Conventional bonds pricing

It is now possible to complete the price tree for the callable bond. Remember that the option in the case of a callable bond is held by the issuer. Its value, given by the tree in figure 11.11, must therefore be subtracted from the conventional bond price, given by the tree in figure 11.10, to obtain the callable bond value. For instance, the current price of the callable bond is 105.875 — 0.76, or 105.115. FIGURE 11.12 shows the tree that results from this process. A tree constructed in this way, which is programmable into a spreadsheet or as a front-end application, can be used to price either a callable or a putable bond. 

A conventional bond has an expected return of 5.875% and a standard deviation of 1 2.50% per annum. The initial price of the bond is 100. From Equation (2.20), the dynamics of the bond price are given by ... 

From market observation we know that index-linked bonds can experience considerable volatility in prices, similar to conventional bonds, and therefore, there is an element of volatility in the real yield return of these bonds. Traditional economic theory states that the level of real interest rates is cmistant however, in practice they do vary over time. In addition, there are liquidity and supply and demand factors that affect the market prices of index-linked bonds. In this chapter, we present analytical techniques that can be applied to index-linked bonds, the duration and volatility of index-linked bonds and the concept of the real interest rate term structure. Moreover, we show the valuation of inflation-linked bonds with different cash flow structures and embedded options. 

Therefore, the break-even analysis allows to determine the spread that equals the price of a conventional bond to the one of an inflation-linked bond. This approach assumes a risk-neutral pricing by which an investor treats conventional and inflation-linked bonds the same. Under break-even hypothesis, both bonds have the same nominal yield. Note if the inflation breakeven is greater than expected inflation, for an investor is favorable to buy a conventional bond. Conversely, the inflation-linked bond is more attractive. If inflation breakeven and expectations are equal, the investor bond s choice will be then indifferent. Figure 6.2 shows the trend of UKGGBEIO and UKGGBE20 Index... 

Using the prices of index-linked bonds, it is possible to estimate a term structure of real interest rates. The estimation of such a curve provides a real interest counterpart to the nominal term structure that was discussed in the previous chapters. More important it enables us to derive a real forward rate curve. This enables the real yield curve to be used as a somce of information on the market s view of expected future inflation. In the United Kingdom market, there are two factors that present problems for the estimation of the real term structure the first is the 8-month lag between the indexation uplift and the cash flow date, and the second is the fact that there are fewer index-linked bonds in issue, compared to the number of conventional bonds. The indexation lag means that in the absence of a measure of expected inflation, real bond yields are dependent to some extent on the assumed rate of future inflatiOTi. The second factor presents practical problems in curve estimation in December 1999 there were only 11 index-linked gilts in existence, and this is not sufficient for most models. Neither of these factors presents an insurmountable problem however, and it is stiU possible to estimate a real term structure. 

The price of an inflation-linked bmid is determined as the present value of future coupon payments and principal at maturity. Like a conventional bond, the valuation depends on the cash flow structure. We can have three main cash flow structures of index-linked bonds. 

If the price of an inflation-linked bond is 104.95 and the price of a conventional bond is 95.42, the difference of value represents the inflation premium, or 9.53. 

The price of a corporate bond is a yield spread for conventional bonds or on an OAS basis for callable or other option-embedded bonds. If an OAS calculation is undertaken in a consistent framework, price changes that result in credit events will result in changes in the OAS. Therefore, we can speak in terms of a sensitivity measure for the change in value of a bond or portfolio in terms of changes to a... 

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) ... 

Conversely, at the lowest node, the hedge ratio is 0 because the option is out of money or 0. This means that in the first case the bond trade like the equity, while in the second case like a conventional bond. Therefore when the share price increases the delta approaches unity, implying that the option is deeply in the money. In contrast, when the share price is low relative to the conversion price, the sensitivity of the convertible and therefore of the embedded option is low. 

The pricing of a floating-rate note at issue does not differ from a conventional bond. In fact, it is the present value of coupon payments and principal repayment and is given by (10.3) ... 

Putable bonds exhibit a positive convexity, although lower than a conventional bond, above aU with rising interest rates. Figure 11.2 shows the changes of prices according to the interest rate. If the interest rates decrease, option free and putable bonds have the same convexity. If the interest rates rise, putable bonds become more valuable. 

Consider the following example. We assume to have two hypothetical bonds, a treasury bond and a callable bond. Both bonds have the same maturity of 5 years and pay semiannual coupons, respectively, of 2.4% and 5.5%. We perform a valuation in which we assume a credit spread of 300 basis points and an OAS spread of 400 basis points above the yield curve. Table 11.1 illustrates the prices of a treasury bond, conventional bond and callable bond. In particular, considering only the credit spread we find the price of a conventional bond or option-free bond. Its price is 106.81. To pricing a callable bond, we add the OAS spread over the risk-free yield curve. The price of this last bond is 99.02. We can now see that the OAS spread underlines the embedded call option of the callable bond. It is equal to 106.81-99.02, or 7.79. In Section 11.2.3, we will explain the pricing of a callable bond with the OAS methodology adopting a binomial tree. 

The value of a callable bond, and therefore of a call option, depends on the interest rate path. Thus, a callable bond has a lower price than the one of a conventional bond due to the embedded option. If the value of a call option increases, the value of a callable bond decreases and vice versa. This happens when interest rates are lower than the ones at issue. 

The pricing of the conventional bond is the same than the one exposed for callable bonds in Figure 11.9. Therefore, the option-free bond is always equal to 106.13. The main difference consists in the estimation of the embedded option (put option rather than call option) and pricing of the putable bond. Thus, we illustrate these two steps ... 

The first effect that we have is that the value of an option-free bond increases with rising coupon payments. The pricing of the conventional bond performed in Figure 11.9 is modified with the step-up coupon payment. The pricing is shown in Figure 11.15. 

FIGURE 11.15 The binomial price tree of a conventional bond with step-up feature. 

How does an investor measure the modified duration of linkers It sounds like a straightforward question and there is an easy answer, but it is sadly not the answer that people generally want. The easy answer is that a linker s modified duration is the (normalised) first derivative of price with respect to real yield, just as a conventional bond s modified duration is that with respect to nominal yield. This answer is a flippant one, because what people really want to know is some empirical rule about the sensitivity of a linker s price with respect to nominal yields, either for hedging purposes or in order to calculate aggregate duration statistics for portfolios holding both nominal and real bonds. 

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. 

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). 

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. 

For a conventional bond, the value of the option component is zero. For a putable one, the option has a positive value. The portfolio represented by a putable bond contains a long position in a put, which, by acting as a floor on the bond s price, increases the bond s attractiveness to investors. Thus the greater the value of the put, the greater the value of the bond. This is expressed in (11.2). 

A callable bond is essentially a conventional bond plus a short position in a call option, which acts as a cap on the bonds price and so reduces its value. If the value of the call option were to increase because of a fall in interest rates, therefore, the value of the callable bond would decrease. This is expressed in (11.3). 

FIGURE 11.14 Projected Prices for Callable and Conventional Bonds with Identical Coupons and Final Maturity Dates ... 

In this situation, the final cash flows are not indexed, and the price-yield relationship is identical to that for a conventional bond. This, then, represents the nonindexed component of the indexed bond. Its yield can be compared with those of conventional bonds, making it possible to quantify the indexation element. This implies a true real yield measure for the indexed bond. 

Essentially, the Fisher identity describes the relationship between nominal and real interest rates. Assuming a value for the risk premium p, the two bond price equations—one for a conventional bond and one for an indexed bond—can be linked using (12.15) and solved as a set of simultaneous equations to obtain values for the real interest rate and the expected inflation rate. 

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. 

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