equivalence calculations under inflation - engineering economics

Applied Software Project Management

Engineering EconomicsEngineering Economics

Equivalence Calculations under Inflation Equivalence Calculations under Inflation

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Road MapRoad Map

Measure of Inflation Actual versus Constant Dollars Equivalence Calculations under Inflation

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Measure of InflationMeasure of Inflation

So far all equivalence calculation under constant condition

Realistic: – purchase power loses value over time or inflation

This chapter:– Define and quantify inflation– Apply it in several equivalence analyses

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Consumer Price Index Consumer Price Index

Measuring inflation effect Eight major groups based on a typical market basket of

goods and services required by the average consumer: (1) food and alcoholic beverages, (2) housing, (3) apparel, (4) transportation, (5) medical care, (6) entertainment, (7) personal care, and (8) other goods and services

CPI compares the cost of the typical market basket of goods and services in a current month with its cost at a previous time, such as 1 month, 1 year, or 10 years ago

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Producer Price Index Producer Price Index

Consumer Price Index: good measure of the general price increase of consumer products, not a good measure of industrial price increases

Performing engineering economic analysis, the appropriate price indices must be selected to accurately estimate the price increases of raw materials, finished products, and operating costs

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Average Inflation Rate (Average Inflation Rate (ff) )

To account for the effect of varying yearly inflation rates over a period of several years.

A single rate that represents an average inflation rate Each year's inflation rate based on the previous year's

rate, these rates have a compounding effect

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Average Inflation Rate (Average Inflation Rate (ff) )

Example: to calculate the average inflation rate for a two-year period. The first year's inflation rate is 4%, and the second year's rate is 8%, with a base price of $100

100(1 + 0.04)(1 + 0.08) = 112.32

100(1+f)2 = 112.32 or 100(F/P,f,2) = 112.32

f = 5.98%

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First Year

Second Year

Applied Software Project ManagementGeneral Inflation Rate (General Inflation Rate (ff) versus ) versus Specific Inflation (Specific Inflation (fjfj))

General inflation rate (f ): this average inflation rate is calculated based on the CPI for all items in the market basket. The market interest rate is expected to respond to this general inflation rate

Specific inflation rate(fi): This rate is based on an index (or the CPI) specific to segment j of the economy

In terms of CPI, the general inflation rate defined asCPIn = CPI0(I + f)n or

Where: CPIn : the consumer price index at the end period n

CPI0 : the consumer price index at the end period 0

f: the general inflation rate

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11/n]CPICPI[f

0

n


Yearly and Average Inflation RatesYearly and Average Inflation Rates

Given: History of utility cost (table below). Find: The yearly inflation rate (fi) and the average inflation

rate over the three-year time period 0 Inflation at year n: in = CPIn/CPIn-1 - 1

Average inflation over 3 years: F = (629,500/504,000)1/3 - 1

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Year Cost Inflation

0 504,000

1 538,400 538,400/504,000 - 1

2 577,000 577,000/538,400 - 1

3 629,500 629,500/577,000 - 1


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Actual versus Constant Dollars Actual versus Constant Dollars

Actual (current) dollars (An): Actual dollars are estimates of future cash flows for year n that take into account any anticipated changes in amount caused by inflationary or deflationary effects.

Actual dollars are the amount of dollars that will be paid or received irrespective of how much these dollars are worth.

Usually, these amounts are determined by applying an inflation rate to base-year dollar estimates

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Actual versus Constant Dollars Actual versus Constant Dollars

Constant (real) dollars (A’n): Constant dollars reflect constant purchasing power independent of the passage of time.

Constant dollars are a measure of worth not an indicator of the number of dollars paid or received.

In situations where inflationary effects were assumed when cash flows were estimated, these estimates can be converted to constant dollars (base-year dollars) by adjustment using some readily accepted general inflation rate


Applied Software Project ManagementConversion from Constant to Conversion from Constant to Actual Dollars Actual Dollars

Constant dollars represent dollar amounts expressed in terms of the purchasing power of the base year

An = A’n(1 + f)n = A’n(F/P,f,n)

where

A’n = the constant-dollar expression for the cash flow occurring at the end of year n and

An = the actual-dollar expression for the cash flow occurring at the end of year n.



Transco Company is considering making and supplying computer-controlled traffic-signal switching boxes to be used throughout Arizona. Transco has estimated the market for its boxes by examining data on new road construction and on deterioration and replacement of existing units. The current price per unit is $550; the before-tax manufacturing cost is $450.The start-up investment cost is $250,000.The projected sales and net before-tax cash flows in constant dollars are as below: Assume that the price per unit and the manufacturing cost keep up with the general inflation rate, which is projected to be 5% annually. Convert the project's before-tax cash flows into the equivalent actual dollars.



Table:

Given: Net cash flows in Constant $, f = 5%. Find: Net cash flows in actual $.


Period Units Sale Net Cash Flow in Constant $

0 -250,000

1 1,000 100,000

2 1,100 110,000

3 1,200 120,000

4 1,300 130,000

5 1,400 120,000


First convert the constant dollars into actual dollars


Period Net Cash Flow in Constant $

Conversion Factor

Cash Flow in Actual $

0 -250,000 1

1 100,000 (1+0.05)1 105,000

2 110,000 (1+0.05)2 121,275

3 120,000 (1+0.05)3

4 130,000 (1+0.05)4

5 120,000 (1+0.05)5

Applied Software Project ManagementConversion from Actual to Conversion from Actual to Constant Dollars Constant Dollars

The reverse of converting from constant to actual dollars A’n = An/(1 + f)n = An(1 + f)-n = An(P/F,f,n)

Where:

A’n = the constant-dollar expression for the cash flow occurring at the end of year n and

An = the actual-dollar expression for the cash flow occurring at the end of year n.



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Equivalence Calculations under Inflation Equivalence Calculations under Inflation

In previous chapters, our equivalence analyses took into consideration changes in the earning power of money. i.e. interest effects.

To factor in changes in purchasing power as well-that is, inflation we may use either (1) constant-dollar analysis or (2) actual-dollar analysis.

Either method produces the same solution. however, each method requires use of a different interest rate and procedure.



Market and Inflation-Free Interest Rates Market and Inflation-Free Interest Rates

Market interest rate (i): commonly known as the nominal interest rate, takes into account the combined effects of the earning value of capital (earning power) and any anticipated inflation or deflation (purchasing power).

Virtually all interest rates stated by financial institutions for loans and savings accounts are market interest rates. Most firms use a market interest rate (also known as inflation-adjusted required rate of return) in evaluating their investment projects.




Inflation-free interest rate (i'): an estimate of the true earning power of money when the effects of inflation have been removed.

This rate is commonly known as the real interest rate, and it can be computed if the market interest rate and the inflation rate are known.

In fact, all the interest rates mentioned in previous chapters are inflation-free interest rates.

As can be seen later, in the absence of inflation, the market interest rate is the same as the inflation-free interest rate.




In calculating any cash flow equivalence, three common cases are as follows:

1. Case 1: All cash flow elements are estimated in constant dollars

2. Case 2: All cash flow elements are estimated in actual dollars

3. Case 3: Some of the cash flow elements are estimated in constant dollars, and others are estimated in actual dollars.

For Case 3, simply convert all cash flow elements into one type-either constant or actual dollars, then proceed with either constant-dollar analysis as for Case 1or actual-dollar analysis as for Case 2.



Constant-Dollar Analysis Constant-Dollar Analysis

Equivalence Calculations When Cash Flows Are Stated in Constant Dollars



Actual-Dollar Analysis Actual-Dollar Analysis

To find the equivalent present worth of this actual dollar amount (An) in year n, to use either:

Deflation method: – Step 1: Bring all cash flows to a common purchasing power.– Step 2: Consider the earning power

Adjusted-discount method:– Combine Steps 1 and 2 into one step


Applied Software Project ManagementEquivalence Calculation When Cash Flows Equivalence Calculation When Cash Flows Are in Actual Dollars: Deflation MethodAre in Actual Dollars: Deflation Method

Applied Instrumentation, a small manufacturer of custom electronics, is contemplating an investment to produce sensors and control systems that have been requested by a fruit-drying company. The work would be done under a proprietary contract that would terminate in five years. The project is expected to generate the following cash flows in actual dollars:


n Net Cash Flow in Actual Dollars

0 -75,000

1 32,000

2 35,700

3 32,800

4 29,000

5 58,000


a) What are the equivalent year-zero dollars (constant dollars) if the general inflation rate (f) is 5% per year?

b) Compute the present worth of these cash flows in constant dollars at i’ = 10%.

Given: Cash flows stated in actual $, f = 5%, and i' = 10%. Find: Equivalent present worth using the deflation method.



a) We convert the actual dollars into constant dollars as follows:



Deflation Factor Net Cash Flow in Constant Dollars

0 -75,000 1

1 32,000 (1+0.05)-1 30,476

2 35,700 (1+0.05)-2 32,381

3 32,800 (1+0.05)-3 28,334

4 29,000 (1+0.05)-4 23,858

5 58,000 (1+0.05)-5 45,445


b) We use i' = 10% to compute the equivalent present worth of constant dollars


N Net Cash Flow in Constant Dollars

Deflation Factor Net Cash Flow in Constant Dollars

0 -75,000 1 -75,000

1 30,476 (1+0.1)-1 27,606

2 32,381 (1+0.1)-2 26,761

3 28,334 (1+0.1)-3 21,288

4 23,858 (1+0.1)-4 16,295

5 45,445 (1+0.1)-5 28,218


The adjusted-discount method performs deflation and discounting in one step

Since the market interest rate i reflects both the earning power and the purchasing power

(1+i)n = (1+f)n(1+i’)n or

i = i’ + f + i’f This equation implies that the market interest rate is a

function of two terms i’ and f


nnn

nn

n

)i'(1f)(1A

)i'(1f)(1

A

nP

nn

n i)(1

Ap

Applied Software Project ManagementEquivalence Calculation When Cash Flows Are Equivalence Calculation When Cash Flows Are in Actual Dollars: Adiusted-Discounted Methodin Actual Dollars: Adiusted-Discounted Method

Consider the cash flows in actual dollars in above Example. Compute the equivalent present worth of these cash flows, using the adjusted-discount method.

Given: Cash flows stated in actual $, f = 5%, and i' = 10%. Find: Equivalent present worth using the adjusted-discounted method.



0 -75,000

1 32,000

2 35,700

3 32,800

4 29,000

5 58,000

Applied Software Project ManagementEquivalence Calculation When Cash Flows Are Equivalence Calculation When Cash Flows Are in Actual Dollars: Adiusted-Discounted Methodin Actual Dollars: Adiusted-Discounted Method

First, to determine the market interest rate i:

i = i’ + f + i’f = 0.1 + 0.05 + 0.1*0.05 = 15.5%


n Net Cash Flow in Constant Dollars

Factor Net Cash Flow in Constant Dollars

0 -75,000 1 -75,000

1 32,000 (1+0.155)-1 27,606

2 35,700 (1+0.155)-2 26,761

3 32,800 (1+0.155)-3 21,288

4 29,000 (1+0.155)-4 16,295

5 58,000 (1+0.155)-5 28,218


Mixed-Dollar Analysis Mixed-Dollar Analysis

A couple wishes to establish a college fund at a bank for their five-year-old child. The college fund will earn 8% interest compounded quarterly. Assuming that the child enters college at age 18, the couple estimates that an amount of $30,000 per year. in terms of today's dollars, will be required to support the child's college expenses for four years. College expenses are estimated to increase at an annual rate of 6%. Determine the equal quarterly deposits the couple must make until they send their child to college. Assume that the first deposit will be made at the end of the first quarter and that deposits will continue until the child reaches age 17. The child will enter college at age 18. and the annual college expense will be paid at the beginning of each college year. In other words. the first withdrawal will be made when the child is 18.

In this problem, future college expenses are expressed in terms of today's dollars, whereas the quarterly deposits are in actual dollars. Since the interest rate quoted for the college fund is a market interest rate: we may convert the future college expenses into actual dollars:




Given: A college savings plan, i = 2% per quarter, N = 12 years. Find: Amount of quarterly deposit in actual $.

First to covert future College expense into actual dollars:


Age College Expense in Today’s Dollar

College Expense (in Actual Dollars)

18 30,000 30,000(F/P,6%,13) = 63,988

19 30,000 30,000(F/P,6%,14) = 67,827

20 30,000 30,000(F/P,6%,15) = 71,897

21 30,000 30,000(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.

Assume the couple deposit an amount of C dollars quarterly. The college expenses as well as the quarterly deposit series now in

actual dollars shown in Figure below:


18 19 20 21

Age 5 6 7 …………………………………...16 17

6398867827

7189776211

C



We first select n = 12, or age 17, as the base period for our equivalence calculation, i.e. to calculate total equivalent money V1 deposit at age 17 and the equivalent worth V2 of total money expense also at age 17, then set V1 = V2 to find C

Interest Rate: APR = 8% or 2% quarterly

V1 = C(F/A,2%,48) (deposit quarterly for 12 years -> total 48 periods)

V1 = 79.3535C

V2 = 63,988(P/F,2%,4) + 67,827(P/F,2%,8) + 71,897(P/F,2%,12) + 76,211(P/F,2%,16)

V2 = 229,211

V1 = V2 → C = 2,888.48



TutorialTutorial

Do end chapter problems: 4.2, 4.6, 4.12, 4.16, 4.18, 4.19,


equivalence calculations under inflation - engineering economics

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