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:

2t

1 2

Dollars

inflation rate between t and t1tDollars =

The Basic Inflation Relationship• Future Dollars = Today’s dollars(1+f)n

• Dollars in period t1 are termed Constant-value or

today’s dollars

• Dollars in time period t2 are termed Future Dollars or

Then-current Dollars

n

future dollarsConstant-Value dollars =

(1+f)

nFuture 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

• Assume i is the real interest rate• Assume F is a future dollar amount with inflation built in,

P is then seen to be:

• Or

• ic is then seen to equal to ic = (i + f + if)

1

(1 )nP F

i

1

(1 ) (1 )n n

FP

f i

1

(1 )nP F

i f if

Relating ic, f, and ir

• The relationship among the three rates is

i c = i r+f +i r f

i r = ( i c - 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 0.040.0577 5.77%

1 0.04i

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 5200 5000 4545 2 5408 5000 4132 3 5624 5000 3757 4 5849 5000 3415

Example Using the Combined Rate

• Using the combined interest rate:Comb. Ratei(f) 14.400%

Cost in PW @Year Future P/F,i(f),n combined

n Dollars i-rate0 $5,000 1.0000 $5,0001 $5,200 0.8741 $4,5452 $5,408 0.7641 $4,1323 $5,624 0.6679 $3,7574 $5,849 0.5838 $3,415

Comparison of $$ Values

Problem 8-4

• Annual expenses for two alternatives have been estimated on different bases as follows

Alt. A Alt. BEOY Annual Expenses Annual Expenses

Estimated In Actual dollars 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 = k=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 five-year 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 yeari f c = 0.26 - 0.06 + (0.02)(-0.06) = 18.44%