Callable bonds

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... 

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. 

In contrast, for putable bonds, the right to exercise the option is held by the bondholder. In fact, putable bonds allow the bondholder to sell the bond back before maturity. Conversely to callable bonds, this happens when interest rates go up (risk-free rate increases, or the issuer s credit quality decreases). In fact, the bondholders may have the advantage to sell the bond and buy another one with higher coupon payments. 

Value of an option free bond Value of a callable bond... 

Determine the price of a callable bond with the current market yield curve ... 

For instance, consider a hypothetical case in which a callable bond has a price equal to 103.78. Suppose that the benchmark yield curve changes by 1 basis point. For a downward parallel shift, the price obtained is 103.83, while for an upward shift the price is 103.73. Therefore, applying Formula (11.1) the bond s effective duration is around 4.93. 

The option-adjusted spread (OAS) is the most important measure of risk for bonds with embedded options. It is the average spread required over the yield curve in order to take into account the embedded option element. This is, therefore, the difference between the yield of a bond with embedded option and a government benchmark bond. The spread incorporates the future views of interest rates and it can be determined with an iterative procedure in which the market price obtained by the pricing model is equal to expected cash flow payments (coupons and principal). Also a Monte Carlo simulation may be implemented in order to generate an interest rate path. Note that the option-adjusted spread is influenced by the parameters implemented into the valuation model as the yield curve, but above all by the volatility level assumed. This is referred to volatility dependent. The higher the volatility, the lower the option-adjusted spread for a callable bond and the higher for a putable bond. 

Consider also that because the OAS is applied over the risk-free yield curve, it includes the credit risk and liquidity risk between defaultable and default-free bonds. Figure 11.3 shows an example of the OAS Bloomberg screen for Mittel s callable bond. 

EXAMPLE 11.1 OAS Analysis for Treasury, Conventional and Callable Bonds... 

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. 

In this section, we illustrate the pricing of bonds with embedded options. The price of a callable bond is essentially formed by an option-free bond and an embedded option. In fact, it is given by the difference between the value of an option-free bond and a call option as follows ... 

P callable bond P option free bond P call option (11 - 3)... 

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. 

TABLE 11.1 OAS analysis for a Treasury Bond, Conventional Bond and Callable Bond... 

Treasury Bond Conventional Bond Callable Bond ... 

The proposed methodology determines the value of a callable bond using the binomial tree. The example assumes a 5-year callable bond with the following conditions (Table 11.2) ... 

Moreover, the callable bond can be called according to the following call schedule (Table 11.3) ... 

In the first step, we determine the interest rate path in which we create a risk-neutral recombining lattice with the evolution of the 6-month interest rate. Therefore, the nodes of the binomial tree are for each 6-month interval, and the probability of an upward and downward movement is equal. The analysis of the interest rate evolution has a great relevance in callable bond pricing. We assume that the interest rate follows the path shown in Figure 11.4. In this example, we assume for simplicity a 2-year interest rate. We suppose that the interest rate starts at time tg and can go up and down following the geometric random walk for each period. The interest rate rg at time tg changes due to two main variables ... 

Coming back with our hypothetical callable bond, the 6-month interest rate has the path illustrated in Figure 11.5. In the example, we assume a volatility of the period equal to 11%. Assuming a drift factor equal to 0%, the interest rate of 2.96% at time tg can reach at time ts a maximum value of 8.56% and a minimum value of 1.03%. In this case, the yield curve is flat and the interest rates in the base scenario at time tg, tj, t2, ts, t4 and ts are always equal (2.96%). 

Several factors affect the decision if exercising the option or not. The first one is the asymmetric profit-loss profile. The potential gain of the option holder is unlimited when the price of the underlying asset rises, and losing only the initial investment if the price decreases. The second one is the time of value. In fact, in callable bonds, usually the price decreases as the bmid approaches maturity. This incentives the option holder to delay the exercise for a lower strike price. However, coupon payments with lower interest rates can favour the early exercise. 

Determine the Value of a Callable Bond Since the option is held by the issuer, the option element decreases the value of the bond. Therefore, the value of a callable bond is found as an option-free bond less the option element according to Formula (11.3). For the hypothetical bond, the price is 106.13-2.31 or 103.82. This is shown in Figure 11.11. The binomial tree shows that at maturity the option free and callable bond have the same price, or 100. Before the maturity, if the interest rates go down, the callable bond s values are less than an option-free bond, and in particular when the embedded option is deeply in the money, the callable values equal the strike price according to the caU schedule. Conversely, when the interest rates go up, the option free and callable bonds have the same price. 

FIGURE 11.11 TTie binomial price tree for a callable bond. 

In other words, if the callable bond price is 103.82, we set this price in the binomial tree shown in Figure 11.9 and through an iterative procedure, we find an option-adjusted spread that matches the price sought. In our case, the option-adjusted spread is around 630 bps over the risk-free yield curve (Figure 11.12). 

To explain the pricing methodology, we suppose a putable bond with the same characteristics of the callable bond. The putable bond can be given back to the issuer with the following put schedule shown in Table 11.4. 

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 ... 

Determine the Value of an Embedded Put Option Conversely to a callable bond, the embedded option of a putable bond is a put option. Therefore, the value is estimated as the maximum between 0 and the difference between the strike price and bond s price. The strike price is defined according to the put schedule, while the bond s price is the value of the option-free bond at each node as shown in Figure 11.9. The value at maturity of a putable option if exercised is given by Formula (11.10) ... 

The embedded option has value when the interest rates go up. Figure 11.13 illustrates the value of a put option. As with callable bonds, the put option is worthless at maturity because the bond is given back in each case. Before the maturity, the put option decreases its value as the bond approaches maturity. The put option value is equal to 0.33. 

Determine the Value of a Putable Bond As exposed in Formula (11.4), the value of a putable bond is the sum of an option-free bond and an embedded put optimi. Therefore, conversely to a callable bond, the embedded option increases the value of the bond. When the option is deeply in the money, the bond matches the values defined in the put schedule. When the option has no value, option free and putable bonds have the same price. The value of our hypothetical putable bond is 106.13 + 0.33 or 106.45. This is illustrated in Figure 11.14. 

Step-up callable notes are a particular type of structured fixed income products. These bonds offer a coupon payment that increase during the bond s life. Moreover, they include a call option, that as we discussed earlier, the issuer has the right to redeem the bond early. The question, whether a callable step-up note will be called or not always depends on the evolution of interests rates. Therefore, the inclusion of these two characteristics makes the bond attractive to investors with higher performance than a conventional bond. The added variable coupon element acts for an investor as cushion compared to a conventional callable bond. In fact, the increasing coupon payment increases the value of a callable bond. However, if interest rates go down and coupon payments increase, the incentive of the issuer to redeem the bond early is greater than a simple callable. 

FIGURE 11.16 The binomial price tree of a callable bond with step-up feature. 

As shown in Figure 11.15, the added step-up feature increases the value from 106.13 to 108.5. However, the inclusion of this feature affects also the values of the embedded call option and callable bond. In practice, increasing the value of the conventional bond at each node increases the value of the embedded option. The call option is now 4.5. As a conventional callable bond, the value of a call option is then subtracted to the one of an option-free bond. Figure 11.16 shows that the value of a step-up callable note is 104. 

The second one, it works as a callable bond and the issuer may call the bond back. This is, therefore, a partial call, in which only a part of the bond may be called. 

Brennan, M.J., Schwartz, E.S., March 1977. Saving bonds, retractable bonds and callable bonds. J. Financ. Econ. 5 (1), 67-88. 

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. 

Callable bonds, putable bonds, mortgage-backed securities, and asset-backed securities are examples of (1). Floating-rate securities and inflation-indexed bonds are examples of (2). Convertible bonds and exchangeable bonds are examples of (3). 

For securities that fall into the first category, a key factor determining whether the owner of the option (either the issuer of the security or the investor) will exercise the option to alter the security s cash flows is the level of interest rates in the future relative to the security s coupon rate. In order to estimate the cash flows for these types of securities, we must determine how the size and timing of their expected cash flows will change in the future. For example, when estimating the future cash flows of a callable bond, we must account for the fact that when interest... 

For callable bonds, the market convention is to calculate a yield to call in addition to a yield to maturity. A callable bond may be called at more than one price and these prices are specified in a call price schedule. The yield to call assumes that the issuer will call the bond at some call date and the call price is then specified in the call schedule... 

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. 

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