Coupons calculation convention

Consider a bond with a dirty price—including the accrued interest the seller is entitled to receive—of 97-89, a coupon of 6 percent, and five years to maturity. FIGURE 1.6 shows the gross redemption yields this bond would have under the different yield-calculation conventions. 

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) ... 

The accrued interest calculation for a bond is dependent on the day-count basis specified for the bond in question. We have already seen that when bonds are traded in the market the actual consideration that changes hands is made up of the clean price of the bond together with the accrued that has accumulated on the bond since the last coupon payment these two components make up the dirty price of the bond. When calculating the accrued interest, the market will use the appropriate day-count convention for that bond. A particular market will apply one of five different methods to calculate accrued interest these are ... 

For settlement purposes there are separate rounding conventions applied to the zero-coupon and coupon-paying issues. For zero-coupon bonds there is no rounding in the calculation, but the settlement price is rounded to the nearest krona. Coupon bonds are rounded once before you get to the settlement price, the clean price is rounded to three decimal places before adding on accrued interest. The settlement price is then rounded to the nearest krona. 

These matching errors are less alarming than they might appear, because all known price information is built into the real yield formula as soon as it is published. Two things really matter, in terms of the achieved real yield relative to the quoted real yield at purchase firstly, the difference between future inflation over the life of the bond and the 3% inflation assumption used in the market convention for calculating yields, and secondly (as with any coupon bond) reinvestment risk. However, we are getting ahead of ourselves—the next section handles the market s yield conventions. 

Calculation of the coupon income is the difference between the accrued interest bought in at the time of purchasing the cash market bond subtracted from the accrued interest received when the bond is sold. Market conventions play an important role here, for example How many days between trade and settlement How many days in a month How many days in a year In the United Kingdom the market convention is actual number of days in a month and a 365 days in a year. In Germany the convention is actual number of days in month and 360 days in a year. 

These figures demonstrate the impact that the coupon-payment and discounting frequencies have on a bonds redemption yield calculation. Specifically, increasing the frequency of discounting lowers the calculated yield, while increasing the frequency of payments raises it. When comparing yields for bonds that trade in markets with different conventions, it is important to convert all the yields to the same calculation basis. 

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. 

All bonds except zero-coupon bonds accrue interest on a daily basis that is then paid out on the coupon date. As mentioned earlier, the formulas discussed so far calculate bonds prices as of a coupon payment date, so that no accrued interest is incorporated in the price. In all major bond markets, the convention is to quote this so-called clean price. 

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. 

Because the future values for the reference index are not known, it is not possible to calculate the redemption yield of an FRN. On the coupon-reset dates, the note will be priced precisely at par. Between these dates, it will trade very close to par, because of the way the coupon resets. If market rates rise between reset dates, the note will trade slightly below par if rates fall, it will trade slightly above par. This makes FRNs behavior very similar to that of money market instruments traded on a yield basis, although, of course, the notes have much longer maturities. FRNs can thus be viewed either as money market instruments or as alternatives to conventional bonds. Similarly, they can be analyzed using two approaches. 

This formula calculates the fair price on a coupon payment date, so there is no accrued interest incorporated into the price. Accrued interest is an accounting convention that treats coupon interest as accruing every day a bond is held this accrued amount is added to the discounted present value of the bond (the clean price) to obtain the market value of the bond, known as the dirty price. The price calculation is made as of the bond s settlement date, the date on which it actually changes hands after being traded. For a new bond issue, the settlement date is the day when the investors take delivery of the bond and the issuer receives payment. The settlement date for a bond traded in the secondary market—the market where bonds are bought and sold after they are first issued—is the day the buyer transfers payment to the seller of the bond and the seller transfers the bond to the buyer. 

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 bond itself may be analyzed—in the first instance— as a conventional fixed-income security, so using its coupon and maturity date we may calculate a current yield (running yield) and yield-to-maturity. The yield advantage is the difference between the current yield and the dividend yield of the underlying share, given by (13 3). 

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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