3.3 PRODUCT MECHANICS AND APPLICATIONS
3.3.1 Callable bonds
A callable bond is a package of a bond and a long issuer call option on bond prices (or long issuer put option on interest rates) that allows the issuer to call the security at particular price and/or time intervals. In fact, we may distinguish between an American callable bond (which is callable at any time before maturity), a European callable bond (which is callable only once, generally one or two periods before final maturity), and a Bermudan callable bond (which is callable on regular, though not continuous, intervals, such as every six or twelve months during the life of the bond). US issuers tend to issue callables primarily in American form, while European and Japanese issuers often opt for the European or Bermudan form.
Investors Issuer Coupons
(enhanced to reflect option premium) +
principal
Proceeds plus option to call bonds at set strike level Figure 3.2 Callable bond
In exchange for granting the issuer the call option, the investor collects a premium in the form of an enhanced coupon. If the issuer chooses to exercise the call at the call strike price implied in the security, the investor delivers the bond and receives a principal repayment amount equal to the price-based strike, times the quantity of bonds. For instance, a bond may be issued at par, in denominations of $10 000/bond, with a call price (strike) of 105. If the price of the bond rises to 107, as an example, and the issuer elects to call, investors will present their bonds for redemption at a price of $10 500 (and not $10 700, which represents the market value of the security). Figure 3.2 illustrates the basic callable bond structure.
In practice, the callability of a security is primarily a function of interest rates. Issuers floating callable bonds will choose to call their outstanding debt when rates are declining and opportunities to refinance in the new lower rate environment expand. Knowing from Chapter 2 that rates and prices on fixed income securities are inversely related, this concept is consistent with our simple example above: as rates fall, the outstanding bonds become more valuable (e.g. prices rise). Returning to the payoff diagram we introduced in the last chapter, Figure 3.3 examines the embedded call option from the issuer’s perspective. Assume that at a current rate of 4 %, the bond is worth par; as rates fall to 3.5 %, the price rises to the call price (strike) of 105. At any point after this, the issuer may find it advantageous to call the bonds and refinance in the market (i.e. at a level of 3.5 % or less). While interest rates dominate the option exercise decision, it is worth remembering that credit spreads can also play a role in the process. Since an issuer’s all-in funding cost (as reflected in the coupon) is a combination of the risk-free interest rate level and the relevant credit spread, it is conceivable that static rates, coupled with a significant tightening of the issuer’s credit spread, may also lead to exercise of the call. If the issuer’s credit has improved dramatically in market terms, it will be able to call the security and reissue new debt at the tighter market spread, so reducing its funding costs.
Valuation
We know that a callable bond is simply a combination of a noncallable bond and a long issuer call option, suggesting that from a valuation perspective, we can examine the price of an identical noncallable bond and deduct the price of the call option to determine the theoretical value. Once the theoretical value of the option is determined, the equation can be inverted to see if the noncallable bond price is accurate (e.g. noncallable bond price=callable bond
{
Profit
Bond call strike price
Premium paid to investors via enhanced
coupon
Increasing refinancing gain to issuer as bond price rises (i.e. as interest rates fall)
Bond price
Loss Figure 3.3 Long issuer call option in callable bond
price+call option price). This exercise is equal to deriving the implied noncallable bond price through observed callable bond prices and theoretical option values.
The value of a callable bond is based, in part, on the value of the option, so we know from the last chapter that option volatility must be one of the pricing inputs. However, determining the callable bond price requires calculating both the noncallable bond price and the call option price, meaning it is difficult to shift from a quoted price on a bond to a yield for a given level of volatility – both variables change as yield changes. Furthermore, since the forward price must be computed, the price of the option depends not on a single yield, but on the entire yield curve. As a result, the market has turned towards the use of option-adjusted yields based on bond price and volatility. The noncallable bond is said to have a fair price when the option- adjusted yield for the callable bond is equal to the yield for a noncallable bond with the same features.1
We know that bond value is the PV of all future cash flows (principal and interest) and, if the bond is riskless, it is the present value of the replicating portfolio of the risk-free benchmark.
The replicating portfolio is valued at the risk-free zero coupon value. If the bond is risky, some spread must be introduced to compensate for the risky nature of the cash flows. The resulting static spread is the spread that makes the PV of the cash flows from the bond, when discounted via the risk-free zero coupon curve, equal to the bond’s price.
Unfortunately, this static spread fails to take account of future interest rate volatility that could affect cash flows on a callable (or puttable) bond: the greater the interest rate volatility, the greater the likelihood that the call (or put) embedded in the security will be exercised (though the decision is still dependent on other factors, including remaining time to maturity, call (or put) strike, and so forth).The future path of interest rates thus determines option exercise
1An investor can compute the yield to call and the yield to maturity, using the differential as a relative valuation measure in order to determine the relative attractiveness of a callable bond. The investor may also compute the yield to each discrete call date; this, along with the yield to maturity, generates the lowest possible yield, or yield to worst.
and, by extension, potential value. Under the option-adjusted yield framework, we can develop a zero coupon curve and assumed spread for each possible interest rate path. The average of all PVs can then be computed; if the average PV is equal to the market price of the bond, then the spread added to the zero coupon rates equals the option-adjusted spread (OAS); if it is not equal, a new path is computed. An OAS can therefore be interpreted as the average spread over the risk-free zero coupon curve based on future interest rate paths. Its value depends critically on assumptions about interest rate volatilities, which are often determined through a simulation process. The procedure is more computationally intensive than standard static spread valuation, but generates a more accurate valuation as it assigns proper value to the embedded optionality.2 That said, OAS relies on certain assumptions of its own: it assumes that the embedded bond is held until the effective maturity date; that cash flows are reinvested at a yield equal to the yield of a noncallable bond plus the OAS; and that interest rate volatility estimates are accurate and option exercise behavior by the issuer (or investor, for puttables) is rational. These assumptions are, however, quite reasonable.
It is possible to compute associated sensitivities, including duration and convexity, once OAS relationships are developed. As indicated in Chapter 2, these provide estimates of how much a bond’s value gains or loses for small/large changes in yield. For instance, a standard duration formula can be recalibrated to take account of the price relationship between the callable bond and its noncallable equivalent, and the delta (price sensitivity) of the embedded call option. Option-adjusted duration can be given as:
OAD = PNCB
PCB *DurNCB* (1 − call) where
PNCBis the price of the noncallable bond PCBis the price of the callable bond
DurNCBis the duration of the noncallable bond
callis the delta of the call option (i.e. price sensitivity to small rate moves).
Similarly, option-adjusted convexity can be computed to determine the degree of price performance, or underperformance, for a large movement in rates, by extending and adjusting the standard convexity calculation:
OAC= PNCB PCB *
CvxNCB* (1− call)−PNCB*call* (DurNCB)2 where
CvxNCBis the convexity of the noncallable bond
callis the gamma of the call (e.g. price sensitivity to large rate movements) all other terms are as defined above.
In fact, convexity is an important dimension of the price performance of callable bonds, particularly for price movements in a declining-rate environment (before the call strike is reached). In practice, the actual amount of upward price movement in the bond tends to be
2Note that we shall revisit the OAS framework in our discussion of mortgage-backed securities in Chapter 4. The same framework can also be applied to other option-embedded securities, including capped floaters, payment-in-kind bonds, and so forth.
Price
Price compression
Noncallable bond
Callable bond not yet called
Callable bond called
Yield
Figure 3.4 Callable and noncallable bonds and price compression
limited given the potential exercise of the call; this phenomenon, known as price compression or negative convexity, means that while the price of a noncallable bond may continue to rise as rates fall, the price of the callable bond will tend to lag. In the extreme, a callable bond that is currently callable (e.g. one where the strike has been passed) but remains outstanding, will feature significant compression, as the probability of the security being called at any moment is very high; this compression is, of course, a manifestation of negative convexity.
We can examine various yield scenarios to understand the impact of price compression on a noncallable bond, a callable bond that has been called, and a callable bond that remains outstanding. Figure 3.4 reflects the fact that as yields decline, the probability of call exercise increases, and price compression begins to set in; however, when yields are positioned above the strike, the callable and noncallable bonds feature similar price/yield movements.
Investors in callable bonds are exposed to reinvestment risk. In granting the call option and receiving enhanced coupons, they may find themselves in a situation where they must reinvest their capital in a lower-rate environment. In the example immediately above, the 4 % yield is attractive when rates decline below 4 %; once they reach 3.5 % and the issuer calls the bonds, investors must reinvest their capital in the new rate environment – which is now 50 bps lower than before, for assets with an equivalent term and credit quality.
In practice, callable bonds may feature call prices (strikes) that change over time. For instance, in order not to dissuade investors from committing capital when rates are declining, an issuer may establish a relatively high call price for the first few years of a multi-year issue, helping ensure that the bonds can be placed successfully. Only after the passage of several years might the call price ratchet down to a lower level, increasing the likelihood of callability.
The reverse may also occur: a step-up callable bond may have increasingly higher strikes to induce the issuer to call sooner, rather than later, in the life of the bond. Naturally, an issuer is under no obligation to call securities, even if it appears optimal to do so (i.e. even if the option is in-the-money). In fact, there are instances where issuers may prefer to preserve a particular type of financing. This may occur when the issuer wants to keep its investor base intact, or when it believes that a particular form of capital (e.g. maturity, market, coupon) may
Coupons (reduced to reflect option
premium) + principal, plus
option to put bonds at set strike price
Issuer Investors
Proceeds Figure 3.5 Puttable bond
be difficult to refinance in a given environment).3In general, however, an investor in callable securities must be prepared for a call event.