Semiannual cash flows

This assumes that the bond in question pays semiannual cash flows. If the bond pays annual cash flows, the cash flows must be reinvested at the annual yield to maturity. 

Note that this situation exposes one party to greater risk that the counterparty will default on payments. For instance, a party paying monthly and receiving semiannual cash flows will have made five interest payments before receiving one in return. 

As noted, the coupon rate is the interest rate the issuer agrees to pay each year. The coupon rate is used to determine the annual coupon payment which can be delivered to the bondholder once per year or in two or more equal installments. As noted, for bonds issued in European bond markets and the Eurobond markets, coupon payments are made annually. Conversely, in the United Kingdom, United States, and Japan, the usual practice is for the issuer to pay the coupon in two semiannual installments. An important exception is structured products (e.g., asset-backed securities) which often deliver cash flows more frequently (e.g., quarterly, monthly). 

For example, consider a 10-year government bond denominated in euros with an 8% coupon rate. Suppose that coupon payments are delivered semiannually and the annual spot rates are shown in the fourth column of Exhibit 3.4. The third column of the exhibit shows the cash flow every six months. The last column shows the present value of each cash flow discounted at the corresponding spot rate. The total in the last column is the arbitrage-free value of the bond, 115.2619. 

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. 

The long end of the swap curve is derived directly from observable coupon swap rates. These are generic plain vanilla interest rate swaps with fixed rates exchanged for floating interest rates. The fixed swap rates are quoted as par rates and are usually compounded semiannually (see Exhibit 20.2). The bootstrap method is used to derive zero-coupon interest rates from the swap par rates. Starting from the first swap rate, given all the continuously compounded zero rates for the coupon cash flows prior to maturity, the continuously compounded zero rate for the term of the swap is bootstrapped as follows ... 

The discount rate used to derive the present value of a bond s cash flows is the interest rate that the bondholders require as compensation for the risk of lending their money to the issuer. The yield investors require on a bond depends on a number of political and economic factors, including what other bonds in the same class are yielding. Yield is always quoted as an annualized interest rate. This means that the rate used to discount the cash flows of a bond paying semiannual coupons is exactly half the bond s yield. 

A bond paying a semiannual coupon has a dirty price of 98.50, an annual coupon of 3 percent, and exactly one year before maturity. The bond therefore has three remaining cash flows two coupon payments of 1.50 each and a redemption payment of 100. Plugging these values into equation (1.20) gives... 

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. 

The true yield measure derived in the previous section is not as straightforward as the one given earlier for the T-bill. Because a T-bill has only a single cash flow, its maturity value is known, so its return is easily calculated as its increase in value from start to maturity. Investors know that money put into a 90-day T-bill with a yield of 5 percent will have grown by 5 percent, compounded semiannually, at the end of three months. No such certainty is possible with coupon-bearing bonds. Consider although the investors in the 90-day T-bill are assured of a 5 percent yield after ninety days, they don t know what their investment will be worth after, say, sixty days or at what yield they will be able to reinvest their money when the bill matures. Such uncertainties don t effect the return of the short-term bill, but they have a critical impact on the return of coupon bonds. 

As noted in chapter 2, a Treasury bond can be seen as a bundle of individual zero-coupon securities, each maturing on one of the bond s cash flow payment dates. In this view, the Treasury s price is the sum of the present values of all the constituent zero-coupon bond yields. Assume that the spot rates for the relevant maturities—ri,r2,rg,.rj f—can be observed. If a bond pays a semiannual coupon computed at an annual rate of C from period 1 to period N, its present value can be derived using equation (16.7). 

The fourth column shows how much the market maker paid for each of the cash flows by buying the entire package of them—that is, by buying the bond at a yield of 8 percent. The 4 coupon payment due in three years, for instance, cost 3.1616, based on the 8 percent (4 percent semiannual) yield. But if the assumptions embodied in the table are correct, investors are willing to accept a lower yield, of 7-30 percent (3-65 percent semiannual), for this maturity and pay 3-2258 for the three-year strip corresponding to the coupon payment. On this one coupon payment, the market maker thus realizes a profit of 0.0645, the difference between 3.2258 and 3.1613. The total profit from selling all the strips is 0.4913 per 100 nominal of the original Treasury. 

The yield to maturity is the discount rate that is used to determine the present value of all future cash flows to be received. The yield is reported on an annual basis but is an add-on interest rate. That is, one half of the reported yield is the correct rate to use per six-month period for coupon bonds with semiannual payments of interest. 

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. 

---