Discounted cash flows

Time is taken into account by discounting the annual cash flow Acf with the rate of interest to obtain the anitual discounted cash flow -Adcf- Thus, at the end of year 1,... 

The sum of the annual discounted cash flows over n years SAdcf is known as the net present value (NPV) of the project ... 

Discounted cash-flow rate of return. Discounted cash-flow rate of return is defined as the discount rate i which makes the NPV of a project zero (curve 3 in Fig. A.2) ... 

The value of i given by this equation is known as the discounted cash-flow rate of return (DCFRR). It may be found graphically or by trial and error. 

Ref 91. Discounted cash-flow models account for use of capital, working capital, income taxes, time value of money, and operating expenses. Real after-tax return assumed to be 12.0%. Short-rotation model used for sycamore and poplar. Herbaceous model used for other species. Costs ia 1990 dollars. Dry tons. 

Fig. 2. (— —) Cumulative and (—° —) 8% discounted cash flow rate for development of a hypothetical agrochemical in constant 1994 U.S. dollars, where A represents the time the patent was first issued B, first registered use C, manufacturing plant start-up D, positive cumulative cash flow and E, patent...

DecoveTj of Capital. In Figure 1, the annual book depreciation is used to retire the fixed capital investment. Whereas this accounting model does not correspond to the typical money flow, it is one possible model for recovery of capital. This model assumes that the investment is reduced each year by the amount of the annual depreciation. Another model (22) assumes that a uniform yearly book depreciation payment is made to an interest-bear sinking fund that accumulates to the depreciable fixed capital amount at the end of the venture. Using this second model, the investment is outstanding throughout the lifetime of the project. This also does not correspond to the actual money flow in most cases. ProfitabiUty analysis utilizes a third model based on discounted cash flows. 

This gives two choices ia interpreting calculated NRR values, ie, a direct comparison of NRR values for different options or a comparison of the NRR value of each option with a previously defined NRR cutoff level for acceptabiUty. The NPV, DTC, and NRR can be iaterpreted as discounted measures of the return, iavestment, and return rate, analogous to the parameters of the earher example. These three parameters characterize a venture over its entire life. Additional parameters can be developed to characterize the cash flow pattern duting the early venture years. Eor example, the net payout time (NPT) is the number of operating years for the cumulative discounted cash flow to sum to zero. This characterizes the early cash flow pattern it can be viewed as a discounted measure of the expected operating time that the investment is at risk. 

Internal Return Rate. Another rate criterion, the internal return rate (IRR) or discounted cash flow rate of return (DCERR), is a popular ranking criterion for profitabiUty. The IRR is the annual discounting rate that makes the algebraic sum of the discounted annual cash flows equal to zero or, more simply, it is the total return rate at the poiat of vanishing profitabiUty. This is determined iteratively. 

The relationships among the various annual costs given by Eqs. (9-1) through (9-9) are illustrated diagrammaticaUy in Fig. 9-1. The top half of the diagram shows the tools of the accountant the bottom half, those of the engineer. The net annual cash flow Acp, which excludes any provision for balance-sheet depreciation Abd, is used in two of the more modern methods of profitability assessment the net-present-value (NPV) method and the discounted-cash-flow-rate-of-return (DCFRR) method. In both methods, depreciation is inherently taken care of by calculations which include capital recoveiy. 

Contribution and Breakeven Charts These can be used to give valuable preliminary information prior to the use of the more sophisticated and time-consuming methods based on discounted cash flow. If the sales price per unit of sales is Cs and the variable expense is Cy per unit of produc tion, Eq. (9-7) can be rewritten as... 

The ways of assessing profitabihty to be considered in this section are (1) discounted-cash-flow rate of return (DCFRR), (2) net present value (NPV) based on a particiilar discount rate, (3) eqmvalent maximum investment period (EMIP), (4) interest-recovery period (IRP), and (5) discounted breakeven point (DEEP). 

Discounted Cash Flow The present value P of a future sum of money F is given by... 

Time is taken into account by using the annual discounted cash flow Adcf, which is related to the annual cash flow Acf and the discount factor by... 

Equation (9-54) may be solved for i either graphically or by an iterative trial-and-error procedure. The value of i given by Eq. (9-54) is known as the discounted-cash-flow rate of return (DCFRR). It is also known as the profitability index, true rate of return, investor s rate of return, and interest rate of return. 

Cash-Flow Curves Figure 9-9 shows the cash-flow stages in a project together with their discounted-cash-flow values for the data given in Table 9-4. In addition to cash-flow and discounted-cash-flow curves, it is also instructive to plot cumulative-cash-flow and cumula-tive-discounted-cash-flowcurves. These are shown in Fig. 9-10 for the data in Table 9-4. 

The cost of capital may also be considered as the interest rate at which money can be invested instead of putting it at risk in a manufacturing process. Let us consider the process data listed in Table 9-4 and plotted in Fig. 9-10. If the cost oi capital is 10 percent, then the appropriate discounted-cash-flow curve in Fig. 9-10 is abcdef. Up to point e, or 8.49 years, the capital is at risk. Point e is the discounted breakeven point (DEEP). At this point, the manufacturing process... 

It is not normally possible to make a comprehensive assessment of profitabihty with a single number. The shape of the cumulative-cashflow and cumulative-discounted-cash-flow curves both before and after the breakeven point is an impoiTant factor. 

TABLE 9-4 Annual Cash Flows and Discounted Cash Flows for a Project... 

C. G. Sinclair [Chem. Process. E/ig., 47, 147 (1966)] has considered similar parameters to the (EMIP) and (IRP) based on a cumulative-discounted-cash-flow curve. 

The discounted-cash-flow rate of return (DCFRR) can readily he obtained approximately hy interpolation of the (NPV) for = 10 percent and = 20 percent ... 

Analysis of Techniques Both the (NPV) and the (DCFRR) methods are based on discounted cash flows and in that sense are vari-... 

Adcf — net annual discounted cash flow. (NPV) = Z Adcf = net present value. 

Discounted-cash-flow rate of return (DCFRR) has the advantage of being unique and readily understood. However, when used alone, it gives no indication of the scale of the operation. The (NPV) indicates the monetary return, but unlike that of the (DCFRR) its value depends on the base year chosen for the calculation. Additional information is needed before its significance can be appreciated. However, when a company is considering investment in a portfoho of projects, individual (NPV)s have the advantage of being additive. This is not true of (DCFRR)s. 

It is possible for some projects to reach a stage at which repairs, replacements, etc., can exceed net earnings in a particular year. In this case the cumulative-discounted-cash-flow or net-present-value curve plotted against time has a genuine maximum. 

No single value for a profitability estimate should be accepted without further consideration. An inteUigent consideration of the cumula-tive-cash-flow and cumulative-discounted-cash-flow curves such as those shown in Fig. 9-10, together with experience and good judgment, is the best way of assessing the financial merit of aprojec t. 

When considering future projects, top management will most likely require the discounted-cash-flow rate of return and the payback period. However, the estimators should also supply management with the following ... 

Cumulative discounted-cash-flow or (NPV) curve for a discount rate of 10 percent per year or other agreed aftertax cost of capital... 

Number of years to reach discounted-cash-flow rates of return of, say, 15 and 25 percent per year respectively... 

Comparisons on the basis of interest can be summarized as (1) the net present value (NPV) and (2) the discounted-cash-flow rate of return (DCFRR), which from Eqs. (9-53) and (9-54) is given formally as the fractional interest rate i which satisfies the relationship... 

These (NPV) data are plotted against the cost of capital, as shown in Fig. 9-12. The discounted-cash-flow rate of return is the value of i that satisfies Eq. (9-5). From Fig. 9-12, (NPV) = 0 at a (DCFRR) of 11.8 percent for project C and 14.7 percent for project D. Thus, on the basis of (DCFRR), project D is more profitable than project C. 

Figure 9-13 is a plot of Eq. (9-61) in the form of the number of years n required to reach a certain discounted-cash-flow rate of return (DCFRR) for a given payback period (PBP). The figure is a modification of plots previously published by A. G. Bates [Hydrocarbon Process., 45, 181-186 (March 1966)], C. Estrup [Br Chem. Eng., 16, 171 (February-March 1971)], and F. A. Holland and F. A. Watson [Process Eng. Eeon., 1, 293-299 (December 1976)]. 

FIG. 9-13 Relationship between payback period and discounted-cash-flow rate of return. 

AAdcf — decrease in net discounted cash flow at income tax rate = 0.5. A(NPV) = 2- AAqcf — decrease in net present value. 

Risk and Uncertainty Discounted-cash-flow rates of return (DCFRR) and net present values (NPV) for future projects can never be predicted absolutely because the cash-flow data for such projects are subject to uncertainty. Therefore, when stating predicted values of (DCFRR) and (NPV) for projects, it is also desirable to give a measure of confidence in the predictions. 

Numerical Measures of Risk Without risk and the reward for successfully accepting risk, there would be no business activity. In estimating the probabilities of attaining various levels of net present value (NPV) and discounted-cash-flow rate of return (DCFRR), there was a spread in the possible values of (NPV) and (DCFRR). A number of methods have been suggested for assessing risks and rewards to be expected from projects. 

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