What is the present value of a cash flow diagram?

14.20.1 Definition and computation of cash flow

Cash flow is basically either receipts of cash (cash inflow) or payments (cash outflow). For the purpose of financial planning and determination of the net cash returns of an investment, it is necessary to distinguish between the financial flow, which is related to financing of an investment, and cash flow (expenditures and revenues), representing the performance or operation of the project (operational cash flow).

Operational cash flow are shown (as discounted cash flow) in Table 14.7:

Table 14.7. Operational Cash Flow

14.20.2 Introduction to discounted cash flow analysis and financial functions

The financial and economic analysis of investment projects is typically carried out using the technique of discounted cash flow (DCF) analysis. This module introduces concepts of discounting and DCF analysis for the derivation of project performance criteria such as NPV, IRR, and benefit-to-cost (B/C) ratios.

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Cash flows, compounding, and discounting

DCF analysis is the technique used to derive economic and financial performance criteria for investment projects. It is important to review some of the basic concepts of DCF analysis before proceeding to topics such as cost-benefit analysis (CBA), financial analysis (FA), linear programming, and the estimation of nonmarket benefits.

Cash flow analysis is simply the process of identifying and categories of cash flows associated with a project or proposed course of action and making estimates of their values. For example, when considering establishment of a forestry plantation, this would involve identifying and making estimates of the cash outflows associated with establishing the trees (e.g., the cost of buying or leasing the land, purchasing seedlings, and planting the seedlings), maintaining the plantation (such as cost of fertilizer, labor, pruning, and thinning), and harvesting. As well, it would be necessary to estimate the cash inflows from the plantation through sales of thinnings and timber at final harvest.

DCF analysis is an extension of simple cash flow analysis and takes into account the time value of money and the risks of investing in a project. A number of criteria are used in DCF to estimate project performance, including NPV, IRR, and B/C ratios. Before discussing criteria to measure project performance, it is necessary to introduce some concepts and procedures with respect to compounding and discounting. Let us begin with the concepts of simple and compound interest. For the moment, consider the interest rate as the cost of capital for the project.

Suppose a person has to choose between receiving $1000 now or a guaranteed $1000 in 12 months' time. A rational person will naturally choose the former, because during the intervening period he or she could use the $1000 for profitable investment (e.g., earning interest in the bank) or desired consumption. If the $1000 were invested at an annual interest rate of 8%, then over the year it would earn $80 in interest. That is, a principal of $1000 invested for 1 year at an interest rate of 8% would have a future value of $1000(1.08) or $1080.

The $1000 may be invested for a second year, in which case it will earn further interest. If the interest again accrues on the principal of $1000 only, it is known as simple interest. In this case the future value after 2 years will be $1160. On the other hand, if interest in the second year accrues on the whole $1080, known as compound interest, the future value will be $1080(1.08) or $1166.40. Most investment and borrowing situations involve compound interest, although the timing of interest payments may be such that all interest is paid before further interest accrues. The future value of the $1000 after 2 years may also be derived as:

(14.4)$1000(1.08)2 =$1166.40

In general, the future value of an amount $a, invested for n years at an interest rate of i is $a(1 + i)n, where it is to be noted that the interest rate i is expressed as a decimal (e.g., 0.08 and not “8” for an 8% rate).

The reverse of compounding—finding the present-day equivalent to a future sum—is known as discounting. Because $1000 invested for 1 year at an interest rate of 8% would have a value of $1080 in 1 year, the present value of $1080 1 year from now, when the interest rate is 8%, is $1080/1.08 = $1000. Similarly, the present value of $1000 to be received 1 year from now, when the interest rate is 8%, is

(14.5)$ 1000/1.08=$925.93

In general, if an amount $a is to be received after n years, and the annual interest rate is i, then the present value is

(14.6)$a/(1+i)n

This discussion has been in terms of amounts in a single year. Investments usually incur costs and generate income in each of a number of years. Suppose the amount of $1000 is to be received at the end of each of the next 4 years. If not discounted, the sum of these amounts would be $4000. But suppose the interest rate is 8%. What is the present value of this stream of amounts? This is obtained by discounting the amount at the end of each year by the appropriate discount factor and then summing:

(14.7)$1000/1.08+$1000/1.082 +$1000/1.083+$1000/1.084=$1000/1.08+$1000/1.1664+$1000/1.2597+$1000/1.3605=$925.93+$857.34+$793.83+$735.03=$3312.13

The discount factors—1/1.08t for t = 1–4—may be calculated for each year or read from published tables. It is to be noted that the present value of the annual amounts is progressively reduced for each year farther into the future (from $925.93 after 1 year to $735.03 after 4 years), and the sum is approximately $700 less than if no discounting (a zero discount rate) had been applied.

14.20.3 Definition of annual net cash flows

DCF analysis is applied to the evaluation of investment projects. Such a project may involve creation of a terminating asset (e.g., a forestry plantation), infrastructure (e.g., a road or plywood plant) or research (including scientific and socioeconomic research). Any project may be regarded as generating cash flows. The term cash flow refers to any movement of money to or away from an investor (an individual, firm, industry, or government). Projects require payments in the form of capital outlays and annual operating costs, referred to as cash outflows. They give rise to receipts or revenues, referred to as project benefits or cash inflows. For each year, the difference between project benefits and capital plus operating costs is known as the net cash flow for that year. The net cash flow in any year may be defined as

(14.8)at=bt−(kt+ct)

where:

bt are project benefits in year t

kt are capital outlays in year t

ct are operating costs in year t

It is to be noted that when determining these net cash flows, expenditure items and income items are timed for the point at which the transactions takes place, rather than the time at which they are used. Thus, for example, expenditure on purchase of an item of machinery rather than annual allowances for depreciation would enter the cash flows. It is to be further noted that cash flows should not include interest payments. The discounting procedure in a sense simulates interest payments, so to include these in the operating costs would be to double-count them.

A project involves an immediate outlay of $25,000, with annual expenditures in each of 3 years of $4000, and generates revenue in each of 3 years of $15,000. These cash flow data may be set out, and annual net cash flow derived, as in Table 14.8.

Table 14.8. Annual Cash Flows for a Hypothetical Project

Two points may be noted about these cash flows. First, the capital outlay is timed for Year 0. By convention this is the beginning of the first year (i.e., right now). On the other hand, only half of the first year's operating costs are scheduled for the Year 0 (the beginning of the first year).

The remaining half of the first year's operating costs plus the first half of the second year's operating costs are scheduled for the end of the first year (or, equivalently, the beginning of the second year). In this way, operating costs are spread equally between the beginning and the end of each year. (The final half of the third year's operating costs are scheduled for the end of Year 3.) In the case of project benefits, these are assumed to accrue at the end of each year, which would be consistent with lags in production or payments.

These within-year timing issues are unlikely to make a large difference to overall project profitability, but it is useful to make these timing assumptions clear. A second point to note about Table 14.1 is that net cash flows (second column less third plus fourth column) are at first negative, but then become positive and increase over time. This is a typical pattern of well-behaved cash flows, for which performance criteria can usually be derived without computational difficulties.

14.20.4 Project performance criteria

Let us now consider a number of project performance criteria, which can be obtained through discounted cash flow analysis. These criteria will be defined and then derived for the cash flow data of the above example.

The NPV is the sum of the discounted annual cash flows. For the example, taking an interest rate of 8%, this is

(14.9)NPV=a0+a1/(1+i)+ a2/(1+i)2+a3/(1+i )3

A project is regarded as economically desirable if the NPV is positive. The project can then bear the cost of capital (the interest rate) and still leave a surplus, or profit. For the example,

(14.10)NPV=−27,000+11,000/(1.08)+11,000/(1.08)2+13, 000/(1.08)3=−27,000+11,000/1.1664+11, 000/1.2597+13,000/1.3605=$2935.73

The interpretation of this figure is that the project can support an 8% interest rate and still generate a surplus of benefits over costs, after allowing for timing differences in these, of approximately $3000.

An alternative to the net present value is the net future value (NFV), for which annual cash flows are compounded forward to their value at the end of the project's life. Once the NPV is known, the NFV may be obtained indirectly by compounding forward the NPV by the number of years of the project life. For Example 1, the net future value is

(14.11)NFV=NPV(1.08)3=$2935.73×1.3605=3994.06

The IRR is the interest rate such that the discounted sum of net cash flows is zero. If the interest rate were equal to the IRR, the net present value would be exactly zero. The IRR cannot be determined by an algebraic formula, but rather has to be approximated by trial-and-error methods. For this example, we know that the IRR is somewhere above 8%. Deriving the NPV with a range of discount rates would reveal that the IRR falls between 13% and 14% but closer to the latter. In practice, a financial function can be called up to perform the trial-and-error calculations. It would be found in this case that the IRR is about 13.8%.

The IRR is the highest interest rate which the project can support and still break even. A project is judged to be worthwhile in economic terms if the IRR is greater than the cost of capital. If this is the case, the project could have supported a higher rate of interest than was actually experienced and still made a positive payoff. In this case, the project would be profitable provided the cost of capital was less than 13.8%.

The IRR as a criterion of project profitability suffers from a number of theoretical and practical limitations. On the theoretical side, it assumes that the same rate of return is appropriate when the project is in surplus and when it is in deficit. However, the cost of borrowed funds may be quite different than the earning rate of the firm. It could be more appropriate to use two rates when determining the IRR. The actual cost of capital could be used when the project is in deficit, and the earning rate (unknown, to be determined by trial-and-error) could be applied when the project is in surplus. This would give a better indication of the earning rate of the project to the firm or government.

From a practical viewpoint, the IRR may not exist or it may not be unique. This problem may be examined in terms of the NPV profile, a graph of NPV versus the rate of interest. When the IRR is well behaved, this profile takes the form as in Figure 14.1. As the interest rate increases the NPV falls, being zero where the NPV curve crosses the interest rate axis; the IRR corresponds with this discount rate.

Consider a project for which the net cash flow in each year (including Year 0) is positive. Regardless of the interest rate, the NPV will never be zero, so it will not be possible to determine an IRR. Similarly, a project with a large initial capital outlay and for which future benefits are relatively small or negative may not have a positive NPV regardless of the interest rate, so again the curve for the NPV profile may never cross the interest rate axis.

If a project generates runs of positive and negative net cash flows, the NPV profile may take the form of a rollercoaster curve, crossing the interest rate axis in several places. This indicates multiple internal rates of return, one at each interest rate where NPV is zero. It is then by no means clear which if any of the rates we should choose to call the IRR. Further, for some sections of the NPV profile (those that are upward sloping), the NPV is increasing as the interest rate increases. This implies that the greater the cost of capital the more profitable the project. Clearly, multiple internal rates of return and perverse relationships between the NPV and the discount rate are not very satisfactory.

14.20.4.4 Benefit-to-cost ratios

A number of benefit-cost ratio concepts have been developed. For simplicity, we will consider only two concepts, referred to as the gross and net B/C ratio and defined respectively as

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Gross B/C ratio = PV of benefits/(PV of capital costs + PV of operating costs)

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Net B/C ratio = (PV of benefits – PV of operating costs)/PV of capital costs

For this project, the present value of capital outlays is $25,000, since outlays are made immediately and as a single amount. The present values of project benefits and operating costs are:

(14.12)PVofbenefits=$15,000/1.08+$15,000/1.082+ $15,000/1.083=$38,656.45

(14.13)PVofoperatingcosts=$2000+$4000/1.08+$4000/1.082+$ 2000/1.083=$9418.38

Hence, the B/C ratios are

(14.14)grossB/Cratio=38,656.4525,000+9418.38=1.12

(14.15)netB/Cratio=38,656.45−9418.3825,000=1.45

A project is judged to be worthwhile in economic terms if it has a B/C ratio is greater than unity; that is, if the present value of benefits exceeds the present value of costs (in gross or net terms). If one of the above ratios is greater than unity, then the other will also be greater than unity. In this example, the ratios are greater than unity, indicating that the project is worthwhile on economic grounds. It is not clear on logical grounds which of the ratios is the most useful. Figure 14.2 shows the NPV profile for a project

FIGURE 14.2. The net present value (NPV) profile for a project.

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The payback period

The payback period (PP) is the number of years for the projects to break even; that is, the number of years for which discounted annual net cash flows must be summed before the sum becomes positive (and remains positive for the remainder of the project's life). The payback period for a project with the above net cash flows can be determined as in the following table. From this table, it is apparent that the sum of discounted net cash flows does not become positive until Year 3, so the payback period is 3 years.

The PP indicates the number of years until the investment in a project is recovered. It is a useful criterion for a firm with a short planning horizon but does not take account of all the information available (i.e., the net cash flows for years beyond the PP).

This is a measure of the greatest amount that the project “owes” the firm or government (i.e., the farther “in the red” it goes). In Table 14.9, the largest negative value is −$27,000, so this is the peak deficit. Peak deficit is a useful measure in terms of financing a project, since it indicates the total amount of finance that will be required.

Table 14.9. Derivation of “Project Balances” and Payback Period

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Review of DCF performance criteria

The most commonly used discounted cash flow performance criteria—NPV, IRR, B/C ratios, and PP—may be summarized as in Table 14.10. The various criteria are closely related, but measure slightly different things. In this respect, they tend to complement one another, so that it is common to estimate and report more than one of the measures.

Table 14.10. Summary of Definitions of Main DFC Performance Criteria

Perhaps the most useful measure, and the one most often reported, is the NPV. This tells the total payoff from a project. A limitation of the NPV is that it is not related to the size of the project. If one project has a slightly lower NPV than another, but the capital outlays required are much lower, then the second project will probably be the preferred one. In this sense, a rate of return measure such as the IRR is also useful.

The PP and peak deficit are useful supplementary project information for decision makers. They have greater relevance for private sector investments, for firms that cannot afford long delays in recouping expenditure, and where careful attention must be paid to the total amount of funds that will need to be committed to the project to remain solvent.

7.14.1 Revenue-dominated cash flow diagram

In a revenue-dominated cash flow diagram, the time value of money for the revenues is considered for each year-end, that is, the cumulative revenues plus the interest rate, either simple or compound as required or specified, are computed and exhibited.

7.14.2 Cost-dominated cash flow diagram

Similarly, in a cost-dominated cash flow diagram, the time value of money for the cost figure is considered for each year-end, that is, the cumulative costs plus the interest rate, are computed and exhibited.

2.1 Projecting Cash Flows

The projected cash flow stream is a composite forecast that results from many separate assumptions concerning physical attributes of the oil field and the economic environment in which it will be produced. The number of wells and size of facilities required to delineate and develop the field, in conjunction with the presumed cost level for drilling services and oil field equipment, will roughly determine the scope and timing of initial expenditures. The magnitude and duration of cash inflows (sales revenue minus operating cost) are determined by a further set of assumptions regarding the flow rate from individual wells (and the rate at which production will decline as the field is depleted), the quality and price of produced oil and gas, necessary operating and maintenance costs required to keep wells and field plant facilities in order, and the level of royalties and taxes that are due to lessors and governmental authorities. Thus, the projection of net cash flow for the field as a whole is the combined result of many interrelated but distinct cash flow streams. Some components are fixed by contract and can typically be projected with relative certainty (e.g., royalty obligations and rental payments), but others require trained guesswork that leaves a wide margin of error (e.g., future production rates and oil price trends). It seems reasonable that those components of the cash flow stream that are known with relative certainty be given greater weight in figuring the overall value of the field, but as discussed further later in this article, properly executing this aspect of the valuation procedure was hardly practical until so-called options-based valuation methods were developed.

2.2 Discounting Cash Flows

A dollar received (or paid) in the future is worth less than a dollar received (or paid) today because of the time value of money. Cash in hand can be invested to earn interest, and therefore will have grown in value to outweigh an equivalent amount of cash to be received at any point in the future. If the relevant periodic rate of interest is represented by the symbol i (e.g., i=10%), then the present value of a dollar to be received t periods hence is given by PV(i, t)=1/(1+i)t. This expression is referred to as the discount factor, and i is said to be the discount rate. The discount factor determines the relative weight to be given to cash flows received at different times during the life of the oil field. Cash flows to be received immediately are given full weight, since PV(i, 0)=1, but the weight assigned to a future receipt declines according to the amount of delay. Thus, the net present value (NPV) of an arbitrary cash flow stream represented by the (discrete) series of periodic receipts {CF0, CF1, CF2,…, CFT} is computed as the sum of individual present values:

NPV =∑t=0TCFt(1+i)t.

It is quite common to perform this computation on the basis of continuous discounting, where the periodic intervals are taken to be arbitrarily short (a day, a minute,…, an instant), in which case the discount factor for cash to be received at future time t declines exponentially with the length of delay: PV(i,t)=e−it. Therefore, when the cash flow stream is expressed as a continuous function of time, NPV is reckoned as the area under the discounted cash flow curve:

(2)NPV=∫0TCFt⋅e−itdt.

It is apparent, whether the problem is formulated in discrete or continuous time, that correct selection of the discount rate is critical to the valuation process. This parameter alone determines the relative weight that will be given to early versus late cash flows. Since exploration and development of oil and gas fields is typically characterized by large negative cash flows early on, to be followed after substantial delay by a stream of positive cash flows, the choice of a discount rate is decisive in determining whether the value of a given property is indeed positive. The extent to which discounting diminishes the contribution of future receipts to the value of the property is illustrated in Fig. 1, which shows the time profile of cash flows from a hypothetical oil field development project. With no other changes to revenues or expense, the property's net present value is reduced by a factor of ten, from nearly $1 billion to roughly $100 million, as the discount rate is raised from 8% (panel a) to 20% (panel b). These panels also illustrate how discounting affects the payback period for the property in question (i.e., the time required before the value of discounted receipts finally offsets initial expenditures): 7 versus 11 years at the respective rates of discount.

With so much at stake, the selection of a discount rate cannot be made arbitrarily. If the discount rate is not properly matched to the riskiness of the particular cash flow stream being evaluated, the estimate of fair market value will be in error. Because no two oil fields are identical, the appropriate discount rate may vary from property to property. A completely riskless cash flow stream (which is rare) should be discounted using the risk-free rate, which is approximated by the interest rate paid to the holders of long-term government bonds. Cash flow streams that are more risky, like the future earnings of a typical oil field, must be discounted at a higher rate sufficient to compensate for the owner's aversion to bearing that risk. The degree of compensation required to adequately adjust for risk is referred to as the risk premium and can be estimated from market data using a framework called the capital asset pricing model.

One important implication of the capital asset pricing model is that diversifiable risks do not contribute to the risk premium. A diversifiable risk is any factor, like the success or failure of a given well, that can be diluted or averaged out by investing in a sufficiently large number of separate properties. In contrast, risks stemming from future fluctuations in oil prices or drilling costs are nondiversifiable because all oil fields would be affected similarly by these common factors. The distinction between diversifiable and nondiversifiable risk is critical to accurate valuation, especially with respect to the exploratory segment of the petroleum industry: although petroleum exploration may be one of the riskiest businesses in the world, a substantial portion of those risks are diversifiable, thus the risk premium and discount rate for unexplored petroleum properties is relatively low in comparison to other industries.

The appropriate discount rate for the type of petroleum properties typically developed by the major U.S. oil and gas producers would be in the vicinity of 8 to 14%. This is the nominal rate, to be used for discounting cash flows that are stated in current dollars (dollars of the day). If future cash flow streams are projected in terms of constant dollars (where the effect of inflation has already been removed), then the expected rate of inflation must be deducted from the nominal discount rate, as well. Cash flow streams derived from properties owned by smaller or less experienced producers who are unable to diversify their holdings, or in certain foreign lands, may be deemed riskier, in which case the discount rate must be increased in proportion to the added risk.

Not only does the appropriate discount rate vary according to property and owner, but the individual components of overall cash flow within any given project are likely to vary in riskiness and should, in principle, be discounted at separate rates. It is fair to say, however, that methods for disentangling the separate risk factors are complex and prone to error, and it is common practice to discount the overall net cash flow stream at a single rate that reflects the composite risk of the entire project. In many applications, the error involved in this approximation is probably not large. Moreover, new insights regarding the valuation of real options (to be discussed later) provide a procedure in which it is appropriate to discount all cash flow streams at the same rate, which circumvents entirely the problem of estimating separate risk-adjusted discount factors.