Coupons formula

In contrast to a conpon rate that remains unchanged for the bond s entire life, a floating-rate security or floater is a debt instrument whose coupon rate is reset at designated dates based on the value of some reference rate. Thus, the coupon rate will vary over the instrument s life. The coupon rate is almost always determined by a coupon formula. For example, a floater issued by Aareal Bank AG in Denmark (due in May 2007) has a coupon formula equal to three month EURIBOR plus 20 basis points and delivers cash flows quarterly. 

Thus far our coverage of valuation has been on fixed-rate coupon bonds. In this section we look at how to value credit-risky floaters. We begin our valuation discussion with the simplest possible case—a default risk-free floater with no embedded options. Suppose the floater pays cash flows quarterly and the coupon formula is 3-month LIBOR flat (i.e., the quoted margin is zero). The coupon reset and payment dates are assumed to coincide. Under these idealized circumstances, the floater s price will always equal par on the coupon reset dates. This result holds because the floater s new coupon rate is always reset to reflect the current market rate (e.g., 3-month LIBOR). Accordingly, on each coupon reset date, any change in interest rates (via the reference rate) is also reflected in the size of the floater s coupon payment. 

We will illustrate this process using a hypothetical 4-year floater that deliver cash flows quarterly with a coupon formula equal to 3-month LIBOR plus 15 basis points and does not possess a cap or a floor. The coupon reset and payment dates are assumed to be the same. For ease of exposition, we will invoke some simplifying assumptions. First, the issue will be priced on a coupon reset date. Second, although floaters typically use an ACT/360 day-count convention, for simplicity we will assume that each quarter has 91 days. Third, we will assume initially that the LIBOR yield curve is flat such that all implied 3-month LIBOR forward rates are the same. (We will relax this assumption shortly.) Note the same principles apply with equal force when these assumptions are relaxed. 

There are several yield spread measures or margins that are routinely used to evaluate floaters. The four margins commonly used are spread for life, adjusted simple margin, adjusted total margin, and discount margin. To illustrate these measures, we will assume a floater that has a coupon formula equal to 3-month LIBOR plus 45 basis points and delivers cash flows quarterly. 

The price process under the new measure Tq, either is used to derive the formula for the zero-coupon bond option (see section (5.2.1)), the characteristic function in (5.2.2), or finally to compute the moments of the underlying random variable (section (5.3.3) and (5.3.4)). 

Thus, we end up with the well known Black and Scholes -like formula for the price of a European call option on a zero-coupon bond... 

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 coupon and/or principal payment is reset periodically based on a formula that depends on one or more market variables (e.g., interest rates, inflation rates, exchange rates, etc.). 

The cash flow is found by multiplying the coupon rate and the maturity value (assumed to be 100). However, the coupon rate (the forward rate in the previous period plus the quoted margin) must be adjusted for the number of days in the quarterly payment period. The formula to do so is... 

As an example, if a 1 million par value position of this 5-year bond is held to maturity, the investor will receive 150,000 in coupon payments over the next five years. The total coupon payments can be found on the left-hand side of the YA screen in Exhibit 3.9 and are labeled Coupon payment. Suppose an investor can reinvest each of these five annual coupon payments at say 2.975% compounded annually. Recall the general formula for the future value of an ordinary annuity when payments occur m times per year is... 

The current yield of a bond is calculated by dividing the security s annual dollar coupon payment by the market price. The formula for the current yield is... 

Current yield possesses a number of drawbacks as a potential return measure. Current yield considers only coupon interest and no other source of return that will affect an investor s yield. To see this, assume the current yield is a yield to perpetuity, the annual euro coupon payment is a perpetual annuity payment, and the security s price is the present value of the perpetual annuity. By rearranging terms such that the price equals the annual coupon payment divided by the current yield, we obtain the present value of a perpetual annuity formula as shown below... 

To verify this, this par bond s total euro return of 96.72 is driven by two sources of euro return coupon payments and reinvestment income. Recall from the beginning of the chapter, the reinvestment income can be determined using the future value of an ordinary annuity formula. Accordingly, the future value of 10 annual payments of 7 to be received plus the interest earned by investing the payments at 7% compounded annually is found as follows ... 

More specifically, this is the formula for the modified duration of a bond on a coupon anniversary date. 

Two measures have been developed to estimate the sensitivity of a floater to each component of the coupon reset formula the index (i.e., reference rate)... 

Here, and in the complexity of the real yield calculation, the United Kingdom suffers perhaps for being the prototype linker market. An eight-month lag is needed in order to always know with certainty the future money, or cash, value of the next coupon, so that the money value of that coupon can be accrued. The major innovation of the Canadian model is the use of a formula that accrues coupons on a real basis, removing the need to know precisely what will be paid in cash terms on the next coupon day. 

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. 

The real yield formula below has been taken from the Debt Management Office s Formulae for Calculating Gilt Prices from Yields, 15 January 2002 update, and it covers bonds with two or more remaining cash flows. The term quasi-coupon date, in the notes that follow the formula, means the theoretical cash flow dates determined by the redemption date—they are quasi dates because weekends and holidays may mean the true payment dates differ. 

With an eight-month lag, we will know the next coupon cash flow and possibly the subsequent one, but no others. The formula takes the latest known RPI value and from that month projects all future monthly RPI values using an inflation assumption (currently 3%), which is a market convention. So an unknown RPI in month t is given by... 

Nicole El Karoui and Jean-Charles Rochet, A Price Formula for Options on Coupon Bonds, SEEDS Discussion Series, Institute de Economica Publica, Spain (1995). 

In the event of a default, there will be no payment of accrued interest by the issuer since generally coupons are not recoverable, so we can ignore that aspect of the valnation. The only component left to the bond is its value upon a default or its recovery value. If a default occurs, the bondholder will lose all future cash flows of the instrument (i.e., they are at risk), but will be left with a nonperforming asset worth R. The same technique is used here as in the default swap—except this time the payout is R rather than 1 - R upon default. Conveniently, this formula is identical to equation (22.11), except that the term (1 - R) is replaced by R. 

Figure 2. Bode-magnitude plots ISPC formula of solvent-free acrylic latex baking enamel on different CRS coupons.

Figure 3. Bode-magnitude plots Control and ISPC formulas of polyester-melamine white paint on 2024 T3 Al coupons after soaking in a 3% NaCl solution for 72 hours.

In cathodic delamination, the delamination rate of an organic coating under a cathodic potential depends upon the applied potential, electrolyte solution, and metal substrate (22,25). Cathodic delamination tests were conducted for all three commercial paints. As an example, Figure 5 shows the cathodic delamination plots of control alkyd enamel and water-reducible ISPC coated on bare CRS coupons. Curves Sa and 5b represent delamination area for the control alkyd enamel and the water-reducible ISPC formulation, respectively curves 5c and 5d are the respective plots of delamination current for the two formulations. The delamination test was conducted in a 3% NaCI solution the alkyd coated CRS coupon served as a cathode and was polarized at -I.IV versus a saturated calomel electrode. A significantly slower delamination rate was obtained for the ISPC formula (curve 5b) as compared to the control alkyd enamel (curve 5a). At 44 hours of delamination time, the entire painted area of the working electrode (almost 20 cm x 20 cm) for the control alkyd paint had been delaminated, whereas the delamination area of the water-reducible ISPC was only as little as 1 cm indicating a remarkable coating adhesion improvement for the alkyd ISPC painted on bare CRS coupon. 

Figure 6. Salt spray (fog) tests (100 hours) for the control and ISPC formulas of water reducible baking enamel on bare CRS, iron phosphated B-IOOO, and iron phosphated plus chromated, B-1000 + P60 coupons.

The fair price of a bond is the sum of the present values of all its cash flows, including both the coupon payments and the redemption payment. The price of a conventional bond that pays annual coupons can therefore be represented by formula (1.12). 

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). 

This formula calculates the fair price on a coupon payment date, so there is no accrued incorporated into the price. Accrued interest is an... 

Equation (I.l4) assumes an even number of coupon payment dates remaining before maturity. If there are an odd number, the formula is modified as shown in (1.15). 

Another assumption embodied in the standard formula is that the bond is traded for settlement on a day that is precisely one interest period before the next coupon payment. If the trade takes place between coupon dates, the formula is modified. This is done by adjusting the exponent for the discount factor using ratio i, shown in (1.16). 

The denominator of this ratio is the number of calendar days between the last coupon date and the next one. This figure depends on the day-count convention (see below) used for that particular bond. Using /, the price formula is modified as (1.17) (for annual-coupon-paying bonds for bonds with semiannual coupons, r/2 replaces r). 

As noted above, the bond market includes securities, known as zero-coupon bonds, or strips, that do not pay coupons. These are priced by setting C to 0 in the pricing equation. The only cash flow is the maturity payment, resulting in formula (1.18)... 

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). 

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 noted earlier, for bonds that are trading ex-dividend, the accrued coupon is negative and is subtracted from the clean price. The negative accrued interest is calculated using formula (1.26). 

Duration is a measure of price sensitivity to interest rates—that is, how much a bond s price changes in response to a change in interest rates. In mathematics, change like this is often expressed in terms of differential equations. The price-yield formula for a plain vanilla bond, introduced in chapter 1, is repeated as (2.1) below. It assumes complete years to maturity, annual coupon payments, and no accrued interest at the calculation date. 

It is possible to shorten the procedure of computing Macaulay duration longhand, by rearranging the bond-price formula (2.1) as shown in (2.10), which, as explained in chapter 1, calculates price as the sum of the present values of its coupons and its redemption payment. The same assumptions apply as for (2.1). 

The unit in which convexity, as defined by (2.18), is measured is the number of interest periods. For annual-coupon bonds, this is equal to the number of years for bonds with different coupon-payment schedules, formula (2.19) can be used to convert the convexity measure from interest periods to years. 

The reason the convexity term is multiplied by one-half is because the second term in the Taylor expansion used to derive the convexity equation contains the coefficient 0.5. The formula is the same for a semiannual-coupon bond. 

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