Continuous cash flow

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. 

Equal payments per year based on continuous cash flow and interest... 

RELATIONSHIPS FOR CONTINUOUS CASH FLOW AND CONTINUOUS INTEREST OF IMPORTANCE FOR PROFITABILITY ANALYSES... 

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. 

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

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. 

Year Predicted after-tax cash flow to project based on total income minus all costs except depreciation with cash flow occurring continuously, (total of continuous cash flow for year indicated)... 

Year Estimated continuous cash flow to project, Trial for = 0.225 Present value, ... 

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. 

Profit before taxes = continuous cash flow - depreciation before taxes... 

Figure 9-4. A continuous cash flow diagram. Source Valle-Riestra [8].

Many cash flows that engineers must consider can be assumed to occur continuously, such as accounts receivable, the cost savings resulting from productivity improvements, or the costs of carrying inventories. The following cash flow diagrams serve to illustrate the differences between discrete and continuous cash flows. 

Again, as with discrete cash flows, to solve for one variable given another, it is only necessary to select the proper interest factor for continuous cash flow, continuous compounding. A listing of these factors can be found in Table 3. 

As an example, consider the equivalent future worth of a uniform series of continuous cash flows totaling 2000 per year for 10 years compounded continuously at 15% nominal annual rate of interest. 

Consider a uniform continuous cash flow, A, which begins at time m and continues for an uncertain duration t. Assume that m and t are statistically independent random variables with known probability functions f m) and f t). It may be shown that... 

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