Present value and discounting

Bond prices are expressed per 100 nominal —that is, as a percentage of the bonds face value. (The convention in certain markets is to quote a price per 1,000 nominal, but this is rare.) For example, if the price of a U.S. dollar—denominated bond is quoted as 98.00, this means that for every 100 of the bond s face value, a buyer would pay 98. The principles of pricing in the bond market are the same as those in other financial markets the price of a financial instrument is equal to the sum of the present values of all the future cash flows from the instrument. The interest rate used to derive the present value of the cash flows, known as the discount rate, is key, since it reflects where the bond is trading and how its return is perceived by the market. All the factors that identify the bond—including the nature of the issuer, the maturity date, the coupon, and the currency in which it was issued— influence the bond s discount rate. Comparable bonds have similar discount rates. The following sections explain the traditional approach to bond pricing for plain vanilla instruments, making certain assumptions to keep the analysis simple. After that, a more formal analysis is presented. 

Another key feature of a bond is its term to maturity the number of years over which the issuer has promised to meet the conditions of the debt obligation. The practice in the bond market is to refer to the term to maturity of a bond simply as its maturity or term. Bonds are debt capital market securities and therefore have maturities longer than one year. This differentiates them from money market securities. Bonds also have more intricate cash flow patterns than money market securities, which usually have just one cash flow at maturity. As a result, bonds are more complex to price than money market instruments, and theit prices are more sensitive to changes in the general level of interest rates. 

A bond s term to maturity is crucial because it indicates the period during which the bondholder can expect to receive coupon payments and the number of years before the principal is paid back. The principal of a bond—also referred to as its redemption value, maturity value, par value, or face value—is the amount that the issuer agrees to repay the bondholder on the maturity, or redemption, date, when the debt ceases to exist and the issuer redeems the bond. The coupon rate, or nominal rate, is the interest rate that the issuer agrees to pay during the bond s term. The annual interest payment made to bondholders is the bond s coupon. The cash amount of the coupon is the coupon rate multiplied by the principal of the bond. For example, a bond with a coupon rate of 8 percent and a principal of 1,000 will pay an annual cash amount of 80. 

A bond s term to maturity also influences the volatility of its price. All else being equal, the longer the term to maturity of a bond, the greater its price volatility. 

There are a large variety of bonds. The most common type is the plain vanilla, otherwise known as the straight, conventional, or bullet bond. A plain vanilla bond pays a regular—annual or semiannual—fixed interest payment over a fixed term. All other types of bonds are variations on this theme. 

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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. 

Exhibit 3.1 depicts this inverse relationship between an option-free bond s price and its discount rate (i.e., required yield). There are two things to infer from the price/discount rate relationship depicted in the exhibit. First, the relationship is downward sloping. This is simply the inverse relationship between present values and discount rates at work. Second, the relationship is represented as a curve rather than a straight line. In fact, the shape of the curve in Exhibit 3.1 is referred to as convex. By convex, it simply means the curve is bowed in relative to the origin. This second observation raises two questions about the convex or curved shape of the price/discount rate relationship. First, why is it curved Second, what is the import of the curvature ... 

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. 

In this formula, P is present worth or present value, F is future value, i is the interest or discount rate, and n is the number of periods. Economically, there is an additional factor at work in present value, and that factor is pure time preference, or impatience. However, this issue is generally ignored in business accounting, because the firm has no such emotions, and opportunities can be measured in terms of financial return. 

First, the project s anticipated benefit and cost are tabulated for each year of the project s lifetime. Then, these values are converted to present values by using the present-value equation, with the company s discount rate plugged in as the discount factor. Finally, the cumulative total of the benefits (at present value) and the cumulative total of the costs (at present... 

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). 

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 ... 

Work sheet for presenting discounted-cash flow, present-value, and net-present-worth determinations ... 

The trial-and-error approach can be simplified by dividing Eq (D) by ei0r and substituting the expression for CPKT0 tjme to give the present-value or discounted-cash-flow equation as follows ... 

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... 

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... 

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 ... 

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. 

A number of researchers have estimated the effective marginal credit rate (the percent reduction in the cost of R D) implicit in the several incarnations of this tax credit using a variety of methods and assumptions they have found effective credit rates that are substantially less than the statutory rates of 20 or 25 percent. The divergence between the effective and statutory rates stems from the way in which the credit is calculated, the interaction of the credit with other provisions of the internal revenue code,11 the rate at which future savings are discounted to their present value, and the fact that not all firms have sufficient tax liability to use credits in the year they are earned. 

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. 

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. 

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. 

What we have calculated is the present value (at a particular reference date) of a future sum of money, using a specified discount rate. In any discounting calculation, it is important to quote the reference date and the discount rate. 

The cashflow discussed in Section 13.2 did not take account of the time value of money, and was therefore an undiscounted cashflow. The discounting technique discussed can now be applied to this cashflow to determine the present value of each annual cashflow at a specified reference date. 

One way of calculating the IRR is to plot the NPV against discount rate, and to extrapolate/ interpolate to estimate the discount rate at which the NPV becomes zero, as in the Present Value Profile in Figure 13.16. The alternative method of calculating IRR is by... 

Wells are worked over to increase production, reduce operating cost or reinstate their technical integrity. In terms of economics alone (neglecting safety aspects) a workover can be justified if the net present value of the workover activity is positive (and assuming no other constraints exist). The appropriate discount rate is the company s cost of capital. 

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