.2 2.6
1973 44.4 6.2 1984 103.9 4.3 1995 152.4 2.8
1974 49.3 11.0 1985 107.6 3.6 1996 156.9 3.0
1975 53.8 9.1 1986 109.6 1.9 1997 160.5 2.3
1976 56.9 5.8 1987 113.6 3.6 1998 163.0 1.6
1977 60.6 6.5 1988 118.3 4.1 1999 166.6 2.2
1978 65.2 7.6 1989 124.0 4.8 2000 172.2 3.4
1979 72.6 11.3 1990 130.7 5.4 2001 177.1 2.8
1980 82.4 13.5 1991 136.2 4.2 2002 179.9 1.6
1981 90.9 10.3 1992 140.3 3.0 2003 184.0 2.3
1982 96.5 6.2 1993 144.5 3.0 2004 188.9 2.7
2005 195.3 3.4
2006 201.6 3.2
2007 207.3 2.9
2008 215.3 3.8
2009 214.5 -0.4
2010 218.1 1.7
2011 224.9 3.2
2012 230.0 2.2
1-24 Slide Sets to accompany Blank & Tarquin, Engineering
Economy, 7th Edition, 2012
© 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
Inflation rate - f
Defined as the percent change in CPI from year to year.
For example, from the previous table: CPI in 2009 is 214.5 CPI in 2010 is 218.1
Inflation rate in 2010 is: 100 x (218.1 – 214.4)/214.5 = approx 1.7%
Slide Sets to accompany Blank & Tarquin, Engineering
Economy, 7th Edition, 2012 1-25 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
Annual
Inflation rate
CPI
1-26 Slide Sets to accompany Blank & Tarquin, Engineering
Economy, 7th Edition, 2012
© 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
1-27 Slide Sets to accompany Blank & Tarquin, Engineering
Economy, 7th Edition, 2012
© 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
Accounting for inflation There are two ways to make meaningful economic
calculations when the currency is changing in value, that is when inflation is considered:
1. Convert the amounts that occur in different time periods into constant value dollars. This is accomplished before any time value of money calculation is made using the real interest rate.
2. Change the interest rate used to account for inflation plus the time value of money. This is called the inflation-adjusted or market interest rate.
Slide Sets to
accompany Blank &
Tarquin, Engineering
Economy, 7th Edition,
2012
28 © 2012 by McGraw-Hill, New
York, N.Y All Rights Reserved
• Future Dollars = Today’s dollars(1+f)n
• Dollars at present time are termed:
– Constant-value or today’s dollars
• Dollars in time period t are termed:
– Future Dollars or,…
– Then-current Dollars.
1) Constant value dollars vs. future dollars
Slide Sets to
accompany Blank &
Tarquin, Engineering
Economy, 7th Edition,
2012
29 © 2012 by McGraw-Hill, New
York, N.Y All Rights Reserved
Critical Relationships to Remember
• Constant-value Dollars (Today’s dollars)
• Future Dollars
n
future dollarsConstant-Value dollars =
(1+f)
nFuture dollars = today's dollars(1+f) .
Slide Sets to
accompany Blank &
Tarquin, Engineering
Economy, 7th Edition,
2012
30 © 2012 by McGraw-Hill, New
York, N.Y All Rights Reserved
Three Important Rates
Inflation-free interest rate.
Denoted as “i”.
Inflation-adjusted interest rate.
Denoted as “if”
Inflation rate.
Denoted as “f”.
2) Adjusting interest rate to include inflation
Slide Sets to accompany Blank & Tarquin, Engineering
Economy, 7th Edition, 2012 1-31 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
Real or Inflation-free Interest Rate - i
• Rate at which interest is earned.
• Effects of any inflation have been removed.
• Represents the actual or real gain received/charged on investments or borrowing.
Slide Sets to
accompany Blank &
Tarquin, Engineering
Economy, 7th Edition,
2012 32 © 2012 by McGraw-Hill, New
York, N.Y All Rights Reserved
Inflation-adjusted rate – if
• The interest rate that has been adjusted to
include inflation.
• Common Term –
– Market Interest Rate.
– Interest rate adjusted for inflation.
• The if rate is the combination of the real
interest rate i, and the inflation rate f.
• Also called the inflated interest rate. Slide Sets to accompany
Blank & Tarquin, Engineering
Economy, 7th Edition, 2012
33 © 2012 by McGraw-Hill, New
York, N.Y All Rights Reserved
Defining if
• if is derived from f and i as follows:
if = (i + f + if )
Or:
1
fi fi
f
Slide Sets to
accompany Blank &
Tarquin, Engineering
Economy, 7th Edition,
2012 34 © 2012 by McGraw-Hill, New
York, N.Y All Rights Reserved
Example
• Assume i = 10%/ year;
• f = 4% per year;
• if is then calculated as:
if = 0.10 + 0.04 + 0.10(0.04) = 0.144 = 14.4%/year
Slide Sets to
accompany Blank &
Tarquin, Engineering
Economy, 7th Edition,
2012 35 © 2012 by McGraw-Hill, New
York, N.Y All Rights Reserved
Which interest rate to use?
• Either rate is correct when the correct dollar
value is selected:
Cash flow in Interest rate to use Today’s dollar (constant value) i Future dollars if
Slide Sets to
accompany Blank &
Tarquin, Engineering
Economy, 7th Edition,
2012
36 © 2012 by McGraw-Hill, New
York, N.Y All Rights Reserved
Example 14.1, Blank (6th ed.), page 477
Slide Sets to
accompany Blank &
Tarquin, Engineering
Economy, 7th Edition,
2012 37 © 2012 by McGraw-Hill, New
York, N.Y All Rights Reserved
Slide Sets to accompany
Blank & Tarquin, Engineering
Economy, 7th Edition, 2012
38 © 2012 by McGraw-Hill, New
York, N.Y All Rights Reserved
Effect of Inflation on the MARR
When inflation is expected during the life of a project, the MARR needs to be increased to avoid accepting poor projects. Inflated MARR = minimum acceptable rate of return when cash
flows are in actual dollars (future dollars).
If investors expect inflation, they require higher actual rates of return on their investments than if inflation were not expected. Inflated MARR = real MARR (without inflation) + upwards
adjustment which reflects the effect of inflation.
So, following from the definition, we have: if = (i + f + if )
MARRinflated = MARRR + f + MARRR * f where MARRinflated(if) is inflated MARR and MARRR (i) is real MARR
(without inflation).
Slide Sets to accompany Blank & Tarquin, Engineering
Economy, 7th Edition, 2012 1-39 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
Abbott Mining Systems wants to determine whether it should buy now or buy
later a piece of equipment used in deep mining operations. If the
company selects plan N, the equipment will be purchased now for $200,000.
However, if the company selects plan L, the purchase will be deferred for
3 years when the cost is expected to rise rapidly to $340,000. Abbott is
ambitious; it expects a real MARR of 12% per year. The inflation rate in the
country has averaged 6.75% per year. From only an economic perspective,
determine whether the company should purchase now or later (a) when inflation
is not considered and (b) when inflation is considered.
Solution
a) Inflation not considered: The real rate, or MARR, is i = 12% per year. The
cost of plan L is $340,000 three years hence. Calculate the FW value for
plan N three years from now:
FWN = -200,000(F/P,12%,3) = $-280,986
FWL = $-340,000
Hence buy now
Example 14.4, Blank (6th ed.), page 483:
Slide Sets to accompany Blank & Tarquin, Engineering
Economy, 7th Edition, 2012 1-40 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved
b) Inflation considered: There is a real rate (12%), and inflation
is 6.75%. First, compute the inflation-adjusted MARR :
MARRf = 0.12 + 0.0675 + 0.12(0.0675) = 0.1956
Compute the FW value for plan N in future dollars.
FWN = -200,000(F/P,19.56%,3) = $-341,812
FWL = $-340,000
Purchasing later is selected, because it requires fewer equivalent future
dollars. The inflation rate of 6.75% per year has raised the equivalent future
worth of costs by 21.6% to $341,812.
Slide Sets to accompany Blank & Tarquin, Engineering
Economy, 7th Edition, 2012 1-41 © 2012 by McGraw-Hill, New York, N.Y All Rights Reserved