MFE8812 Bond Portfolio Management

Transcription

1 MFE8812 Bond Portfolio Management William C. H. Leon Nanyang Business School January 23, / 118 William C. H. Leon MFE8812 Bond Portfolio Management 1 Overview Shapes of the Term Structure Evolution of the Term Structure The Pure Expectations Theory The Risk Premium Theory The Market Segmentation Theory 2 Overview 2 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

2 Overview The term structure of interest rates, also called the yield curve, is defined as the graph mapping interest rates corresponding to their respective maturity. The term structure of interest rates can take at any point in time several shapes that can be divided into four standard types: quasi-flat, increasing, decreasing and humped. Historical studies of the evolution of the yield curve over time enable us to identify five important stylized facts: (1) interest rates are not negative; (2) interest rates are mean-reverting; (3) changes of interest rates are not perfectly correlated; (4) the volatility of short-term rates is higher than the volatility of long-term rates; and (5) three main factors explain more than 95% of the changes in the yield curve. Classical economic theories of the term structure can be gathered in three categories: (1) the pure expectations theory; (2) the risk premium theory (including the liquidity premium theory and the preferred habitat theory); and (3) the market segmentation theory. 3 / 118 William C. H. Leon MFE8812 Bond Portfolio Management of the Term Structure The term structure of interest rates is the series of interest rates ordered by term-to-maturity at a given time. The nature of interest rates determines the nature of the term structure. Depending on which of the above rates we are interested in, we can get several different types of yield curves. We focus on the following types: (1) yield to maturity curve; (2) zero-coupon yield curve; (3) par yield curve; (4) forward rate curve; and (5) instantaneous forward rate curve. Note that yields to maturity are market rates, whereas zero-coupon, forward and par yields are constructed implicitly using market data. So the first distinction to be made is the one between market curves and implied curves. Besides, the zero-coupon yield curve permits to derive forward curves and the par yield curve. 4 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

3 of the Term Structure We focus on the following: τ y(τ), which is the yield to maturity curve at date t with maturity τ. τ R(t,τ), which is the zero-coupon yield curve at date t with maturity τ. τ c(τ), which is the par yield curve at date t with maturity τ. τ F (t, s,τ), which is, at date t, the curve of forward rates starting at date s with residual maturity τ. s f (t, s), which is the instantaneous forward term structure at date t, starting at date s with infinitesimal maturity. 5 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Shapes of the Term Structure The term structure of interest rates can take several varied shapes that can be divided into four standard types: Quasi-flat. Increasing. Decreasing. Humped decreasing on the short end, and then increasing; or, increasing on the short end, and then decreasing. 6 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

4 Quasi-Flat Shape (US Treasury Par Yield Curve, 3 Nov 1999). 7 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Increasing Shape (Japanese Government Par Yield Curve, 27 Apr 2001). 8 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

5 Decreasing Shape (UK Government Par Yield Curve, 19 Oct 2000). 9 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Humped Shape: Decreasing, Then Increasing (Euro Government Par Yield Curve, Made of French and German Government Bonds, 4 Apr 2001). 10 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

6 Humped Shape: Increasing, Then Decreasing (US Treasury Par Yield Curve, 29 Feb 2000). 11 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Evolution of the Term Structure (Monthly evolution of US government yield curves between January 1999 and March 2001). 12 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

7 Characteristics of the Term Structure Evolution The historical studies of the evolution of the yield curve over time emphasize thefollowingfivepoints: Interest rates are not negative. Interest rates are affected by mean-reversion effects. Changes of interest rates are not perfectly correlated. The volatility of short-term rates is higher than the volatility of long-term rates. Three main factors explain more than 95% of the changes in the yield curve. 13 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Interest Rates Are Not Negative While real interest rates may become negative, generally in a context when the inflation rate is rising under the effect of external shocks, such as a petroleum crisis, nominal interest rates cannot be negative. Indeed it seems crazy to lend money at a negative interest rate. That is why, in particular, interest rates cannot be assumed to be normally distributed. 14 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

8 Paying to Save (Japan & Germany Government Bond Yields, 30 June 2016). 15 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Mean Reversion in Interest Rates (Fed fund rate compared to the trend followed by Dow Jones Index). 16 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

9 Mean-Reverting Behavior of Interest Rates Historical observation shows that when rates reach high levels, they subsequently tend to decline rather than rise still further. Reversion has also been observed when rates fall to unusually low levels. For this reason, interest rates are often modeled using a mean-reversion process. 17 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Correlation Matrix for Zero-Coupon Rates Daily Change (Correlations between implied zero-coupon rates with various maturities, derived from the French swap market in 1998). 18 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

10 Changes in Interest Rates Are Not Perfectly Correlated A statistical analysis typically shows that correlations between interest-rate movements are clearly not equal to 1 (see Table on previous slide). All correlation coefficients are positive and they are decreasing as the difference in maturity is increasing. In some cases, mostly for the short maturities, correlation coefficients are very close to 1. Hence, when pricing and hedging short-term fixed-income products (e.g., contingent claims on a bond with maturity less than 1 year), a single-factor model (which would involve a correlation matrix for interest-rate movements in which all the terms are equal to 1) based on the short-rate dynamics may be used with minimal mispricing risk. When dealing with contingent claims on fixed-income securities with longer maturity, one is better-off using multi-factor models. This holds, in particular, when one attempts to price an asset involving different segments of the yield curve, such as an option on a short- to long-rate spread. 19 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Standard Term Structure of Volatility 20 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

11 Short-Term Rates More Volatile than Long-Term Rates Historically, the term structure of volatility is a decreasing function, or an increasing function until the 1-year maturity and a decreasing function for longer maturities, which is referred to as the humped form. Furthermore, there seems to exist some positive correlation between interest-rate volatility and interest rate level. 21 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Three Factors Explain Quasi-Totality of Rates Changes Using a principal components analysis (PCA) to study movements of the term structure allows one to aggregate interest rate risks in a non-arbitrary way. The motivations are fairly simple: Various interest rates for different maturities are highly correlated variables. Even though they are not perfectly correlated (we know this precisely because we witness nonparallel shifts of the yield curve), various interest rates along the yield curve are affected by a limited set of common economic, monetary and financial shocks. As a result, interest rates for various maturities tend to move in the same direction. Highly correlated variables provide redundant information of one with respect to another. As a consequence, it is tempting to try and identify a set of independent factors that would account for most of the information contained in the time-series of interest-rate variations. 22 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

12 Three Factors Explain Quasi-Totality of Rates Changes Every empirical investigation using PCA shows that the variance of the term structure of interest rates is explained to more than 90% using only the three first components. Below is a typical sensitivity of variations in zero-coupon rates to variations in these factors: 23 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Three Factors Explain Quasi-Totality of Rates Changes These three factors have nice interpretations as being related, respectively, to parallel movement, slope oscillation and curvature of the term structure. The parallel movement component can be interpreted as an average rate over shorter and longer maturities. It always explains more than 60% of the variations in the curve, and provides some justification for simple hedging methods that rely on the assumption of parallel movements. The slope oscillation component is associated with the steepness of the interest-rate curve, and it can be regarded as a short-term vs. long-term spread. It accounts for 5 to 30% of the changes of the yield curve. The curvature component has a different impact on each of the three segments of the yield curve (short, medium and long term). It brings more or less concavity to the intermediate segment of the curve. It accounts for 0 to 10% of the yield-curve changes. 24 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

13 Upward & Downward Parallel Movements 25 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Flattening & Steepening Twist Movements 26 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

14 Concave & Convex Butterfly Movements 27 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Theories of the Term Structure Studying the term structure of interest rates boils down to wondering about market participants preferences for curve maturities. Indeed, if they were indifferent to these, interest-rate curves would be invariably flat, and the notion of term structure would be meaningless. Market participants preferences can be guided by their expectations, the nature of their liability or asset and the level of the risk premiums they require for offsetting their risk aversion. Term structure theories attempt to account for the relationship between interest rates and their residual maturity. They can be gathered in three categories: The pure expectations theory. The risk premium theories, including the liquidity premium theory and the preferred habitat theory. The market segmentation theory. 28 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

15 Theories of the Term Structure The pure expectations theory and risk premium theories are based on the existence of a close link between interest rates with different maturities. Recall that long-term rates can be expressed as the average of the current short-term rate and the forward short-term rates as follows: 1+R(t,τ)=( (1+R(t, 1) )( 1+F (t, t +1, 1) )( 1+F (t, t +2, 1) ) ( 1+F (t, t + τ 1, 1) )) τ 1. The pure expectations theory postulates that forward rates exclusively represent future short-term rates as expected by the market. The risk premium theory postulates that forward rates exclusively represent the risk premium required by the market to hold longer-term bonds. 29 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Theories of the Term Structure The market segmentation theory postulates that there exists no relationship between short-, medium- and long-term interest rates. It postulates that each of the two main market investor categories (the one preferring short bonds, the other long bonds) is invariably located on the same curve portion (short, long). As a result, short and long curve segments are perfectly impermeable. To the three main types we can add the biased expectations theory, which combines the pure expectations theory and risk premium theories. 30 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

16 Pure Expectations Theory The term structure of interest rates reflects at a given time the market expectations of future short-term rates. An increasing (respectively, flat, respectively, decreasing) structure means that the market expects an increase (respectively, a stagnation, respectively, a decrease) in future short-term rates. Formally, 1+R(t,τ)=( (1+R(t, 1) )( 1+F a (t, t +1, 1) )( 1+F a (t, t +2, 1) ) ( 1+F a (t, t + τ 1, 1) )) τ 1, where R(t,τ) is the current rate with maturity τ years observed at time t, F a (t, t + k, 1) is the future short-term rate with maturity 1 year, anticipated by the market at time t and starting at time t + k. 31 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Example Suppose that the interest-rate curve is flat at 5%, and investors expect a 100 bps increase in rates within 1 year. For simplicity, assume that the short (respectively, long) segment of the curve is the 1-year (respectively, 2-year) maturity. Then, under these conditions, the interest-rate curve will not remain flat but will be increasing. To see this, consider a long investor with a 2-year investment horizon. His objective consists in maximizing his return in the period. He is indifferent to the strategy that will allow him to reach it. He has to choose one of the following two alternatives: Invest in a long 2-year security; or Invest in a short 1-year security, then reinvest in 1 year the proceeds in another 1-year security. 32 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

17 Example Before interest rates adjust to the 6% level, the first alternative provides him with an annual return of 5% over 2 years, whereas the second one returns him 5% the first year and, according to his expectations, 6% the second year, that is to say 5.5% on average per year over 2 years. This is the most profitable. The investor will thus buy short bonds (1 year) rather than long bonds (2 years). Any investor having the same investment horizon will act in the same way. As a result, the price of the 1-year bond will increase (its yield decrease) and the price of the 2-year bond will decrease (its yield will increase). The curve will steepen. 33 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Mechanism & Limits of Pure Expectations Theory The pure expectations theory mechanism is as follows: Market participants formulate expectations about the evolution of short-term rates using all the available information, and make their decisions rationally in accordance with it. Their expectations guide their curve position, and hence the long-term rate level. As these expectations prove to be correct on average, future short-term rates equal forward short-term rates. As a result, zero-coupon rate returns equal each other at a given time, whatever their maturity. Market participants behave in accordance with their expectations for the unique purpose of maximizing their investment return. They are risk-neutral. They do not take into account the fact that their expectations may be wrong. The pure expectations theory has two limits: It does not take bond price risk into account. It does not take bond coupon reinvestment risk into account. 34 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

18 Risk Premium Theory The pure risk premium theory accounts for the contingency that market participants expectations may be wrong. Indeed, if forward rates were perfect predictors of future rates, the future bond prices would be known with certainty. An investment return would be certain and independent of the invested bonds maturity as well as the date on which the investor wishes to liquidate his position. Unfortunately, it is practically not the case. As future interest rates are unknown and hence also future bond prices, the latter are risky because their return in the future is unknown. According to the risk premium theory, the term structure of interest rates reflects at a given time the risk premium that is required by the market for holding long bonds. However, the two versions of this theory liquidity premium theory and preferred habitat theory differ about the risk premium shape. 35 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Liquidity Premium Theory The liquidity premium theory postulates the risk premium increases with maturity in a decreasing proportion. In other words, an investor will be interested in holding all the longer bonds as their return contains a substantial risk premium, offsetting their higher volatility. Formally, 1 + R(t,τ) = ( (1 )( )( ) ( ) ) τ 1 + R(t, 1) 1 + L2 1 + L Lτ, where R(t,τ) is the current yield observed at time t with maturity τ years, k j=2 L j is the liquidity premium required by the market to invest in a bond maturing in k years, and 0 < L 2 < L 3 < < L τ L 2 > L 3 L 2 > L 4 L 3 > > L τ L τ / 118 William C. H. Leon MFE8812 Bond Portfolio Management

19 Liquidity Premium Theory This theory simultaneously takes into account investors preference for liquidity and their aversion against the short-term fluctuations of asset prices. Its limits lie in the impossibility of explaining decreasing and humped curves, as well as its ignorance of coupon reinvestment risk. 37 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Preferred Habitat Theory The preferred habitat theory is an improvement of the liquidity premium theory because it postulates that risk premium is not uniformly increasing. Indeed, investors do not all intend to liquidate their investment as soon as possible, as their investment horizon is dictated by the nature of their liabilities. Nevertheless, when bond supply and demand on a specific curve segment do not match, some lenders and borrowers are ready to move to other curve parts where there is the inverse disequilibrium, provided that they receive a risk premium that offsets their price or reinvestment risk aversion. Thus, all curve shapes can be accounted for. 38 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

20 Market Segmentation Theory In the market segmentation framework, there exist different investor categories, each systematically investing on a given curve segment in accordance with its liabilities, without ever moving to other segments. The shape of term structure is then determined by the law of supply and demand on each segment of the curve. Two categories of investor are usually underlined, because they carry on their own such an overwhelming weight that the other investors behaviors only have a small impact: Commercial banks, which invest on a short- to medium-term basis, and Life insurance companies and pension funds, which invest on a longto very long-term basis. Thus, the curve shape is determined by the law of supply and demand on the short-term bond market on the one hand, and on the long-term bond market on the other hand. 39 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Market Segmentation Theory While life insurance companies and pension funds are structural buyers of long-term bonds, the behavior of commercial banks is more volatile. Indeed, banks prefer to lend money directly to corporations and individuals than to invest in bond securities. Their demand for short-term bonds is influenced by business conditions: During growth periods, banks sell bond securities in order to meet corporations and individuals strong demand for loans, hence the relative rise in short-term yields compared to long-term yields. During slowdown periods, corporations and individuals pay back their loans, thus increasing bank funds and leading them to invest in short-term bond securities, hence the relative fall in short-term yields compared to long-term yields. 40 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

21 Market Segmentation Theory The main limit of this theory lies in its basic assumption, i.e., the rigidity of investors behaviors. Moreover, the assumption of a market segmentation due to regulations is not acceptable in an increasingly internationalized financial environment. The market segmentation theory can be viewed as an extreme version of the pure risk premium theory, where risk premia are infinite. Although the different theories discussed above differ from each other, they do not exclude each other. On the contrary, in practice, the effects they emphasize may jointly apply to interest rates. 41 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Biased Expectations Theory The biased expectations theory is a combination of the pure expectations theory and the risk premium theory. It postulates that the term structure of interest rates reflects market expectations of future interest rates as well as permanent liquidity premia that vary over time. Thus, all curve shapes can be accounted for. Formally, 1+R(t,τ)=( (1+R(t, 1) )( 1+F a (t, t +1, 1)+L 2 )( 1+F a (t, t +2, 1)+L 3 ) ( 1+F a ) ) τ 1 (t, t + τ 1, 1) + L τ, where R(t,τ) is the current yield observed at time t with maturity τ years, F a (t, t + k, 1) is the future short-term rate with maturity 1 year, anticipated by the market at time t and starting at time t + k, k j=2 L j is the liquidity premium required by the market to invest in a bond maturing in k years, 0 < L 2 < < L τ and L 2 > L 3 L 2 > > L τ L τ / 118 William C. H. Leon MFE8812 Bond Portfolio Management

22 Illustration of Theories 43 / 118 William C. H. Leon MFE8812 Bond Portfolio Management 44 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

23 45 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Overview Bond pricing is performed by taking the discounted value of the bond cash flows, and information about discount rates is extracted from market sources. This is known as relative pricing: the price of a bond is obtained in such a way that it is consistent with prices of other bonds in a reference set. There is a direct method to fit a default-free yield curve, known as the bootstrapping method, but it somewhat lacks in robustness; thus, indirect methods are preferred. The common character of all indirect models is that they involve fitting data to a pre-specified form of the zero-coupon yield curve. In the context of interest-rate risk management, fitting the zero-coupon yield curve is generally better than fitting the discount function because it allows for a clearer interpretation of the parameters, and it usually requires a smaller number of parameters. 46 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

24 Why Derive a Zero-Coupon Yield Curve Deriving the zero-coupon yield curve is very important in practice because it enables investors at a date t: To know the discount factor curve, and consequently to price at this date t any fixed-income security delivering known cash flows in the future (e.g., afixedcouponbond). To obtain implicit curves such as the forward rate curve (with forward rates beginning at a future date T > t), the instantaneous forward rate curveandtheparyieldcurve. 47 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Methods to Derive a Zero-Coupon Yield Curve Methods to derive zero-coupon yield curves may be categorized into direct and indirect methods. Direct methods extract the zero-coupon yield curve from market data and allow the investor to recover exactly the prices of the selected bonds. If there is an abundance of zero-coupon bonds traded in the market, one can extract directly the zero-coupon yield curve from the bond prices. Under very limited circumstances, there is a simple direct method for extracting zero-coupon prices from the current fixed coupon bearing bond prices. For real-life situations, bootstrapping method is developed to derive the implied zero-coupon bond prices. 48 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

25 Methods to Derive a Zero-Coupon Yield Curve Indirect methods allow for a very efficient derivation of forward and par yield curves and are very useful for implementing rich-cheap analysis. The common character of all indirect models is that they involve fitting data to a pre-specified form of the zero-coupon yield curve. The general approach is as follow: First, we select a reference set of bonds with market prices and cash flows taken as given. Then, we postulate a specific form of the discount function or the zero-coupon rates, where the function is usually defined in a piecewise manner in order to allow different sets of parameters for different types of maturities (short, medium and long). Finally, the set of parameters is estimated as the one that best approximates given market prices. 49 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Building a General Information Database To ensure the selection of a coherent basket of bonds, we first need to generate a database gathering a list of information for each bond. About the issuer: name, country of issuance, S&P and Moodys ratings. About the issue: amount issued, amount outstanding, issue date, issue price, maturity, currency, type of coupon, coupon frequency, coupon rate, day-count basis, first coupon date, redemption value, presence of option features, such as convertibility, callability, putability, pricing source, ask yield to maturity (YTM), bid YTM, ask clean price, bid clean price, mid clean price, daily volume. 50 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

26 Building a General Information Database Using this information, the idea is to select bonds that are not concerned with the following: Option features (convertibility, callability, putability, etc.): the presence of options can make the price of the bond higher or lower, and not homogeneous with bonds that do not contain options. Pricing errors: pricing errors are typically due to an error in the input database. To detect bonds with pricing errors, draw the yield to maturity (YTM) curve and identify the outlier bonds. Illiquidity or over-liquidity: illiquid bonds must be excluded from the reference baskets, because such bonds typically imply misprices. Some bonds may be overliquid at certain periods of time because they are the cheapest-to-deliver bonds of futures contracts or simply benchmark bonds. Assessing the degree of liquidity of a given bond is a challenging task. The idea is to take into account the size of the issue, the nature of the issue (on-the-run or off-the-run), the daily traded volume and so on. 51 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Building a General Information Database For example, JP Morgan publish liquidity states in their Global Bond Index on a monthly basis. They distinguish between three liquidity levels: benchmark, active and traded. A benchmark issue is an issue recognized as a market indicator, a recent sizable new issue or reopened issue, and a current coupon issue. An active issue is an issue with significantly daily turnover, a previous benchmark issue. A traded issue is an issue with prices that change regularly, and for which two-way markets exist. On the basis of these categories, we can define a rule that depends on the size of the local market that is annualized: active bonds are used if there are enough such bonds to cover the whole maturity spectrum. But a mix of active and traded bonds is used if there are not enough active bonds. 52 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

27 A Theoretical Direct Method The direct methodology extracts zero-coupon implied prices from the coupon bond market prices. If one needs n distinct zero-coupon rates, one needs to first collect the prices of n bonds. Default-free coupon bonds (like US Treasury bonds) are usually preferred, because they provide information about the risk-free structure of interest rates. Let P t = ( ) P (1) t P (2) t... P (i) t... P (n) t be an n-dimensional vector of coupon bond prices at time t, andlet ( ) F = F (i) t j 1 i,j n be the n n matrix of the cash flows corresponding to these n bonds (assuming that different bonds have the same cash flow dates t j ). 53 / 118 William C. H. Leon MFE8812 Bond Portfolio Management A Theoretical Direct Method Let B t = ( B(t, t 1 ) B(t, t 2 )... B(t, t n ) ) be the n-dimensional vector of the zero-coupon bond prices at time t that we want to determine. In the absence of arbitrage opportunities, P (i) t = n j=1 F (i) t j B(t, t j ), for 1 i n, i.e., P t = FB t. Suppose there is no linear dependence in the cash flows of the bonds. Then F is invertible, and B t = F 1 P t. 54 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

28 A Theoretical Direct Method At date t, the annual compounded zero-coupon rate R(t, t j t) with maturity t j t is R(t, t j t) = ( ) 1 t B(t, t j ) j t 1, and its continuously compounded equivalent R c (t, t j t) is R c (t, t j t) = ln B(t, t j). t j t 55 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Exercise Annual Maturity Bond Coupon (Years) Price 1 5.0% % % % Using the above bond data, derive the zero-coupon curve until the 4-year maturity. 56 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

29 Answer 57 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Remarks Note that the prices B t we derived using the direct method are not some real market values but, rather, implied zero-coupon bond values consistent with the set of market prices P t. The direct method is fairly simple in theory and not computationally intensive. Unfortunately, finding many distinct linearly independent bonds with the same coupon dates is quasi-impossible in practice. This is why practitioners use a common approach known as the bootstrapping method. 58 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

30 The Bootstrapping Method Bootstrapping is the term for generating a zero-coupon yield curve from existing market data such as bond prices. Bootstrapping can be viewed as a repetitive double-step procedure: Firstly, we extract directly zero-coupon rates with maturity less than or equal to 1 year from corresponding zero-coupon bond prices. We use a linear or cubic interpolation to draw a continuous zero-coupon yield curve for maturities less than or equal to 1 year. Secondly, we consider the bond with nearest maturity between 1 and 2 years. The bond has two cash flows and its price is logically the discounted value of these two cash flows. We know the first discount factor needed to obtain the discounted value of the first cash flow. The unknown variable is the second discount factor. Solving a nonlinear equation we get this discount factor and the corresponding zero-coupon rate. We begin again with the same process considering the next bond with nearest maturity between 1 and 2 years. 59 / 118 William C. H. Leon MFE8812 Bond Portfolio Management The Bootstrapping Method. We use a linear or cubic interpolation to draw a continuous zero-coupon yield curve for maturities between 1 and 2 years, using the zero-coupon rates obtained from market prices. Thirdly, consider the bond with nearest maturity between 2 and 3 years and repeat the same process, and so on. The unknown discount factors are always reduced to one and solving one equation enables to determine it as well as the corresponding zero-coupon rate. 60 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

31 Example Suppose we know from market prices the following Treasury bills with maturities less than or equal to 1 year: Maturity Zero-Coupon Rate 1 Day 4.40% 1 Month 4.50% 2 Months 4.60% 3 Months 4.70% 6 Months 4.90% 9 Months 5.00% 1 Year 5.10% We can draw the continuous zero-coupon yield curve for maturities less than or equal to 1 year using a linear interpolation. 61 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Example Zero-Coupon Rate R(0, t) [in%] Maturity t [in Year] 62 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

32 Example In addition, consider the following Treasury bonds until the 3-year maturity: Maturity Annual Coupon Gross Price 1 Year & 2 Months 5.0% Year & 9 Months 6.0% Years 5.5% Years 5.0% We shall extract the zero-coupon rate in the following order: The one-year-and-two-month zero-coupon rate. The one-year-and-nine-month zero-coupon rate. The 2-year zero-coupon rate. The 3-year zero-coupon rate. 63 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Example In the absence of arbitrage, the price of the one-year-and-two-month bond is the sum of its future discounted cash flows: = 5 ( 1+4.6% ) ( 1+r ) where r is the one-year-and-two-month zero-coupon rate that we want to determine. Solving this equation we obtain r = %. 64 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

33 Example Applying the same procedure with the one-year-and-nine-month and the 2-year bonds we get: 102 = 99.5 = 6 ( 1+5% ) ( 1+r ( ) 2 + ( ) % 1+r2 ) , Solving these equations we obtain r = % and r 2 = %. Using again a linear interpolation, we draw the continuous zero-coupon yield curve for maturities between 1 and 2 years. 65 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Example Zero-Coupon Rate R(0, t) [in%] Maturity t [in Year] 66 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

34 Example To extract the 3-year zero-coupon rate, we solve the equation 97.6 = 5 ( 1+5.1% ) + 5 ( ) 2 + ( 105 ) % 1+r3 Thus, r 3 = %, and we obtain the zero-coupon curve for maturities less thanorequalto3years. Using again a linear interpolation, we draw the continuous zero-coupon yield curve for maturities between 1 and 3 years. 67 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Example Zero-Coupon Rate R(0, t) [in%] Maturity t [in Year] 68 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

35 Kinds of Interpolation Suppose that we know the 3-year and 4-year maturity zero-coupon rates and if we want to get the three-and-a-half-year maturity zero-coupon rate, we have to interpolate it. There exist many kinds of interpolation. In practice, we use: Linear Interpolation. Cubic Interpolation. 69 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Linear Interpolation Consider that we know R(0, x) andr(0, z), respectively, the x-year and the z-year zero-coupon rates, and that we need R(0, y) the y-year zero-coupon rate with x y z. Using the linear interpolation, R(0, y) isgivenbytheformula R(0, y) = z y z x R(0, x)+ y x z x R(0, z). Linear interpolations are computed using first two points to construct the first part of the curve, then two other points to construct the second part of the curve and so on, this imply that the slope of the curve is not smooth. 70 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

36 Exercise Suppose you have the following information on par bonds that pay annual coupons: Maturity Par Yield 1 Year 3.30% 2 Years 3.90% 3 Years 4.70% 5 Years 5.20% 10 Years 6.00% Draw a continuous zero-coupon yield curve for maturities less than or equal to 10 years using linear interpolation. 71 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Answer 72 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

37 Cubic Interpolation Cubic interpolation requires four zero-coupon rates, respectively, R(0, v), R(0, x), R(0, y) andr(0, z) to implement the cubic interpolation with the condition v < x < y < z. The interpolated rate R(0, w) withv w z, verifies the three-order polynomial equation R(0, w) =aw 3 + bw 2 + cw+ d where a, b, c and d satisfy the system R(0, v) =av 3 + bv 2 + cv+ d R(0, x) =ax 3 + bx 2 + cx+ d R(0, y) =ay 3 + by 2 + cy+ d R(0, z) =az 3 + bz 2 + cz+ d. 73 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Cubic Interpolation Note that a v 3 v 2 v 1 b c = x 3 x 2 x 1 y 3 y 2 y 1 d z 3 z 2 z 1 1 R(0, v) R(0, x) R(0, y) R(0, z) Cubic interpolations are computed using first four points to construct the first part of the curve, then four other points to construct the second part of the curve and so on, which may imply that the slope of the curve is not smooth. Besides, the curve may be concave on one maturity segment and convex on the other. 74 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

38 Exercise Given the following zero-coupon rates: Maturity Zero-Coupon Rate 1Year 3.0% 2 Years 5.0% 3 Years 5.5% 4 Years 6.0% Find the 2.5-year zero-coupon rate using a cubic interpolation. 75 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Answer 76 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

39 The common trait of all indirect models is that they involve fitting the data to a pre-specified form of the zero-coupon yield curve. It should be noted that, while indirect models help one to avoid some practical difficulties, they suffer from the risk of possible mis-specification, i.e. if one chooses a very bad model for the zero-coupon yield curve, then fitting the data with this model will not give a very reliable framework. The general approach is as follows. First, one selects a reference set of n default-free bonds with market prices and cash flows F s (i) for some s t. P (i) t They will be used later to estimate the zero-coupon yield curve. 77 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Next, one postulates a specific form of the discount function B(t, s) f (s t; β 1 ) or the zero-coupon function R c (t, s t) g(s t; β 2 ), where β 1 and β 2 are vectors of parameters. The function f is usually defined piecewise in order to allow different sets of parameters for different types of maturities (short, medium and long), under the form of a polynomial or exponential spline functional. The function g is often defined in such a way that the parameters are usually easily interpretable. Finally, the set of parameters β is estimated as the one that best approximates the given market prices. 78 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

40 . In other words, the one that is a solution to the optimization program β = arg min β n ( i=1 P (i) t ) 2 (i) P t where P (i) t are the theoretical prices from the model P (i) t = s t F (i) s f (s t; β) or P (i) t = s t F (i) s e (s t) g(s t;β). 79 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Parametrization of Discount Function To facilitate the discussion some of the more popular methods, namely, polynomial and exponential splines, we recap and introduce the following notations: n is the number of bonds used for the estimation of the zero-coupon yield curve. P (i) t P (i) t is the market price at date t of the i-th bond. is the theoretical price at date t of the i-th bond. ( ) The price vectors are P t = and P ) t = respectively. P (i) t T i is the maturity of the i-th bond (in years). ( P(i) t F (i) s is the cash flow of the i-th bond at time s t. B(t, s) is the discount factor (price at date t of a zero-coupon bond paying $1 at date s). Note that B(t, t) = 1 is a constraint for the minimization program. 80 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

41 Parametrization of Discount Function In the absence of arbitrage opportunities, the following must hold: P (i) t = F s (i) B(t, s). t s T i The model is expressed as where the residuals ɛ satisfy for all 1 i, j n and i j. P t = P t + ɛ E (ɛ i )=0, Var (ɛ i )=σ 2 ωi 2, Cov (ɛ i,ɛ j )=0, 81 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Parametrization of Discount Function The variance-covariance matrix of the residuals is σ 2 Ω, where ω Ω= 0 ω ωn 2 The specification of ωi 2 may be used to over-weigh or under-weigh some bonds in the minimization program. Many authors simply consider the homoscedastic case ω 2 i =1 forall1 i n, where all bonds have the same weight in the minimization program. In that case, however, the short-term end of the curve (1 day to 6 months) is fitted with only approximate precision. 82 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

42 Parametrization of Discount Function In an attempt to provide a more accurate specification, Vasicek and Fong (1982) have overweighted short bonds in the minimization program. They suggested the following specification ω 2 i = ( ) dp (i) 2 ( t D (i) t = dy (i) t P (i) t 1+y (i) t where y (i) t and D (i) t are the yield to maturity and the Macaulay duration of the i-th bond at date t, respectively. The rationale for this choice follows the intuition that the longer the maturity of a given bond, the more difficult is its price estimation. This difficulty comes from the fact that a bond with maturity shorter than 1 year is priced using a single pure discount rate. On the other hand, pricing a bond with maturity 15 years requires the use of 15 pure discount rates (30 rates if payments are semiannual). ) 2 83 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Parametrization of Discount Function If one wishes to improve the quality of the method for short-term bonds, various specifications for ωi 2 may be used, such as and ω 2 i = T 2 i ω 2 i = dp(i) t dy (i) t = D(i) t P (i) t 1+y (i) t. 84 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

43 Estimation of Discount Function Denote β as the vector of parameters for the function we need to estimate; β as the estimator of β in the absence of the constraint; β as the estimator of β in the presence of the constraint; Z as the matrix of the coefficients of the parameters for each bond, i.e., P t = Zβ. The minimization program is as follow: min β n i=1 ( P (i) t ω i P (i) t ) 2 such that B(t, t) = 1. In matrix notation, write this as C β =1. 85 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Estimation of Discount Function A direct application of the constrained generalized least squares method gives β = β + ( ) ( 1C Z Ω 1 Z C ( ) ) 1C 1( ) Z Ω 1 Z 1 C β where β = ( Z Ω 1 Z) 1Z Ω 1 P t. 86 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

44 Proof Let L(β) = 1 2 ( Pt Zβ ) Ω 1( P t Zβ ). The unconstrained minimization program min β n i=1 ( P (i) t ω i P (i) t The first order condition (FOC) for optimality is ) 2 min L(β). β L β = Z Ω 1( P t Zβ ) =0. Denote the solution of the FOC by β, i.e., Z Ω (P 1 t Z β ) =0. Then Z Ω 1 Z β = Z Ω 1 P t ( 1Z β = Z Ω Z) 1 Ω 1 P t. 87 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Proof The Lagrange function of the constrained minimization program is ( ) L (β,λ) =L(β)+λ 1 C β, where λ isthelagrangemultiplier.thefocsare L β = Z Ω 1( P t Zβ ) λ C =0 and L λ =1 C β =0. Denote the solutions of the FOCs by β and λ.then β = β ( ) 1C + Z Ω 1 Z λ 1=C β ( ) + C 1C Z Ω 1 Z λ ( λ = C ( ) ) 1C 1( ) Z Ω 1 Z 1 C β. 88 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

45 Example Suppose that the discount factors are given by the following polynomial splines B 0 (s) =d 0 + c 0 s + b 0 s 2, for 0 s 5; B(0, s) = B 5 (s) =d 1 + c 1 s + b 1 s 2, for 5 < s 10; B 10 (s) =d 2 + c 2 s + b 2 s 2, for 10 < s 20. Then d 0 c 0 b 0 d 1 β = c 1 b 1 d 2 c 2 b z 1,1 z 1,2... z 1,n. Z = z 2,1 z.. 2,1.... z n 1,n z n,1... z n,n 1 z n,n n C = / 118 William C. H. Leon MFE8812 Bond Portfolio Management Example If the first of the n bonds is a zero-coupon bond maturing 6 years from now, the discount factor is B(0, 6) = B 5 (6) = d 1 +6c 1 +36b 1, and z 1,1 z 1, z 1,n 1 z 1,n Z = z 2,1 z 2, z n 1,n z n,1 z n, z n,n 1 z n,n n 9 What is Z if the bond pays an annual coupon of 5%? 90 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

46 Answer 91 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Smoothness of Discount Function The choice of the spline function stems from the requirement of smoothness in the discount function. More precisely, we attempt to obtain B(t, s) asap 1 times differentiable function with continuous derivatives if the polynomial used for fitting it is of order p. Hence, for p = 3, the function will be continuous and will be twice differentiable. The latter condition ensures a smoothness property in the slope of the instantaneous forward rates. Henceforth, we consider date t =0anddenoteB(0, s) =B(s). 92 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

47 Polynomial Splines Polynomial splines have been introduced by McCulloch (1971, 1975). It is of great importance to carefully select the order of the polynomial. A parsimonious choice here is a polynomial spline of order 3, since a spline of order 2 generally implies a discontinuity of the second derivative. Furthermore, choosing a greater order (four or five) leads to an increase in complexity with no real justification about the continuity of the third or fourth derivative. There are two ways of modeling polynomial splines, the standard modelization and the expression in the B-spline basis. 93 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Standard Modelization It is common to consider discount function as cubic splines. For example, B 0 (s) =d 0 + c 0 s + b 0 s 2 + a 0 s 3, for 0 s 5; B(s) = B 5 (s) =d 1 + c 1 s + b 1 s 2 + a 1 s 3, for 5 s 10; B 10 (s) =d 2 + c 2 s + b 2 s 2 + a 2 s 3, for 10 s 20. In that case, the discount factor function has 12 parameters. Note that the constraints of smoothness of the function and its derivatives further require B 0 (0) = 1, B [i] 0 (5) = B[i] 5 (5), B [i] 5 (10) = B[i] 10 (10), where B [i] ( ) isthei-th derivative of the function B ( ), for i =0, 1, / 118 William C. H. Leon MFE8812 Bond Portfolio Management

48 Standard Modelization Using the first constraint, we may reduce the number of independent parameters by 1: B 0 (0) = 1 = d 0 =1. Using the next three constraints, we may further reduce the number of independent parameters by 3: B 0 (5) = B 5 (5) = δ +5ζ +5 2 β +5 3 α =0, B [1] 0 (5) = B[1] 5 (5) = 5ζ +2 5β α =0, B [2] 0 (5) = B[2] 5 (5) = 2β +6 5α =0, where δ = d 1 d 0, ζ = c 1 c 0, β = b 1 b 0 and α = a 1 a 0. Solve these equations to obtain β = 3 5α, ζ =3 5 2 α and δ = 5 3 α. 95 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Standard Modelization Substitute these values into B 5 (s) toget B 5 (s) =(δ +1)+(ζ + c 0 )s +(β + b 0 )s 2 +(α + a 0 )s 3 =1+c 0 s + b 0 s 2 + a 0 s 3 + α ( s 3 5s 2 + s 3) }{{} (s 5) 3 =1+c 0 s + b 0 s 2 + a 0 ( s 3 (s 5) 3) + a 1 (s 5) 3. Similarly, using the last three constraints, we may further reduce the number of independent parameters by 3 and show that B 10 (s) =1+c 0 s + b 0 s 2 + a 0 ( s 3 (s 5) 3) + a 1 ( (s 5) 3 (s 10) 3) + a 2 (s 10) / 118 William C. H. Leon MFE8812 Bond Portfolio Management

49 Standard Modelization Using the 7 constraints, we may reduce the number of independent parameters to 5: B 0 (s) =1+c 0 s + b 0 s 2 + a 0 s 3, for 0 s 5; B(s) = B 5 (s) =1+c 0 s + b 0 s 2 ( + a 0 s 3 (s 5) 3) + a 1 (s 5) 3, for 5 s 10; B 10 (s) =1+c 0 s + b 0 s 2 ( + a 0 s 3 (s 5) 3) ( + a 1 (s 5) 3 (s 10) 3) + a 2 (s 10) 3, for 10 s / 118 William C. H. Leon MFE8812 Bond Portfolio Management Standard Modelization This may be written as B(s) =1+c 0 s+b 0 s 2 + a 0 s 3 +(a 1 a 0 ) ( s 5 ) 3 + +(a 2 a 1 ) ( s 10 ) 3 +, for 0 s 20, where ( s τ ) 3 + = ( max(0, s τ) )3. The basis of the space under consideration is {1, s, s 2, s 3, (s 5) 3 +, (s 10) 3 +}, and its dimension is equal to six. We may also consider another basis for this space to model the discount factor function, e.g., the B-spline basis and the exponential spline basis. 98 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

50 Exponential Splines Exponential splines have been introduced by Vasicek and Fong (1982) as a potentially efficient model for the zero-coupon yield curve. Consider the following third-order exponential spline B 0 (s) =d 0 + c 0 e us + b 0 e 2u s + a 0 e 3u s, for 0 s 5; B(s) = B 5 (s) =d 1 + c 1 e us + b 1 e 2u s + a 1 e 3u s, for 5 s 10; B 10 (s) =d 2 + c 2 e us + b 2 e 2u s + a 2 e 3u s, for 10 s 20. The discount factor function B has 13 parameters. The 6 smoothness constraints as in the case of polynomial splines apply, and they may be used to express B as a function of 7 parameters (i.e., d 0, c 0, b 0, a 0, a 1, a 2 and u) under the further constraint that B 0 (0) = / 118 William C. H. Leon MFE8812 Bond Portfolio Management Exponential Splines Using exponential splines requires one additional step in the estimation procedure. Note that B(s) isalsoafunctionofu. Even though u can be regarded as just one of the parameters, it also has an interesting economic interpretation. Vasicek and Fong (1982) have shown that u = lim s f (0, s). Thus, u can be considered as the instantaneous forward rate for an infinite horizon. The optimization procedure needs to be altered for exponential splines in the following way: Fix u = f (0, ) initially to be some reasonable value. Compute β using this u. Then optimize over u (each value of u corresponds to a different value of β ). 100 / 118 William C. H. Leon MFE8812 Bond Portfolio Management

51 Some Technical Issues Optimal number of splines. The question of how many splines should be used is very rarely addressed. The greater the number of splines, the better the fit in terms of the variance of the residuals, but the curve is less smooth. As the number of splines increases, the curve tends to be much more sensitive to abnormal data. The difference in prices between the actual and the theoretical prices becomes increasingly small. It is then impossible to detect bonds with abnormal prices that one should take out of the reference set. The lower the number of splines, the smoother the curve. On the other hand, when a small perturbation is introduced, errors become significant, which tends to imply that the quality of fit is not great. 101 / 118 William C. H. Leon MFE8812 Bond Portfolio Management Some Technical Issues Optimal choice of pasting (or knot) points. Changing the choice of pasting points may have significant impact on the forward rate curve level (Deacon and Derry (1994) show that it can change by up to 13 basis points). One may strive to have the same numbers of bonds for each spline, which implicitly defines where the pasting points should be. One may also argue that the choice should reflect a natural segmentation of the bond market. 102 / 118 William C. H. Leon MFE8812 Bond Portfolio Management