Bond Pricing and Yield Curve Modeling

A Structural Approach

Search within full text

Summary

If all we ever did was wind and unwind the tangled threads of learning without ever getting any further – then what an unhappy fate we would have!

Herder

The men of experiment are like the ant, they only collect and use; the reasoners resemble spiders, who make cobwebs out of their own substance. But the bee takes a middle course: it gathers its material from the flowers of the garden and of the field, but transforms and digests it by a power of its own. Not unlike this is the true business of philosophy.

Sir Francis Bacon, Novum Organon

Computers are useless. They only give us answers.

Picasso

WHAT HAVE WE LEARNT?

The Road Followed

By the end of a book, the writer is pretty tired, and the reader rarely feels much perkier. Therefore I will try to keep these concluding remarks short, sweet and to the point.

We have looked at affine yield curve modelling from a structural perspective. We started from a really beautiful and simple model (the Vasicek model), that we liberally employed to build our intuition about the workings of more complex affine models. We examined carefully up to which point this entry-level approach could take us in our structural understanding.

Alas, when we looked at the recent empirical information about excess returns and term premia, we concluded that, for all its elegance and beauty, we had to graft a pretty substantial extension onto the Vasicek structure. Above all, we decided that for a model to be worth its predictive salt it would have to have a non-constant market price of risk. More precisely, as a bare minimum, the market price of risk would have to be state-dependent, and would have to capture the dependence of the expected excess returns on the slope of the yield curve (and perhaps on more exotic return-predicting factors).

We therefore gathered the tools to analyze, and perhaps even build, new models that could incorporate this evidence. We clambered with greater determination than elegance over the analytical and notational difficulties of the more modern approaches, and we compared their predictions about term premia and rate expectations with what has been found empirically in the last 10-or-so years. What did we find?

Summary

THE PURPOSE OF THIS CHAPTER

In this chapter we look at the risks to which bonds are exposed. Risk comes from not knowing for certain how things will pan out. Therefore saying that we are going to look at the risks to which bonds are exposed is the same as saying that we are going to consider those stochastic drivers that affect the prices of bonds.

The price of bonds is also affected by non-stochastic quantities, such as the residual time tomaturity, or their coupons.However, these deterministic quantities are perfectly known, and investors demand no ‘uncertainty compensation’. We are interested in the risks to which a bond is exposed, because it is for risk, and for risk only, that investors can ask compensation.

This chapter is therefore important for two different but related reasons. The first is rather obvious: if we want to pursue a structural approach to yield curve modelling, a clear understanding of the nature of the risks to which bonds are exposed is essential. For instance, if we speak of risk premia, it is important to understand what risk investors get compensated for taking, and why the bearing of certain risks should command a greater or smaller compensation than the bearing of others.

The second reason for looking at the ultimate sources of risk in some detail is a bit subtler.We may or we may not choose as state variables the risk factors. It may seem reasonable to do so, but sometimes considerations such as analytical tractability, or availability of empirical knowledge, will point in the direction of state variables other than the ‘ultimate’ risk factors that drive the yield curve. Using these ‘convenient coordinates’ is fine, as long as we understand clearly which risk factors ‘really’ drive the yield curve, and the nature of the ‘mapping’ from these ‘true’ risks to our chosen state variables.

We shall see in this chapter that the risks to which a (perfectly liquid and creditworthy) bond is exposed are inflation and real-rate risk. In order to understand the compensation that investors can expect for bearing these risks, we give a first ‘informal’ presentation in Section 4.3 of risk-neutral and real-world expectations.

Summary

I have not failed. I have just found 10,000 ways that won't work.

– Thomas A. Edison

I refuse to recognize that there are impossibilities. I cannot discover that anyone knows enough about anything on this earth definitely to say what is and what is not possible.

– Henry Ford

THE PURPOSE OF THIS CHAPTER

The ‘value of convexity’ (ie, the difference between the market yields and the yields that would prevail in the absence of convexity) is substantial, as Figure 21.1 clearly shows. Yet one finds in the literature relatively little discussion of the fair value of convexity.

Of course, speaking of fairness makes reference to a modelling approach, and, probably herein lies the problem: in studies of this type, one always tests a joint hypothesis, that the convexity is correctly priced and the model correctly specified. Given the great uncertainty in model specification, any possible rejection must therefore always include a caveat of particularly lurid health warnings.

It is for this reason that we adopt in this chapter a quasi-model-agnostic approach. By this we mean that we do place ourselves in an affine-modelling framework, but we make our results as independent as possible from the specifics of any affine model (such as the number of factors, or the nature of the state variables – latent, specified, yield-curve-based, macroeconomic, etc). We

only require that some affine model should exist, capable of recovering themarket yield curve and yield covariance matrix with the precision required by our study. We show in Chapter 33 that at least one such model exists, and we then rely in our reasoning on a simple equivalence result that we state and briefly prove in Section 2.

We put our quasi-model-agnostic methodology to the test by examining whether it is possible to systematically make money by means of a trading strategy based on immunizing against the exposure to ‘level’ and ‘slope’ risk, on estimating the yield volatility over the next time interval, and on comparing this estimate with the volatility ‘implied’ by the curvature in the yield curve. On purpose, we do not employ a sophisticated volatility estimate, because we want to ascertain whether easily available information about future volatility is correctly embedded in the shape of the yield curve.

Summary

If the attractiveness of an economic hypothesis is measured by the number of papers which statistically reject it, the expectations theory of the term structure is a knockout.

Froot (1989)

THE PURPOSE OF THIS CHAPTER

In this chapter we define precisely the local expectation hypothesis and excess returns, and we present several convenient expressions for this latter quantity. We do so both for nominal and for real returns.We show how the existence (or otherwise) of systematic excess returns from the strategy of ‘investing long and funding short’ is linked to the validity of the (local) expectation hypothesis.

THE (LOCAL) EXPECTATION HYPOTHESIS

As Sangvinatsos (2008) points out in his helpful review paper, the expectation hypothesis has spawned an immense literature, as the tongue-in-cheek quote by Froot that opens the chapter indicates. This is not surprising, because the expectation hypothesis relates to the evolution of yields, and is of direct relevance for the investment decisions of firms, for monetary policy, for portfolio allocation, and, less directly but not less importantly, for derivatives pricing and hedging. Above all, as Sangvinatsos (2008) says, the expectation hypothesis matters for our economic understanding.

Given its importance, researchers have begun looking into its validity as early as the late 1930s (see, eg, Macaulay (1938), Hicks (1939) and Lutz (1940)). Not surprisingly, the definition of the hypothesis provided in these early days (when Arbitrage Theory and stochastic discount factors were decades in the making) bears little apparent resemblance to the ‘modern’ definition. As Sangvinatsos (2008) says, ‘[t]hese [early versions of the hypothesis] were developed as a need of understanding the returns and yields on long versus short-term bonds, and the time series movements of the term structure. Later, researchers developed theoretical models that give rise to some of the hypothesized equations associated with the expectations hypothesis’.

Despite the apparent differences, the underlying concept (and, more importantly, the underlying economic concern) can be expressed in terms of the Local (or Pure) Expectation Hypothesis, and the Weak Expectation Hypothesis. Confusion is created because some authors speak of the Expectation Hypothesis with any qualification when they refer to the weak version, and other authors when they mean the local variety.

Summary

“Right now it's only a notion, but I think I can get money to make it into a concept, and later turn it into an idea.”

A movie producer at a Hollywood party in Woody Allen's Annie Hall

THE PURPOSE OF THIS CHAPTER

In this chapter we put some flesh around the theoretical bones of Chapter 10. More precisely, earlier in the book I have argued that convexity is intrinsically different from risk premia (whose value can be realized with essentially static strategies). I made the point that the value of convexity can be harvested only by engaging in a very active hedging strategy, akin to the practice of gamma trading in the derivatives domain. In this chapter, I provide an expression for the value of convexity and discuss the actual hedging strategies that must be established in order to capture the value of convexity.

I do so first by looking at the stylized (and not very realistic) case of the Vasicek model, and then by employing more general and realistic affine models. The more general treatment presented in Sections 20.4 and to 20.5 requires some of the formalism and of the vector notation that is introduced in Chapter 18. The reader who is not familiar with vector notation may want to go back to Chapter 18 and refresh her memory.

By the end of the chapter we will know what the value of convexity should be, if the model of our choice is correct, and if it has been properly calibrated. These are big if s. We will therefore explore in the next chapter to what extent we can obtain (quasi-) model-independent results about convexity.

BREAK-EVEN VOLATILITY – THE VASICEK SETTING

A sketch of the map of the road ahead is as follows. We start by placing ourselves in a Vasicek world. We are going to use two bonds to create a risk-less portfolio – a portfolio, which, as we know, qua riskless can only earn the riskless rate. Since, as we also know, working with bond prices is unpleasant, we swiftly change coordinates and work in terms of their yields. This is handy, but conceptually irrelevant.

Summary

THE PURPOSE OF THIS CHAPTER

The discussion about risk factors presented in the previous chapter may have seemed a bit abstract. In this chapter we therefore put some empirical flesh on the theoretical bones.We do so by looking in some detail at the US yield curve after the Great Recession, and in the summer of 2013 in particular. The reason for doing so is that what happened in this period gives a clear illustration of ‘risk factors in action’.

In Chapter 4 we argued that expectations about inflation and real rates, and the attending risk factors, are in principle all embedded in the observed bond yields. Can we say something more precise? For instance, are investors currently reaping a greater compensation for bearing inflation risk, or real-rate risk? Has the ratio of the inflation-to-real-rate risk compensation changed over time?

To answer these questions we examine the joint evidence coming from recent changes in prices of nominal and real Treasury bonds. We will draw the conclusion that, in the twenty-first century, real rates and real-rate risk premia have been the main drivers of yield curve changes. (This may not have been the case in periods such as the 1970s when inflation was a major concern, but, unfortunately, we do not have real-bond data for that period to test this hypothesis.)

EXPECTATIONS AND RISK PREMIA DURING AN IMPORTANT MARKET PERIOD

An Account of What Happened

Economics and finance are plagued by the fact that controlled and repeatable experiments are hardly, if ever, possible. Fortunately, sometimes market conditions and external events occur in combinations not too dissimilar from what a keen experimentalist would have set up if she had been allowed to play God. If we want to understand the interplay between expectations and risk premia (both for real and for nominal rates), the summer of 2013 provided one of these rare ‘real-life laboratory experiments’.

To understand why, let's begin by looking at Figure 5.1, which shows the 10- year, 5-year and 2-year Treasury yields. The first (rather obvious) observation is that, as a response to monetary actions in the wake of the Great Recession of 2007–2009, all these nominal Treasury yields came down (bonds became more expensive) very significantly from late 2009 until early summer of 2013.

Summary

A common mistake that people make when trying to design something completely foolproof is to underestimate the ingenuity of complete fools.

– Douglas Adams, Mostly Harmless

THE PURPOSE OF THIS CHAPTER

In this chapter we tease out some of the consequences of the dry definitions provided in the previous chapter. In particular, we

Summary

THE PURPOSE OF THIS CHAPTER

This is a short but important chapter. It brings together several strands of empirical and theoretical evidence to answer two questions: first, what type of variables are required to predict future yields or excess returns; and, second, whether affine models are in principle up to this predictive task. The first is an empirical/econometric question, the second a theoretical/modelling one.

We bring together evidence and arguments developed in the previous chapters of this part of the book, and broaden considerably the discussion by including several recent contributions to the debate.

WHAT IS THE SPANNING PROBLEM?

This chapter is about the so-called spanning problem. It is therefore important to be clear what we are talking about.

Informally speaking, a set of variables, ﹛x﹜, ‘span’ another set of variables, ﹛y﹜, if the variables in ﹛y﹜ can be fully accounted for (explained, ‘spanned’) by ﹛x﹜. What does this mean?

Suppose that we are interested in explaining the cross-sectional changes in a set of market yields ﹛y﹜. Let's call this our ‘target set’. Let's take a few yields, ﹛x﹜, as our explanatory variables. We want to see whether we can explain the yield changes ﹛y﹜ by means of the few selected yields, ﹛x﹜. So, for instance, we know that to predict excess returns all we have to predict are changes in yields. We also know that the level of the yield curve has hardly any predictive power for yield changes. So, the time series of any one single yield will not even begin to span the observed yield changes.

Suppose then that we take two yields. With these we can already create a yield-curve slope of sorts. As we well know, we can now explain a fair bit of excess return.

We saw in Part VI that the second principal component has a better predicting factor than a simple-minded difference between a long and a short yield. And we also saw that recent research has suggested that other combinations of yields may have even better predicting power. (See Chapters 26 and 27.) Creating these more powerful return-predicting factors requires more ﹛x﹜ variables. So, more yields explain (span) the yield changes better.

Summary

A basic rule of [research]: if the universe hands you a hard problem, try to solve an easier one instead, and hope the simpler version is close enough to the original problem that the universe doesn't object.

Ellenberg (2014), How Not To Be Wrong

My dad once gave me the following advice: “If you have a tough question that you can't answer, first tackle a simple question that you can't answer.”

Tegmark (2015), Our Mathematical Universe

THE PURPOSE OF THIS CHAPTER

In this chapter we are going to put the cart before the horse.Out of metaphor,we are going to introduce without any proofs the simplest affine model (theVasicek model), and show its qualitative behaviour. Methodically minded readers may prefer to go through Chapters 12 to 16 first, and revisit this chapter once the theoretical foundations are firmly in place. Either way, this chapter should not be skipped.

The reason for this unseemly haste in presentation is that the Vasicek model is unsurpassed when it comes to building one's intuition about how affine models work. Whenever we look at more complex models, and our intuition begins to falter, translating the problem in Vasicek-like term (asking the question: ‘What would Vasicek do?’) is almost invariably illuminating.We therefore take a first look in this chapter at how the Vasicek model juggles our three fundamental building blocks – expectations, risk premia and convexity – to determine shape of the yield curve.

More generally, keeping the results of this chapter in mind will put some flesh around the dry bones of the theoretical treatment to follow – even if the particular Vasicek flesh may later have to be flayed away to be replaced with the more muscular structure of the later-generation affine models.

THE VASICEK MODEL

In the Vasicek (1977) model, the full yield curve is driven by a single state variable (a single factor): the short rate. It may seem that, by making this assumption, we are wagging the yield curve dog from its short-end tail.

Summary

In this part of the book we look again at convexity. We want to understand if the value of convexity is fairly reflected in the shape of the yield curve. This means that we must first estimate what convexity ‘should be’ and then compare our theoretical estimates with the market's.

We carry out this comparison from three different angles.

The first (Chapter 20) is model driven, in the sense that we look at the output from affine models to gauge the value of convexity.

The second (dealt with in Chapter 21) is (almost) model agnostic, and looks at what the shape of the yield curve (its ‘carry’ and its ‘roll-down’) tells us about what the value of convexity should be.

The third (Chapter 22) is empirical: given what we have learnt about what convexity should be, we explore whether any discrepancies between the theoretical and the observed curvature of the yield curve can be effectively exploited.

As we shall see, extracting the value of convexity boils down to crystallizing the difference between what yield volatilities should be in theory and what they will be in practice. The infinitely wise words of Yogi Berra's that provide the opening quote to Chapter 22 act as a healthy reminder of the width of the chasm that, more often than not, separates theory and reality.

Summary

In this part of the book we look at whether any excess returns can be reaped by bearing ‘duration’ risk – or, for that matter, any other yield curve–related risk. If the price of a risky bond reflects a compensation (a premium) over the actuarial expectation of its payoff, then investing in a long bond while financing the position by borrowing at the risk-less short rate should, in the long run, earn a positive compensation – an ‘excess return’. Is this true? If it is, what does the magnitude compensation depend on? What risks are investors paid to bear? Are there no-peek-ahead variables that can tell us when it is a good idea to engage in the buy-long–fund-short strategy?

These are the questions we look at in this part of the book. They have obvious practical relevance. Their answers can also tell us something very important about such fundamental issues as the (local) expectation hypothesis, (ie, the statement that forward rates are unbiased predictors of future rates – and Figure 23.1 explains what we mean when we say that a predictor is biased or unbiased), term premia, whether expectations of long-term inflation are ‘well-anchored’, and the market price of risk in general.

The empirical estimation of excess returns is also extremely important for a structural modelling of the yield curve, because excess returns are directly related to the market price of risk, and the market price of risk is the bridge between the real-world and the risk-neutral description of rates. Structural models are the (sometimes oddly shaped) vessels into which the empirical information about excess returns can be poured.

So, Chapter 23 defines precisely the topic at hand, and introduces the standard notation that will be encountered in the literature. It does so by looking at nominal and real returns. Chapter 24 derives some important implications of the definitions provided in Chapter 23. In Chapter 25 we present our own empirical results on excess results for nominal and real bonds. Chapters 26 and 27 review the recent literature in the area. Next, Chapter 28 looks at different possible explanations of why the return-predicting factor(s) that we identify – such as the slope – should predict at all. Finally, in Chapter 29 we revisit an important topic touched upon in Chapter 27, namely the number and nature of the variables needed to ‘span’ (account for) the observed excess returns.

Summary

It is a capital mistake to theorize before one has data. Insensibly one begins to twist facts to suit theories, instead of theories to suit facts.

Sir Arthur Conan Doyle, “A Scandal in Bohemia”, The Adventures of Sherlock Holmes

The experiments are so sophisticated these days that they can match the theory almost perfectly.

A former colleague at the high-flux research nuclear reactor at the Institut Laue Langevin (Grenoble), commenting on the power of the local experimental facilities.

THE PURPOSE OF THIS CHAPTER

We have presented in the previous chapter the results of some basic regressions to explain excess returns. We looked at nominal and real excess returns. What we found is broadly in line with earlier work by Fama and Bliss (1987) and Campbell and Shiller (1991) (discussed in Section 26.2), but seems to be at odds with the more recent findings. To understand the origin of these differences, in this chapter we look in considerable detail at the results obtained by Cochrane and Piazzesi (2005). We do so because their work has become the standard against which subsequent regression studies (such as those by Cieslak and Povala (2010a, 2010b), Ludvigson and Ng (2009), and Radwanski (2010) examined in the next chapter) have pitted themselves. Once these literature results have been presented, we will try in Chapter 28 to reconcile, to the extent possible, the traditional, slope-based, results (we include ours in the traditional class) with the new findings.

We devote a lot of time to understanding the empirical findings well, because we think that a solid anchor with empirical reality is essential, lest we get carried away by the beauty of the models and confuse what the world should look like (if the models were true) with how the world actually behaves. Hence the first quote.

Furthermore, newer-generation affine models are endowed with such richness and flexibility that they can now perform tricks and stunts totally beyond the capabilities of the early-generation models. This is good and bad, as richer models must be kept in check by correspondingly richer and more ‘constraining’ empirical evidence. Hence the second quote.

Bond Pricing and Yield Curve Modeling

A Structural Approach

Refine listing

Refine listing

Actions for selected content:

50 results in Bond Pricing and Yield Curve Modeling

36 - Conclusions

Summary

4 - Bonds: Their Risks and Their Compensations

Summary

21 - A Model-Independent Approach to Valuing Convexity

Summary

Part I - The Foundations

14 - No-Arbitrage Conditions for Real Bonds

Index

23 - Excess Returns: Setting the Scene

Summary

Symbols and Abbreviations

20 - The Value of Convexity

Summary

5 - The Risk Factors in Action

Summary

Dedication

12 - No-Arbitrage in Continuous Time

24 - Risk Premia, the Market Price of Risk and Expected Excess Returns

Summary

29 - The Spanning Problem Revisited

Summary

10 - A Preview: A First Look at the Vasicek Model

Summary

17 - First Extensions

Part V - The Value of Convexity

Summary

Part VI - Excess Returns

Summary

9 - Convexity: A First Look

26 - Excess Returns: The Recent Literature – I

Summary

A Structural Approach

Refine listing

Refine listing

Actions for selected content:

Save Search

50 results in Bond Pricing and Yield Curve Modeling

Summary

Summary

Summary

Summary

Summary

Summary

Summary

Summary

Summary

Summary

Summary

Summary