Maturity bonds

Companies in need of more capital can also raise it by selling bonds. A bond is a certificate, with a face value usually much larger than that of a share, which has a fixed lifetime, usually 10 years at least, at the end of which time the bond matures and the company pays back its face value (by then, of course, much decreased in real value). However, the company also guarantees to pay interest at a fixed percentage of the bond s face value for this lifetime, usually twice a year, and this rate is usually quite an attractive one. 

This is convenient because this means that the price at time t of a zero-coupon bond maturing at T is given by Equation (3.7), and forward rates can be calculated from the current term structure or vice versa. 

The Ho-Lee (1986) model was one of the first arbitrage-free models and was presented using a binomial lattice approach, with two parameters the standard deviation of the short-rate and the riskpremium of the short-rate. We summarise it here. Following Ho and Lee, let ( ) be the equilibrium price of a zero-coupon bond maturing at time T under state i. That is F( ) is a discount... 

In order to calculate the range of implied forward rates, we require the term stmcture of spot rates for all periods along the continuous discount function. This is not possible in practice, because a bond market will only contain a finite number of coupon-bearing bonds maturing on discrete dates. While the coupon yield curve can be observed, we are then required to fit the observed curve to a continuous term structure. Note that in the United Kingdom gilt market, for example there is a zero-coupon bond market, so that it is possible to observe spot rates directly, but for reasons of liquidity, analysts prefer to use a fitted yield curve (the theoretical curve) and compare this to the observed curve. 

The technique proposed by McCulloch (1975) used aregression cubic spline to approximate the discount function, and he suggested that the number of node points that are used be roughly equal to the square root of the number of bonds in the sample, with equal spacing so that an equal number of bonds mature between adjacent nodes. A number of writers have suggested that this approach produces accurate results in practice. The discount function is constrained to set v(0) = 1. Given these parameters the discount function chosen is the one that minimises the function (5.14). As this is a discount function and not a yield curve, Equatimi (5.14) can be solved using the least squares method. 

The approach used by Fisher et al. (1995) is a smoothed cubic spline that approximates the forward ciuve. The number of nodes to use is recommended as approximately one-third of the number of bonds used in the sample, spaced apart so that there is an equal number of bonds maturing between adjacent nodes. This is different to the theoretical approach, which is to have node points at every interval where there is a bond cash flow however, in practice using the smaller number of nodes as proposed by Fisher et al. produces essentially an identical forward rate curve, but with fewer calculations required. The resulting forward rate curve is the cubic spline that minimises the function (5.17) ... 

Simply put, this number tells us how much the coupon payments contribute to the bond s valne. In addition, the bondholder receives the maturity value when the bond matures so the present value of the maturity value must be added to the present value of the coupon payments. The present value of the maturity value is... 

To illustrate this calculation, suppose that a German government bond maturing 17 June 2005 is purchased with a settlement date of 8 July 2003. This bond s coupon rate is 2% and has a coupon date of 17 June. As a result, the next coupon payment date is 17 June 2004, while the previous coupon payment was 17 June 2003. There are two cash flows remaining 17 June 2004 and 17 June 2005. 

Any capital gain (or capital loss, which is a negative euro return) when the bond matures, is sold by the investor, or is called by the issuer. 

The investor s tenure as a bond s owner ends as a result of one of the following circumstances. First, the investor may simply sell the bond and will receive the bond s prevailing market price plus accrued interest. Next, the issuer may call the bond in which case the investor receives the call price plus accrued interest or the investor may put the bond and receive the put price plus accrued interest. Lastly, if the bond matures, the investor will receive the maturity value plus the final coupon payment. Regardless of the reason, if the proceeds received are greater than the investor s initial purchase price, a capital gain is generated, which is an additional source of dollar return. Similarly, if the proceeds received are less than the investor s initial purchase price, a capital loss is gener-... 

To illustrate the various yield to call measures, consider a callable bond with a 5.75% coupon issued by DZ Bank. The Security Description screen from Bloomberg is presented in Exhibit 3.12. The bond matures on 10 April 2012 and is callable on coupon anniversary dates until maturity at a call price of 100. Exhibit 3.13 present the Yields to Call screen. Using a settlement date of 22 July 2003, the various yield to call measures are presented. 

Bond 1 4% coupon Italian government bond maturing 1 March 2005 Full Price 104.1947 Yield 2.147%... 

Bond 2 5.75% coupon Italian government bond maturing 1 February 2033 Full Price 119.1198 Yield 4.727%... 

Inside the Eurozone, the Spanish exchange MEFF Renta Fija in Barcelona offers a notional 10-year, 4% coupon Spanish government bond future and covers the 7.5-10.5-year cash market bond maturity range. It has a March, June, September, and December expiry cycle, a nominal contract value of 100,000, and a tick size of 1 basis point with a value of 10. The average open interest over the period 1 November 2002-29 November 2002 was relatively small at approximately 1,260. 

Trading in these bond options contracts is extensive across the bond maturity spectrum, averaging at over 200,000 contracts per day dnring January 2003, as the monthly statistics in Exhibit 17.5 shows. The total open interest in bond options, amounting to around 2 million contracts, represents an underlying position of 200 billion. 

At the time, the June 2003 Euro-BUND future is trading at 116.42, and the cheapest-to-deliver (CTD) bond for the June 2003 futures contract is the 5% bond maturing on 4 January 2012, and having a conversion factor of 0.9341. The PVBP (or DVOl) for the CTD bond is 0.0788, while that for the bond actually held is 0.0840. The hedge ratio for the bond portfolio is therefore... 

Initially the first formulas on pricing options on pure discount bonds used the Vasicek model for the term structure of interest rates. Thus, given that r follows equation (18.6), the price of a European call option with maturity Tq with exercise price fC on a discount bond maturing at T(Tq < T) is... 

Here we demonstrate that the credit spread is related to risk of default (as represented by the hazard rate) and the level of recovery of the bond. We assume that a zero-coupon risky bond maturing in a small time element At where ... 

The belief that markets will change in structure and composition over time is almost axiomatic among market participants. These changes are often more pronounced in fixed-income market than in the equities market, primarily due to the fact that the number of securities in fixed income is much larger, and the churn associated with new issues and bonds maturing is significant. 

Say a trader holds a long position of 1 million of the 8 percent bond maturing in 2019. The bond s modified duration is 11.14692, and its price is 129.87596. Its basis point value is therefore 0.14477. The trader decides to protect the position against a rise in interest rates by hedging it using the zero-coupon bond maturing in 2009, vi/hich has a BPV of 0.05549. Assuming that the yield beta is 1, what nominal value of the zero-coupon bond must the trader sell ... 

As discussed in chapter 1, there are two types of fixed-income securities zero-coupon bonds, also known as discount bonds or strips, and coupon bonds. A zero-coupon bond makes a single payment on its maturity date, while a coupon bond makes interest payments at regular dates up to and including its maturity date. A coupon bond may be regarded as a set of strips, with the payment on each coupon date and at maturity being equivalent to a zeto-coupon bond maturing on that date. This equivalence is not purely academic. Before the advent of the formal market in U.S. Treasury strips, a number of investment banks traded the cash flows of Treasury securities as separate zero-coupon securities. 

Say an investor at time t simultaneously buys one unit of a zero-coupon bond maturing at time T that is priced at P t, T) and sells P t, T) P t, T + 1) units of zero-coupon bonds maturing at T+ 1. Together these two transactions generate a zero cash flow The investor receives a cash flow equal to one unit at time Tand pays out P t, T)IP t, T+ 1) at time 7+ 1. These cash flows are identical to those that would be generated by a loan contracted at time t for the period T to T + 1 at an interest rate of P t, T) P t, T + ). Therefore P t, T) P t, T+ 1) is the forward rate. This is expressed formally in (3.25). 

The left-hand side of (3-38) specifies the yield-to-maturity at time t of the zero-coupon bond maturing at time 77 The equation states that the expected holding-period yield generated by continually rolling over a series of one-period bonds will be equal to the yield guaranteed by holding a long-dated bond until maturity. 

P t, T) = the price at time r of a zero-coupon bond maturing at time T f = the forward price of the underlying asset with maturity T f = the forward price at time t X = the strike price of the option N = normal distribution... 

FIGuRE 7.11 PVBPof a 5- Year Swap and Fixed-Rate Bond Maturity Period ... 

The tools discussed so far in this chapter are the building blocks of a simple model for pricing callable bonds. To illustrate how this model works, consider a hypothetical bond maturing in three years, with a 6 percent semiannual coupon and the call schedule shown in FIGURE 11.8. 

U.S. Treasury price quotes are in ticks, or thirty-seconds of a price point. A half tick is denoted by a plus sign. On May 10, 1994, the 10.25 percent Treasury bond maturing July 21, 1995, was quoted at 104-28+— in other words, an investor would pay 104.28625 for every 100 in face value. It pays coupons on January 21 and July 21. On May 11, 1994, the settlement date, it will have accrued 109 days of interest, for a total of 10.25 X 109/365 x 0.5, or 1.53048 for every 100 of face value. The dirty price of the bond on this date is thus 104-28+ plus 1.53048, or 106.421105. 

The first bond matures in precisely six months and thus has no intermediate cash flow before redemption. It can therefore be treated as a zero-coupon bond, and its yield of 6 percent taken as the 6-month spot rate. Using this, the 1-year spot rate can be derived from the price of a 1-year coupon Treasury. The principle of no-arbitrage pricing requires that the price of a 1-year Treasury strip equal the sum of the present value of the coupon Treasury s two cash flows ... 

Example A 9 8 government bond is purchased for delivery on June 22, 1992. The face value of this bond is 100,000 and the current yield is 6 percent, and it pays interest semiannually on May 15 and November 15. If this bond matures on November 15, l994, what is the quoted price of this bond and what price is actually paid ... 

Step I. There are 184 days from the last coupon payment (March 5) to the date the bond matures. There are 107 days from the last coupon payment to the delivery or settlement date and 77 days from the settlement date to the maturity date. 

Example On April 1, 1992, an investor takes delivery of a 5 percent coupon bond. This bond has a face or par value of 1,000 and pays interest on April 1 and October 1. This bond matures on October 1,1996, and the current yield to maturity is 6 percent. What is the amount of discount for this bond ... 

Note that the discount factors in Figure 1.3 decrease as the bonds maturity increases. This makes intuitive sense, since the present value of something to be received in the future diminishes the further in the future the date of receipt lies. 

The discount factor for f = 0, that is, for a bond maturing right now, is 1, i.e., the present value of a cash flow received right now is simply the value of the cash flow. Therefore Oq=, and (5.1) can then be rewritten as (5-2). 

P(t, T) = the price at time r of a zero-coupon bond maturing at time T... 

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