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Time Value of Money and Investment Analysis

Explanations and Spreadsheet Applications

for Agricultural and Agribusiness Firms

Part II.

by

Bruce J. Sherrick

Paul N. Ellinger

David A. Lins

V 1.2, September 2000

The Center for Farm and Rural Business Finance

Department of Agricultural and Consumer Economics

and

Department of Finance

University of Illinois, Urbana-Champaign


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Time Value of Money and Investment Analysis

Table of Contents

Part II: INVESTMENT ANALYSIS AND CAPITAL BUDGETING . . . . . . . . . . . . . . . . . . 17

Information Needs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Expected Net After-Tax Cash Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Discount Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Planning Horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Terms of Financing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Marginal Tax Bracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Net Present Value Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Special Problems in Measuring Cash Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Working Capital Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Sunk Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Opportunity Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Synergies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Diversions and cannibalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Internal Rate of Return (IRR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Problems with IRR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Different Lengths of Life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32


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1Bierman, H., Capital Budgeting in 1992: A Survey, Financial Management, Fall 1993: 24.

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TIME VALUE OF MONEY AND INVESTMENT ANALYSIS

PART II: INVESTMENT ANALYSIS AND CAPITAL BUDGETING

This section develops the necessary background for investment analysis and capital budgeting

techniques using the methods discussed and described in Part I. Building on the materials from Part I,

the impacts of taxes, differential financing terms, depreciation and other items affecting cash flows, are

each addressed. In addition, methods for dealing with unequal lives or different sizes of competing

projects are discussed, as are special problems in ranking competing projects.

The two basic issues associated with an investment decision are its feasibility and its desirability.

Feasibility refers to the ability to actually access the necessary capital and complete the project. Clearly

an investment could promise highly desirable levels of returns, but simply be infeasible in terms of the

initial cash flow requirements -- the simplest illustration of a budget constraint. Alternatively, an

investment may be affordable and completely feasible, but the returns could be so low as to make the

investment alternative unattractive. In the examples which follow, it is shown how investments can be

evaluated in the context of both returns and cash flow feasibility.

There are a variety of ways by which one can evaluate returns on investment options. Five of

the most common are: (1) net present value method, (2) internal rate of return methods, (3) profitability

index or Q methods, (4) payback or breakeven period methods, (5) the average rate of return on

investment. A study inFinancial Management indicated that the capital budgeting practices employed

most by large firms to make decisions were internal rate of return methods (88%) and net present value

methods (63%).1 Both the payback and the average rate of return approaches fail to account for the

time value of money. And, the profitability index simply a variant of the NPV approach that is used to

control for size effects. Consequently, the following discussion focuses on net present value and internal

rate of return methods of evaluating investments. These two methods are the most commonly


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employed in practice, both account for the time value of money, and together they provide the most

meaningful decision making information in investment analysis and capital budgeting situations.

Information Needs

The following information is needed to evaluate investments using time value of money

concepts:

(1) the expected net after-tax cash flows (NATCF, or ATNCF) for the investment by

period including a salvage value, if any,

(2) an appropriate interest rate or discount rate,

(3) the length of planning horizon,

(4) terms of financing if borrowed funds are used,

(5) the marginal tax bracket of the borrower, and the taxability status for each cash flow.

A brief discussion of each of these items follows:

Expected Net After-Tax Cash Flow: Note that net cash flows from the business rather than

accounting profits are used in capital budgeting applications. The net cash flow is the stream of cash

that is either required as an outflow, or that could be withdrawn by the owner or reinvested in the

business. The cash inflows include all flow values generated by the investment, and must be considered

at each point in time at which the flows occur, whether they are actually withdrawn or not. Cash

outflows include the original payment for the investment itself, taxes, and all other expenses to

implement the investment. These net cash flows for every period must then be put on an after tax basis,

before discounting.


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The net after tax flow per period can normally be calculated by the following procedure:

i). Cash inflow - cash outflow = net before tax flow

ii). Net before tax flow - depreciation = taxable income

iii). Taxable income * marginal tax rate = tax paymentiv). Cash inflow - cash outflow - tax payment = net after tax cash flow

Consider the following example described in Table 1 below. A firm is considering an

investment which costs $50,000 at inception and is expected to last 10 years. The investment is

expected to generate additional cash income of $13,000 per year and $2,000 of additional cash

expenses per year. The investment can also be depreciated to zero on straight-line basis over 5 years,

generating a noncash deduction against taxable income of $10,000 per year. The investment is

expected to have a salvage value of $2,000 at the end of 10 years and the firm is in a 30% marginal tax

bracket. Given this situation, the after-tax cash flows can be shown in Table 1 below: Note that if

borrowed money were used to purchase the original investment, the interest portion of any loan

payments would have been an expense to reduce taxable income, and the principal payments represent

additional negative cash flows, and some of the initial negative cash flow would have been offset by

borrowed funds. The additional issues created by debt financing are treated later.


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Table 1. After Tax Cash Flow Computations

Year 0 (Now) -$50,000 (Outflow for Investment)

Years 1-5 $13,000

-$2,000$11,000

-$10,000

1,000

x .30

$300

Added Cash Income

Added Cash ExpensesCash Income Before Depreciation

Depreciation

Taxable Income

Marginal Tax Rate

Taxes Paid

Cash Flow/Year = $11,000 - $300 = $10,700/yr.

Year 6-10 $13,000

-$2,000

$11,000-0

$11,000

x .30

$3,3000

Added Cash Income

Added Cash Expenses

Cash IncomeDepreciation

Taxable Income

Marginal Tax Rate

Taxes Paid

Cash Flow/Year = $11,000 - $3,300 = $7,700/yr.

Year 10 $2000

x .30

$600

Salvage Value

Marginal Tax Rate

Taxes Paid

Cash Flow = $2,000 - $600 = $1,400

Discount Rate: The discount rate reflects the appropriate cost of capital or rate of return on

the investment (often an interest rate, or cost of capital calculation). Several different concepts underlie

the choice of an appropriate discount rate. The first concept is a "consistency principle" -- the idea that

the discount rate should be consistent with the cash flows being discounted. For example, if nominal

after-tax cash flows are being discounted, then a nominal after-tax discount rate should be used. If realcash flows are being discounted, then a real discount rate should be used. In other words, the discount

rate should be in the same form as the cash flows being discounted.


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A second concept in determining discount rates relates to the issue of risk. The discount rate

chosen should reflect the degree of risk associated with the investment under consideration. For

example, suppose you are considering two investments--A and B. Suppose further that the expected

cash flows for investment A are quite low, but very certain, while expected cash flows for B are muchhigher, but much more uncertain. Using the same discount rate would not reflect this difference in risk

and could bias decisions toward acceptance of risky projects.

A third concept in identifying the discount rates relates to the effects of the investment on the

capital structure--the proportion of debt capital versus equity capital--of the firm. For large diversified

firms that can easily adjust capital structure through debt and equity issuances, it is quite common to

measure the discount rate as the weighted average cost of debt and equity capital, adjusted for the

degree of risk if the new project has a different degree of risk than the firms overall assets. For smaller

firms, such as many farms and smaller agribusiness firms, it is common to use the cost of equity capital

as a measure of the discount rate. This approach recognizes that for small firms, a new investment may

significantly alter the capital structure of the firm. Under this approach, the discount rate reflects what

equity capital could earn in its best alternative use.

Planning Horizon: The planning horizon is the length of time over which the project is being

evaluated. It is normal to use the length of time over which the project is expected to last or until it is

expected to be sold. For example, suppose you plan to buy a machine, use it for 5 years and then sell

or trade it in on a new machine. The length of planning horizon in this case is 5 years even if machine

purchase were financed with a three year loan.

In evaluating several investment alternatives, care must be taken if the investments have unequal

lives or different planning horizons. A subsequent section identifies procedures for comparing

investments which have different planning horizons.


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Terms of Financing: The terms of financing include the amount of downpayment, the length of

loan, frequency of payments, interest rate, and loan servicing charges. In evaluating investment

alternatives it is important to recognize that differences in terms of financing between two investment

options can have a significant impact on the ranking of the two investment alternatives. This issue is ofparticular importance on investments involving the purchase of land where the buyer may have an

alternative between seller financing and financing provided by a commercial lender. An example

provided later shows how to evaluate the trade-offs between price and interest rate terms on land

purchases.

Marginal Tax Bracket: The tax bracket of an investor is often a crucial variable which

influences investment decisions. The relevant tax bracket in investment analysis is the marginal tax rate,

or what percentage of taxes would be paid on additional or incremental taxable income associated with

the project. Use of a marginal tax rate rather than an average tax rate is consistent with the economic

principle of evaluating investments according to what they add or subtract from an existing business.

Net Present Value Approach

The following example provided in Table 2 illustrates the appropriate procedures for solving

simple investment-type problems. Consider an investment that costs $6,000 initially. It is expected to

yield a net cash inflow of $1,500 per year for 5 years and have an expected salvage value of $1,500 at

the end of 5 years. Assume an 8 percent after-tax discount rate.


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Table 2. Net Present Value Calculations

Period

Net After-Tax Cash

Flow

8% Present

Value Factor*

Present Value

(Rounded)

0 -$6,000 1.0 -$6,000

1-4 +$1,500 3.31213 +$4,968

5 +$3,000 .68058 +$2,042

Net Present Value (NPV) = $1,010

*For single payments, the PV factor is the same as the SPPV factor per $1 payment, or (1+r)-n. For a series, as in the

middle row labeled Period 1-4", it contains the results of the USPV formula per $1 needed to generate the value as of

the initial time period.

A negative cash flow reflects the outflow (in this example, the initial payment for the purchase)

while the positive signs indicate net inflows. A common criterion for acceptance or rejection of a given

project, abstracting from risk considerations and budget constraints, is to accept projects with positive

NPV values and reject those with NPV values less than zero. At NPV = 0, the return is equal to the

cost of capital, and the investor would be indifferent to making the investment or not. In the example

above, note the technique used to discount the equal $1,500 flow per year for four years. The USPV

factor can be used to discount an entire stream of income with one calculation. In year five, there is a

salvage value of $1,500 plus the $1,500 net operating inflow, for a total of $3,000. For amounts

different than the uniform flow, the SPPV factor must be applied. A positive NPV of $1,010 for the

sample problem suggests the investment should be made.

It is important to recognize that the NPV of $1,010 is not the amount of profit made by

undertaking this investment. Rather, it is the amount by which this investment exceeds the return from

the next best investment. The next best investment generates an 8% after-tax return as reflected by the

discount rate. Remember, net present value is a measure of net cash returns expressed in today's

dollars, and is not a measure of profit made on an investment.


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Special Problems in Measuring Cash Flows

In discounting after-tax cash flows, it is important to include all cash flow items that belong in the

analysis and exclude those that do not belong. As a general rule, all incremental cash flows should be

included, i.e., all cash flows which result from making the investment. In cases where two competingprojects are made, many of the incremental cash flows or benefits are equivalent between the two

options and can be canceled by evaluating the NPV of one project minus the other. But in most cases,

all flow values that differ with or without the project should be included. Sometimes these incremental

cash flows are complicated to isolate, or appear to exist when they really do not. Below, some of the

common pitfalls in measuring cash flows are provided along with guidance on how they should be

treated.

Working Capital Requirements Most investment opportunities for agricultural businesses, or

other businesses require some capital expenditure including the purchases of land, buildings or

machinery. For example, a hog producer may build new confinement facilities to expand the size of the

hog operation, or a meat packer might build and equip a new slaughter house. In doing a net present

value calculation of these investment alternatives, it is important to remember that working capital

requirements will also likely increase as a result of the expansion. For example, the hog producer will

likely have more feed and livestock inventory as a result of the expansion. Likewise, the packer will

have a larger inventory of slaughtered hogs. These increases in working capital must be accounted for

as a cash outflow to do a proper net present value calculation of the expansion option. If this working

capital is sold or returned at the end of investment period, an inflow of cash occurs from the sale of

inventories or other items that return working capital at the end of the investment, much like a salvage

value. In any case, the investment of additional working capital should be treated in terms of its

incremental cash flows.

Sunk Costs Suppose a farmer spends $1,000 over a 6 month period searching for a property

to expand the size of the operation. A parcel is now found and the farmer wants to do a net present


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value analysis of the purchase. However, the $1,000 spend searching for the property should not be

included as a cash outflow in this calculation. The reason is that it is now a sunk cost and will exist

whether the farmer does or does not buy the current property under consideration. Thus, it has no

bearing on the financial rewards from the decision at hand. Although it is a real expense, it is not anincremental cash flow associated with the investment because it is not affected by the acceptance or

rejection decision about the project.

Opportunity Costs Suppose an agribusiness firm is considering a new product line that will

use up existing excess capacity in the firm's manufacturing plant. As a result, the firm will need to

expand the size of their plant at the end of the second year rather than at the end of the fourth year.

Should a net present value analysis of the new product line include a charge for using up existing excess

capacity. The answer here is yes, because it causes the firm to rebuild in year two versus the current

plans which call for expansion in the fourth year.

As another example, suppose a farmer is doing a net present value analysis of a slurry store

handling system versus a lagoon system. To construct the lagoon, the farmer will need to take 2 acres

of land out of production. In doing the net present value analysis, one should account for the

opportunity cost of using the land for something other than a lagoon. Perhaps more importantly in this

case, there is an important contingency cost that can be thought about as the cost of insuring against

liability created by the lagoon. Many companies explicitly budget to fund a legal reserve associated

with new projects.

Synergies Suppose you were considering the addition of a new project that utilized products

produced elsewhere in your operation, or otherwise contributed to the profitability of your operation.

For example, suppose you run a pick your own orchard and are considering the addition of pony

rides and Halloween hayrides. In addition to measuring the direct cash flows from the additional

projects, the potential increase in apple sales while the new customers are on site should be included.


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Diversions and cannibalism Suppose you now running a pick your own orchard and are

considering the addition of a cider press to expand your product line. Clearly the value of the apples

used in production should be accounted for in the NPV calculation, but so should any reduction in fresh

apple sales that might occur if the availability of cider reduces the demand for fresh apples. (Theexample might be more obvious if prunes were replaced by apples, except that the authors know of no

prune orchards). Cannibalism effects have been frequently noted in food manufacturing businesses.

For example, when Post introduced Dino Pebbles to compete with Kelloggs Marshmallow Krispies,

the main effect was in the reduction of sales of Posts other similar line -- Fruity Pebbles. This case

illustrates the importance of including any lost revenues associated with other projects owned by the

same person or firm as a cost of the project being evaluated.

Internal Rate of Return (IRR)

NPV analysis results in a dollar-valued answer based on discounting cash inflows and cash

outflows. Given no capital constraints, all projects generating positive NPV values would be accepted.

However, because the NPV is simply a dollar value, it does not provide a measure of the rate of return

generated by the project. And, NPV results are not always sufficient to evaluate the desirability of two

very different sized projects. For example, suppose project A has NPVA = $100, and project B has

NPVB = $105. If they were otherwise equal, project B would be more desired. However, if project B

had twice as large an initial cost, and its acceptance prohibited other positive NPV projects, then

project A may be more desirable. Thus, complementary information about the effective yield provided

per dollar of cash flow in the project is also useful to know. That measure is the internal rate of return

or IRR. Its solution uses the same principles that are employed in NPV calculations except that the

discount rate being solved for is the one that results in the project having an NPV of zero. In this sense,

it is simply the highest costs of capital that would make the project exactly break even in terms of theNPV. If the actual discount rate or cost of capital is less than the IRR, then the project is viewed as a

desirable project because it generates more than it costs. If the IRR is lower than the costs of capital or

negative, then the project is not desirable and should be rejected.


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To illustrate, the example problem used above to illustrate the basic NPV approach is solved for

its IRR. The IRR is the discount rate that results in the present value of the inflows and the present

value of the outflows being exactly equal and thus resulting in NPV equal 0. Because the discount rate

in the time value of money formulas is inside an equation that often has an exponent greater than one, itoften cannot be algebraically isolated and analytically solved. Fortunately, most financial calculators can

be used to compute an IRR for the basic types of problems. Moreover, trial and error approaches

converge fairly rapidly toward a solution, and simple interpolation methods are easily used at any stage

of a trial and error process. Finally, the spreadsheet supplied with this booklet also contains a simple

facility to calculate the IRR for most analyses.

In attempting to solve for the IRR by hand, a trial and error process can be used to bracket the

answer fairly quickly. Depending on the precision needed, the search can then be stopped and any two

answers used to interpolate (or extrapolate) an approximate answer. To do so, the common steps are:

(1) guess a discount rate and solve for the NPV. If the NPV is positive, then increase the discount rate

and try again. (2) If the trial NPV is negative, decrease the trial discount rate and recalculate. (3)

Repeat this search until you have two trial discount rates that bracket the answer (one negative NPV

and one positive NPV). Then, (4) a manual method of bisection search can be used to locate the exact

solution. The bisection involves the following approach. First take the average of the two rates that are

known to bracket the IRR and recompute the NPV at the average rate. If the NPV is positive, use the

average and the higher rate for the next bracket. If the NPV at the average rate is negative, then use

the average and the lower rate for the next bracket. Then, re-average the rates from the new bracket

and repeat the process until the NPV is suitably close to zero. At any time, the two rates that bracket

the IRR can also be used in a process known as interpolation to approximate the answer. The

interpolation process is illustrated below using the same example that was used to demonstrate the

NPV technique above.


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Table 3. Interpolation Calculation Inputs

Present Value Factors Present Value

Period

Net After

Tax Cash Flow 12% 14% at 12% at 14%

0 -$6,000 1.0 1.0 -$6,000 -$6,000

1-4 +1,500 3.03735 2.91371 +$4,556 +$4,371

5 +3,000 .56743 .51937 +$1,702 +$1,558

Net Present Value (NPV) = +$258 - $71

Table 3 above shows that at a 12 percent discount rate, the NPV is still positive, but at 14

percent it is negative, an thus the IRR is bracketed between 12 and 14 percent. To illustrate the idea of

interpolation, first refer to the figure below.

Interpolation relies on simple proportionality arguments as follows. If items a, b, c, d, andfare known,

then e can be found by noticing that e is, in a proportion sense, at the same relative position between d

andfas b is between a and c. Thus, even if the scale for the items below the line differs from the items

above the line, the fractions of the distances will be the same. It is an identity that b = a + (b-a), and

multiplying the term in parenthesis by (c-a)/(c-a) leaves it unchanged, thus:

[13]b a

b a

c ac a= +

( )

( )*( )

And, (b-a)/(c-a) equals (e-d)/(f-d), and thus, by proportionality:

[14] e db a

c a f d = +

( )

( )* ( )


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To use this technique to solve for the IRR, note that two discount rates are known and three NPVs are

known, namely $258, -$71, and $0 the point for which the IRR is needed. Graphically,

Thus, IRR = 14% + ($0-(-$71))/($258-($-71)) * (12%-14%), or in this case13.57%. Be careful not

to be confused by the signs and negative numbers, and make certain to keep the proper discount rates

associated with the proper NPVs when doing these calculations.

The interpolation process is only a close approximation of the true IRR. Using the previous

example, a more accurate numeric search produces a resulting IRR of 13.55%. The relationship

between the discount rate and the NPV is not linear, and thus the linear approximation provided by the

interpolation will not be exact. Of course, the wider the range of values over which you interpolate, the

greater the potential degree of inaccuracy in your answer. And, surprisingly, the error is usually greatest

when the IRR is approximately evenly bracketed by the endpoints, with the approximation improving

the closer the sought value is to one of the endpoints. Thus, it is more important to get at least one of the

endpoints to have an NPV near zero than to find similar sized positive and negative values.

Problems with IRR

In addition to problems associated with calculating an IRR, there are several other issues with

which the user should be aware. First, if the series of cash flows has more than one sign reversal

(changes from a positive to a negative cash flow, or vise versa) then there are multiple solutions. For

example, there are two sign changes in the following series of cash flows (Table 4 below) and thus we

have two IRRs, in this case equal to 25% and 400%.


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Table 4. Multiple IRRs Illustrated

Time Cash Flow

0 - 4,000

1 + 25,000

2 - 25,000

In fact, according to Descartes rule of signs, there are as many roots (solutions) as there are

changes in signs, so a problem with 4 sign reversals would have 4 different solutions. To deal with this

issue, a modified internal rate of return, or MIRR, is often used. Under this approach, all negative cash

flows are first treated as a single problem and placed into an equivalent negative single present value.

Then, all positive cash flows are treated as a single problem and represented as a single positive future

value. Finally, NPV methods are applied to the two values the negative single initial value and the

positive single future value as though these were the only two cash flows and therefore having only one

solution. Note that to use the MIRR approach, a discount rate is needed in the first stage to compute

the single positive and negative values. It is suggested that the firms cost of capital be used for this

stage to compute both the present value of the cash outflows and the future value of the positive cash

flows. Denoting the present value of the cash outflows as P0-

and the future value of the positive cashflows as Pn

+, the MIRR is the discount rate, r, that solves:

[15] P0-(1+r)n = Pn

+

The MIRR is also used in cases where the IRR is unrealistically influenced by the assumption that the

cash flows can be reinvested in the project at the same long-run rate of return. In this case, using the

MIRR approach is useful because a lower, often more realistic cost of capital is used to first convert

cash flows to a future value that is then used to solve for an MIRR assuming that cash flows are taken

out of the project and employed elsewhere in the firm at a return equal to the discount rate.


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A second potential problem is if that if all cash flows are all of the same sign, then no solution

exists. For example, if an investment generates only positive cash flows, then the NPV of the

investment can easily be calculated, but an IRR does not exist as it represents an infinite return. The

example below in Table 5 illustrates this point (however, few projects generate only positive cash flowswith no initial investment, so in practice this problem is rare).

Table 5. Illustration of Non Existent IRR

10% Present Present

Time Cash Flow Value Factor Value

1 1,000 .90909 909.09

2 3,000 .82645 2,479.35

3 3,000 .75131 2,253.93

Net Present Value 5,642.37

IRR NA

A third issue in interpreting IRR is that of borrowing versus lending or investing. Consider the

following projects A and B, with cash flows as shown in Table 6.

Table 6. IRRs under Lending versus Borrowing

Cash Flows at: NPV at:

Project t=0 t=1 IRR 10 percent

A -2,000 +2,400 +20% $181.82

B +2,000 -2,400 +20% -181.82

Notice each project has an IRR of 20%, but A has a positive NPV while B has a negative NPV.

Project A represents an investment which is, in essence, lending money at a rate of 20%. When lending


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money, the highest IRR is preferred. For Project B is in essence a case of borrowing with a positive

cash flow today followed by the repayment at time 1. In this case, the IRR indicates an interest rate on

borrowed money of 20%. When borrowing, the lowest IRR is preferred. The point is that the IRR

alone does not indicate whether the project is a borrowing or lending style investment, a fact whichcan lead to confusion on how to interpret the IRR number.

Different Lengths of Life

Frequently, otherwise comparable investments have different lengths of life. With the simplest

NPV approach, the net present value of two investments cannot be directly compared if different time

horizons are involved. There are two alternatives to correctly solves such problems, as illustrated in the

following example.

Assume you have two alternative machinery investment programs which would perform equally

for you. The goal is to compare the present value of the cost streams for these two alternatives,

allowing one to choose the alternative that offers the lowest cost stream. The specific characteristics

are in Table 7 below.

Table 7. Unequal Length of Life Comparisons

Machine X Machine Y

Original Cost $10,000 $12,000

Projected Economic Life (planning horizon) 4 years 6 years

Projected After Tax Annual Cash Outflows $3,200 $3,500

Neither machine has a salvage value. The discount rate is 10 percent.

Machine X Machine Y

10% 10%

Period Flow Factor P.V. Period Flow Factor P.V.

0 -$10,000 1.0 -$10,000 0 -$12,000 1.0 -$12,000

1-4 -$3,200 3.16987 -$10,144 11-6 -$3,500 4.35526 -$15,243

NPV = -$20,144 NPV = -$27,242


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It is incorrect to compare the two net present values since the machines have different lengths of

life. The easiest way to put the two analyses on a comparable basis is to put each on an annuity

equivalent (AE) basis. Under the AE approach, the size of annual annuity is determined for theeconomic life of the investment that could be provided by a sum equal to the present value of its

projected cashflow stream. To use this approach, the capital recovery formula is applied to determine

the annual equivalent cash flow to be able to compare investments with different lives. Comparing the

two options, Machine X has an annuitized value of its NPV = -$20,144 of -$6,355 per year for 4

years and Machine Y has an annuitized value of its NPV = -$27,242 of -$6,255 per year for 6 years.

Note that these values are equivalent to the size of loan payments needed to pay off a 4 year $20,144

loan and a 6-year $27,242 loan respectively. In this case, Machine Y is preferred, as it has the lowest

annuity equivalent, even though the NPV of its cost was greater.

The second possible alternative for solving problems with different lives is to use a least

common denominator of time and figure more than one life span. This approach is known as a

replacement chain and simply involves repeating both investments until they end at the same point in

time. In the above problem, twelve years is the first point in time that repeated investments in each

machine would end. Machine X would be replaced at the end of year 4 and year 8. Machine Y would

be replaced at the end of year 6 as shown in Table 8.

Table 8. Replacement Chain Comparison of Unequal Length of Life Investments

Machine X Machine Y

10% 10%

Period Flow Factor P.V. Period Flow Factor P.V.

0 -$10,000 1.0 -$10,000 0 -$12,000 1.0 -$12,0004 -$10,000 .68301 -$ 6,830 6 -$12,000 .56447 -$ 6,774

8 -$10,000 .46651 -$ 4,665 1-12 $ 3,500 6.81369 -$23,848

1-12 -$ 3,200 6.81369 -$21,804

NPV = -$43,299 NPV = -$42,622


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This process puts the two investments on an equal time horizon and generates conceptually

correct answers. The NPVs shown above can be directly compared this time since the replacement

chains are of equal length. Again, Machine Y is shown to be the best investment, since it has the lowerNPV of the cost stream.

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