1 Chapter 4 Equivalence Calculations Under Inflation

2 What is Inflation?  Value of Money  Earning Power  Purchasing Power Earning Power Purchasing power Investment Opportunity Decrease in purchasing power (inflation) Increase in purchasing power (deflation)

3 Purchasing Power 1990 $ $100 You could buy 50 Big Macs in year You can only buy 40 Big Macs in year $2.00 / unit $2.50 / unit 25% Price change due to inflation The $100 in year 2003 has only $80 worth purchasing power of 1990

$ $100 You could purchase gallons of unleaded gas a year ago. You can now purchase 80 gallons of unleaded gas. $1.57 / gallon$1.25 / gallon Price change due to deflation 20.38%

5 Inflation Terminology - I Producer Price Index: a statistical measure of industrial price change, compiled monthly Consumer Price Index: a statistical measure of change, over time, of the prices of goods and services in major expenditure groups—such as food, housing, apparel, transportation, and medical care—typically purchased by urban consumers Average Inflation Rate (f): a single rate that accounts for the effect of varying yearly inflation rates over a period of several years. General Inflation Rate ( ): the average inflation rate calculated based on the CPI for all items in the market basket.

6 Measuring Inflation Consumer Price Index (CPI): the CPI compares the cost of a sample “market basket” of goods and services in a specific period relative to the cost of the same “market basket” in an earlier reference period. This reference period is designated as the base period. Market basket Base Period (1982) 2002 $100$179.9 CPI for 2002 = 179.9

7 Average Inflation Rate ( f ) Fact: Base Price = $100 (year 0) Inflation rate (year 1) = 4% Inflation rate (year 2) = 8% Average inflation rate over 2 years? Step 1: Find the actual inflated price at the end of year 2. $100 ( ) ( ) = $ Step 2: Find the average inflation rate by solving the following equivalence equation. $100 ( 1+ f) 2 = $ f = 5.98% $100 $

8 Average Inflation Rate ( f ) The average inflation rate is a geometric average, not an arithmetic average. The price increase in the last two years are equivalent to an average percentage rate of 5.98% per year. We need the average inflation rate to estimate the future prices on the basis of the hstorical data.

9 Example - Average Inflation Rate Item2006 Price 2000 Price Average Inflation Rate (%) Consumer price index (CPI) Postage Homeowners Insurance Private college tuition and fees22,21815, Gasoline Haircut Car (Toyota Camry)22,90021, Natural gas (MBTU) Baseball tickets (family of four) Movies (average ticket)

10 General Inflation Rate ( f ) Average inflation rate based on the CPI

11 Example: Yearly and Average Inflation Rates YearCost 0$504, , , ,500 What are the annual inflation rates and the average inflation rate over 3 years? Solution Inflation rate during year 1 (f 1 ): ($538,400 - $504,000) / $504,000 = 6.83%. Inflation rate during year 2 (f 2 ): ($577,000 - $538,400) / $538,400 = 7.17 %. Inflation rate during year 3 (f 3 ): ($629,500 - $577,000) / $577,000 = 9.10%. The average inflation rate over 3 years is

12 Inflation Terminology – II Actual Dollars (A n ): Estimates of future cash flows for year n that take into account any anticipated changes in amount caused by inflationary or deflationary effects. Constant Dollars (A n ): Estimates of future cash flows for year n in constant purchasing power, independent of the passage of time (or base period).

13 Conversion from Constant to Actual Dollars $1,000 ( ) 3 = $1,260 Constant Dollars $1,000 3 Actual Dollars $1,260 3

14 Example - Conversion from Constant to Actual Dollars PeriodUnit SalesNet Cash Flow in Constant $ 0-$250,000 (initial investment) 11, ($100)=100,000 21, ($100)= 110,000 31, ($100)= 120,000 41, ($100)= 130,000 51, ($100)= 120,000 The current price/unit is $550 and cost/unit is $450. Price and cost will keep up with the general inflation rate, 5%. Profit/Unit = $550 - $450 = $100

15 Example - Conversion from Constant to Actual Dollars PeriodNet Cash Flow in Constant $ Conversion Factor due to inflation Cash Flow in Actual $ 0-$250,000(1+0.05) 0 -$250, ,000(1+0.05) 1 105, ,000(1+0.05) 2 121, ,000(1+0.05) 3 138, ,000(1+0.05) 4 158, ,000(1+0.05) 5 153,154

$250,000 $105,000 $121,275 $138,915 $158,016 $153,154 Years (b) Actual dollars $250,000 $100,000 $110,000 $120,000 $130,000 $120,000 Years (a) Constant dollars $250,000(1+0.05) 0 $100,000(1+0.05) $110,000(1+0.05) 2 $120,000(1+0.05) 3 $130,000(1+0.05) 4 $120,000(1+0.05) 5

17 Conversion from Actual to Constant Dollars Constant Dollars $1,260 ( ) -3 = $1,000 $1,000 3 Actual Dollars $1,260 3

18 Example - Conversion from Actual to Constant Dollars End of period Cash Flow in Actual $ Conversion at f = 5% Cash Flow in Constant $ Loss in Purchasing Power 0$20,000(1+0.05) 0 $20,0000% 120,000(1+0.05) -1 19,048 ( )/20000 = 4.76% 220,000(1+0.05) -2 18,141 ( )/20000 = 9.30% 320,000(1+0.05) -3 17, % 420,000(1+0.05) -4 16, %

19 Practice Problem - How to Compare the Winning Prizes in Two Different Points in Time Jack Nicklaus won his first Master Tournament in The prize was $20,000. Phil Mickelson won his first Master Tournament in The prize amount was $1.17M Price Index in 2003 (base year 1967) = Price Index in 1963 (base year 1967) = 91.7 Inflation-free interest rate = 5.65%

20 Consumer Price Index inflation rate = 4.525% ( / 91.7) (1/41) =

21 What is the worth of $1.17M in terms of purchasing power in 1963? The inflation rate between 1963 and 2004 is about 4.525% per year. $1.17M in 2004 would have a purchasing power of $190,616 in M(P/F,4.525%,41)=$190,616 (in constant 1963 dollars)

22 If Jack invested his prize money in 1963 at 5.65% (inflation-free interest rate), the prize money would grow to match Phil’s 2004 prize $20,000 $190,

23 Equivalence Calculation Under Inflation 1.Types of Interest Rate 2.Types of Cash Flow 3.Types of Analysis Method Market Interest rate (i) Inflation-free interest rate (i) In Actual Dollars (A n ) In Constant Dollars (A n ) Constant Dollar Analysis Actual Dollar Analysis Deflation Method Adjusted-discount method

24 Inflation Terminology - III Inflation-free Interest Rate (i ): an estimate of the true earning power of money when the inflation effects have been removed (also known as real interest rate). Market interest rate (i): interest rate which takes into account the combined effects of the earning value of capital and any anticipated changes in purchasing power (also known as inflation-adjusted interest rate).

25 Inflation and Cash Flow Analysis  Constant Dollar Analysis Estimate all future cash flows in constant dollars. Use i as an interest rate to find equivalent worth.  Actual Dollar Analysis Estimate all future cash flows in actual dollars. Use i as an interest rate to find equivalent worth.

26 Constant Dollar Analysis In the absence of inflation, all economic analyses up to this point is, in fact, constant dollar analysis. Constant dollar analysis is common in the evaluation of many long-term public projects, because government do not pay income taxes. For private sector, income taxes are computed based on taxable income in actual dollars, actual dollar analysis is more common.

27 Example Consider the constant- dollar flows given in Example 4.3. If the managers want the company to earn a 12% inflation-free rate of return on any investment, what would be the present worth of this project? PeriodNet Cash Flow in Constant $ 0-$250, , , , , ,000

28 Solution i=12% P=-250, ,000 (P/A,12%,5) +10,000 (P/G,12%,4) +20,000 (P/F,12%,5) = $163,099 (in year zero dollars) PeriodNet Cash Flow in Constant $ 0-$250, , , , , ,000

29 Actual Dollars Analysis  Method 1: Deflation Method Step 1:Bring all cash flows to have common purchasing power. Step 2:Consider the earning power.  Method 2: Adjusted-discount Method Combine Steps 1 and 2 into one step.

30 Example Step 1: Convert actual dollars to Constant dollars nCash Flows in Actual Dollars Multiplied by Deflation Factor Cash Flows in Constant Dollars 0-$75,0001-$ ,000(1+0.05) -1 30, ,700(1+0.05) -2 32, ,800(1+0.05) -3 28, ,000(1+0.05) -4 23, ,000(1+0.05) -5 45,445

31 Step 2: Convert Constant dollars to Equivalent Present Worth nCash Flows in Constant Dollars Multiplied by Discounting Factor Equivalent Present Worth 0-$75, ,476(1+0.10) -1 27, ,381(1+0.10) -2 26, ,334(1+0.10) -3 21, ,858(1+0.10) -4 16, ,445(1+0.10) -5 28,218 $45,268

32 Deflation Method: Converting actual dollars to constant dollars and then to equivalent present worth -$75,000$30,476 $32,381 $28,334$23,858 $45,455 -$75,000 $32,000 $35,700 $32,800 $29,000 $58,000 -$75,000 $27,706 $26,761$21,288$16,295$28,218 $45,268 Actual Dollars Constant Dollars Present Worth n = 0 n = 1n = 2n = 3n = 4n = 5

33 Adjusted-Discount Method Step 1 Step 2

34 Example - Adjusted-Discounted Method nCash Flows in Actual Dollars Conversion factor Equivalent Present Worth 0-$75, ,000( ) -1 27, ,700( ) -2 26, ,800( ) -3 21, ,000( ) -4 16, ,000( ) -5 28,217 $45,268

35 Example - Equivalence Calculation with Composite Cash Flow Elements College fund for 5-year-old child, that will earn 8% compounded quarterly. Deposits will be made every quarter until the child is 17. First withdrawal will be at age 18. College expenses are estimated to be $30,000/year in today’s dollars for 4 years, starting at age 18. The inflation rate is 6%/year. Find the amount of quarterly deposits in actual dollars.

36 Example - Equivalence Calculation with Composite Cash Flow Elements AgeCollege expenses (in today’s dollars) College expenses (in actual dollars) 18 (Freshman)$30,000$30,000(F/P,6%,13) = $63, (Sophomore)30,00030,000(F/P,6%,14) = 67, (Junior)30,00030,000(F/P,6%,15) = 71, (senior)30,00030,000(F/P,6%,16) = 76,211 Approach: Convert any cash flow elements in constant dollars into actual dollars. Then use the market interest rate to find the equivalent present value.

37 V 1 = C(F/A, 2%, 48) V 2 = $229,211 Let V 1 = V 2 and solve for C: C = $2, Required Quarterly Contributions to College Funds

38 Example A company wants to buy computers. The maintenance cost in today’s dollars will be $25,000, $30,000, $32,000, $35,000 and $40,000. General inflation rate is 8% per year and market interest rate is 15% per year. If the company wants to pay maintenance cost in equal annual payments for these years, what is the payment amount?

39 Solution

40 Solution CostConversion factorPresent value 25,000( ) -1 23, ,000( ) -2 26, ,000( ) -3 26, ,000( ) -4 27, ,000( ) -5 29, ,894 A=132,894 (A/P,15%,5)=$39,644.29

41 Example Suppose that $1,000 is deposited each year for five years in an account earning 8% per year. During this period, general inflation is expected to remain at 3% per year. At the end of five years, what is the dollar value of the account in terms of today’s purchasing power?

42 Solution

43 Example Suppose that your salary is $45,000 in year one and will increase at 4% per year until year four. The general inflation rate is expected to be 6% per year. What is the present worth of the four-year actual- and real-dollar salary cash flows at the end of year one (base year) if your personal MARR is 10% per year?

44 Solution End of yearSalary (A) 145,000 2 (45000*1.04=) 46,800 [46800*(1.06) -1 =] 44,151 3 (46800*1.04=) 48,672 [48672*(1.06) -2 =] 43,318 4 (48672*1.04=) 50,619 [50619*(1.06) -3 =] 42,500

45 Solution

46 Solution

47 Example Annual expenses for two alternatives have been estimated on different bases as follows: If the general inflation rate is expected to be 6% per year and the real interest rate is 9% per year, which alternative should be selected according to the equivalent worth in year 0? End of yearAlternative A (Expenses in actual dollars) Alternative B (Expenses in real  constant dollars) 1120,000100, ,000110, ,000120, ,000130,000

48 Solution

49 Solution

50 Summary The Consumer Price Index (CPI) is a statistical measure of change, over time, of the prices of goods and services in major expenditure groups—such as food, housing, apparel, transportation, and medical care—typically purchased by urban consumers. Inflation is the term used to describe a decline in purchasing power evidenced in an economic environment of rising prices. Deflation is the opposite: An increase in purchasing power evidenced by falling prices.

51 The general inflation rate (f) is an average inflation rate based on the CPI. An annual general inflation rate ( ) can be calculated using the following equation: Specific, individual commodities do not always reflect the general inflation rate in their price changes. We can calculate an average inflation rate for a specific commodity (j) if we have an index (that is, a record of historical costs) for that commodity.

52 Project cash flows may be stated in one of two forms Actual dollars (A n ): Dollars that reflect the inflation or deflation rate. Constant dollars (A’ n ): Year 0 dollars Interest rates for project evaluation may be stated in one of two forms: Market interest rate (i): A rate which combines the effects of interest and inflation; used with actual dollar analysis Inflation-free interest rate (i’): A rate from which the effects of inflation have been removed; this rate is used with constant dollar analysis

53 To calculate the present worth of actual dollars, we can use a two-step or a one-step process: Deflation method—two steps: 1. Convert actual dollars by deflating with the general inflation rate of 2. Calculate the PW of constant dollars by discounting at i’ Adjusted-discount method—one step 1. Compute the market interest rate. 2. Use the market interest rate directly to find the present value.