MFE8812 Bond Portfolio Management

Transcription

1 MFE8812 Bond Portfolio Management William C. H. Leon Nanyang Business School January 8, / 87 William C. H. Leon MFE8812 Bond Portfolio Management 1 Overview Building an Interest-Rate Tree Calibrating an Interest-Rate Tree 2 Overview Institutional Aspects Pricing Option Adjusted Spread Analysis Effective Duration & Convexity Institutional Aspects Terminology Valuation of Options 2 / 87 William C. H. Leon MFE8812 Bond Portfolio Management

2 Overview Managing fixed-income securities and interest-rate derivatives is a challenging task. It requires not only the use of the current term structure of interest rates, but also a model that describes how the term structure is going to evolve over time. Pricing and hedging of assets paying cash flows that are known with certainty at the present date boils down to a computation of the sum of these cash flows, discounted at a suitable rate. The challenge for the bond portfolio manager is therefore limited to being able to have access to a robust methodology for extracting implied zero-coupon prices from market prices. 3 / 87 William C. H. Leon MFE8812 Bond Portfolio Management Overview Pricing and hedging fixed-income securities that pay uncertain cash flows (e.g., options on bonds) is more involved. It requires not only the knowledge on discount factors at the present date but also some kind of understanding of how these discount factors (i.e., the term structure of pure discount rates) are going to evolve over time. In particular, one needs to account for potential correlations between the discount factor and the promised payoff. Some dynamic model of the term structure of interest rates is therefore needed to describe the explicit nature of the variables of interest in the valuation formula. 4 / 87 William C. H. Leon MFE8812 Bond Portfolio Management

3 Early Merton s (1973) model and Vasicek s (1977) model dr(t) =μ 1 dt + σ 1 dw (t) dr(t) = ( μ 1 + μ 2 r(t) ) dt + σ 1 dw (t) are the first examples of attempts to model the yield-curve dynamics. Since then, many models have been introduced in the literature, and it has become a somewhat daunting task for the practitioner to decide which to use. 5 / 87 William C. H. Leon MFE8812 Bond Portfolio Management Criteria of a Good Model In order to price and hedge fixed-income derivatives efficiently, an interest rate model should be: flexible enough to capture a variety of situations that might be encountered in practice (in particular, a variety of possible shapes of the term structure of interest rates); well specified, in the sense that the model inputs should be observable or at least easy to estimate; consistent with reference market prices; simple enough to allow for fast numerical computation; realistic (e.g., it excludes the possibility of negative values for the interest rate); and coherent, from a theoretical standpoint (e.g., it excludes the possibility of arbitrage opportunity). 6 / 87 William C. H. Leon MFE8812 Bond Portfolio Management

4 Binomial Interest-Rate Tree The binomial way of modeling interest-rate dynamics is most frequently used in the market place. The interest-rate tree concept it is based on is a simple description of the evolution of the short-term interest rate over time. 7 / 87 William C. H. Leon MFE8812 Bond Portfolio Management Building an Interest-Rate Tree In a binomial interest-rate tree model, it is assumed that, given today s level of the short-term interest rate r, the next-period short interest rate, in period 1, can take on only two possible values: alowervaluer l (where the subscript l stands for lower), and an upper value r u (where the subscript u stands for upper), with equal probability. In period 2, the short-term interest rate can take on four possible values: r ll, r lu, r ul,andr uu. More generally, in period n, the short-term interest rate can take on 2 n values, i.e., this is a non-recombining tree. 8 / 87 William C. H. Leon MFE8812 Bond Portfolio Management

5 Non-Recombining Interest-Rate Tree The implementation of such non-recombining tree is very time-consuming and computationally inefficient. For instance, pricing a 30-year callable bond with annual coupon payments would require the computation of over 1 billion (i.e., 10 9 ) interest-rate values in period 30. For a 30-year bond with semiannual coupon payments, the number of values to be calculated in period 60 wouldbeevengreaterthan10 18! 9 / 87 William C. H. Leon MFE8812 Bond Portfolio Management Recombining Interest-Rate Tree In practice, market participants prefer to use a recombining binomial interest-rate tree, which means that a downward-upward sequence leads to the same result as an upward-downward sequence. For example, r lu = r ul. All in all, the short-term interest rate takes only 3, as opposed to 4, values on period 2. There is a 0.25 probability of reaching the node r ll or the node r uu, while there is a probability to reach the intermediate state, since r lu = r ul. This is a binomial distribution. In period n, itcantakeonn + 1 values with the probability of reaching node r l k u n k given by Ck n ( ) 1 k ( ) 1 n k, 0 k n, 2 2 where C n k = n! k!(n k)! is the number of paths with k down moves and n k up moves in n steps. 10 / 87 William C. H. Leon MFE8812 Bond Portfolio Management

6 Period 1 Period 2 Period 3 Period 4 Time: r uuuu r uuu r uu r luuu r r u r lu r luu r lluu r l r llu r ll r lllu r lll r llll 11 / 87 William C. H. Leon MFE8812 Bond Portfolio Management Interest-Rate Process Changes in the short-term interest rate r are generated by the model: where Δr t = r t+δt r t = μ Δt + σ Δt ɛ t Δt is the change in time from one period to another; Δr is the change in the short-term interest rate over one period; μ is the expected absolute change of the short-term interest rate per time unit, which can be a function of time t, the short-term interest rate r and potentially some other state variables; σ is the standard deviation of the absolute change in the short-term interest rate per time unit, which can be a function of time t, the short-term interest rate r and potentially some other state variables; ɛ designates independent Bernoulli distributed variables that take on the values +1 and 1 with equal probabilities. 12 / 87 William C. H. Leon MFE8812 Bond Portfolio Management

7 Interest-Rate Process We sometimes prefer to use the following model: Δlnr t =lnr t+δt ln r t = μ Δt + σ Δt ɛ t. (1) This is similar to the earlier model, except that it is written on relative changes of interest rates (proxied by Δ ln r t =ln r t+δt r t r t+δt r t 1), as opposed to focusing on absolute changes on interest rates (Δr t = r t+δt r t ). The reason we generally prefer to model the changes in the natural log of the interest rates is that it prevents interest rates from becoming negative with positive probability. 13 / 87 William C. H. Leon MFE8812 Bond Portfolio Management Calibrating an Interest-Rate Tree The determinant characteristic of an arbitrage-free interest-rate tree is that it is calibrated to the current zero-coupon fitted term structure of Treasury bonds or interbank instruments. We shall explain how to perform this calibration exercise in the context of a model written on relative changes of interest rates, as in equation (1). Let σ denotes the assumed volatility of the short-term interest rate; T the maturity (expressed in periods) of the bond to be priced; and n the number of periods of the tree. From equation (1), we have at date 0 Δlnr 0 =lnr Δt ln r 0 = μ Δt + σ Δt ɛ / 87 William C. H. Leon MFE8812 Bond Portfolio Management

8 Calibrating an Interest-Rate Tree ln r u =lnr 0 + μ Δt + σ Δt, ln r l =lnr 0 + μ Δt σ Δt. Therefore, ln r u ln r l =2σ Δt r u = r l e 2 σ Δt. More generally, we have on period t where 0 < k t. r l t k u k = r l t e2 kσ Δt, 15 / 87 William C. H. Leon MFE8812 Bond Portfolio Management Calibrating an Interest-Rate Tree We may now proceed to the calibration of the interest-rate tree. For simplicity, assume that the time span between two periods is Δt =1year. The risk-free interest-rate tree is constructed by using constant maturity Treasury-bond zero-coupon yields derived from highly liquid bonds. We denote by y τ the par Treasury-bond yield maturing in τ years. The process is an iterative one. First, the values of the 1-year risk-free rate a year from now are determined using a 2-year Treasury par yield bond. Then, the values of the 1-year risk-free rate 2 years from now are determined using a 3-year Treasury par yield bond and so on. 16 / 87 William C. H. Leon MFE8812 Bond Portfolio Management

9 Calibrating an Interest-Rate Tree We must choose a value for the 1-year interest-rate volatility σ. We can, for example, use some estimate of the historical volatility over the last 1 year. Since r 0 = y 1, we start by determining r u, the value of the short-term interest rate one step ahead in case of an up move, and r l,thevalueof the short-term interest rate one step ahead in case of a down move. The price 1 year from now of the 2-year par Treasury can take 2 values: P u = 100(1 + y 2) 1+r u and P l = 100(1 + y 2) 1+r l. 17 / 87 William C. H. Leon MFE8812 Bond Portfolio Management Calibrating an Interest-Rate Tree The current price of the 2-year par Treasury bond is equal to 100. Thus, ( 1 Pu + 100y 2 + P ) l + 100y 2 = y 1 1+y 1 Substitute the equations for r u, P u and P l in to obtain ( 100(1+y2 ) y 1+r l exp(2σ) y 1 100(1+y 2 ) ) 1+r l + 100y 2 = y 1 From this equation, we find the value of r l,withy 1, y 2 and σ being known. 18 / 87 William C. H. Leon MFE8812 Bond Portfolio Management

10 Calibrating an Interest-Rate Tree To compute r uu, r ul and r ll, consider the 3-year par Treasury bond. The price 2 years from now of the 3-year par Treasury bond can take 3 values: P uu = 100(1 + y 3), 1+r uu P lu = 100(1 + y 3) 1+r lu and P ll = 100(1 + y 3) 1+r ll. 19 / 87 William C. H. Leon MFE8812 Bond Portfolio Management Calibrating an Interest-Rate Tree The price 1 year from now of the 3-year par Treasury bond can take 2 values: P u = 1 ( ) P uu + 100y 3 + P lu + 100y r u 1+r u P l = 1 ( ) P lu + 100y 3 + P ll + 100y r l 1+r l The current price of the 3-year par Treasury bond is equal to 100. Thus, ( ) 1 P u + 100y 3 + P l + 100y 3 = y 1 1+y 1 20 / 87 William C. H. Leon MFE8812 Bond Portfolio Management

11 Calibrating an Interest-Rate Tree Substitute the equations for r uu, r lu, P u, P l, P uu, P lu and P ll in to obtain ( 100(1+y3 ) 1+r ll exp(4σ) +100y 3 1+r u ) 100(1+y 3 ) 1+r ll exp(2σ) +100y 3 1+r u + 100y 3 1+y 1 ( 100(1+y3 ) 1+r ll exp(2σ) +100y 3 1+r l + ) 100(1+y 3 ) +100y 1+r 3 ll 1+r l 1+y y 3 = 100. From this equation, we find the value of r ll,withy 1, y 2, y 3, r u, r l and σ being known. 21 / 87 William C. H. Leon MFE8812 Bond Portfolio Management Exercise Suppose that the today s Treasury-bond par yield curve is as follow: Maturity (in years) Par yield (%) Calibrate a 3-year binomial interest-rate tree, assuming a volatility of 1% for the 1-year interest rate. 22 / 87 William C. H. Leon MFE8812 Bond Portfolio Management

12 Answer 23 / 87 William C. H. Leon MFE8812 Bond Portfolio Management Overview Most traded bonds issued by corporations are not straight bonds; they contain all kinds of embedded options. A callable bond contains an embedded call option. A putable bond contains an embedded put option. A convertible bond contains embedded call and/or put options. Options are also written on bonds just as they are written on stocks. There exists a specific relationship, known as the put-call parity relationship, between the price of a call option and the price of a put option on the same underlying debt instrument, with the same strike price and the same expiration date. Pricing these bonds with embedded options and options on bonds requires the use of a model for the short-term rate dynamics, for example, a discrete-time binomial interest rate model. 24 / 87 William C. H. Leon MFE8812 Bond Portfolio Management

13 Definition & Characteristics Callable and putable bonds contain embedded call options and embedded put options, respectively. 1 A callable bond is a bond that can be redeemed by the issuer before its maturity date. Hence, buying a callable bond comes down to buying an option-free bond and selling a call option to the issuer of the bond. 2 A putable bond is a bond that can be sold by the bond holder before its maturity date. Hence, buying a putable bond comes down to buying an option-free bond as well as a put option. The embedded call or put option can be exercised from a specified date onwards or on a specified date(s), depending on the bond, and at a specified price. 25 / 87 William C. H. Leon MFE8812 Bond Portfolio Management Example The UK Treasury bond with coupon 5.5% and maturity date 10 Sep 2012 can be called in full or part from 10 Sep 2008 onwards, at a price of GBP100. The US Treasury bond with coupon 7.625% and maturity date 15 Feb 2007 can be called on coupon dates only, at a price of USD100, from 15 Feb 2002 onwards. Such a bond is said to be discretely callable. The Bayerische Landesbank bond with coupon 6% and maturity date 10 Mar 2020 can be put on 10 Mar 2010 only, at a price of EUR / 87 William C. H. Leon MFE8812 Bond Portfolio Management

14 Advantages & Drawbacks of Callable Bond A callable bond allows its issuer to buy back his debt at a specified value prior to maturity in case interest rates fall below the coupon rate of the issue. So, he will have the opportunity to issue a new bond at a lower coupon rate. Thus, the issuer will be willing to sell such a bond at a lower price than a comparable option-free bond. A callable bond has two disadvantages for an investor. First, if it is effectively called, the investor will have to invest in another bond yielding a lower rate than the coupon rate of the callable issue; hence, he incurs a loss in interests. Second, a callable has the unpleasant property for an investor to appreciate less than a normal similar bond when interest rates fall. This property is called negative convexity. In order to be compensated for these drawbacks, an investor will only be willing to buy such a bond at a lower price than a comparable option-free bond. 27 / 87 William C. H. Leon MFE8812 Bond Portfolio Management Advantages & Drawbacks of Putable Bond A putable bond allows its holder to sell the bond at par value prior to maturity in case interest rates exceed the coupon rate of the issue. So, he will have the opportunity to buy a new bond at a higher coupon rate. Hence, a putable bond trades at a higher price than a comparable option-free bond. On the other hand, the issuer of this bond will have to issue another bond at a higher coupon rate if the put option is exercised. In order to be compensated for this drawback, the issuer will only be willing to sell such a bond at a higher price than a comparable option-free bond. 28 / 87 William C. H. Leon MFE8812 Bond Portfolio Management

15 Pricing Unlike an option-free fixed coupon bond, a bond with an embedded option is a contingent claim, i.e., its future cash flows are not known with certainty, because they are dependent on the future values of interest rates. To price such a bond, we need to use a model that accounts for the fact that future interest rates are uncertain. This uncertainty is described by volatility (the annualized standard deviation of the relative changes in interest rate). Traditionally, bonds with an embedded options are priced with the yield-to-worst approach. This approach is fairly popular amongst investors, because it gives a quick measure of the potential return that can be earned on a bond with an embedded option. 29 / 87 William C. H. Leon MFE8812 Bond Portfolio Management Yield-to-Worst Consider a bond with an embedded call option trading over its par value. This bond can be redeemed by its issuer prior to maturity from its first call date onwards. Apart from yield-to-maturity, we can compute a yield-to-call on all possible call dates. The yield-to-call is the internal rate of return that equalizes the sum of all discounted cash flows of a bond until the call date to its trading gross price. For example, the cash flows until the first call date of a bond with 5% coupon, 10 years maturity and callable at 100 after 5 years are F 1 =5,F 2 =5,F 3 =5,F 4 =5andF 5 = 105. Thus, the yield-to-call is the internal rate of return that equalizes the sum of these discounted cash flows to its trading gross price. The yield-to-worst is the lowest of the yield-to-maturity and all yields-to-call. 30 / 87 William C. H. Leon MFE8812 Bond Portfolio Management

16 Example An investor wants to assess the potential return he can earn on a 10-year bond bearing an interest coupon of 5%, discretely callable after 5 years and trading at 102. There are five possible call dates before maturity. The corresponding yields-to-call and the yield-to-maturity of the bond are as follow: Yield-to-Call (%) Yield-to-Maturity (%) Year Year Year Year Year Year So, the yield-to-worst of this bond is equal to 4.54%. 31 / 87 William C. H. Leon MFE8812 Bond Portfolio Management Yield-to-Worst The yield-to-worst is the worst yield that an investor can achieve on a callable bond at the current market price, given that the issuer of the callable bond has the choice to call or not to call the bond. For a putable bond, the yield-to-worst concept is meaningless since the put option is exercised by the investor. 32 / 87 William C. H. Leon MFE8812 Bond Portfolio Management

17 Limitations The yield-to-worst measure has the following drawbacks: Like the yield-to-maturity measure, it is based upon the following unrealistic assumptions: An investor will hold the bond until its call exercise date. The coupons of the bond will be reinvested at one single rate, the yield-to-worst. It neither takes into account price risk (in case the investor sells the bond before the call exercise date), nor reinvestment risk (in case the bond is called by the issuer during the investment period). It assumes that the issuer will call the bond on the call date, which of course depends on the level of interest rates on that date. If the issuer redeems the bond on another date, the yield-to-worst measure will be of course irrelevant. It is a static and simplistic return measure, the yield-to-worst cannot be reasonably used for accurate valuation of bonds with embedded options. 33 / 87 William C. H. Leon MFE8812 Bond Portfolio Management What is Needed To accurately value bonds with embedded options, we need to use a methodology taking into account the uncertain nature of future interest rates. Such methodologies have at least the following common features: Interest rates are assumed to be log-normal, so that changes in interest rates are proportional to the level of interest rates. Moreover, this property ensures that interest rates are never negative. The models are built so as to be free of arbitrage opportunities: they must perfectly reflect todays market prices of risk-free bonds (this is a matter of calibration). 34 / 87 William C. H. Leon MFE8812 Bond Portfolio Management

18 Two Ways to Price a Bond To understand the nature of the valuation methodologies, one must first note that a bond can be priced in two different ways leading to exactly the same results: by using spot interest rates; or by using forward interest rates. 35 / 87 William C. H. Leon MFE8812 Bond Portfolio Management Example Consider a bond with maturity 2 years and annual coupon 5%. We denote by R(0, t) the zero-coupon rate with maturity t years and by F (0, s,τ)the forward rate currently determined, starting on year s, with maturity τ years. The todays price P of the bond can be determined by using spot interest rates, P = or by using forward interest rates, 5 1+R(0, 1) ( ) 2 ; 1+R(0, 2) P = 1+R(0, 1) F (0, 1, 1) ( )( ) =. 1+R(0, 1) 1+F (0, 1, 1) 1+R(0, 1) 36 / 87 William C. H. Leon MFE8812 Bond Portfolio Management

19 Example That is, the price discovery process of the bond can be viewed as a backward process. First, the price of the bond 1 year from now is equal to the sum of its cash flows on year 2 (principal plus coupon) discounted at the forward rate starting in 1 year and with maturity 1 year. Then, the current price of the bond is equal to the sum of its cash flows on year 1 (price on year 1 previously computed plus coupon) discounted at the 1-year spot rate. We shall use this backward pricing process in the binomial interest-rate tree methodology, except that forward interest rates are replaced by assumed future short-term interest rates derived from a volatility assumption about these rates. 37 / 87 William C. H. Leon MFE8812 Bond Portfolio Management Assume that a binomial interest-rate tree has been built and calibrated to the current zero-coupon rates. Consider a callable bond and adopt the following notations: V is its redemption value, T k is its call date(s), CP Tk is its call price on call date T k, C is its annual coupon rate, T is its maturity, and P l i u j is its price at time i + j and at node ( l i, u j). 38 / 87 William C. H. Leon MFE8812 Bond Portfolio Management

20 Time: T 2 T 1 T P u T P u T 1 P P u P l P u T 2 P lu T P lu T 1 P l 2 u T 2. P l T 1 u P l T 1 P l T 39 / 87 William C. H. Leon MFE8812 Bond Portfolio Management At time T,for0 i T,notethat P l i ut i = V + C. We infer the callable bond price by beginning with the last cash flows and going back step by step to the first ones. At time T 1, for 0 i T 1, we have ) P l i u (CP T 1 i =min V + C T 1,. 1+r l i u T 1 i At time T 2, for 0 i T 2, we have ( P l i u T 2 i =min CP T 2, ( P l i+1 u T 2 i + P ) ) l i u T 1 i + C. 1+r l i u T 2 i 40 / 87 William C. H. Leon MFE8812 Bond Portfolio Management

21 The backward formula applies till time 1. At time 0, the price P of the callable bond is P = ( ) P l + P u + C. 1+r For a putable bond, the formulas are exactly the same except that instead of taking at each node the minimum value of the computed price of the bond and its call price, we take the maximum value of the computed price of the bond and its put price. 41 / 87 William C. H. Leon MFE8812 Bond Portfolio Management Exercise Consider a callable bond with maturity 2 years, annual coupon 5%, callable in 1 year at 100. The interest-rate tree is as follow: 5.03% 4.66% 4.00% 4.93% 4.57% 4.83% What is the price of the callable bond? 42 / 87 William C. H. Leon MFE8812 Bond Portfolio Management

22 Answer 43 / 87 William C. H. Leon MFE8812 Bond Portfolio Management Practical Issues In practice, implementing the interest-rate tree methodology raises the following issues: The period frequency must be greater than 1 year; at least semiannual. This is of course due to the fact that bond maturities are very rarely multiples of 1 year. The higher the period frequency, the more accurate the valuation method, but also the more demanding its implementation. There is clearly a trade-off between complexity and efficiency. For the purpose of pricing a very long-term bond, you need not choose as high a period frequency as for pricing a short-term bond, because the number of computation steps is significantly higher. Many bonds with embedded options contain American options that can be exercised at any time after the first exercise date, which requires a fine node spacing and significantly complicates the models implementation. 44 / 87 William C. H. Leon MFE8812 Bond Portfolio Management

23 Practical Issues Very often, cash flows fall between two nodes, which does not make the computation easy. The volatility parameter is fixed arbitrarily. One can take an estimate of the historical volatility of the short-term interest rate over a given period (to be chosen) or consider the implied volatility of an option on an interest rate with similar maturity and behavior. Anyway, the higher the volatility parameter, the higher the value of the embedded call option (put option, respectively) and the lower the price of the callable bond (the higher the price of the putable bond, respectively). In the presentation of the methodology, we have assumed a single interest-rate volatility for all periods. This may seem rather simplistic. It is in fact possible to incorporate a term structure of volatility. 45 / 87 William C. H. Leon MFE8812 Bond Portfolio Management Option Adjusted Spread As market participants are familiar with yield spreads, a market habit has developed, which consists in determining the yield spread that, when added to all short-term interest rates, equalizes the theoretical price of the bond to its market price. This spread is called the option-adjusted spread (OAS). The denomination option adjusted refers to the fact that the cash flows of the bond to be priced are adjusted for the option exercise price(s). In other words, unlike the traditional yield spread, the OAS takes the optional feature of the bond into account. The OAS depends on the volatility parameter assumed in the valuation model. The higher the volatility of the short-term interest rate, the lower (the higher, respectively) the theoretical price of a callable (putable, respectively) bond, and hence, the lower (the higher, respectively) the OAS of the callable (putable, respectively) bond. 46 / 87 William C. H. Leon MFE8812 Bond Portfolio Management

24 Option Adjusted Spread As the valuation models are mostly calibrated to the term structure of government interest rates, the OAS of a bond with an embedded option mainly represents the liquidity premium and/or the credit premium attached to it. 47 / 87 William C. H. Leon MFE8812 Bond Portfolio Management Example Consider the callable bond in the previous exercise where we have computed its theoretical price to be Suppose that the market price of the bond is 100. Then the OAS of this bond is such that ( min ( ) ( )) , %+OAS +min 100, %+OAS +5 = % + OAS We find that OAS = +43bps. 48 / 87 William C. H. Leon MFE8812 Bond Portfolio Management

25 Effective Duration & Convexity The cash-flows of a bond with an embedded option is affected by the level of interest rates. Thus, the traditional modified duration and convexity measures are not relevant for such a bond. Instead, market participants use what is known as effective duration and effective convexity defined by the following formulas: Effective Duration = P P + 2 P Δy Effective Convexity = P + P + 2 P P ( Δy ), 2 where P is the current price of the bond, Δy is absolute change in the Treasury bond yield curve, and P + (P, respectively) is price of the bond after shifting upwards (downwards, respectively) the yield curve by Δy. 49 / 87 William C. H. Leon MFE8812 Bond Portfolio Management Effective Duration & Convexity In the binomial interest-rate tree framework, P + (respectively, P )is determined as follow: 1 Compute the OAS of the bond. 2 Shift the Treasury yield curve upwards (respectively, downwards) by the amount Δy. 3 Determine the resulting new interest-rate tree. 4 Add the OAS to the new interest-rate tree. 5 Price the bond backwards along the option-adjusted interest-rate tree to get P + (respectively, P ). 50 / 87 William C. H. Leon MFE8812 Bond Portfolio Management

26 Example Consider the callable bond in the previous exercise where the market price of the bond is 100 and the OAS of the bond is +43 bps. Suppose Δy = 10 bps. Step 1: The OAS of the callable bond is equal to +43 bps. Step 2: Shift the initial yield curve upwards and downwards by Δy. Maturity Par Yield (%) (in years) Intial Shift by +10 bps Shift by 10 bps / 87 William C. H. Leon MFE8812 Bond Portfolio Management Example Step 3: Compute the new interest-rate trees. +10 bps 4.76% 10 bps 4.56% 4.10% 3.90% 4.67% 4.47% Step 4: Compute the option-adjusted interest-rate trees. +10 bps 5.19% 10 bps 4.99% 4.53% 4.33% 5.10% 4.90% 52 / 87 William C. H. Leon MFE8812 Bond Portfolio Management

27 Example Hence, Step 5: Find P + and P. P + = ( min ( 100, P = ( min ( 100, ) ( )) % +min 100, % % % ) +min ( 100, % )) % = = Effective Duration = Effective Convexity = % = (0.10%) 2 = / 87 William C. H. Leon MFE8812 Bond Portfolio Management Convertible bond is a security that gives the bondholder the right to exchange the par amount of the bond for common shares of the issuer at some fixed ratio during a particular period. Convertible bond offers the investor the safety of a fixed-income instrument coupled with participation in the upside of the equity market. As a debt security, convertible bond has an advantage over the common stock in case of distress or bankruptcy. There is less risk in holding the convertible because it has seniority in payment. In addition, it has termination value that must be paid at maturity. Convertible bond is similar to a normal coupon bond plus a call option on the underlying stock. It is therefore priced as a function of the price of the underlying stock, expected future volatility of equity returns, risk-free interest rates, call provisions, supply and demand for specific issues, issue-specific corporate/treasury yield spread and expected volatility of interest rates and spreads. 54 / 87 William C. H. Leon MFE8812 Bond Portfolio Management

28 Convertible bond may be issued with call and/or put options allowing prepayment before the maturity date. A call exercisable by the issuer can force the convertible bondholder to sell her the bond when the underlying equity exceeds a given value. A put exercisable by the bondholder can force the issuer to buy the bond at a determined price when the underlying equity falls below a given value. 55 / 87 William C. H. Leon MFE8812 Bond Portfolio Management Terminology The following are some terms that are essential to understanding what a convertible bond is: The convertible price is the price of the convertible bond. The bond floor or investment value is the price of the bond if there is no conversion option. The conversion ratio is the number of shares a bond is exchanged for. The conversion price is the share price at which the face value of the bond may be exchanged for shares; it is the strike price of the embedded equity option, and it is given by the following formula: Conversion Price = Par Value of Convertible Bond. Conversion Ratio 56 / 87 William C. H. Leon MFE8812 Bond Portfolio Management

29 Terminology The conversion value is equal to Conversion Value = Current Share Price Conversion Ratio. The conversion premium is equal to Conversion Premium = Convertible Price Conversion Value. Conversion Value The income pickup is the amount by which the yield-to-maturity of the convertible bond exceeds the dividend yield of the share. 57 / 87 William C. H. Leon MFE8812 Bond Portfolio Management Example AXA has issued in the Euro zone a convertible bond paying a 2.5% coupon rate and maturing on 1 Jan 2014 (see figure on next slide). The amount issued initially was EUR billion. The coupon frequency is annual and the day-count basis is actual/actual. The conversion ratio is The conversion of the bond into 4.04 shares can be executed on any date before the maturity date. On 13 Dec 2001, the market share price was EUR and the bid-ask convertible price was / The conversion value (called parity) was equal to The conversion premium calculated with the ask price was 61.73%. The redemption value on 1 Jan 2014 is equal to EUR (= % 165). 58 / 87 William C. H. Leon MFE8812 Bond Portfolio Management

30 59 / 87 William C. H. Leon MFE8812 Bond Portfolio Management Advantages & Drawbacks of Convertible Bond For the Issuer. Issuing convertible bonds enables a firm to obtain better financial conditions. In fact, the coupon rate of such a bond is always inferior to that of a bullet bond with the same characteristics in terms of maturity and coupon frequency. This comes directly from the conversion advantage that is attached to this product. Besides, the exchange of bonds for shares diminishes the liabilities of the firm issuer and increases its equity in the same time so that its debt capacity is improved. For the Bondholder. A convertible bond is a defensive security, very sensitive to a rise in the share price and protective when the share price decreases. If the share price increases, the convertible price will react in the same manner and it will not be inferior to the bond floor if the share price decreases. Recall that the bond floor is the price of the bullet bond with no conversion option. 60 / 87 William C. H. Leon MFE8812 Bond Portfolio Management

31 Price of a convertible bond as a function of the price of the underlying stock. 61 / 87 William C. H. Leon MFE8812 Bond Portfolio Management Valuation of A convertible bond is similar to a normal coupon bond plus a call option on the underlying stock, with an important difference: the effective strike price of the call option will vary with the price of the bond, which itself varies with interest rates. Because of this characteristic, it is difficult to price the conversion option using a BlackScholes (1973) formula or other similar models that assume constant interest rates. In practice, a popular method for pricing convertible bonds is the component model, also called the synthetic model. The convertible bond is divided into a straight bond component and a call option on the conversion price, with strike price equal to the value of the straight bond component. The fair value of the two components can be calculated with standard formulas, such as the famous BlackScholes (1973) valuation formula. 62 / 87 William C. H. Leon MFE8812 Bond Portfolio Management

32 Valuation of The synthetic model pricing approach, however, has several drawbacks: First, separating the convertible into a bond component and an option component relies on restrictive assumptions, such as the absence of embedded options. Callability and putability, for instance, are convertible bond features that cannot be considered in the above separation. Second, convertible bonds contain an option component with a stochastic strike price. This is in contrast to standard call options, where the strike price is known in advance. It is stochastic because the value of the bond to be delivered in exchange for the shares is usually not known in advance unless conversion is certain not to occur until maturity. In effect, the future strike price depends on the future development of interest rates and the future credit spread. 63 / 87 William C. H. Leon MFE8812 Bond Portfolio Management Valuation of We will use again a binomial setup to the problem of pricing convertible securities in the presence of embedded options. Since the problem involves the dynamics of the stock price, we actually need to also model the stochastic evolution of stock returns. 64 / 87 William C. H. Leon MFE8812 Bond Portfolio Management

33 The Binomial Model for Stock Prices Assume that d < (1 + r) < u where r is the one-period risk-free rate. Let the price of the risky stock S and that of a portfolio C evolve as follow: Stock us Portfolio C u p p S C 1 p ds 1 p C d Then where the risk-neutral probability C = 1 ) (pc u +(1 p) C d 1+r p = (1 + r) d. u d 65 / 87 William C. H. Leon MFE8812 Bond Portfolio Management Application to the Valuation of Convertible Securities Since a convertible bond is nothing but an option on the underlying stock, we expect to be able to use the binomial model to price it. At each node, we test 1 whether conversion is optimal; 2 whether the position of the issuer can be improved by calling the bonds. It is a dynamic programming procedure given by max (min ( Q 1,, Q 2, ), Q3,, Q 4, ), where Q 1, = value given by the rollback (neither converted nor called back), Q 2, = call price, Q 3, =conversionvalue, Q 4, = put value. 66 / 87 William C. H. Leon MFE8812 Bond Portfolio Management

34 Example Consider a stock and a convertible bond with the following characteristics: The underlying stock price trades at $50.00 with a 30% annual volatility. The convertible bond has a a 4% annual coupon, $1,000 face value, 9-month maturity, a conversion ratio of 20 and a call price is $1, Assume that the risk-free rate is a continuously compounded 10%, while the yield-to-maturity on straight bonds issued by the same company is a continuously compounded 15%. What is the price of the convertible bond? 67 / 87 William C. H. Leon MFE8812 Bond Portfolio Management Example We use a 3-period binomial tree for stock price dynamics to value that convertible bond and assume a constant continuously compounded interest rate. Set the elementary time step Δt =0.25 year, i.e., 3 months. Let and d =1/u = ( u =exp σ ) ( Δt =exp 0.3 ) 0.25 = The risk-neutral probability p = (1 + r) d u d = e = / 87 William C. H. Leon MFE8812 Bond Portfolio Management

35 69 / 87 William C. H. Leon MFE8812 Bond Portfolio Management Example Start with the terminal nodes. At node G, the bondholder optimally chooses to convert, since what is obtained under conversion ($1,568.31) is higher than the payoff under the assumption of no conversion ($1,000.00). The same applies to node H ($1,161.83). On the other hand, at nodes I and J, the value under the assumption of conversion is lower than if the bond is not converted to equity. Therefore, bondholders optimally choose not to convert, and the payoff is simply the nominal value of the bond, plus the interest payments ($1,040.00). 70 / 87 William C. H. Leon MFE8812 Bond Portfolio Management

36 Example Work your way backward through the tree to node D, E and F. At node D, the value of the convertible bond as the discounted expected value, using risk-neutral probabilities of the payoffs at nodes G and H, is $1, = e 10% 0.25( 466 1, , ). Since this rollback value exceeds the call price ($1,100.00), the bond will be called by the issuer. Bondholders are left with the choice between not converting and getting the call price, or converting and getting the stock ($1, = ), which is what they optimally choose. Now, this is less than $1,388.87, the value of the convertible bond if it were not called, and this is precisely why it is called by the issuer. Note that at node D, the bond is already essentially equity. 71 / 87 William C. H. Leon MFE8812 Bond Portfolio Management Example At node F, the same principle applies, except for one difference. Because the bond will not be converted at subsequent nodes I and J, it can be regarded at node F as a standard bond. The valuation of a standard bond cannot be made by discounting the cash flows at the risk-free rate of 10% because that would assume away default risk. We therefore use the rate of return on a comparable, nonconvertible bond issued by the same company as a discount rate. We have assumed that the defaultable rate is 15% and therefore, the value is $1, = e 15% , / 87 William C. H. Leon MFE8812 Bond Portfolio Management

37 Example At node E, the situation is more interesting because the convertible bond willendupasastockincaseofanupmove(conversion),andasabond in case of a down move (no conversion). One may use a weighted average of the risk-free and risky interest rate in the computation, where the weighting is performed according to the (risk-neutral) probability of an up versus a down move, i.e., Hence, the value at node E is % % = %. $1, = e % 0.25( 466 1, , ). 73 / 87 William C. H. Leon MFE8812 Bond Portfolio Management Example Work your way backward through the tree to node B and C. At node B, the convertible bond will end up as a bond in case of two down move (no conversion), and as a stock otherwise (conversion). The weighted average of the risk-free and risky interest rate in this case is ( ) 10% % = %. Hence, the rollback value at node B is $1, = e % 0.25( 466 1, , ). It is thus optimal for the issuer to call the bond. Bondholders are left with the choice between not converting and getting the call price ($1,100.00), or converting and getting the stock ($1, = ), which is what they optimally choose. Now, this is less than $1,200.48, the value of the convertible bond if it were not called, and this is precisely why it is called by the issuer. 74 / 87 William C. H. Leon MFE8812 Bond Portfolio Management

38 Example At node C, the convertible bond will end up as a stock in case of two up move (conversion), and as a bond otherwise (no conversion). The weighted average of the risk-free and risky interest rate in this case is % + ( ) 15% = %. Hence, the value at node C is $1, = e % 0.25( 466 1, , ). 75 / 87 William C. H. Leon MFE8812 Bond Portfolio Management Example Work your way backward through the tree finally to node A. At node A, the convertible bond will end up as a stock in case of three up move or two up plus 1 down move (conversion), and as a bond otherwise (no conversion). The probability of three up move or two up plus 1 down move is 698 = The weighted average of the risk-free and risky interest rate in this case is % + (1 698) 15% = %. Hence, the present fair value of the convertible bond is $1, = e % 0.25( 466 1, , ). 76 / 87 William C. H. Leon MFE8812 Bond Portfolio Management

39 Remark This valuation model discussed is based upon the assumption of optimal conversion from the bondholder. If the bondholder fails to convert optimally, then the theoretical value computed by the model is higher than the actual value of the security. Similarly, there is ample evidence that issuers call back their bond late. This would tend to increase the value of convertible bonds compared to the model predictions. 77 / 87 William C. H. Leon MFE8812 Bond Portfolio Management Valuing Convertibles in the Presence of Interest-Rate Risk The previous approach for valuing convertible bonds has a shortcoming, however. It is based on the assumption of a constant interest-rate process. Such a simplistic assumption might prove to be problematic, especially in the valuation of long-term convertible bonds. One may actually easily generalize the previous model by allowing for the presence of two intersecting recombining binary trees, an interest-rate tree and a stock price tree. Both trees terminate at the maturity date of the bond. This model assumes that the bond yield and the stock price are the two most important factors in valuation and that these factors are independent of each other. This formula is applied at each maturity node of the tree. Working backwards through the tree the intrinsic value of the conversion option is compared with the wait value of the conversion option, that is, the option value from subsequent nodes discounted at the appropriate interest rate. 78 / 87 William C. H. Leon MFE8812 Bond Portfolio Management

40 Valuing Convertibles in the Presence of Interest-Rate Risk 79 / 87 William C. H. Leon MFE8812 Bond Portfolio Management 5.0%,$14 4.5%,$12 4.0%,$14 p 1 p 2 3.2%,$14 3.6%,$12 5.0%,$11 4.0%,$10 4.0%,$11 p 4 p 3 4.5%,$9 3.2%,$11 5.0%,$8 3.6%,$9 4.0%,$8 3.2%,$8 80 / 87 William C. H. Leon MFE8812 Bond Portfolio Management

41 Definition An option is a contract in which the seller (i.e., writer) of the option grants the buyer of the option the right to purchase from, or sell to, the seller a designated instrument at a specified price within a specified period of time. The seller grants this right to the buyer in exchange for a certain sum of money called the option price or option premium. The price at which the instrument may be bought or sold is called the exercise or strike price. The date after which an option is void is called the expiration date. An American option may be exercised any time up to and including the expiration date. A European option may be exercised only on the expiration date. 81 / 87 William C. H. Leon MFE8812 Bond Portfolio Management Definition When an option seller grants the buyer the right to purchase the designated instrument, it is called a call option. When the option buyer has the right to sell the designated instrument to the seller, the option is called a put option. The buyer of any option is said to be long the option; the seller is said to be short the option. 82 / 87 William C. H. Leon MFE8812 Bond Portfolio Management

42 Binomial Model Not only do some bonds contain embedded option features but standard options are also written on bonds just as they are written on stocks. To price an option on bond, we need an interest-rate model calibrated to the current zero-coupon fitted term structure of Treasury bonds, e.g., the binomial interest-rate model. 83 / 87 William C. H. Leon MFE8812 Bond Portfolio Management Exercise The figure below shows a tree for the 1-year rate of interest that is calibrated to the current zero-coupon fitted term structure of Treasury bonds. 7.0% 7.5% 6.5% 6.5% 6.0% 6.0% 5.5% 5.5% 5.0% 4.5% Consider a 3-year zero-coupon $100-par bond. What is the price of a 2-year European call on this bond struck at 93.50? 84 / 87 William C. H. Leon MFE8812 Bond Portfolio Management

43 Answer 85 / 87 William C. H. Leon MFE8812 Bond Portfolio Management