basic methods for making economy study (written report)

8/13/2019 Basic Methods for Making Economy Study (Written Report)

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Chapter 5

BASIC METHODS FOR MAKING ECONOMY STUDIES

Objective: To demonstrate the mechanics of calculations for six basic methods for making economy study

and to briefly describe underlying assumptions and interrelationships of those methods

INTRODUCTION

All engineering economy studies of capital projects should be made so as to include consideration of the

return that a given project will or should produce. Since the patterns in capital investment, revenue or

saving flows, and cost flows are quite different in various projects, there is no ideal method for making

economy studies. Consequently, several methods, or patterns are commonly used in practice, and all will

produce equally satisfactory results and will lead to the same decision in cases where the inherentassumptions of each are applicable. There are six basic methods or patterns for making economy studies.

These are:

Annual Worth (A.W.); Present Worth (P.W.); Future Worth (F.W.); Internal Rate of Return (I.R.R.); External Rate of Return (E.R.R.); and Explicit Reinvestment Rate of Return (E.R.R.R.).

The first three methods convert all financial happenings into equivalent worths at some point or points intime using an interest rate equal to the cost of capital , or the minimum attractive rate of return

(M.A.R.R.). The last three methods are different ways to calculate a rate of profit or savings (annual

return as per cent of investment) so that this can, in turn, be compared against the M.A.R.R.

ANNUAL WORTH METHOD

Annual Worth (A.W.) refers to a uniform annual series of net cash flows for a certain period of time that

is equivalent in amount to a particular schedule of cash inflows (receipts or savings) and/or cash outflows

(disbursements or opportunity cost) under consideration. The criterion for this method is that as long asthe net annual worth (i.e., annual equivalent of inflows minus outflows) is ≥ 0 the project is economically

justified; otherwise, it is not justified.

Calculati on of Capital Recovery Cost

The capital recovery cost (C.R.) for a project is the equivalent uniform annual cost of the capital

invested. It is annual amount which covers the Depreciation (loss in value of the asset) and Interest

(minimum required profit) on invested capital.


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There are several convenient formulas to solve for capital recovery cost to obtain the same

answer as above. One of which is to find the annual equivalent of the investment and then subtract the

annual equivalent of the salvage value. Thus

eq. 1 where P = investment at beginning of life

F = salvage value at end of life

N = life of project

Applied to example above,

Another way to find the C.R. cost is to add the annual sinking fund depreciation charge (ordeposit) to the interest on original investment (sometimes called minimum required profit). Thus

eq. 2 Applying to above problem,

Yet another equation is to add the equivalent annual cost of the depreciable portion of the investment and

the investment and the interest on the nondepreciable portion (salvage value). Thus

eq. 3 Applying again to the above problem,

Example 5- 1: An investment of $10,000 can be made in a project that will produce uniform annual

revenue of $5,310 for 5 years and then have a salvage value of $2,000. Annual disbursements will be

$3,000 each year for operation and maintenance costs. The company is willing to accept any project that

will earn 10% or more, before income taxes, on all invested capital. Show whether this is a desirable

investment using the annual worth method.


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Solution:

Annual WorthAnnual Revenue $5,310Annual Disbursements -$3,000C.R Cost = ($10,000 -

$2,000)(A/P,10%,5) +$2,000(10%) - 2,310

Total -$5,310 Net A.W. $ 0

Since net A.W. =$0, the project earns exactly 10% and is thus barely justified.

PRESENT WORTH METHOD

The present worth (P.W.) method for economy studies is based on the concept of equivalent worth of allcash flows relative to some base or beginning point in time called the present . That is, all cash inflows

and outflows are discounted back at an interest rate that is generally the M.A.R.R. The criterion for this

method is that as long as the net present worth (i.e., annual equivalent of inflows minus outflows) is ≥ 0

the project is economically justified; otherwise, it is not justified.

Example 5-2: Consider the same project as in Example 5-1, show whether it is justified using the P.W.

method.

Solution:

Present WorthAnnual Revenue: $5,310(P/A,10%,5) $20,125Salvage Value: $2,000(P/F,10%,5) 1,245Investment -$10,000Annual disbursements:$3,000(P/A,10%,5) - 11,370

Total -$21,370 Net A.W. $ 0

Since net P.W. = $0, the project is once again shown to be barely justified.

FUTURE WORTH METHOD

The future worth (F.W.) method for economy studies is exactly comparable to the present worth method

except that all cash inflows and outflows are compounded forward to a reference point in time called the


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Eq.4

future. The criterion for this method is that as long as the net future worth (i.e., annual equivalent of

inflows minus outflows) is ≥ 0 the project is economically justified; otherwise, it is not justified.

Example 5-3: Consider the same project as in Example 5-1, show whether it is justified using the P.W.

method.

Solution:

Future WorthAnnual Revenue: $5,310(F/A,10%,5) $32,420Salvage Value 2,000Investment: $10,000(F/P,10%,5) -$16,105Annual disbursements:$3,000(F/A,10%,5) - 18,315

Total -$34,420 Net A.W. $ 0

Since the net F.W. = $0, the project is again, not surprisingly, shown to be barely justified.

Rate of return which is also known as rate of investment (ROI), rate of profit or just rate, is defined as the

ratio between the money gained or lost on an investment and the amount of money invested. ROI is a

measure of investment profitability, not a measure of investment size. While compound interest and

dividend reinvestment can increase the size of the investment (thus potentially yielding a higher dollar

return to the investor), Return on Investment is a percentage return based on capital invested. The other

three methods for making economy studies are based on the investment’s rate of return.

INTERNAL RATE OF RETURN METHOD

The internal rate of return or I.R.R. is one of the basic methods for making economy studies in which the

net present value or net present worth of all cash flows from a particular investment is equal to zero. To

make it more specific, the I.R.R. of an investment is the discount rate at which the net negative cash flows

(or the net present worth of costs) of the investment is equal to the net positive cash flows (or the net

present worth of benefits) of the investment.

I.R.R. method is the most widely used rate of return method for making economic studies. It is commonly

called by others as the investor’s method, discounted cash flow method, receipts versus disbursements

methods or profitability index. It is used to evaluate the desirability of a project by calculating its I.R.R.

and making sure that this value is greater than the minimum acceptable rate of return (M.A.R.R.) of the

project. Mathematically speaking, the internal rate of return is expressed as:

∑

∑


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Eq.5

where Rk = net receipts for k th year

Dk = net disbursements for k th year

N = project life or maximum number of years for study

= Single Payment Present Worth Factor Or, as stated above, the I.R.R. is the i’ % at which the present worth of cash inflow minus the present

worth of cash outflow equals 0 or expressed mathematically is:

∑

∑

There are two ways to solve the I.R.R. (i’ %) from equation 2. The first method is through the use of

advanced calculator and computing software: inputtting the whole equation then wait for the solution of

the equation which is very accurate in terms of its iterative calculation algorithm. The second method is

substituting initial values of the I.R.R. to equation 2 making sure that the two values would result to a positive and negative output, graphing this value through a I.R.R. versus net present worth graph then

obtaining the y-intercept (when the net P.W. is equal to zero) of the line by linear interpolation as shown

in the figure below:

Figure 1. Use of linear interpolation to find the approximation of I.R.R.

As one can see, the interpolation method is not that accurate to determine the I.R.R. of an investment

mainly because of the selection of the two points to generate the line. The error of this method is due tothe nonlinearity of the net P.W. function and it would be lessen if the range between the interest rates

used in the interpolation would be smaller.

0

5

10

15

20

25

30

-4000 -3000 -2000 -1000 0 1000 2000

I n e r n a l R a t e o f R e t u r n

Net Present Worth

I.R.R. vs. N.P.W.


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Example 5-4. Considering the same project as in Example 5-1, determine whether it is justified using the

I.R.R. method.

Solution: Expressing the Present Worth of involved cash flow (same as example 5-2 except that the i’ %

is unknown) and setting it equal to zero yields:

Or simplfying the factors would generate an equation of:

(

) Using scientific calculators, the rate of return is equal to 9.99% or approximately equal to 10% which also

conforms to the result of Example 5-1. Thus the project is shown to be marginally justified.

Using the interpolation method needs an initial i’% values. For this example problem, set i’% values to

5% and 25%. Substitution of this value to the above equation would give a net present worth of $1568and -$3132 respectively. Graphing these data on a interest versus net present worth would generate figure

2 as shown below:

Figure 2. Linear interpolation of Example 5-4

Thus, the I.R.R. of the problem is located at the y-intercept or at N.P.W. is equal to zero. The value of theI.R.R. based on this method is 11.7% which is a little bit off compared to the true I.R.R. which is equal to

9.99%.

0

5

10

15

20

25

30

-4000 -3000 -2000 -1000 0 1000 2000

I n t e r s

e t ( % )

Net Present Worth ($)

I.R.R vs N.P.W


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Eq.6

EXTERNAL RATE OF RETURN

The external rate of return or E.R.R. method involves the assumption that all recovered funds or the net

cash flows can be reinvested at some specified rate of return (usually the M.A.R.R.) until the life or study

period for the project. The calculation of E.R.R. for a single project involves merely finding the interet

rate at which the future worth of the outflows equals the future worth of the inflows. Expressed in general,

the E.R.R. is the i’ % at which:

∑

∑

where Rk = net inflow (excess of receipts over disbursements) for k th year

Dk = net outflow (excess of disbursements over receipts) for k th year

N = project life or maximum number of years for study

= Single Payment Present Worth Factor = Single Payment Compound Amount Factore = reinvestment rate

Example 5-5: Considering the same project as in Example 5-1, determine whether it is justified using the

E.R.R. method assuming that funds can be reinvested at the M.A.R.R. = 10%.

Solution: Expressing the Future Worth of involved cash flows (same as example 5-3 except that the i% is

unknown and e = 10%) gives the following equation:

Or simplifying the factors would genrate the equation:

Thus 10% is the E.R.R. and the project is shown to be marginally justified.

EXPLICIT REINVESTMENT RATE OF RETURN METHOD

The explicit reinvestment rate of return (E.R.R.R.) method is a simple way to solve for the rate of return

when there is a single lump sum investment and uniform cash savings or returns at the end of each period

throughout the life N of the project. The concept behind E.R.R.R. method is to divide the net profit by its

initial investment, where the net profit is calculated using a depreciation charge based on the sinking fund

method of depreciation. The sinking fund depreciation charge is obtained by multiplying the depreciable


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Eq.7

investment by the sinking fund factor (A/F, i%, N). The interest rate used for the sinking fund factor is the

rate at which recovered depreciation funds are assumed to be reinvested and this is usually the same as the

M.A.R.R. Mathematically speaking:

where R = uniform annual receipts or savings

D = uniform annual costs or disbursements

P = investment

F = salvage value

e = reinvestment rate

= sinking fund factor

Example 5-6. Considering the same project as in Example 5-1, determine whether it is justified using the

E.R.R.R. method assuming that depreciation funds can be reinvested st M.A.R.R. = 10%.

Solution:

Annual revenue $5310

Annual disbursements -$3000

Depreciation: ($10000-$2000)(A/F,10%,5) -$1310Total -$4310

Net annual profit $1000

Thus once again, the project is barely justified because the E.R.R.R. just meet the M.A.R.R.

SUMMARY COMPARISON OF ECONOMY STUDY METHODS

Any of the six basic methods (A.W., P.W., F.W., I.R.R., E.R.R. and E.R.R.R.) could be used for economy

study of a certain investment but selecting one of these methods depends on the preference between the

analyst and the decision maker and also the fact that one method is simplier compared to other methods.

In general, the equivalence method such as A.W., P.W. and the F.W. are easier to use in terms of

computation compared to the rate of return methods but for many decision makers, they prefer to analyze

the investment through the rate of returns.


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The three equivalence method ar directly related by the following relations:

A.W. = P.W.( A/P, i%, N ) = F.W.( A/F, i%, N )

P.W. = A.W.( P/A, i%, N ) = F.W.( P/F, i%, N )

F.W. = A.W.( F/A, i%, N ) = P.W.( F/P, i%, N )

These equivalence methods all have assumptions that funds can be reinvested at the i% which is normally

at the M.A.R.R. The same assumption is true for the E.R.R. and E.R.R.R. except that in the E.R.R.R. the

reinvenstment applies only to recovered depreciation funds. For the I.R.R., it assumes that all the funds

are reinvested at a particular I.R.R. rate computed for the project generating those funds.

basic methods for making economy study (written report)

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