Swaps calculating interest

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. 

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 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... 

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. 

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 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. 

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. 

The final component of the default swap is the accrued premium that may be payable by the buyer to the seller. If a default occurs somewhere in between two premium payment dates, which is likely considering there are only four payment dates a year on a quarterly default swap, then it is standard market practice for the buyer of protection to pay the accrued premium from the most recent premium payment date to and including the date of default. The value of this accrued on default is calculated in a similar manner to the value of the default protection above. However, instead of receiving 1 - R upon a default, the buyer will be paying a certain amount of accrued interest. 

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. 

Although for the purposes of explaining swap structures both parties are said to pay and receive interest payments, in practice only the net difference between both payments changes hands at the end of each interest period. This makes administration easier and reduces the number of cash flows for each swap. The final payment date falls on the maturity date of the swap. Interest is calculated using equation (7.1). 

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. 

The primary risk measure required when using a swap to hedge is the present value of a basis point. PVBP, known in the U.S. market as the dollar value of a basis point, or DVBP indicates how much a swap s value will move for each basis point change in interest rates and is employed to calculate the hedge ratio. PVBP is derived using equation (7 23). 

If the system is reciprocal, the swapping of the recording and current carrying electrode pairs shall give the same transfer impedance. It is also possible to have the eleetrode system situated into the volume of interest, for example, as needles or catheters. Sueh volume calculation, for example, of cardiae output, is used in some implantable heart pacemaker designs (see Seetion 10.12.3). 

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. 

The zero-coupon curve is used in the asset swap valuation. This curve is derived from the swap curve, so it is the implied zero-coupon curve. The asset swap spread is the spread that equates the difference between the present value of the bond s cash flows, calculated using the swap zero rates, and the market price of the bond. This spread is a function of the bond s market price and yield, its cash flows, and the implied zero-coupon interest rates. ... 

The conventional approach for analyzing an asset swap uses the bonds yield-to-maturity (YTM) in calculating the spread. The assumptions implicit in the YTM calculation (see Chapter 2) make this spread problematic for relative analysis, so market practitioners use what is termed the Z-spread instead. The Z-spread uses the zero-coupon yield curve to calculate spread, so is a more realistic, and effective, spread to use. The zero-coupon curve used in the calculation is derived from the interest-rate swap curve. 

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