Maturity date, bonds

Bond A long-term debt-type of security generally issued by corporations or governments to generate cash. The coupon rate is the interest rate paid to the bondholder. The maturity date is when the face value of the bond will be paid to the bondholder. 

The continuously compounded gross redemption yield at time ton a default-free zero-coupon bond that pays 1 at maturity date 7 is x. We assume that the movement in X is described by... 

We can define forward rates in terms of the short rate. Again for infinitesimal change in time from a forward date TitoT (for example, two bonds whose maturity dates are very close together), we can define a forward rate for instantaneous borrowing, given by... 

This was first described by McCulloch (1975) and is referred to in Deacon and Derry (1994). We assume the maturity term structure is partitioned into q knot points with qwhere qi = 0 and q is the maturity of the longest dated bond. The remaining knot points are spaced such that there is, as far as possible, an equal number of bonds between each pair of knot points. With j < q, we employ the following functions ... 

One final point regarding duration is that it is possible to calculate a tax-adjusted duration for an index-linked bond in markets where there is a different tax treatment to indexed bonds compared to conventional bonds. In the United States market, the returns on indexed and conventional bonds are taxed in essentially the same manner, so that in similar fashion to Treasury strips, the inflation adjustment to the indexed bond s principal is taxable as it occurs, and not only on the maturity date. Therefore, in the US-indexed bonds do not offer protection against any impact of after-tax effects of high inflation. That is, Tips real yields reflect a premium for only pretax inflation risk. In the United Kingdom market however, index-linked gilts receive preferential tax treatment, so their yields... 

The drawbacks of each of these approaches are apparent. A rather more valid and sound approach is to constmct a term structure of the real interest rates, which would indicate, in exactly the same way that the conventional forward rate curve does for nominal rates, the market s expectatimis rat future inflation rates. In countries where there are liquid markets in both conventional and inflation-indexed bmids, we can observe a nominal and a real yield curve. It then becomes possible to estimate both a conventional and a real term structure using these allows us to create pairs of hypothetical conventional and indexed bonds that have identical maturity dates, for any point on the term structure. We could then apply the break-even approach to any pair of bonds... 

Expression (7.1) states that the price of a zero-coupon bond is equal to the discount factor from time t to its maturity date or the average of the discount factors under all interest-rate scenarios, weighted by their probabilities. It can be shown that the T-maturity forward rate at time t is given by... 

Maturity Generally default spreads are larger for longer date bonds and narrow for short-term bonds. 

Generally, the theoretical default spread is almost exactly proportional to the default probability, assuming a constant default probability. Generally, however, the default probability is not constant over time, nor do we expect it to be. In Figure 8.3, we show the theoretical default spread for triple-B-rated bonds of various maturities, where the default probability rises from 0.2% to 1 % over time. The longer dated bonds, therefore, have a higher aimual default risk and so their theoretical default spread is higher. Note that after around 20 years the expected default probability is constant at 1%, so the required yield premium is also fairly constant. 

Convertible instruments are usually issued with attached call or put options. Such features can be implemented into the valuation model. If a soft call feature has been implemented, it enables the issuer to force the conversion when the share price overcomes a percentage or trigger level above the conversion price. However, this option cannot be called in the first years hard call . Differently, after the protection period, the issuer can exercise the option. This second time is referred to soft call . Using the same example shown in Section 9.3.1, we assume that the bond may be redeemed in whole but not in part at their principal amount plus accrued interest on the last 2 years, in which the maturity date is at 20 February 2019. On and after this call date , if the share price exceeds 130% of the conversion price the issuer can force the conversion. Figure 9.23 shows the stock price tree in which at years 4 and 5 the stock price is above the threshold. 

Bonds with embedded options are instruments that give the option holder the right to redeem the bond before its maturity date. For callable bonds, this right is held by the issuer. The main reason for an issuer to issue these debt instruments is to get protection from the decline of interest rates or improvement of issuer s credit quahty. In other words, if interest rates fall or credit quality enhances, the issuer has convenience to retire the bond from the market in order to issue again another bond with lower interest rates. 

Yield to maturity This measure assumes that the bond is not called until the maturity date. Therefore, it is calculated as the yield of a straight bond ... 

A bond is therefore a financial contract from the person or body that has issued the bond, that is, the borrowed funds. Unlike shares or equity capital, bonds carry no ownership privileges. The bond remains an interest-bearing obligation of the issuer until it is repaid, which is usually on its maturity date. 

Some bonds include a provision in their offer particulars that gives either the bondholder and/or the issuer an option to enforce early redemption of the bond. The most common type of option embedded in a bond is a call feature. A call provision grants the issuer the right to redeem all or part of the debt before the specified maturity date. An issuing company may wish to include such a feature as it allows it to replace an old bond issue with a lower coupon rate issue if interest rates in the market have declined. As a call feature allows the issuer to change the maturity date of a bond it is considered harmful to the bondholder s interests therefore the market price of the bond at any time will reflect this. A call option is included in all asset-backed securities based on mortgages, for obvious reasons. 

As noted, a bond may contain an embedded option which permits the issuer to call or retire all or part of the issue before the maturity date. The bondholder, in effect, is the writer of the call option. From the bondholder s perspective, there are three disadvantages of the embedded call option. First, relative to bond that is option-free, the call option introduces uncertainty into the cash flow pattern. Second, since the issuer is more likely to call the bond when interest rates have fallen, if the bond is called, then the bondholder must reinvest the proceeds received at the lower interest rates. Third, a callable bond s upside potential is reduced because the bond price will not rise above the price at which the issuer can call the bond. Collectively, these three disadvantages are referred to as call risk. MBS and ABS that are securitized by loans where the borrower has the option to prepay are exposed to similar risks. This is called prepayment risk, which is discussed in Chapter 11. 

To illustrate the process, let s value a 4-year, 6% coupon bond with a maturity value of 100. The coupon payments are 6 for the next four years. In addition, on the maturity date, the investor receives the repayment of principal ( 100). The value of a nonamortizing bond can be divided in two components (1) the present value of the coupon payments (i.e., an annuity) and (2) the present value of the maturity value (i.e., a lump sum). Therefore, when a single discount rate is employed, a bond s value can be thought of as the sum of two presents values—an annuity and a lump sum. 

As a bond moves towards its maturity date, its value changes. More specifically, assuming that the discount rate does not change, a bond s value ... 

At the maturity date, the bond s value is equal to its par or maturity value. So, as a bond s maturity approaches, the price of a discount bond will rise to its par value and a premium bond will fall to its par value— a characteristic sometimes referred to as pull to par value. ... 

The gilts market is primarily a plain vanilla market, and the majority of gilt issues are conventional fixed interest bonds. Conventional gilts have a fixed coupon and maturity date. By volume they made up 82% of the market in June 2002. Coupon is paid on a semi-annual basis. The coupon rate is set in line with market interest rates at the time of issue, so the range of coupons in existence reflects the fluctuations in market interest rates. Unlike many government and corporate bond markets, gilts can be traded in the smallest unit of currency and sometimes nominal amounts change hands in amounts quoted down to one penny ( 0.01) nominal size. 

A repo-to-maturity is a classic repo where the termination date on the repo matches the maturity date of the bond in the repo. We can discuss this trade by considering the Bloomberg screen used to analyse repo-to-maturity, which is REM. The screen used to analyse a reverse repo-to-maturity is RRM. 

Rule 2 As the maturity date of a bond approaches the size of the bond s discount/premium decreases. 

Rule 5 For high coupon bonds with a maturity date greater than one year, the percentage change in a price will be smaller than for low coupon bonds with the same maturity. 

Probably the most popular euro-denominated contracts available on European bonds are those offered by Eurex Deutschland and, in particular, those on longer-dated bonds issued by the German Federal Government the Buxl, Bnnd, and Bobl. Also popular but at the shorter end of the maturity spectrum are short-term notes issued by either the Federal Government or the Treuhandanstalt the Euro-Schatz notes. 

A specialised depositary will hold definitive notes representing aggregate investor positions held in a particular issue on coupon and maturity dates it presents the coupons or bond to the paying agent and passes the proceeds on to the clearing system. 

States, counties, and cities issue bonds to raise money to pay for various projects, such as schools, highways, convention centers, and stadiums. Corporations also issue bonds to raise money to expand or to modernize their facilities. There are many different types of bonds, but basically, they are loans that investors make to government or corporations in return fi r some gain. When a bond is issued, it will have a maturity date (a year or less to 30 years or longer), par value (the amount originally paid for the bond and the amount that will be repaid at maturity date), and an interest rate (percentage of par value that is paid to bond holder at regular intervals). 

Bond prices are expressed per 100 nominal —that is, as a percentage of the bond s face value. (The convention in certain markets is to quote a price per 1,000 nominal, but this is rare.) For example, if the price of a U.S. dollar-denominated bond is quoted as 98.00, this means that for every 100 of the bond s face value, a buyer would pay 98. The principles of pricing in the bond market are the same as those in other financial markets the price of a financial instrument is equal to the sum of the present values of all the future cash flows from the instrument. The interest rate used to derive the present value of the cash flows, known as the discount rate, is key, since it reflects where the bond is trading and how its return is perceived by the market. All the factors that identify the bond—including the nature of the issuer, the maturity date, the coupon, and the currency in which it was issued—influence the bond s discount rate. Comparable bonds have similar discount rates. The following sections explain the traditional approach to bond pricing for plain vanilla instruments, making certain assumptions to keep the analysis simple. After that, a more formal analysis is presented. 

Up to this point the discussion has involved plain vanilla bonds. But duration applies to all bonds, even those that have no conventional maturity date, the so-called perpetual, or irredeemable, bonds (also known as annuity bonds), which pay out interest for an indefinite period. Since these make no redemption payment, the second term on the right side of the duration equation disappears, and since coupon payments can stretch on indefinitely, n approaches infinity. The result is equation (2.12), for Macaulay duration. 

EXAMPLE Calculating the Modified and Macaulay Durations as of 1999 of a Hypothetical Bond Having an Annual Coupon of 8 Percent and a Maturity Date of 2009... 

Duration increases as coupon and yield decrease. The lower the coupon, the greater the relative weight of the cash flows received on the maturity date, and this causes duration to rise. T ong the non—plain vanilla types of bonds are some whose coupon rate varies according to an index, usually the consumer price index. Index-linked bonds generally have much lower coupons than vanilla bonds with similar maturities. This is true because they are inflation-protected, causing the real yield required to be lower than the nominal yield, but their durations tend to be higher. 

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