Present value of the cash flow

To illustrate the method for determining net present worth, consider the example presented in Table 1 for the case where the value of capital to the company is at an interest rate of 15 percent. Under these conditions, the present value of the cash flows is 127,000 and the initial investment is 110,000. Thus, the net present worth of the project is... 

For investments 2 and 3, the present values of the cash flows to the projects are determined from the first two equations under part (c) of this problem, with i 0.15. The resulting net present worth are ... 

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 ... 

The present value of the cash flow in year n can be found by multiplying by (1 -E i) ", as described in equation 6.40. The net present value up to year n is the cumulative sum of all the present values of cash flow up to that year. 

The values of the general parameters we have assumed are given in table 5. The present values of the cash flows in the year 0 for the various rates of social discount and the general parameter values are given in table 6. With these data the various calculations are executed, results are given in table 7 and Figure 3. 

Some of the revenue and cost assumptions underlying this analysis were very uncertain, so OTA analyzed the sensitivity of the estimated returns to changes in critical assumptions. The results are somewhat sensitive to the ratio of global sales (about which we know relatively little) to U.S. sales (about which we know much more). If the ratio of global sales to U.S. sales is much greater than 2, as we have reason to believe it may be, the present value of the cash flows would be even more (after taxes) than is necessary to repay the R D investment. 

The results were not very sensitive to changes in the speed with which originator brand sales decline after patent expiration. If the average sales per compound were to decline by 20 percent per year after patent expiration, the present value of the cash flows would be 311 million before taxes and 209 million after taxes, still above the full after-tax cost of R D. Fully 6 years after the... 

In order to value a bond with the settlement date between coupon payments, we must answer three questions. First, how many days are there until the next coupon payment date From Chapter 1, we know the answer depends on the day count convention for the bond being valued. Second, how should we compute the present value of the cash flows received over the fractional period Third, how much must the buyer compensate the seller for the coupon earned over the fractional period This is accrued interest that we computed in Chapter 1. In the next two sections, we will answer these three questions in order to determine the full price and the clean price of a coupon bond. 

The sum of the present values of the cash flows is 101.8466. This price is referred to as the full price (or the dirty price). 

It is from the assumed values of 3-month LIBOR (i.e., the current spot rate and the implied forward rates) and the required margin in Column (6) that the discount rate that will be used to determine the present value of the cash flows will be calculated. The discount factor is found as follows ... 

The zero-volatility spread, also referred to as the Z-spread or static spread, is a measure of the spread that the investor would realize over the entire benchmark spot rate curve if the bond were held to maturity. Unlike the nominal spread, it is not a spread at one point on the yield curve. The Z-spread is the spread that will make the present value of the cash flows from the nongovernment bond, when discounted at the benchmark rate plus the spread, equal to the nongovernment bond s market price plus accrued interest. A trial-and-error procedure is used to compute the Z-spread. 

Suppose we select a spread of 100 basis points. To each benchmark spot rate shown in column 3 of Exhibit 3.15, 100 basis points are added. So, for example, the 1-year spot rate 5.33% (4.33% plus 1%). This spot rate is used to calculate the present values shown in the fourth column. Because the present value is not equal to the nongovernment issue s price of 101.9141, the Z-spread is not 100 basis points. If a spread of 120 basis points is tried, it can be seen from the next-to-last column of Exhibit 3.15 that the present value is 103.1835 again, because this is not equal to the nongovernment issue s price, 120 basis points is not the Z-spread. The last column shows the present value of the cash flows is equal to the nongovernment issue s price. Accordingly, 150 basis points is the Z-spread, compared to the nominal spread of 148.09 basis points. 

Step 4. Compare the present value of the cash flows as calculated in Step 3 to the price. If the present value is equal to the security s price, the discount margin is the margin assumed in Step 2. If the present value is not equal to the security s price, go back to Step 2 and select a different margin. 

PVCF = present value of the cash flow in period t discounted at the yield to maturity where t = 1,2,..., n... 

The zero-volatility or static spread is the spread that when added to the government spot rate curve will make the present value of the cash flows equal to the bond s price plus accrued interest. When spread is defined in this way, spread dnration is the approximate percentage change in price for a 100 basis point change in the zero-volatility spread holding the government spot rate curve constant. 

Bond prices are expressed per 100 nominal —that is, as a percentage of the bond s 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. 

This technique of calculating discount factors, known as bootstrapping, is conceptually neat, but may not work so well in practice. Problems arise when you do not have a set of bonds that mature at precise six-month intervals. Liquidity issues connected with individual bonds can also cause complications. This is true because the price of the bond, which is still the sum of the present values of the cash flows, may reflect liquidity considerations (e.g., hard to buy or sell the bond, difficult to find) that do not reflect the market as a whole but peculiarities of that... 

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. 

As explained in chapter 3, zero-coupon, or spot, rates are true interest rates for their particular terms to maturity. In zero-coupon swap pricing, a bank views every swap, even the most complex, as a series of cash flows. The zero-coupon rate for the term from the present to a cash flows payment date can be used to derive the present value of the cash flow. The sum of these present values is the value of the swap. 

As noted earlier,a newly transacted interest rate swap denotes calculating the swap rate that sets the net present value of the cash flows to zero. Valuation signifies the process of calculating the net present value of an existing swap by setting its fixed rate at the current market rate. Consider a plain vanilla interest rate swap with the following terms ... 

One economic evaluation method that is used is based on the internal return rate (IRR), which is considered suitable for an initial estimate of economic feasibility of industrial processes (Di Lucdo et al, 2002). This method is based on evaluating the discount rate that causes the present value of the cash flow, projected for the plant life, to be equal to the invested capital. The basic idea for the IRR evaluation is to obtain a single value that synthesizes the merits of the project for its lifetime. This value is not determined by market interest rates, and hence it is labelled internal return rate. The IRR is intrinsic to the project and does not depend on anything but the project cash flow. 

To find the present value of the cash flows, the following steps may be used. 

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... 

We illustrate the Z-spread calculation at FIGURE 19.6. This is done using a hypothetical bond, the XYZ PLC 5 percent of June 2008, a three-year bond at the time of the calculation. Market rates for swaps. Treasury, and CDS are also shown. We require the spread over the swaps curve that equates the present values of the cash flows to the current market price. The cash flows are discounted using the appropriate swap rate for each cash flow maturity. With a bond yield of 5-635 percent, we see that the I-spread is 43-5 basis points, while the Z-spread is 19.4 basis points. In practice, the difference between these two spreads is rarely this large. 

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