Lecture # 3 compounding factors effects of inflation

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)



The IRR and RATE Spreadsheet Functions



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



Determination of Unknown Number of Years



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



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%







0 1 2 3 4

$70,000 $70,000

$35,000 $25,000

7% 7% 7% 7%

P =







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)





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)



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!



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”.




To obtain the equivalent uniform series A

over all n years, substitute the symbol A for

each Fi, then solve for A:




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



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.



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.



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!



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)



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”.



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.



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.




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




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%



Annual

Inflation rate

CPI




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

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2012



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) .

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accompany Blank &



2012



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



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.

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2012 32 © 2012 by McGraw-Hill, New


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




Defining if

• if is derived from f and i as follows:

if = (i + f + if )

Or:

1

fi fi

f

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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

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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

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2012



Example 14.1, Blank (6th ed.), page 477

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Blank & Tarquin, Engineering




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).



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:



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.



Lecture # 3 compounding factors effects of inflation

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Transcript of Lecture # 3 compounding factors effects of inflation