Net present value and discounted cash flow

If we have a present sum of money P, its value n years in the future (F) will be F = F(1 + iy where i is the accepted compound interest rate fraction. Conversely, the present-day value (P) of a sum of money (F) to be received n years in the future is given by the expression 

Thus the individual cash flows in future years can be discounted to a common base (the present) and directly compared. From the form of the expression for present value it will be seen that a sum of money received in two years time is worth more than the same sum received ten years hence. This reduces the effect of the less certain cash flows in the long-term future and emphasizes the importance of early positive cash flows. 

The net present value of a project or investment is the sum of the present values of each individual cash flow. To reduce the arithmetic involved the net cash flow in each year is determined and the present value of the annual cash flows is summed. For a project with net annual cash flow C, in year t, the net present value  

---

The other indices to be described, net present value and discounted cash flow return, are more comprehensive because they take account of the changing pattern of project net cash flow with time. They also take account of the time value of money. 

There are a variety of objective functions that are used for economic optimization. Some are quite elegant and incorporate the concept of the time value of money. Examples are net-present-value and discounted cash flow. These methods are preferred by business majors, accountants, and economists because they are more accurate measures of profitability over an extended time period. However, a lot of assumptions must be made in applying these methods, and the accuracy of these assumptions is usually quite hmited. The prediction of future sales, prices of raw materials and products, and construction schedule is usually a guessing game made by marketing and business managers whose track record for predicting the future is almost as poor as the weather man. 

The use of depreciation as an allowance against tax forms part of net present value and discounted cash flow measures of profitability to be considered later. 

The same methods that were explained and applied earlier in this chapter are applicable for replacement analyses. Net-present-worth and discounted-cash-flow methods give the soundest results for maximizing the overall future worth of a concern. However, for the purpose of explaining the basic principles of replacement economic analyses, the simple rate-of-retum-on-investment method of analysis is just as effective as those methods involving the time value of money. Thus, to permit the use of direct illustrations which will not detract from... 

Calculation of the after-tax ROI is complicated if the depreciation term is less than the plant life and if an accelerated method of depreciation such as MACRS is used. In such cases, it is just as easy to calculate one of the more meaningful economic criteria such as net present value or discounted cash flow rate of return, described later. Because of this complication, a pre-tax ROI is often used instead ... 

If the existing production plant has not yet reached the end of its economic life, the total costs CjNx must also include the sunk-cost. This is the basic economic principle. In the case of investment this is modified by use of the following parameters Net present value (NPV = discounted cash flow in ten years minus investment), internal rate of return (IRR) and pay back period (PBP). Generally should be NPV > investment, IRR > cost of capital and PBP < 5 years. 

The ideas discussed in Chapter 9 are extended to evaluate the profitability of chemical processes. Profitability criteria using nondiscounted and discounted bases are presented and include net present value (NPV), discounted cash flow rate of return (DCFROR), and payback period (PBP). A discussion of evaluating equipment alternatives using equivalent annual operating costs (EAOC) and other methods is presented. Finally, the concept of evaluating risk is covered and an introduction to the Monte Carlo method is presented. 

It is clear from the bond price formula that a bonds yield and its price are closely related. Specifically, the price moves in the opposite direction from the yield. This is because a bonds price is the net present value of its cash flows if the discount rate—that is, the yield required by investors— increases, the present values of the cash flows decrease. In the same way, if the required yield decreases, the price of the bond rises. The relationship between a bond s price and any required yield level is illustrated by the graph in FIGURE 1.5, which plots the yield against the corresponding price to form a convex curve. 

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. 

The same questions may then be asked for different values of the probabilities p and po. The answers to these questions can give an indication of the importance to the company of P at various levels of risk and are used to plot the utility curve in Fig. 9-25. Positive values are the amounts of money that the company would accept in order to forgo participation. Negative values are the amounts the company woiild pay in order to avoid participation. Only when the utihty value and the expected value (i.e., the straight line in Fig. 9-25) are the same can net present value (NPV) and discounted-cash-flow rate of return (DCFRR) be justified as investment criteria. 

The method of allocating overheads can seriously affect the assigned costs of a project and hence the apparent cash flows for that project . Since these cash flows are used to assess profitability by the net-present-value (NPV) and discounted-cash-flow-rate-of-return (DCFRR) methods, unfair allocation of overhead costs can result in a wrong choice between alternative projec ts. 

Various other evaluation schemes based on the concept of time value of money are also sometimes used. These, together with the Net Present Value and Rate of Return methods, are all grouped together under the title of discounted cash flow methods. 

Three methods are used to assess the value of a capital investment. They are cash payback, net present value, and internal rate of return (also known as Discounted Cash Flow-Rate of Return). 

Discounted cash flow diagram can determine the profitability criteria in terms of the payback period, net present value, and rate of return from. In the discounted cash flow diagram each of the annual cash flow is discounted to time zero for the latent heat storage system. The payback period is the time required, after construction, to recover the fixed capital investment. The net present value shows the cumulative discounted cash value at the end of useful life. Positive values of net present value and a shorter payback period are preferred. The rate of return is the interest rate at which all the cash flows must be discounted to obtain zero net present value. If rate of return is greater than the internal discount rate, then the latent heat storage system is considered feasible. 

Investment, discounted cash flow and internal rate of return, net present value, and break-even... 

Calculating the present values of the cash flows from the previous example requires adding two columns to the spreadsheet. We first calculate the discount factor (1 + and then multiply this by the cash flow in year n to give the present value of the cash flow The present values can then be summed to give the net present value ... 

For a given discount rate, calculate the discount factor during the operating life of the project. Multiply the discount factor by the cash flow and obtain the net present value and the net return rate. Valle-Riesta [8] has listed alternative methods of calculating cash flow, as shown in Table 9-6. 

Be able to compute cash flows and depreciation, and use them to project the net present value and investor s rate of return (IRR) (also known as the discounted cash-flow rate of return, DCFRR), two measures that account for projections of revenues and costs over the life of the proposed process, and the time value of money. 

To calculate the net present value (NPV) of an investment, a firm s accounting department estimates the firm s minimum desired rate of return. This is used to discount the cash flows above. For the cash flow example above, if we assume that the minimum desired rate of return for that firm is 10%, then the cash flow for each period is multiplied by the present value of a dollar for that period (i.e., P = S/(l -F r)° where S = 1, r is the interest rate or the desired rate of return, and n is the number of periods in the future) to find the present value of that cash flow. This is shown in Table 3.2. 

The DCFR method is based on finding the interest rate that satisfied the conditions implied by the method. Here, we provide a value for i that is an acceptable rate of return on the investment and then calculate the discounted value (present value) of the cash flow using this i. The net present value is then given by... 

Net Present Va.Iue, Each of the net annual cash flows can be discounted to the present time using a discount factor for the number of years involved. The discounted flows are then all at the same time point and can be combined. The sum of these discounted net flows is called the net present value (NPV), a popular profit criterion. Because the discounted positive flows first offset the negative investment flows in the NPV summation, the investment capital is recovered if the NPV is greater than zero. This early recovery of the investment does not correspond to typical capital recovery patterns, but gives a conservative and systematic assumption for investment recovery. 

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. 

---