Semiannual coupon payment

The valuation of a conventional bond can be performed also using a binomial tree. On maturity, the bond must be priced at par value plus the semiannual coupon payment equal to 2.75. Therefore, the value of a conventional IxHid at maturity tio must be equal to 102.75. The value of the bond in other nodes prior to maturity is calculated using the semi-aimual discount rate of 4.02%. For instance, at node the pricing is given by Equation (9.6) ... 

There are several features about floaters that deserve mention. First, a floater may have a restriction on the maximum (minimum) coupon rate that be paid at any reset date called a cap (floor). Second, while a floater s coupon rate normally moves in the same direction as the reference rate moves, there are floaters whose coupon rate moves in the opposite direction from the reference rate. These securities are called inverse floaters. As an example, consider an inverse floater issued by the Republic of Austria. This issue matures in April 2005 and delivers semiannual coupon payments according to the following formula ... 

The semiannual coupon payment, or interest, on a particular dividend date is calculated using equation (12.3). 

The current yield takes into account the coupon and bond s price and is calculated by taking the coupon and dividing that by the market price of the bond. Though widely quoted in newspapers and by brokers, it is relatively useless since it ignores the interest you receive on your coupon payments. Also, how do you calculate the return when you reinvest your semiannual coupon payments ... 

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. 

Consider a hypothetical 10-year bond selling at par ( 100) with a coupon rate of 7%. Assume the bond delivers coupon payments aimually. The yield to maturity for this bond is 7%. Suppose an investor buys this bond, holds it to maturity, and receives the maturity value of 100. In addition, the investor receives 10 annual coupon payments of 7 and can reinvest them every year that they received at an annual rate of 7%. What are the total future euros assuming a 7% reinvestment rate As demonstrated above, an investment of 100 must generate 196.72 in order to generate a yield of 7% compounded semiannually. Alternatively, the bond investment of 100 must deliver a total euro return of 96.72. 

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. 

Note that 2A is now the power to which the discount factor is raised. This is because a bond that pays a semiannual coupon makes two interest payments a year. It might therefore be convenient to replace the number of years to maturity with the number of interest periods, which could be represented by the variable n, resulting in formula (1.14). 

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... 

Calculating the redemption yield of bonds that pay semiannual coupons involves the semiannual discounting of those payments. This approach is appropriate for most U.S. bonds and U.K. gilts. Government bonds in most of continental Europe and most Eurobonds, however, pay annual coupon payments. The appropriate method of calculating their redemption yields is to use annual discounting. The two yield measures are not directly comparable. 

It is possible to make a Eurobond directly comparable with a U.K. gilt by using semiannual discounting of the formers annual coupon payments or using annual discounting of the latter s semiannual payments. The formulas for the semiannual and annual calculations appeared above as (1.13) and (1.12), respectively, and are repeated here as (1.22) and (1.23). 

The market convention is sometimes simply to double the semiannual yield to obtain the annualized yields, despite the fact that this produces an inaccurate result. It is only acceptable to do this for rough calculations. An annualized yield obtained in this manner is known as a hand equivalent yield. It was noted earlier that the one disadvantage of the YTM measure is that its calculation incorporates the unrealistic assumption that each coupon payment, as it becomes due, is reinvested at the rate rm. Another disadvantage is that it does not deal with the situation in which investors do not hold their bonds to maturity. In these cases, the redemption yield will not be as great. Investors might therefore be interested in other measures of return, such as the equivalent zero-coupon yield, considered a true yield. 

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. 

Expression (7.8) is for bonds paying semiannual coupons. It can be generalized to apply to bonds whose coupon frequency is N, where = 1 (for an annual coupon payment) to 12, and replacing 2 in the discount factors denominator with TV. Solving (7.8) thus modified for the th discount factor results in equation (7.9). 

Semiannual coupon Time to maturity Bond price volatility Coupon payments... 

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. 

Example Assume that a 9 percent coupon bond with a face value of 1,000 has just been purchased. Interest on this bond is paid semiannually, and it has been 45 days since the last coupon payment. What is the accrued interest on this bond ... 

Note that the swap PVBP, 425, is lower than that of the 5-year fixed-coupon bond, which is 488.45. This is because the floating-rate bond PVBP reduces the risk exposure of the swap as a whole by 63.45. As a rough rule of thumb, the PVBP of a swap is approximately the same as that of a fixed-rate bond whose term runs from the swaps next coupon reset date through the swap s termination date. Thus, a 10-year swap making semiannual payments has a PVBP close to that of a 9.5-year fixed-rate bond, and a swap with 5.5 years to maturity has a PVBP similar to that of a 5-year bond. 

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. 

The principal is the amount of money that you lend to the issuer of the bond, most of the time, this principal is set at a relatively simple 1,000 so that institutions can more easily market an issue (it makes it easier for us individual investors, tool). Also included in the indenture is the coupon, which is the stated interest rate that the borrower promises to pay you. To make things easier, many bond investors think of coupons as the annual rate of interest expressed as a percentage of the face value of the bond. This rate is fixed when the bond is first issued (although there are some bonds with floating coupons), most issuers make semiannual payments based on this fixed rate. 

U.S. Treasury notes and bonds are coupon bonds that pay interest semiannually. For example, if the bond s coupon rate is 10 percent, a 1,000 investment will give the investor 100, paid out in two semiannual payments of 50. This represents a 10 percent return on investment. 

Because bonds typically have a predictable stream of payments of interest and repayment of principal, many people invest in them to receive interest income or to preserve and to accumulate capital. If you are looking for current income, you will most likely be interested in bonds that pay an interest rate that stays fixed until maturity with interest that is paid semiannually. However, if you are saving for retirement or a child s education or other capital accumulation goal, you may wish to consider investing in zero coupon bonds which do not have periodic interest payments. Instead, they are sold at a substantial discount from their face amount and the investor receives one payment— at maturity—that is equal to the purchase price (principal) plus the total interest earned, compounded semiannually at the original interest rate. 

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