Redemptions dates

Bonds with embedded options are debt instruments that give the right to redeem the bond before maturity. As we know, the yield to maturity represents the key measure of bond s return (although, of course, it is an anticipated return that is seldom realised in practice). The calculation of the return is particularly easy for conventional bonds because the redemption date is known with certainty, as their value. In contrast, for callable bonds, but also for other bonds such as putable and sinking fund bonds, the redemption date is not known with certainty because the bonds can be redeemed before maturity. If we want to calculate... 

EXHIBIT 5.5 Euro Government Bond Market Outstanding by Redemption Date ( million)... 

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. 

The master trust transactions are largely insensitive to prepayment rates. The only requirement is that the principal receipts in the trust are sufficient for it to accumulate the bullet payments to meet the scheduled redemption dates. The principal payment rate, measured as the proportion of collateral redeemed or repurchased, has been running at an average rate of 4% per month. 

The typical credit card transaction structure has three different cash flow periods revolving, accumulation, and early amortisation. Each period performs a distinct function and allocates cash flows differently. Credit card transactions are usually structured as soft bullets in order to mimic a traditional corporate bond, that is, investors receive monthly or quarterly payments of interest with one single payment of principal on the scheduled redemption date. 

The investor interest is not reduced to zero on the scheduled redemption date. 

A bond s term to maturity is crucial because it indicates the period during which the bondholder can expect to receive coupon payments and the number of years before the principal is paid back. The principal of a bond—also referred to as its redemption value, maturity value, par value, or face value—is the amount that the issuer threes to repay the bondholder on the maturity, or redemption, date, when the debt ceases to exist and the issuer redeems the bond. The coupon rate, or nominal rate, is the interest rate that the issuer agrees to pay during the bond s term. The annual interest payment made to bondholders is the bond s coupon. The cash amount of the coupon is the coupon rate multiplied by the principal of the bond. For example, a bond with a coupon rate of 8 percent and a principal of 1,000 will pay an annual cash amount of 80. 

Expression (3.13) states that the bond price is a function of the continuously compounded interest rate. The r ht-hand side is the discount factor at time t. At t= T—that is, on the redemption date—the discount factor is 1, which is the redemption value of the bond and hence the price of the bond at that time. 

The yield calculation for conventional bonds is relatively straightforward. This is because their redemption dates are fixed, so their total cash flows—the data required to calculate yield to maturity— are known with certainty. Less straightforward to analyze are bonds with embedded options—calls, puts, or sinking funds—so called because the option element cannot be separated from the bond itself The difficulty in analyzing these bonds lies in the fact that some aspects of their cash flows, such as the timing or value of their future payments, are uncertain. 

Because a callable bond has more than one possible redemption date, its future cash flows are not clearly defined. To calculate the yield to maturity for such a bond, it is necessary to assume a particular redemption date. The market convention is to use the earliest possible one if the bond is priced above par and the latest possible one if it is priced below par. Yield calculated in this way is sometimes referred to as yield to worst (the Bloomberg term). 

If a bond s actual redemption date differs from the assumed one, its return computed this way is meaningless. The market, therefore, prefers to use other methods to calculate the return of callable bonds. The most common method is option-adjusted spread, or OAS, analysis. Although the discussion in this chapter centers on callable bonds, the principles enunciated apply to all bonds with embedded options. 

To calculate the modified duration of a bond with an embedded option, the bondholder must assume a fixed maturity date based on the bond s current price. When it is unclear what redemption date to use, modified duration may be calculated to both the first call date and the final maturity date. This is an unsatisfactory compromise, however, since neither date, and so neither measure, may be appropriate. The problem is more acute for bonds that are continuously callable or putable from the first call or put date until maturity. 

CPIm- = the CPI level three months before the bond s redemption date... 

If the bond has just paid the last coupon before its redemption date, (12.13) reduces to (12.14). 

FRNs can have additional features, such as flbors, which specify minimum levels below which the coupon cannot fall caps, which specify maximum rates and calls, which specify possible redemption dates before maturity. Perpetual FRNs also exist. As in other markets, borrowers frequently issue floating notes with specific, even esoteric, terms to meet particular requirements or customer demands. For example. Citibank issued a series of U.S. dollar—denominated FRNs indexed to the Euribor rate and another set of notes whose day count was linked to a specified LIBOR range. 

The PO bond is similar to a zero-coupon in that it is issued at a discount to par value. The PO bondholders return is a function of the rapidity at which prepayments are made the quicker the prepayment, the higher the return. This is like the buyer of a zero-coupon bond receiving the maturity payment ahead of the redemption date. The highest possible return for the bondholder would occur if all the mortgages were prepaid the instant after the PO bond was bought. A low return occurs if all the mortgages are held until maturity, so that there are no prepayments. 

The modified duration and convexity methods we have described are only suitable for use in the analysis of conventional fixed-income instruments with known fixed cash flows and maturity dates. They are not satisfactory for use with bonds that contain embedded options such as callable bonds or instruments with unknown final redemption dates such as mortgage-backed bonds. For these and other bonds that exhibit uncertainties in their cash flow pattern and redemption date, so-called option-adjusted measures are used. The most common of these is option-adjusted spread (OAS) and option-adjusted duration (OAD). The techniques were developed to allow for the uncertain cash flow structure of non-vanilla fixed-income instruments, and model the effect of the option element of such bonds. 

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