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Model for valuing bonds and embedded options
(Source: Andrew J Kalotay, George O. Williams, Frank J. Fabozzi, Financial Analysts Journal, May-June 1993
Copyright  1991, Association for Investment Management & Research, Charlottesville, Va.)

Background

One can value a bond by discounting each of its cash flows at its own zero-coupon ("spot") rate.  This procedure if equivalent to discounting the cash flows at a sequence of one-period forward rates.  When a bond has one of more embedded options, however, its cash flow is uncertain.  If a callable bond is called by the issuer, for example, its cash flow will be truncated.

To value such a bond, one must consider the volatility of interest rates, as their volatility will affect the possibility of the call option being exercised.  One can do so by constructing a binomial interest rate tree that models the random evolution of future interest rates.  The volatility-dependent one-period forward rates produced by this tree can be used to discount the cash flows of any bond in order to arrive at a bond value.

Given the values of bonds with and without an embedded option, one can obtain the value of the embedded option itself.  The procedure can be used to value multiple or interrelated embedded options, as well as stand-alone risk control instruments such as swaps, swaptions, caps and floors.

Introduction

In the good old days, bond valuation was relatively simple.  Not only did interest rates exhibit little day-to-day volatility, but in the long run they inevitably drifted up, rather than down.  Thus the ubiquitous call option on long-term corporate bonds hardly ever required the attention of the financial manager.  Those days are gone. Today, investors face volatile interest rates, a historically steep yield curve, and complex bond structures with one or more embedded options.  The framework used to value bonds in a relatively stable interest rate environment is inappropriate for valuing bonds today. This article sets forth a general model that can be used to value any bond in any interest rate environment.

A brief history of bond valuation

The value of any bond is the present value of its expected cash flows.  This sounds simple: Determine the cash flows and then discount those cash flows at an appropriate rate.   In practice, it's not so simple for two reasons.  First, holding aside the possibility of default, it is not easy to determine the cash flows for bonds with embedded options.  Because the exercise of options embedded in a bond depends on the future course of interest rates, the cash flow is a priori uncertain.  The issuer of a callable bond can alter the cash flows to the investor by calling the bond, while the investor in a putable bond can alter the cash flows by putting the bond.  The future course of interest rates determines when and if the party granted the option is likely to alter the cash flows.

A second complication is determining the rate at which to discount the expected cash flows.  The usual starting point is the yield available on Treasury securities.  Appropriate spreads must be added to those Treasury yields to reflect additional risks to which the investor is exposed.  Determining the appropriate spread is not simple, and is beyond the scope of this article.  The ad hoc process for valuing an option-free bond (i.e., a bond with no options) once was to discount all cash flows at a rate equal to the yield offered on a new full-coupon bond of the same maturity.   Suppose,  for example, that one needs to value a 10-year option-free bond.  If the yield to] maturity of an on-the-run 10-year bond of given credit quality is 8%, then the value of the bond under consideration would be taken to be the present value of its cash flows, all discounted at 8%.

According to this approach, the rate used to discount the cash flows of a 10-year current-coupon bond would be the same rate as that used to discount the cash flow of a 10-year zero-coupon bond.  Conversely, discounting the cash flows of bonds with different maturities would require different discount rates.  This approach makes little sense because it does not consider the cash flow characteristics of the bonds.  Consider, for example, a portfolio of bonds of similar quality but different maturities.  Imagine two equal cash flows occurring, say, five years hence, one coming from a 30-year bond and the other coming from a 10-year bond.   Why should these two cash flows have different discount rates and hence different present values?

Given the drawback of the ad hoc approach to bond valuation, greater recognition has been given to the fact that any bond should be thought of as a package of cash flows, with each cash flow viewed as a zero-coupon instrument maEagleTraders.comg on the date it will be received.  Thus, rather than using a single discount rate, one should use multiple discount rates, discounting each cash flow at its own rate.

One difficulty with implementing this approach is that there may not exist zero-coupon securities from which to derive every discount rate of interest.  Even in the absence of zero-coupon securities, however, arbitrage arguments can be used to generate the theoretical zero-coupon rate that an issuer would have to pay were it to issue zeros of every maturity.  Using these theoretical zero-coupon rates, more popularly referred to as theoretical spot rates, the theoretical value of a bond can be determined.   When dealer firms began stripping of full-coupon Treasury securities in August 1982, the actual prices of Treasury securities began moving toward their theoretical values. 

Another challenge remains, however - determining the theoretical value of a bond with an embedded option.  In the early 1980s, practitioners came to recognize that an option-bearing bond should be viewed as a package of cash flows (i.e., a package of zero-coupon instruments) plus a package of options on those cash flows.  For example, a callable bond can be viewed as a package of cash flows plus a package of call options on those cash flows.  As such, the position of an investor in a callable bond can be viewed as:

Long a Callable Bond = Long an Option-Free Bond + Short a Call Option on the Bond.

In terms of the value of a callable bond, this means:

Value of Callable Bond = Value of an Option-Free Bond - Value of a Call Option on the Bond.

But this also means that

Value of an Option-Free Bond = Value of Callable Bond + Value of a Call Option on the Bond.

An early procedure to determine the fairness of a callable bond's market price was to isolate the implied value of its underlying option-free bond by adding an estimate of the embedded call option's value to the bond's market price.    The former value could be estimated by applying option pricing theory as applied to interest rate options.  

This insight led to the first generation of valuation models that sought to value a callable bond by estimating the value of the call option.  However, estimation of the call option embedded in callable bonds is not that simple.  For example, suppose a 20-year bond is not callable for five years after which time it becomes callable at any time on 30-days notice.  Suppose also that the schedule for the call price begins at 105 and declines to par by the 15th year.  This is an extremely complicated call option.  It is a deferred American call option for the first five years.  Also, its striking, or exercise, price (the call price in the case of a callable bond) is not constant but declines over time. Or consider a corporate sinking fund bond whose indenture grants the issuer the right to accelerate the repayment of the principal.  This is a partial European call option struck at par and is extremely difficult to value with an option pricing model.     When a bond has multiple or interrelated embedded options (e.g., both call and put options) the valuation becomes complicated.

The valuation model presented in this article does not rely on an explicit option pricing model.  Rather, it is based on a consistent framework for valuing any bond-option-free or with embedded options.  It focuses on discounting each cash flow at an appropriate volatility-dependent one-period forward rate.  These rates are derived in a natural, self-consistent manner.  Once the value of a bond with embedded options and the value of its underlying option-free bond are known, the value of the options themselves can be determined.

Spot Rates and Forward Rates

The relationship among the yields to maturity on securities selling at par with the same credit quality but different maturities is called the on-the-run yield curve.  The yield curve is typically constructed using the maturities and observed yields of Treasury securities.  As market participants do not perceive these securities to have any default risk, this government yield curve reflects the effect of maturity alone on yield.   However, a theoretical yield curve can be constructed for any issuer.

Interest Rate Volatility

Once we allow for embedded options, consideration must be given to interest rate volatility.   This can be done by introducing a binomial interest rate tree. This tree is nothing more than a discrete representation of the possible evolution over time of the one-period rate based on some assumption about interest rate volatility.  The construction of this tree is illustrated below.

Constructing a Binomial Interest Rate Tree

To see how to construct a binomial interest rate tree, assume on-the-run yields are as given in Table I and that the volatility, s, is 10%.  We construct a two-year model that correctly values a two-year bond with a 4% coupon at 100.

Figure C shows a binomial interest rate tree that gives the cash flow at each node.  The root rate for the tree, r0, is simply the current one-year rate, 3.5%.  This rate was chosen because it will value a one-year bond with a 3.5% coupon at 100.

There are two possible one-year rates one year forward-one if rates fall and one if rates rise.  What we want to find are the two forward rates consistent with the volatility assumption that result in a value of 100 for a 4% full-coupon, two-year bond.  While one can construct a simple algebraic expression for these two rates, the formulation becomes increasingly complex beyond the first year.  Furthermore, because one may want to implement this procedure on a computer, the natural approach is to find these rates by an iterative process (i.e., trial-and -error).  The steps are described below:

  1. Select a value for r1. Recall that r1 is the one-year rate one year forward if rates fall.  In this first trial we arbitrarily selected a value of 4.5%.

  2. Determine the corresponding value for the one-year rate one year forward if rates rise.  As explained earlier, this rate is given by r1*exp(2s).  Since r1 is 4.5%, the forward one-year rate if rates rise is 5.496% (= 4.5%*exp (2*0.10)).  This value is reported in Figure C at node NU.

  3. Compute the bond's value in each of the two interest rate states one year from now, using the following steps.

  • Determine the bond's value two years from now.  Since we are using a two-year bond, the bond's value is its maturity value of $100 plus its final coupon payment of $4, or $104.

  • Calculate the bond's present value at node NU.  The appropriate discount rate is the forward one-year rate assuming rates rise, or 5.496% in our example.  The present value is $98.582 (= $104/1.05496).  This is the value of VU referred to earlier.

  • Calculate the bond's present value at node NU.  The appropriate discount rate is the forward one-year rate assuming rates fall, or 4.500%.  The present value is $99.522 (= $104/1.04500) and is the value of VD.

  1. Add the coupon to the cash flows VU and VD computed in Step 3 to get the total cash flow at NU and ND, respectively.  Then calculate the average of the present values of these two cash flows using the assumed value of r0, 3.5%, for r*.

                  _               _
           1 | VU + C   VD + C |
       V = - | ------ + ------ |
           2 |_1 + r0   1 + r0_|
    
              _                                  _
           1 |$98.562 + $4.000   $99.522 + $4.000 |
         = - |---------------- + ---------------- |
           2 |_     1.035               1.035    _|
    
    
           $99.113 + $100.021
         = ------------------ = $ 99.567
                   2       
  2. Compare the value in obtained Step 4 with the target value of $100.  If the two values are the same, then the r1 used in this trial is the one we seek.  It is the forward one-year rate to be used in the binomial interest rate tree for a decline in rates as well as the rate to be used if rates rise.  If, however, the value found in Step 4 is not equal to the target value of $100, our assumed value is not consistent with the volatility assumption and the yield curve.  In this case, repeat the five steps with a different value for r1.

If we use 4.5% for r1, a value of $99.567 results in Step 4.  This is smaller than the target value of $100.  Therefore, 4.5% is too large.  The five steps must be repeated, with a smaller value for r1.  It turns out that the correct value for r1 is 4.074%.  The corresponding binomial interest rate tree is shown in Figure D.  Steps 1 through 5, using the correct rate, are as follows.

  1. Select a value of 4.074% for r1.

  2. The corresponding value for the forward one-year rate if rates rise is 4.976% (= 4.074%* exp(2*0.10)).

  3. The bond's value one year from now is determined from the bond's value two years from now-$104, just as in the first trial; the bond's present value at node NU-VU, or $99.071 (= $104/1.04976); and the bond's present value at node ND-VD, or $99.929 (= $104/1.04074).

  4. Add the coupon to both VU and VD and average the present values of the resulting cash flows:

                  _               _
           1 | VU + C   VD + C |
       V = - | ------ + ------ |
           2 |_1 + r0   1 + r0_|
    
              _                                  _
           1 |$98.071 + $4.000   $99.929 + $4.000 |
         = - |---------------- + ---------------- |
           2 |_     1.035               1.035    _|
    
    
           $99.586 + $100.414
         = ------------------ = $ 100.000
                   2       
    
     
  5. Since the average present value is equal to the target value of $100, r1 is 0.04074 = 4.074%.  We're not done.  Suppose that we want to "grow" this tree for one more year-that is, we must determine r2.  We will use a three-year on-the-run 4.5% coupon bond to get r2.  The same five steps are used in an iterative process to find the one-year rates two years forward.  Our objective now is to find the value of r2 that will produce a value of $100 for the 4.5% on-the-run bond and will be consistent with (1) a volatility assumption of 10%, (2) a current one-year forward rate of 3.5%, and (3) the two forward rates one year from now of 4.074% and 4.976%.  The desired value of r2 is 4.530%.  Figure E shows the completed binomial interest rate tree.  Using this binomial interest rate tree, we can value a one-year, two-year, or three-year bond, whether option-free or not.

Using the Binomial Tree

To illustrate how to use the binomial interest rate tree, consider an option-free 5.25% bond with two years remaining to maturity.  Assume that the issuer's on-the-run yield curve is the one given in Table I so that the appropriate binomial interest rate tree is the one in Figure EFigure F shows the various values obtained in the discounting process.  If produces a bond value of $102.075.  It is important to note that this is value is identical to the bond value found earlier by discounting at either the zero-coupon rates or the forward one-year rates.  We should expect to find this result, since our bond is option-free.  This clearly demonstrates that, for an option-free bond, the valuation model is consistent with the standard valuation model.

The Challenge of Implementation

To transform the basic interest rate tree into a practical tool requires several refinements.   For one thing, the spacing of the node lines in the tree must be much finer, particularly if American options are to be valued.  However, the fine spacing required to value short-dated securities becomes computationally inefficient if one seeks to value, say, 30-year bonds.  While one can introduce a time-dependent node spacing, caution is required;  it is easy to distort the term structure of volatility.  Other practical difficulties include the management of cash flow dates that fall between two node lines. 

Recommended further reading:
Bonds
Corporate Bonds
Example of a bond