Spot rates, calculating

In order to calculate the range of implied forward rates, we require the term stmcture of spot rates for all periods along the continuous discount function. This is not possible in practice, because a bond market will only contain a finite number of coupon-bearing bonds maturing on discrete dates. While the coupon yield curve can be observed, we are then required to fit the observed curve to a continuous term structure. Note that in the United Kingdom gilt market, for example there is a zero-coupon bond market, so that it is possible to observe spot rates directly, but for reasons of liquidity, analysts prefer to use a fitted yield curve (the theoretical curve) and compare this to the observed curve. 

The traditional approach to yield curve fitting involves the calculation of a set of discount factors from market interest rates. From this, a spot yield curve can be estimated. The market data can be money market interest rates, futures and swap rates and bond yields. In general, though this approach tends to produce ragged spot rates and a forward rate curve with pronounced jagged knot points, due to the scarcity of data along the maturity structure. A refinement of this technique is to use polynomial approximation to the yield curve. 

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 ... 

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. 

A Z-spread can be calculated relative to any benchmark spot rate curve in the same manner. The question arises what does the Z-spread mean when the benchmark is not the euro benchmark spot rate curve (i.e., default-free spot rate curve) This is especially true in Europe where swaps curves are commonly used as a benchmark for pricing. When the government spot rate curve is the benchmark, we indicated that the Z-spread for nongovernment issues captured credit risk, liquidity risk, and any option risks. When the benchmark is the spot rate curve for the issuer, for example, the Z-spread reflects the spread attributable to the issue s liquidity risk and any option risks. Accordingly, when a Z-spread is cited, it must be cited relative to some benchmark spot rate curve. This is essential because it indicates the credit and sector risks that are being considered when the Z-spread is calculated. Vendors of analytical systems such Bloomberg commonly allow the user to select a benchmark. 

The third and final step is to calculate the percentage change in the bond s portfolio value when each key rate and neighboring spot rates are changed. There will be as many key rate durations as there are preselected key rates. Let s illustrate this process by calculating the key rate duration for a coupon bond. Our hypothetical 6% coupon bond has a maturity value of 100 and matures in five years. The bond delivers coupon payments semiannually. Valuation is accomplished by discounting each cash flow using the appropriate spot rate. The bond s current value is 107.32 and the process is illustrated in Exhibit 4.27. The initial hypothetical (and short) spot curve is contained in column (3). The present values of each of the bond s cash flows is presented in the last column. 

To compute the key rate duration of the 5-year bond, we must select some key rates. We assume the key rates are 0.5, 3, and 5 years. To compute the 0.5-year key rate duration, we shift the 0.5-year rate upwards by 20 basis points and adjust the neighboring spot rates between 0.5 and 3 years as described earlier. (The choice of 20 basis points is arbitrary.) Exhibit 4.28 is a graph of the initial spot curve and the spot curve after the 0.5-year key rate and neighboring rates are shifted. The next step is to compute the bond s new value as a result of the shift. This calculation is shown in Exhibit 4.29. The bond s value subsequent to the shift is 107.30. To... 

Calculating the Forward Rate from Spot-Rate Discount Factors... 

A swap s fixed-rate payments are known in advance, so deriving their present values is a straightforward process. In contrast, the floating rates, by definition, are not known in advance, so the swap bank predicts them using the forward rates applicable at each payment date. The fotward rates are those that are implied from current spot rates. These are calculated using equation (7.6). 

The relationship between discount factors and the spot rates for the same periods can be used to calculate forward rates. Say the spot rate for period 1 is known. The corresponding discount rate can be derived using (7.9), which reduces to (7.11). 

The theoretical 1-year spot rate is twice 2, or 0.06308, for an annualized bond-equivalent yield of 6.308 percent. This figure can now be used to calculate the theoretical 1.5-year spot rate. The cash flows for the 7 percent 1.5-year coupon Treasury are September 1, 1999 3.50... 

We have shown then that the price of a CDS contract can be calculated from the spot rates and default probability values given earlier. In this example, we assume that the credit event (default) occurs halfway through the premium period, thus enabling us to illustrate the calculation of the present value of the receipt in event of default (the second part of the left-hand side of the original no-arbitrage equation (10.11), the accrual factor) in more straightforward fashion. 

Using the spot rate structure at Table 12.1, the price of this bond is calculated to be 98.21. This would be the bonds fair value if it were liquid and default free. Assume, however, that the bond is a corporate bond and carries an element of default risk, and is priced at 97.00. What spread over the risk-free price does this indicate We require the spread over the implied forward rate that would result in a discounted price of 97.00. Using iteration, this is found to be 67.6 basis points. The calculation is... 

The approach just described is in essence the OAS methodology. However, it applies only to an environment in which the future path of interest rates is known with certainty. The spread calculated is the OAS spread under conditions of no uncertainty. We begin to appreciate that this approach is preferable to the traditional one of comparing redemption yields whereas the latter uses a single discount rate, the OAS approach uses the correct spot rate for each period s cash flow. Our interest, though, lies with conditions of interest-rate uncertainty. 

Our starting point is the redemption yield curve, from which we calculate the current spot rate term structure. This was done using RATE software and is shown in column four. Using the spot rate structure, we calculate the present value of the Treasury security s cash flows, which is shown in column seven. We wish to calculate the OAS that equates the price of the Treasury to that of the corporate bond. By iteration, this is found to be 110.81 basis points. This is the semiannual OAS spread. The annualized OAS spread is double this. With the OAS spread added to the spot rates for each period, the price of the Treasury matches that of the corporate bond, as shown in column nine. The adjusted spot rates are shown in column eight. Figure 12.3 illustrates the yield curve for the Treasury security and the corporate bond. 

Computer Models, The actual residence time for waste destmction can be quite different from the superficial value calculated by dividing the chamber volume by the volumetric flow rate. The large activation energies for chemical reaction, and the sensitivity of reaction rates to oxidant concentration, mean that the presence of cold spots or oxidant deficient zones render such subvolumes ineffective. Poor flow patterns, ie, dead zones and bypassing, can also contribute to loss of effective volume. The tools of computational fluid dynamics (qv) are useful in assessing the extent to which the actual profiles of velocity, temperature, and oxidant concentration deviate from the ideal (40). 

Catalyst Effectiveness. Even at steady-state, isothermal conditions, consideration must be given to the possible loss in catalyst activity resulting from gradients. The loss is usually calculated based on the effectiveness factor, which is the diffusion-limited reaction rate within catalyst pores divided by the reaction rate at catalyst surface conditions (50). The effectiveness factor E, in turn, is related to the Thiele modulus,

first-order rate constant, a the internal surface area, and the effective diffusivity. It is desirable for E to be as close as possible to its maximum value of unity. Various formulas have been developed for E, which are particularly usehil for analyzing reactors that are potentially subject to thermal instabilities, such as hot spots and temperature mnaways (1,48,51). 

With a reactive solvent, the mass-transfer coefficient may be enhanced by a factor E so that, for instance. Kg is replaced by EKg. Like specific rates of ordinary chemical reactions, such enhancements must be found experimentally. There are no generalized correlations. Some calculations have been made for idealized situations, such as complete reaction in the liquid film. Tables 23-6 and 23-7 show a few spot data. On that basis, a tower for absorption of SO9 with NaOH is smaller than that with pure water by a factor of roughly 0.317/7.0 = 0.045. Table 23-8 lists the main factors that are needed for mathematical representation of KgO in a typical case of the absorption of CO9 by aqueous mouethauolamiue. Figure 23-27 shows some of the complex behaviors of equilibria and mass-transfer coefficients for the absorption of CO9 in solutions of potassium carbonate. Other than Henry s law, p = HC, which holds for some fairly dilute solutions, there is no general form of equilibrium relation. A typically complex equation is that for CO9 in contact with sodium carbonate solutions (Harte, Baker, and Purcell, Ind. Eng. Chem., 25, 528 [1933]), which is... 

D-Xylulose 5-phosphate (ii-threo-2-pentulose 5-phosphate, XP) stands as an important metabolite of the pentose phosphate pathway, which plays a key fimction in the cell and provides intermediates for biosynthetic pathways. The starting compound of the pathway is glucose 6-phosphate, but XP can also be formed by direct phosphorylation of D-xylulose with li-xylulokinase. Tritsch et al. [114] developed a radiometric test system for the measurement of D-xylulose kinase (XK) activity in crude cell extracts. Aliquots were spotted onto silica plates and developed in n-propyl alcohol-ethyl acetate-water (6 1 3 (v/v) to separate o-xylose/o-xylulose from XP. Silica was scraped off and determined by liquid scintillation. The conversion rate of [ " C]o-xylose into [ " C]o-xylulose 5-phosphate was calculated. Some of the works devoted to the separation of components necessary while analyzing enzyme activity are presented in Table 9.8. 

Chen et al. [70] suggested that temperature gradients may have been responsible for the more than 90 % selectivity of the formation of acetylene from methane in a microwave heated activated carbon bed. The authors believed that the highly nonisothermal nature of the packed bed might allow reaction intermediates formed on the surface to desorb into a relatively cool gas stream where they are transformed via a different reaction pathway than in a conventional isothermal reactor. The results indicated that temperature gradients were approximately 20 K. The nonisothermal nature of this packed bed resulted in an apparent rate enhancement and altered the activation energy and pre-exponential factor [94]. Formation of hot spots was modeled by calculation and, in the case of solid materials, studied by several authors [105-108],... 

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