Compounded cash flow

At the end of each year, the compounded cash flow to the project, with continuous uniform flow and continuous compounding, gives, by Eq. (37) of... 

After 3 years, the compounded cash flow, CCF, will be as follows ... 

If money is borrowed, interest must be paid over the time period if money is loaned out, interest income is expected to accumulate. In other words, there is a time value associated with the money. Before money flows from different years can be combined, a compound interest factor must be employed to translate all of the flows to a common present time. The present is arbitrarily assumed often it is either the beginning of the venture or start of production. If future flows are translated backward toward the present, the discount factor is of the form (1 + i) , where i is the annual discount rate in decimal form (10% = 0.10) and n is the number of years involved in the translation. If past flows are translated in a forward direction, a factor of the same form is used, except that the exponent is positive. Discounting of the cash flows gives equivalent flows at a common time point and provides for the cost of capital. 

Numerically, the difference between continuous and annual compounding is small. In practice, it is probably far smaller than the errors in the estimated cash-flow data. Annual compound interest conforms more closely to current acceptable accounting practice. However, the small difference between continuous and annual compounding may be significant when apphed to very large sums of money. 

The money earned in any year can be put to work (reinvested) as soon as it is available and start to earn a return. So money earned in the early years of the project is more valuable than that earned in later years. This time value of money can be allowed for by using a variation of the familiar compound interest formula. The net cash flow in each year of the project is brought to its present worth at the start of the project by discounting it at some chosen compound interest rate. 

For the accumulated costs and resources devoted to the development of a new chemical entity (NCE) or new molecular entity (NME) to make sense financially, the commercial potential of the compound must be evaluated in a rigorous manner. Compounds whose expected financial performance does not warrant these high investment costs must be abandoned or out-licensed as soon as possible so as to direct resources toward more profitable endeavors. By operating effectively, a well-designed drug discovery and development process can focus its efforts to operate efficiently on the compounds that will maximize cash flow to the pharmaceutical firm. 

Pi is the cash flow for the year i, and r is the relevant discount rate, which can be thought of either as the interest of borrowing money to start the company, or as the return from an alternate opportunity for a safe investment. This equation considers that 1 next year is worth only 1/(1 + r) dollars this year. The value of the NPV is the sum of all the cash flows for the n year, discounted by the power of compound interest. We give here the values of NPV for a number of values of r ... 

Continuous Compound Interest In some companies, namely petroleum, petrochemical, and chemical companies, money transactions occur hourly or daily, or essentially continuously. The receipts from sales and services are invested immediately upon receipt. The interest on this cash flow is continuously compounded. To use continuous compounding when evaluating projects or investments, one assumes that cash flows continuously. 

Discounted Cash Flow In the discounted cash flow method, all the yearly after-tax cash flows are discounted or compounded to time zero depending upon the choice of time zero. The following equation is used to solve for the interest rate i, which is the discounted cash flow rate of return (DCFROR). 

The expression for the case of continuous cash flow and interest compounding, equivalent to Eq. (21) for discrete cash flow and interest compounding, is developed as follows ... 

Example 6 Application of annuities in determining amount of depreciation with continuous cash flow and interest compounding. Repeat Example 5 with continuous cash flow and nominal annual interest of 6 percent compounded continuously. 

The fundamental relationships dealing with continuous interest compounding can be divided into two general categories (1) those that involve instantaneous or lump-sum payments, such as a required initial investment or a future payment that must be made at a given time, and (2) those that involve continuous payments or continuous cash flow, such as construction costs distributed evenly over a construction period or regular income that flows constantly into an overall operation. Equation (12) is a typical example of a lump-sum formula, while Eqs. (23) and (25) are typical of continuous-cash-flow formulas. 

Fot illustrations of the applications of continuous interest compounding and continuous cash flow to cases of profitability evaluation, see Examples 2 and 3 in Chap. 10. 

Discount and compounding factors for continuous interest and cash flows ... 

Compounding factors to give future worths for cash flows which ... 

C, = Compounding factor to give future worth for cash flows which occur in an instant at a point in time before the reference point. 

Compounding factors to give future worths for cash flows which occur uniformly before the reference point. The basis for these factors is a uniform and continuous flow of cash amounting to a total of one dollar during the given time period of T years, such as for construction of a plant. The factor converts this one dollar to the future worth at the reference time and is based on Eq. (23). 

Tables of interest and cash-flow factors, such as are illustrated in Tables 1, 5, 6, 7, and 8 of this chapter, are presented in all standard interest handbooks and textbooks on the mathematics of finance as well as in appendices of most textbooks on engineering economy. Exponential functions for continuous compounding are available in the standard mathematical tables. The development of tables for any of the specialized factors is a relatively simple matter with the ready availability of digital computers, as is illustrated in Example 3 of this chapter.

In the tabulation of factors for continuous interest compounding and continuous cash flow, the nominal interest rate r is used for calculating the... 

At the end of five years, the cash flow to the project, compounded on the basis of end-of-year income, will be... 

Some of the tedious and time-consuming calculations can be eliminated by applying a discount factor to the annual cash flows and summing to get a present value equal to the required investment. The discount factor for end-of-year payments and annual compounding is... 

Example 2 Discounted-cash-flow calculations based on continuous interest compounding and continuous cash Row. Using the discount factors for continuous interest and continuous cash flow presented in Tables 5 to 8 of Chapter 7, determine the continuous discounted-cash-flow rate of return r for the example presented in the preceding section where yearly cash flow is continuous. The data follow. 

Because the assumed trial value bf r = 0.225 discounted all the cash flows to the present worth of 110,000, the continuous interest rate of 22.5 percent represents the discounted-cash-flow rate of return for this example which can be compared to the value of 20.7 percent shown in Table 1 for the case of discrete interest compounding and instantaneous cash flow. 

Example 3 Determination of profitability index with continuous interest compounding and prestartup costs. Determine the discounted-cash-flow rate of return (i.e., the profitability index) for the overall plant project described in the following, and present a plot of cash position versus time to illustrate the solution. 

Solution. The procedure for this problem is similar to that illustrated in Table 1 in that a trial-and-error method is used with various interest rates until a rate is found which decreases the net cash position to zero at the end of the useful life. Let r represent the profitability index or discounted-cash-flow rate of return with continuous cash flow and continuous interest compounding. 

---