CHAPTER 8
PRICE CHANGES and ECHANGE
RATES
Definitions
• Inflation
– An increase in the average price paid for goods and
services bringing about a reduction in the purchasing
power of money
– Value of the currency changes downward in value
• Deflation
– A decrease in the average price paid for goods in
services, resulting in an increase in the purchasing
power of money
– Less amounts of the currency can purchase more
goods and services
– Not commonly seen……any more!
Importance of Inflation Impacts
• In most countries, inflation is from 2% to 8% per year;
• Some countries with weak currencies, political
instability, and poor balance of payments can have
hyperinflation (as high as 100% per year)
• Firms should set their MARR rate to:
– Cover the cost of capital;
– Cover or buffer the inflationary aspects perceived to exist;
– Account for risk.
The Inflation rate – f
• The inflation rate, f, is a percent per time period;
• Similar to an interest rate.
• Let n represent the period of time between t1 and t2 then
• Money in time period t1 can be related to money in time
period t2 by the following:
Dollars t1 =
Dollars t 2
inflation rate between t1 and t 2
The Basic Inflation Relationship
• Future Dollars = Today’s dollars(1+f)n
• Dollars in period t1 are termed Constant-value or
today’s dollars
future dollars
Constant-Value dollars =
(1+f) n
• Dollars in time period t2 are termed Future Dollars or
Then-current Dollars
n
Future dollars = today's dollars(1+f) .
Relating Actual Dollars To Real Dollars
• Actual dollars as of any time period (e.g., year), k, can
be converted into real dollars (as of any base time
period, b) by using
• (R$)K = (A$)K [1/ (1+f)]K-b = (A$)K(P / F, f %, k-b)
• The equation changes as follows for a specific type cash
flow (i.e. specific good or service “j ”)
• (R$)Kj = (A$)Kj [1/ (1+f)]K-b = (A$)Kj (P / F, f %, k-b)
• In the base period, purchasing power of actual dollar
and real dollar are the same
Example
• Suppose a firm desires to purchase a productive asset
that costs $209,000 in today’s dollars
• If the inflation rate is 4% per year, in 10 years the same
piece of equipment would cost
– 209,000(1.04)10 = $309,371!
• Does not count an interest rate or rate of return
consideration
• Notice that at a modest 4% rate of inflation the future
impact on cost can be and is significant!
• Have not considered the time value of money!
• Consider both inflation and the time value of money
Consumer Price Index (CPI)
•
•
•
•
One measure of price changes in our economy
An estimate of general price inflation
Tabulated by the U S Government
A composite price index that measures price changes in
food, shelter, medical care, transportation, apparel, and
other selected goods and services used by average
individuals and families
• See Figure 8-1
Producer Price Index (PPI)
• While the CPI shows how the prices that consumers pay
change from year to year or month to month
• The PPI (Producer Price Index) shows how the prices
paid to producers change from year to year or month to
month
• Another measure of price changes in our economy
• Also tabulated by the U S Government
• A composite price index that measures price changes in
raw materials, intermediate materials, and finished
goods
• Also tabulated monthly
Calculating Inflation Rate
• Inflation Rate =[(CPIk - CPIk-1)/CPIk-1]x100
– CPIk = Consumer price index for the current year
– CPIk-1 = Consumer price index for the preceding year
• Example: Annual inflation rate of 1996 (See Table 8-1)
(CPI)1996-(CPI)1995
• Inflation rate of 1996 =---------------------------------- x100
CPI(1995)
• =[(158.6 –153.5)/153.5]x100=3.32%
Example
• Suppose that your salary is $35,000 in year one, will
increase at 6% per year through year four, and is
expressed in actual dollars as follows:
EOY
Salary (A$)
Salary (R$, b=1)
1
$35,000
$35,000(P/F, 8%,0) = $35,000
2
$37,100
37,100(P/F, 8%,1) = 34,351
3
$39,362
39,362(P/F, 8%,2) = 33,714
4
$41,685
41,685(P/F, 8%,3) = 33,090
• If the f=8% per year, what is the real dollar equivalent of
these actual dollar salary amount?
Three Important Rates
• Real or inflation-free interest rate (ir)
– 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.
• Inflation-adjusted interest rate (ic)
– The interest rate that has been adjusted for inflation;
– The ic rate is the combination of the real interest rate – i, and
the inflation rate – f
– Also called the inflated interest rate or Market Interest Rate
• Inflation rate (f)
– The measure of the rate of change in the value of a currency.
Derivation of a Combine Interest Rate
• We now derive ic – the inflation adjusted interest rate
• Start with
1
P  F
(1  i ) n
• Assume i is the real interest rate
• Assume F is a future dollar amount with inflation built in,
P is then seen to be:
F
1
P
(1  f ) n (1  i ) n
• Or
1
PF
(1  i  f  if ) n
• ic is then seen to equal to ic = (i + f + if)
Relating ic, f, and ir
• The relationship among the three rates is
i c = i r+f +i r f
ir=(ic-f)/(1+f)
• Similarly, current-dollar internal rate of return is related
to the real rate of return in the following way:
IRR r = (IRR c - f ) / ( 1 + f )
• Given ic = 10% and f = 4%, find the real interest rate
0.10  0.04
i 
 0.0577  5.77%
1  0.04
Present Worth and Inflation
• In previous chapters, all cash flows were calculated in
constant value dollars
• Calculate the PW of $5,000 inflated at 4% per year with
a discount rate of 10% per year
Year
Cost in Future
Future cost in
Constant
Value
PW @ real int.
rate
0
5000
5000
5000
1
2
3
4
5200
5408
5624
5849
5000
5000
5000
5000
4545
4132
3757
3415
Example Using the Combined Rate
• Using
i(f)
Year
n
0
1
2
3
4
Comb. Rate
the14.400%
combined
Cost in
Future
Dollars
$5,000
$5,200
$5,408
$5,624
$5,849
interest rate:
P/F,i(f),n
1.0000
0.8741
0.7641
0.6679
0.5838
PW @
combined
i-rate
$5,000
$4,545
$4,132
$3,757
$3,415
Comparison of $$ Values
Problem 8-4
• Annual expenses for two alternatives have been
estimated on different bases as follows
Alt. A
Alt. B
EOY Annual Expenses
Estimated In Actual dollars
Annual Expenses
Estimated In Real dollars with b=0
1
-$120,000
-$100,000
2
-132,000
-110,000
3
-148,000
-120,000
4
-160,000
-130,000
• If the average general price inflation rate is expected to be 6%
per year and the rate of interest is 9% per year, show that which
alternative has the least negative equivalent worth in the base
time period.
Solution
•
•
•
•
The combined (Market) interest rate is:
i c = i r + f + f *i r
= 0.09 +0.06 + (0.09)(0.06)
= 0.1554 or 15.15.54%
• PW (15.54%) = -$120,000(P/F, 15.54, 1)+…+$160,000(P/F, 15.54, 4)=-$388,466
• PW (9%) = -$100,000(P/A, 9%, 4)+-$10,000(P/G, 9%,
4)=-$369,080
Fixed and Responsive Annuities
• Cash flows predetermined by contract -- bonds or
fixed annuities -- do not respond to general price
inflation
• Future amounts that are not predetermined may,
by varying degrees, respond to general price
inflation
• See Table 8-3
Calculating An Effective General
Price Inflation Rate
• f = An (estimated) effective general price
inflation rate for a period of N years
• f = PNk=1( 1 + f k ) 1 / N- 1
• See Table 8-5
Differential Price Inflation
• Inflation may not be the best estimate of future price
changes
• Variation between general price inflation rate and the
best estimate of future price changes for specific goods
and services are called Differential Price Inflation
• Caused by: changes in supply, changes in demand,
technological improvements, productivity changes,
regulatory requirements
• e `j -- The increment ( % ) of price change above or
below the general price inflation rate for a given time
period for good or service “ j “
Total Price Escalation
• Price changes caused by some combination of general
price and differential price inflation
• e j -- The total rate (%) of price change during a time
period for good or service “ j “
• Includes the effects of both the general price inflation
rate ( f ) and the differential price inflation rate (e ´j ) on
price changes
• e ´j = ( e j - f ) / ( 1 + f ) or e j = e ´j + f + f * e ´j
• Where
– e ´j is the increment ( % ) of price change
– e j -- The total rate (%) of price change
– f -- General price inflation
Determining A Convenience Rate For
Geometric Cash Flow Sequences
• Geometric series involves CF’s increasing at an constant
rate per period
• This rate is equal to e ´j in a R$ analysis
• This rate is equal to e j in a A$ analysis
• Then, convenience rate (iCR) in A$ analysis
iCR = ( ic - e j ) / ( 1 + e j )
• In R$ analysis
´
´
iCR = ( ir - e j ) / ( 1 + e j )
Example
• The prospective maintenance expenses for a HVAC system
are estimated to be $12,200 per year in year base year
dollars (assume b=0). The total price escalation rate is
estimated to be 7.6% for the next three years (e 1,2,3 = 7.6%),
and for years 4 and 5 it is estimated to be 9.3% (e 4,5
=9.3%). The General price inflation rate (f) for this fiveyear period is estimated to be 4.7% per year. Develop the
maintenance expense estimates for years one through five in
actual dollars and in real dollars, using ej, e’j values,
respectively.
Solution
Year
A$
Adjustment
Main. Exp.
A$
R$
Adjustment
Main. Exp
R$
1
12,200 (1.076)1
13127
12200(1.0277)1
12,538
2
12,200 (1.076)2
14125
12200(1.0277)2
12,855
3
12,200 (1.076)3
115198
12200(1.0277)3
13242
4
12,200 (1.076)3(1.093)
115198
12200(1.0277)3(1.0439)
13823
5
12,200 (1.076)3(1.093)2
115198
12200(1.0277)3(1.0439)2
14430
Problem 8-29, pp385
Foreign Exchange Rates
• When domestic corporations make foreign investments, the
resultant cash flow are in a different currency from U.S.
Dollars
• The main question is that “What is the PW (or IRR) that our
company will obtain by investing in a foreign country?”
– i US = market int. rate of return relative to US dollars
– i f c = market int. rate of return relative to foreign country currency
– fe = Annual rate of change in exchange rate -- annual devaluation
rate -- between foreign country and the U.S. dollar
• i f c = i US + f e + f e ( i US ) or i US = ( i f c - f e )/( 1 + f e )
– fe +: foreign currency devalued relative to dollar
– fe - : dollar devalued relative to foreign currency
Problem 8-31, pp385
• a) f e = 8% per year
i f c = 0.26 + 0.08 + (0.02)(0.08) = 36.08%
• b) f e = -6% per year
i f c = 0.26 - 0.06 + (0.02)(-0.06) = 18.44%