Jones D.S.J., Pujado P.R. Handbook of Petroleum Processing

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866 CHAPTER 17

The basis for the analysis is identical to what has been said before, so there is no need

to repeat it here. All that needs to be discussed is the discounting effect over time.

A cash ﬂow analysis is a net present value analysis. If we assume an initial outﬂow

or investment at time zero, the total of the net present values of all future cash ﬂows

must exceed the initial investment for the project to be attractive.

Obviously, the result will be highly dependent on the rate of discounting used to bring

future cash ﬂows to time zero. What rate of discounting should be used? We should

use a rate of discounting equivalent to the money market rate; namely, the rate of

return that we would obtain if we were to grant a loan to others. On this basis, then,

the appropriate rate of discounting is equivalent to the long-term interest rate that we

would be charged for any debt incurred in connection with the construction of the plant.

The rate of discounting should be increased above this value if we feel that the project

involves risks such as market potential, future obsolescence of the project, etc.

If, when we use this rate of discounting, the net present value of all future cash

ﬂows exceeds the net present value of all investments (initial and future, if any), the

project is regarded as proﬁtable relative to a passive interest generating activity. If

the net present value of all future cash ﬂows is less than the net present value of the

investments, the project is relatively less attractive than a passive interest generating

activity.

One may ask, if a project is attractive at a certain rate of discounting, what would

be the rate of discounting that renders the project unattractive? The break-even rate

of return at which the summation of the net present values of all future cash ﬂows

(inﬂows) just equals the net present value of the outﬂows (investments, etc.) is the

so-called internal rate of return (IRR). If, therefore, the IRR exceeds the discounting

rate, the project may be regarded as proﬁtable and economically attractive. If it does

not, the project is not attractive or less attractive. The IRR has no real meaning of its

own; it must always be viewed relative to the appropriate discount rate.

The IRR is also called the discounted cash ﬂow rate of return or DCF rate of return or

DCFRR. Everything else being the same, one can raise or lower the IRR by adjusting

the sale price of the product and, therefore, the projected revenue. If we then ﬁx

the desired IRR, we can calculate the necessary sale price. The desired IRR should

eventually be higher than the discounting rate as explained above.

There is a further complexity. Sometimes, one wishes to examine the overall return

over an extended period—say, 10 or 20 years. Whether we consider an active activity

(i.e., a plant investment) or a passive activity (i.e., a loan to others) the capital is

always returned in discrete parcels over this period. What becomes of this capital

once it is returned? Logically, one can expect that it will be reinvested but, at what

ECONOMIC ANALYSIS 867

rate? Treatments found in the literature often assume that the capital that is returned

can be reinvested at the same rate of return as the original project. This tends to be

unrealistic, principally for highly proﬁtable projects; a more conservative approach

is to assume that the capital that is returned can be reinvested at a then prevailing

rate (constant or varied, depending on the desired complexity) so as to generate a

fund at the end of the economic analysis timeframe. This fund can then be discounted

to time zero to add to the overall cash ﬂow. Depending on the choice of rates and

discount rates, this adjustment can have a positive or negative effect on the overall

analysis.

It is often customary to assume a gradual sales build-up. For example, 60% of capacity

the ﬁrst year, 80% the second year, and 100% the third year and thereafter. A simple

computer program can easily keep track of this and any other assumptions. For the

sake of simplicity, however, we normally assume a uniform production capacity over

the entire period of 10 or 20 years.

Tax accounting, whenever desired, should be done on an extended discounted cash

ﬂow basis. In this format it is easier to keep track of such terms as ﬁnancial deprecia-

tion, tax depreciation, tax liability, taxes paid, carryover losses, investment tax credits,

etc. so as to be able to determine the actual cash ﬂow every year. Even if done on a

constant-dollar basis, the annual cash ﬂows can be expected to vary from year-to-year.

A discounted cash ﬂow analysis can easily keep track of such variations. Again, for

simplicity, the concept can be explained equally well if we assume a uniform stream

of cash ﬂows.

It is helpful to visualize expenses and cash ﬂows as a series of arrows: positive arrows

for positive cash ﬂows (net inﬂows) and negative arrows for outﬂows such as capital

investment and negative cash ﬂows. We start with a fully capitalized investment ﬁgure

at time zero, the start of production.

Let n be the total number of periods. For simplicity, let us assume that n are years and

that we consider either 10 or 20 years.

Let I represent the initial investment at time zero and CF the annual cash ﬂow that is

assumed uniform on a constant-dollar basis (i.e., no inﬂation).

The internal rate of return (IRR) is that at which the cash ﬂows over n years just pay

back the invested capital, or, equivalently, when the capital equals the stream of net

present values:

I =

1 + r

(1 + r )

+···+

(1 + r )

(a)

868 CHAPTER 17

Then, by using the summation formula (Appendix 17.2.2), we can write:

I = CF

(1 + r )

− 1

r(1 +r )

or CF = I

r(1 +r )

(1 + r )

− 1

(b)

Internal rate of return

In the formulas above, given I, n, and CF, ﬁnd r. The calculated rate of return, r,

is the “internal rate of return”of the project over a period of n years. Note that it

varies with the number of years and it increases with the number of years. Thus,

the IRR calculated for 20 years will be slightly higher than the IRR calculated for

10 years.

Computers use an iterative built-in function to calculate the IRR, often in the form:

@irr(0.15, −I, CF

, CF

,...,CF

)

Where “0.15”is an initial guess that should be modiﬁed if the function does not

converge in some 20 iterations. −I is the initial investment expressed as a negative

number and CFs are the values of the annual cash ﬂows.

Some software packages require that the cash ﬂow numbers be entered in a separate

array of cells; others allow for direct input.

The summation formula (b) derived above only applies if the annual cash ﬂows are

uniform. The net present value summation (a) applies however even if the CFs vary

from year-to-year. The @irr function also accepts variable annual CFs.

Discounted pay-back time

Given I , r , and CF, ﬁnd n. This calculated n is a discounted pay-back time. The

simple pay-back time (I /CF) is equivalent to a discounted pay-back time when r = 0

(i.e., with no discounting). Likewise, if we calculate IRR for, say, n = 20, and then

we set r = IRR, we will just back-calculate n = 20. Therefore, for 0 < r < IRR,

the calculated pay back time will be (I /CF) < n < 20. Likewise, if r > IRR, then

n > 20.

If the cash ﬂows, CF, are uniform, we can solve for n as follows:

n =



CF − Ir



ln (1 +r )

ECONOMIC ANALYSIS 869

Sales price required to obtain a given discounted cash ﬂow rate or return

Given I , n, and a desired rate of return, r, we can calculate the required cash ﬂow,

CF, according to formula (b). This cash ﬂow can be used in the economic analysis to

calculate the product sales price that will be necessary to generate this cash ﬂow. If this

cash ﬂow is obtained, then we have the desired discounted cash ﬂow rate of return, r.

This calculated sales price is a function of both r and n.

For the calculation of the cash ﬂow, CF, the appropriate computer function, if available,

is:

@pmt(I, r, n)

where r must be used in the decimal form for the corresponding period.

APPENDIX 17.2.1

Background for economic calculations

The following summary provides the background necessary for most economic cal-

culations and should be used for reference as needed.

First of all, let us consider the following terms:

P present value of a sum of money or a series of sums

F future value of a sum of money or a series of sums

C capital (investment or loan) at one given time

A sum of money every compounding period

n number of compounding periods

i annual rate of interest

r rate of interest per compounding period

L salvage value

The rate of interest per compounding period (r) should be used in all cases.

The interest rate stated legally is the nominal annual rate (i). If the compounding

period is less than one year, the rate of interest per compounding period is obtained

by dividing the nominal annual rate, i, by the number of compounding periods in a

year. It is done this way by law.

A commercial year is understood to consist of 360 days. Likewise, a generic commer-

cial month consists of 30 days. Thus, if the compounding is done daily, the annual

870 CHAPTER 17

rate of interest, i, will have to be divided by 360; if monthly, the annual rate will be

divided by 12, irrespective of the month. Note, however, that the amount compounded

over a period of time will be calculated on the basis of the actual days. Thus, 28 days

for a normal February, 31 days for May, 365 days for a normal year or 366 days for a

leap year.

Future value

= P(1 +r )

, or simply F, represents the value at the end of the nth period of a sum of money,

P, at time zero (e.g., present time).

The factor, (1 + r )

, is sometimes called the (F/P) factor or the “compound amount

factor for a single sum,”(CAFS). Thus,

F = P(F/P)

F = P(CAFS)

Present value

The present value, P, of a future sum of money, F

, represents the reciprocal of the

future value calculation.

P = F

(1 + r )

The reciprocal factor is also called the (P/F) factor or the “present worth factor for a

single sum,”(PWFS). Thus,

P = F(P/F)

P = F(PWFS)

In all these formulas the compounding becomes effective at the end of the period.

This is true in all the expressions used in economic calculations; modiﬁed formulas

are needed if the compounding becomes effective at the beginning of the period,

but this is seldom, if ever, the case. The formula modiﬁcation, if needed, involves a

(1 + r ) factor.

Future value of a uniform series of payments (made at the end of each period)

See Appendices B and C for justiﬁcation.

ECONOMIC ANALYSIS 871

If the amount paid at the end of each period is A, the future value, F

, of the sum is

given by:

= A

(1 + r )

− 1

The multiplier is also known as the (F/A) factor or the “compound amount factor for

a uniform series,”(CAFUS).

F = A(F/A)

F = A(CAFUS)

Sinking funds

A sinking fund consists of a series of payments made for the purpose of accumulating

a given amount of money, F, at some future time. As usual, we assume a uniform

series of payments, A being the amount paid at the end of each period. The calculation

assumes that the rate of interest does not change over time; trial-and-error would be

required otherwise.

In order to accumulate an amount F

at the end of n periods, the amount A to be

deposited at the end of each period is:

A = F

(1 + r )

− 1

This is the reciprocal of the future value of a series of uniform payments.

The multiplier is at times called the (A/F) factor or the “sinking fund factor”(SFF).

Thus,

A = F( A/F)

A = F(SFF)

As used in economic estimates, depreciation and amortization are essentially sinking

funds at zero rate of interest. Again, to repeat, no item should ever be treated both as

a sinking fund amortization and as an expense at the same time.

Loan repayments

The loan repayment calculation, or “mortgage formula,”is one of the most useful

expressions in economic calculations. It is a combination of the previous cases. It

represents the series of uniform payments, A, to be made at the end of each period to

repay a loan, P, taken out at time zero. In this situation, the present value, P, often is

just denoted by “capital” C.

872 CHAPTER 17

The derivation of the formula is given in Appendices B and C.

A = P

r(1 +r )

(1 + r )

− 1

The multiplier is also variously known as the (A/P) factor or the “capital recovery

factor”(CRF).

A = P( A/P)

A = P(CRF)

It ought to be very clear what these formulas mean. A, the uniform series of pay-

ments, represents a constant amount paid to the lender at the end of each compound

period. This quantity, A, is made up of two variable parts: a capital return part and

an interest payment part. The capital return part represents a very small fraction at

the beginning of the loan repayment period, but it increases slowly (and nonlinearly)

until it becomes the dominant term toward the end. Conversely, the interest payment

term is proportionately very large at the beginning of the loan repayment period and

decreases progressively until it becomes small at the end.

This formula is predicated on the idea of making uniform payments, as is often done

with mortgages; hence the name. Naturally, there are many other ways of repaying

loans and expressions could be derived for other situations too, if desired.

An offshoot of the mortgage formula is the concept of an “average rate of interest,”

sometimes used in economic calculations to reﬂect the average cost of long-term debt.

See Appendix 17.2.4 for this concept.

Present value of a uniform series

Logically, this represents the reciprocal of the loan repayment schedule.

P = A

(1 + r )

− 1

r(1 +r )

The multiplier is also called the (P/A) factor or the “present worth factor for a uniform

series,”(PWFUS).

P = A(P/ A)

P = A(PWFUS)

The above expressions represent most of the situations encountered in economic

evaluations. There are other expressions available for other situations (e.g., when

the series of payments increases or decreases progressively at a uniform rate—the

“gradient”method), but these are far rarer and need not be discussed here.

ECONOMIC ANALYSIS 873

APPENDIX 17.2.2

Progressions

Though trivial, the concept of arithmetic and geometric progressions is useful in

economic calculations. These concepts are outlined in as a refresher.

Arithmetic progressions

An arithmetic progression (or an arithmetic series) is a series of terms such that each

is made up of the previous term plus an additive (positive or negative) constant.

The following series is an arithmetic progression:

Term: 1 2 3 4 ... n

Value: aa+ qa+ 2qa+ 3q ... a + (n − 1)q

where a is the ﬁrst term and q is the constant addendum.

As seen, the nth term of an arithmetic progression is given by:

b = a + (n − 1)q

The sum of the ﬁrst n terms of an arithmetic progression is:

S =

(a + b)n

There is a spurious anecdote attributed to child Gauss who, having been punished by

his teacher along with his school mates to add the ﬁrst 100 numbers, he immediately

came up with the answer, 5,050. Supposedly, he had just observed that 1 + 100 = 101,

2 + 99 = 101, and so on to 50 + 51 = 101. Since there are 50 such pairs, the answer

is immediate.

Geometric progressions

Geometric progressions (or geometric series) are of more interest for economic cal-

culations.

A geometric progression is a series of terms such that each term is the previous one

multiplied by a constant factor.

Term: 1 2 3 4 ... n

Value aaqaq

... aq

(n−1)

874 CHAPTER 17

Therefore, the nth term of a geometric progression is given by:

b = aq

(n−1)

More importantly, the sum of the ﬁrst n terms is:

S =

bq − a

q − 1

S =

− a

q − 1

= a

− 1

q − 1

Often q < 1. In this case, both the numerator and the denominator will be negative

and, for comfort, we can write:

S = a

1 − q

These expressions are the basis for the mortgage formula and many other economic

calculations.

APPENDIX 17.2.3

Loan repayments (mortgage formula)

Let us assume that we have borrowed a certain amount of money, C, that we plan to

return over a total of n periods such that at the end of each period we repay the same

amount, A. We further assume that the rate of interest remains constant and is r for

each period.

Time zero

Take the loan out. We owe C.

End of the ﬁrst period

We owe C(1 +r ). We repay an amount A. Therefore, we now owe C(1 + r ) − A.

End of the second period

We owe [ C(1 + r ) − A](1 +r) = C(1 + r )

− A(1 +r). Again we repay an amount

A. Therefore, we now owe C(1 +r )

− A(1 +r) − A.

End of the third period

After repaying the amount A,weowe[C(1 +r )

− A(1 +r) − A](1 + r ) − A,or

C(1 + r )

− A[(1 +r)

+ (1 +r ) +1].

ECONOMIC ANALYSIS 875

Keep repeating the reasoning for all other intermediate periods.

End of the nth period

After making the last payment, A,weoweC(1 +r )

− A[(1 +r)

(n−1)

+ (1 +r )

(n−2)

+···+(1 +r) + 1]. But since we are repaying the loan in n periods, at this time the

balance must be zero!

By applying the summation formulas in Appendix 17.2.2, we can rewrite this as:

C(1 + r )

− A

(1 + r )

− 1

(1 + r ) −1

= 0

C(1 + r )

= A

(1 + r )

− 1

or, solving for A:

A = C

r(1 +r )

(1 + r )

− 1

which is the “mortgage”formula as seen in Appendix 17.2.1. All other formulas in

Appendix A can be derived in a like manner.

APPENDIX 17.2.4

Average rate of interest

The “average rate of interest”is a concept that arises when a loan is repaid over a

number of time periods, but we wish to conduct a preliminary economic analysis

over one of the time periods only. We saw in Appendix 17.2.1, Item 5, that, if a

uniform series of payments is used, the actual amounts paid as interest vary from a

high at the beginning to relatively small amounts toward the end. The question then

is: what is the “average”amount of interest paid over a typical period? This value is

normally calculated by adding up all the interest payments and simply prorating them

by dividing the sum over the total number of periods. Calculation of this value has

interest in cases when the capital may be strongly leveraged (i.e., a high percentage

of debt) since, obviously, the average interest paid will be less than the interest paid

over the ﬁrst period on the basis of the total amount of debt outstanding.

How much interest is paid over the life of the loan? Since we have borrowed a capital

C and we have made a total of n payments of A dollars each, the difference between

the amount paid and the capital borrowed must be the amount paid as interest.

Total interest paid = nA − C