Interest compounding-discounting

Compounding-Discounting When money is moved forward in time from the present to a future time, the process is called compounding. The effect of compounding is that the total amount of money increases with time due to interest. Discounting is the reverse process, i.e., a sum of money moved backward in time. Figure 9-8 is a... 

For profitability analyses, certain discounting or compounding factors based on continuous interest compounding are of sufficient importance that tables have been prepared which give values of the factors for various interest rates and time periods. Table 3 gives examples of tabulated factors for the following cases t ... 

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. 

In the above example, the discount rate used was the annual compound interest rate offered by the bank. In business investment opportunities the appropriate discount rate is the cost of capital to the company. This may be calculated in different ways, but should always reflect how much it costs the oil company to borrow the money which it uses to invest in its projects. This may be a weighted average of the cost of the share capital and loan capital of a company. 

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. 

With a disbursement of 1000 in Year 0, the discounted breakeven point (DEEP) will be reached in 3 years at a compound-interest rate of 30 percent if the annual net profit Avp = 550.63 per year. Thus, a... 

Bills are due on monthly account with a 2 percent discount for cash. Overdraft and deferred-tax interest are compounded daily at nominal annual interest rates of 15 and 9 percent respectively. Corporation tax, capital gains tax, and personal income tax rates are 50, 40, and 30 percent respectively. The current rate of inflation is at 8 percent per year. The traditional return expected hy investors is 7 percent per year net of all taxes in real terms. 

The interest-rate equivalent of the cash discounts is 2 percent per month, since this discount could he obtained every month if payment were to he made at the beginning of the month rather than, as at present, at its end. Since the hills are settled monthly, the notional interest is paid monthly and should not he compounded. The discount is equivalent to 12 monthly simple-interest payments per year. Hence, from Eq. (9-31) the effective annual interest rate on discounts = (12)(0.02) = 0.24 = 24 percent. It would, therefore, he a good use of surplus cash to reduce this debt as quickly as possible. This would require cash equivalent to one-sixth of the annual hills due, or 16,700, to he avadahle. It can, therefore, he assumed that this level of liquidity is not available for capital projects, either as working capital to reduce the debt or for fixed-capital projects. Further, since the new project will not increase sales, it cannot generate further debt of this kind. Hence, this source is not available to capitahze the new project. 

A shortage of cash may prevent a company from taking advantage of large discounts available for bulk purchase of raw materials. The importance of the availability of adequate cash or near cash can be seen by considering an account payable within 28 days, with a 2 percent discount allowed if paid within 7 days. If cash is not available to pay the account within 7 days, this is then equivalent to paying 2 percent interest on the money for the remaining 21-day period, or an annual compound-interest rate of more than 41 percent. 

The selection of the discount factor depends on the financial policy of the business, but is usually 2-3 per cent above the current interest rates. Use of discounting methods will determine whether the project cost will produce a better return than by simply investing the capital involved at the highest compound interest rate or, if the capital cost has to be borrowed, whether the rate of return is much higher than the cost of borrowing. 

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. 

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 ... 

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 photochromic compounds of potential interest, based on the 2//-chromene ring system, are the 2//-benzopyrans (1.18) or the three isomeric naphthopyrans (1.19-1.21). However, 2H-naphtho[2,3-( ]pyrans (1.21) show little or no useful photochromic behaviour and can be discounted from any further discussion. Although R and can be part of a carbocyclic spiro ring, they are more commonly unconnected substituents such as gem dialkyl or aryl groups. 

Time value of money has been integrated into investment-evaluation systems by means of compound-interest relationships. Dollars, at different times, are given different degrees of importance by means of compounding or discounting at some preselected compound-interest rate. For any assumed interest value of money, a known amount at any one time can be converted to an equivalent but different amount at a different time. As time passes, money can be invested to increase at the interest rate. If the time when money is needed for investment is in the future, the present value of that investment can be calculated by discounting from the time of investment back to the present at the assumed interest rate. 

Example 4 Determination of present worth and discount. A bond has a maturity value of 1000 and is paying discrete compound interest at an effective annual rate of 3 percent. Determine the following at a time four years before the bond reaches maturity value ... 

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

As indicated in the footnote for Eq. (12), the common interest expressions can be written in simplified form by using discount-factor and compound-interest-factor notation. Following is a summary showing the significance and meaning of the compounding factors presented in Table 3, Derivations of the factors are presented in the text. 

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