(4) Callable bonds
What is the interest-rate model?
An interest-rate model is a probabilistic description of how interest rates can change over the life of a financial instrument, by making an assumption about the relationship between the level of short-term interest rates and interest-rate volatility. The interest-rate models commonly used are arbitrage-free models based on how short-term interest rates can evolve (i.e., change) over time. The interest-rate models based solely on movements in the short-term interest rate are referred to as one-factor models.
What is an interest-rate lattice?
The figure shows an example of the most basic type of interest-rate lattice or tree, a binomial interest-rate tree, known as the binomial model. The model assumes that interest rates can realise one of two possible rates in the next period.
What two disadvantages exist to the bondholder in the presence of a call option?
(1) Callable bonds expose bondholders to reinvestment risk. (2) The price appreciation-potential for a callable bond in a declining interest-rate environment is limited. This phenomenon for a callable bond is referred to as price compression. If the investor receives sufficient potential compensation in the form of a higher potential yield, an investor would be willing to accept call risk.
What is a callable bond?
A callable bond is a bond in which the bondholder has sold the issuer a call option that allows the issuer to repurchase the contractual cash flows of the bond from the time the bond is first callable until the maturity date. Effectively, the owner of a callable bond is entering into two separate transactions: (1) She buys a non-callable bond from the issuer for which she pays some price. (2) She sells the issuer a call option for which she receives the option price.
What is the difference between modified duration and effective duration?
Duration Interpretation: Generic description of the sensitivity of a bond's price (as a percent of initial price) to a parallel shift in the yield curve. Modified Duration: Duration measure in which it is assumed that yield changes do not change the expected cash flow. Effective Duration: Duration measure in which recognition is given to the fact that yield changes may change the expected cash flow.
What are the different notations for the binomial model?
Each node represents a time period that is equal to one year from the node to its left. Each node is labelled with an N, representing node, and a subscript that indicates the path that 1-year forward rates took to get to that node. H represents the higher of the two forward rates and L the lower of the two forward rates from the preceding year. For example, we get to node N(HH) by 1-year rate realised is the higher of the two rates in the first year and then the higher of the 1-year rates in the second year.
How do we determine the value at the node?
First, calculate the bond's value at the two nodes to the right of the node where we want to obtain the bond's value. The cash flow at a node will be either: (1) The bond's value if the short rate is the higher rate plus the coupon payment. OR (2) The bond's value if the short rate is the lower rate plus the coupon payment. The bond's value at a node is the present value of the expected cash flows. The appropriate discount rate to use is the 1-year forward rate at the node.
What is negative convexity?
For yields below y*, there is price compression—that is, there is limited price appreciation as yields decline. The portion of the callable bond price-yield relationship below y* is said to be negatively convex. Negative convexity means that the price appreciation will be less than the price depreciation for a large change in yield of a given number of basis points. For a bond that is option-free and displays positive convexity, the price appreciation will be greater than the price depreciation for a large change in yield.
What is the price-yield relationship between a non-callable bond and a callable bond?
Non-Callable Bond: The price-yield relationship for an option-free bond (non-callable bond) is convex, curve a-a'. Callable Bond: The unusual shaped curve denoted by a-b is the price-yield relationship for the callable bond.
What does a comparison of traditional yield spreads and static spreads display for various bonds?
Maturity: The shorter the maturity of the bond, the less the static spread will differ from the traditional yield spread. Yield Curve: The steeper the yield curve, the more the difference for a given coupon and maturity. Corporate Bonds: Corporate bond makes a bullet payment at maturity. The difference between the traditional yield spread and the static spread will be considerably greater for mortgage-backed securities in a steep yield curve environment.
How can we simplify the notation?
Simplify the notation by letting r(t) be the lower 1-year forward rate t years from now because all the other forward rates t years from now depend on that rate.
How do we construct the binominal interest rate tree? (Step 1 & 2)
Step 1: Select a value for r(1). Recall that r(1) is the lower 1-year forward rate one year from now. Step 2: Determine the corresponding value for the higher 1-year forward rate. This rate is related to the lower 1-year forward rate as r(1)e^(2σ). This value is reported at node N(H).
How do we construct the binominal interest rate tree? (Step 3)
Step 3: Compute the bond's value one year from now. This value is determined as follows: 3a. The bond's value two years from now must be determined. 3b. Calculate the present value of the bond's value found in 3a using the higher rate. This value is V(H). 3c. Calculate the present value of the bond's value found in 3a using the lower rate. This value is V(L). 3d. Add the coupon to V(H) and V(L) to get the cash flow at N(H) and N(L), respectively. 3e. Calculate the present value of the two values using the 1-year forward r(*). Therefore, we use the attached formula.
How do we construct the binominal interest rate tree? (Step 4 & 5)
Step 4: Calculate the average present value of the two cash flows in step 3. This is the value of the bond at the node found as the attached formula. Step 5: Compare the value in step 4 with the bond's market value. If the two values are the same, the r(1) used in this trial is the one we seek. This is the 1-year forward rate that would be used in the binomial interest-rate tree for the lower rate, and the corresponding rate would be for the higher rate. If, instead, the value found in step 4 is not equal to the market value of the bond, this means that the value r(1) in this trial is not the one-period forward rate that is consistent with (1) the volatility assumption of 10%, (2) the process assumed to generate the 1-year forward rate, and (3) the observed market value of the bond. In this case the five steps are repeated with a different value for r(1).
What is a puttable bond? How is the puttable bond priced?
The bondholder has the right to sell the bond to the issuer at a designated price and time. A puttable bond can be broken into two separate transactions: (1) The investor buys a non-puttable bond. (2) The investor buys an option from the issuer that allows the investor to sell the bond to the issuer. The price of a puttable bond is then: puttable bond price = non-puttable bond price + put option price.
What is the option-adjusted spread?
The option-adjusted spread (OAS) was developed as a measure of the yield spread (in basis points) that can be used to convert dollar differences between value and price. Thus, the OAS is used to reconcile value with market price. The OAS is a spread over the spot rate curve or benchmark used in the valuation. The reason that the resulting spread is referred to as option-adjusted is because the cash flows of the security whose value we seek are adjusted to reflect the embedded option. In the case of the binomial method, the OAS is a spread over the binomial interest-rate tree.
How is a callable bond priced?
The price of a callable bond is equal to the price of the two components parts: callable bond price = non-callable bond price -call option price. Graphically, this can be seen in the figure. The difference between the price of the noncallable bond and the callable bond at any given yield is the price of the embedded call option
What is the proper way to compare non-Treasury bonds of the same maturity but with different coupon rates? (Static spread)
The proper way to compare non-Treasury bonds of the same maturity but with different coupon rates is to compare them with a portfolio of Treasury securities that have the same cash flow. The corporate bond's value is equal to the present value of all the cash flows. The corporate bond's value, assuming that the cash flows are riskless, will equal the present value of the replicating portfolio of Treasury securities. In turn, these cash flows are valued at the Treasury spot rates.
Why is convexity inappropriate for bonds with embedded options?
The standard convexity measure may be inappropriate for a bond with embedded options because it does not consider the effect of a change in interest rates on the bond's cash flow.
How is the standard deviation in the binomial model determined?
The standard deviation of the 1-year forward rate is equal to r(0)σ. For example, if σ is 10% and r(0) is 4%, then r(0)σ =4% × 10% = 0.4% or 40 basis points.
What is the static spread?
The static spread, also referred to as the zero-volatility spread, is a measure of the spread that the investor would realise over the entire Treasury spot rate curve if the bond is held to maturity. The static spread is calculated as the spread that will make the present value of the cash flows from the corporate bond, when discounted at the Treasury spot rate plus the spread, equal to the corporate bond's price. A trial-and error procedure is required to determine the static spread.
What is modelling risk?
The user of any valuation model is exposed to modelling risk. This is the risk that the output of the model is incorrect because the assumptions upon which it is based are incorrect. Consequently, it is imperative that the results of a valuation model be stress tested by altering the assumptions. A critical assumption in the valuation model is the volatility assumption. For a callable bond, a higher volatility assumption lowers its value.
How is a callable corporate bond valued?
The valuation process for a callable corporate bond proceeds in the same fashion as in the case of an option-free bond but with one exception. When the call option may be exercised by the issuer, the bond value at a node must be changed to reflect the lesser of its value if it is not called (i.e., the value obtained by applying the recursive valuation formula described previously) and the call price.
What are the drawbacks of traditional yield spread analysis?
Traditional yield spread analysis fails to take into consideration the term structure of interest rates. Additionally, in the case of callable and/or puttable bonds, expected interest-rate volatility may alter the cash flow of a bond.
How can we grow the binomial tree to one more year?
We will use the three-year on-the-run issue to get r(2). The same five steps are used in an iterative process to find the 1-year forward rate two years from now. But now we find the value for r(2) that: (1) Will produce an average present value at node N(H) equal to the bond value at that node. (2) Will also produce an average present value at node N(L) equal to the bond value at that node. When this value is found, we know that given the forward rate we found for , the bond's value at the root—the value of ultimate interest to us—will be the observed market price. The binomial interest-rate tree constructed is said to be an arbitrage-free tree. It is so named because it fairly prices the on-the-run issues.
What are the fundamental assumptions of callable bonds?
When a bond is callable, the practice has been to calculate a yield to worst, which is the smallest of the yield to maturity and the yield to call for all possible call dates. The yield to call, like the yield to maturity, assumes that all cash flows can be reinvested at the computed yield to call until the assumed call date. Moreover, the yield to call assumes that: (1) The investor will hold the bond to the assumed call date. (2) The issuer will call the bond on that date. Often, these underlying assumptions about the yield to call are unrealistic because they do not take into account how an investor will reinvest the proceeds if the issue is called.
What is effective duration?
When the approximate duration formula is applied to a bond with an embedded option, the new prices at the higher and lower yield levels should reflect the value from the valuation model. Duration calculated in this way is called effective duration or option-adjusted duration.
Why do we need to account for interest-rate volatility?
When we allow for embedded options, consideration must be given to interest-rate volatility. This can be done by introducing an interest-rate tree (lattice). The tree is a graphical depiction of the one-period forward rates over time based on some assumed interest-rate model and interest-rate volatility.
