Swap rates calculation

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. 

The volatility value used can be estimated in two ways. We can estimate volatility separately and then use this to calculate what the approximate convexity adjustment should be, or we may observe the convexity bias directly and derive a volatility value from this. This would require an examination of market swap rates and bond yields, and use these to estimate the volatility implied by these rates. 

At the initiation of an interest rate swap, the counterparties are agreeing to exchange future payments and no upfront payments by either party are made. This means that the swap terms must be such that the present value of the payments to be made by the counterparties must be at least equal to the present value of the payments that will be received. In fact, to eliminate arbitrage opportunities, the present value of the payments made by a party will be equal to the present value of the payments received by that same party. The equivalence (or no arbitrage) of the present value of the payments is the key principle in calculating the swap rate. 

Let s apply the formula to determine the swap rate for our 3-year swap. Exhibit 19.7 shows the calculation of the denominator of the formula. The forward discount factor for each period shown in Column (5) is obtained from Column (4) of Exhibit 19.6. The sum of the last column in Exhibit 19.7 shows that the denominator of the swap rate formula is 281,764,282. We know from Exhibit 19.6 that the present value of the floating-rate payments is 14,052,917. Therefore, the swap rate is... 

EXHIBIT 19.7 Calculating the Denominator for the Swap Rate Formula... 

The calculation of the swap rate for all swaps follows the same principle equating the present value of the fixed-rate payments to that of the floating-rate payments. 

To illustrate this, consider the 3-year swap used to demonstrate how to calculate the swap rate. Suppose that one year later, interest rates change as shown in Columns (4) and (6) in Exhibit 19.8. In Colnmn (4) shows the current 3-month EURIBOR. In Column (5) are the EURIBOR futures price for each period. These rates are used to compute the forward rates in Column (6). Note that the interest rates have increased one year later since the rates in Exhibit 19.8 are greater than those in Exhibit 19.3. As in Exhibit 19.3, the current 3-month EURIBOR and the forward rates are used to compute the floating-rate payments. These payments are shown in Column (8) of Exhibit 19.8. 

As discussed above, vanilla swap rates are often quoted as a spread that is a function mainly of the credit spread required by the market over the risk-free government rate. This convention is logical, because government bonds are the principal instrument banks use to hedge their swap books. It is unwieldy, however, when applied to nonstandard tailor-made swaps, each of which has particular characteristics that call for particular spread calculations. As a result, banks use zero-coupon pricing, a standard method that can be applied to all swaps. 

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 ... 

A forward-start swap s effective date is a considerable period—say, six months—after the trade date, rather than the usual one or two days. A forward start is used when one counterparty, perhaps foreseeing a rise in interest rates, wants to fix the cost of a future hedge or a borrowing now. The swap rate is calculated in the same way as for a vanilla swap. 

Some of the newer models refer to parameters that are difficult to observe or measure direcdy. In practice, this limits their application much as B-S is limited. Usually the problem has to do with calibratii the model properly, which is crucial to implementing it. Galibration entails inputtii actual market data to create the parameters for calculating prices. A model for calculating the prices of options in the U.S. market, for example, would use U.S. dollar money market, futures, and swap rates to build the zero-coupon yield curve. Multifactor models in the mold of Heath-Jarrow-Morton employ the correlation coefficients between forward rates and the term structure to calculate the volatility inputs for their price calculations. 

A more accurate approach m ht be the one used to price interest tate swaps to calculate the present values of future cash flows usit discount tates determined by the markets view on where interest rates will be at those points. These expected rates ate known as forward interest rates. Forward rates, however, are implied, and a YTM derived using them is as speculative as one calculated using the conventional formula. This is because the real market interest rate at any time is invariably different from the one implied earlier in the forward markets. So a YTM calculation made using forward rates would not equal the yield actually realized either. The zero-coupon rate, it will be demonstrated later, is the true interest tate for any term to maturity. Still, despite the limitations imposed by its underlying assumptions, the YTM is the main measure of return used in the markets. 

In effect, this is the standard bond price equation with the discount rate adjusted by whatever the Z-spread is it is an iterative calculation. The appropriate maturity swap rate is used, which is the essential... 

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. 

Configurational-bias Monte Carlo in the Gibbs ensemble has been successfully applied to the calculations of single-component vapor-liquid phase equilibria of linear and branched alkanes [61,63-67], alcohols [68,69], and a fatty-acid Langmuir monolayer [70]. The extension to multicomponent mixtures introduces a case in which the smaller molecules (members of a homologous series) have a considerably higher acceptance rate in the swap move than do larger molecules. In recent simulations for alkane mixtures... 

Z-spread is an alternative spread measure to the ASW spread. This type of spread uses the zero-coupon yield curve to calculate the spread, in which in this case is assimilated to the interest-rate swap curve. Z-spread represents the spread needful in order to obtain the equivalence between the present value of the bond s cash flows and its current market price. However, conversely to the ASW spread, the Z-spread is a constant measme. 

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. 

It was useful to show the basic features of an interest rate swap using quick calculations for the payments such as described above and then explaining how the parties to a swap either benefit or hurt when... 

The swap will specify the frequency of settlement for the fixed-rate payments. The frequency need not be the same as the floating-rate payments. For example, in the 3-year swap we have been using to illustrate the calculation of the floating-rate payments, the frequency is quarterly. The frequency of the fixed-rate payments could be semiannual rather than quarterly. 

We will refer to the present value of 1 to be received in period t as the forward discount factor. In our calculations involving swaps, we will compute the forward discount factor for a period using the forward rates. These are the same forward rates that are used to compute the floating-rate payments—those obtained from the EURIBOR futures contract. We must make just one more adjustment. We must adjust the forward rates used in the formula for the number of days in the period (i.e., the quarter in our illustrations) in the same way that we made this adjustment to obtain the payments. Specifically, the forward rate for a period, which we will refer to as the period forward rate, is computed using the following equation ... 

Once the swap transaction is completed, changes in market interest rates will change the payments of the floating-rate side of the swap. The value of an interest rate swap is the difference between the present value of the payments of the two sides of the swap. The 3-month EURIBOR forward rates from the current EURIBOR futures contracts are used to (1) calculate the floating-rate payments and (2) determine the discount factors at which to calculate the present value of the payments. 

To derive the swap term structure, observed market interest rates combined with interpolation techniques are used also, dates are constructed using the applicable business-day convention. Swaps are frequently con-strncted nsing the modified following bnsiness-day convention, where the cash flow occurs on the next business day unless that day falls in a different month. In that case, the cash flow occurs on the immediately preceding business day to keep payment dates in the same month. The swap curve yield calculation convention frequently differs by currency. Exhibit 20.2 lists the different payment frequencies, compounding frequencies, and day count conventions, as applicable to each currency-specific interest rate type. 

Although the pricing of a credit default swap can be numerically reduced to a model, the inputs to that model still remain subjective. How can one calculate an exact valne for R, the recovery value of an issuer s assets post-default Or, more importantly, how can one calculate the hazard rate X for an issuer What is the probability that a particular issuer will default in five years Determining the true credit risk of an issuer has been a topic of intense focus in recent years and, as a result, quite a variety of methods and models have surfaced. 

Further, if there were a variety of bonds of a particular issuer outstanding, with different maturities, a term structure of hazard rates could be constructed—which in turn could be used to price default swaps of any maturity. By reducing everything to the hazard rate X, we are able to calculate correctly the prices of different instruments regardless of their interest or premium payment frequencies and daycount conventions. Similarly each instrument s mechanics are stripped away (e.g., a default swap versus a bond) to reveal the true hazard rate. 

An interest rate swap is an agreement between two counterparties to make periodic interest payments to one another during the life of the swap. These payments take place on a predetermined set of dates and are based on a notional principal amount. The principal is notional because it is never physically exchanged—hence the off-balance-sheet status of the transaction—but serves merely as a basis for calculating the interest payments. 

An interest rate swap is thus an agreement between two parties to exchange a stream of cash flows that are calculated hy applying different interest rates to a notional principal. For example, in a trade between Bank A and Bank B, Bank A may agree to pay fixed semiannual coupons of 10 percent on a notional principal of 1 million in return for receiving from Bank B the prevailing 6-month LIBOR rate applied to the same principal. The known cash flow is Bank As fixed payment of 50,000 every six months to Bank B. 

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). 

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