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50 results in Bond Pricing and Yield Curve Modeling

Page 3 of 3

  • 15 - Links with an Economics-Based Description of Rates

  • from Part III - The Conditions of No-Arbitrage
  • Riccardo Rebonato
  • Book:
    Bond Pricing and Yield Curve Modeling
    Published online:
    25 May 2018
    Print publication:
    07 June 2018, pp 241-260

  • Summary

    Financial economists like ketchup economists […] are concerned with the interrelationships between the prices of different financial assets. They ignore what seems to many to be the more important question of what determines the overall level of prices.

    – Summers, 1985, On Economics and Finance

    THE PURPOSE OF THIS CHAPTER

    If this book were a novel, the risk premium would clearly be one of its central characters, and this chapter would be a pause in the fast-paced plot, where the author takes a breather and regales the reader with illuminating flashbacks about the childhood of the hero. So, this chapter will not change the plot of the book, but will help the reader understand why its main character (the risk premium) behaves the way it does. More specifically, recall that Chapter 11 gave a very ‘constructive’ derivation of the no-arbitrage conditions, which were obtained by building a very tangible risk-less portfolio, invoking no-arbitrage, and reasoning from there what the ability to do so entailed. Chapter 12 then employed a much more abstract (and general) treatment that starts from the ‘virtual’ prices that the representative investor would assign to some special hypothetical securities. Finally, we looked at the condition of no-arbitrage with real (inflation-linked) bonds. In all of this, we have strictly engaged in ketchup economics (see the quote by Summers, 1985 that opens the chapter). Now we want to take a brief look at what the price of ketchup should be, not which price two bottles of ketchup should fetch, given the price of one (or three) bottles.

    What does this mean in practice? When it comes to risk premia, we can try to establish what consistency relationships they should obey (this is the one-bottle/two-bottles bit); we can describe them empirically; or we can try to explain them – which means, we can try to explain why there should be a risk premium at all, whether it should be positive or negative, how large it should be, to which quantities it may be related, and why so.

  • Part VII - What the Models Tell Us

  • Riccardo Rebonato
  • Book:
    Bond Pricing and Yield Curve Modeling
    Published online:
    25 May 2018
    Print publication:
    07 June 2018, pp 557-558

  • Summary

    Models are to be used, not believed.

    – Henri Theil, Principles of Econometrics

    Our mistake is not that we do not our theories too seriously, but that we do not take them seriously enough.

    – Steven Weinberg

    In this part of the book we look in detail at what a number of significant Gaussian affine models tell us about expectations, risk premia and the behaviour of the yield curve in general. Some of the models are well known; some are introduced here for the first time.

    Our intention is not to present an encyclopaedia of models. Rather, we want to discuss the strengths and the shortcomings of a handful of models, each of which is a salient representative of an interesting class. In a way, each model we present in this part of the book tries to answer the questions left open by the Vasicek approach that we have examined in detail in Parts I to II of this book.

    As explained in the Introduction, we do not consider affine models as (just) mathematical entities endowed with formal properties (axioms and theorems). Rather, we look at them as restricted-view windows on the financial reality we would like to understand. Therefore the calibration of a model, and its congruence with the empirical evidence about how yields behave, take centre stage in this part of the book.

    Broadly speaking, we will look at two classes of models – which, for lack of a better description, we will refer to as the ‘financial-story’ and the ‘statistically motivated’ models. With the first class, one starts from a simplified picture of how the world works (a ‘financial story’), and one translates as best one can this picture into a set of relationships among the chosen variables: a simple model. One then makes the model speak (ie, one extracts predictions from the model), and compares the model predictions with reality. If the match is acceptable, this is corroborating evidence that the financial story we started with was not entirely fanciful, and we can use the model for predictions about quantities that we would like to know, but cannot observe – such as for instance, the risk premium.

  • 2 - Definitions, Notation and a Few Mathematical Results

  • from Part I - The Foundations
  • Riccardo Rebonato
  • Book:
    Bond Pricing and Yield Curve Modeling
    Published online:
    25 May 2018
    Print publication:
    07 June 2018, pp 24-48

  • Summary

    Boswell: “He says plain things in a formal and abstract way, to be sure; but his method is good: for to have clear notions about any subject, we must have recourse to analytick arrangement.”

    Johnson: “Sir, it is what every body does, whether they will or not. But sometimes things can be made darker by a definition. I see a cow. I define her, Animal quadrupes ruminans. But a goat ruminates, and a cow may have no horns. Cow is plainer.”

    Boswell (1791) – The Life of Samuel Johnson

    THE PURPOSE OF THIS CHAPTER

    This is not going to be a chapter that will set pulses racing, but it is a useful one. In it we define the financial quantities that we are going to use in the book, we fix our vector notation and we try to clarify possible sources of notational confusion.

    We also present (in the appendices at the end of the chapter) some elementary mathematical results that we shall use throughout the book.

    THE BUILDING BLOCKS

    Arbitrage

    For the purposes of this book, an arbitrage is a strategy that is too good to be true – and, therefore, one that in an efficient market does not exist.

    A bit more precisely, one says that a strategy is an arbitrage if it costs nothing to set up, if it never produces a negative payoff, and if in at least one possible state of the world it produces a strictly positive payoff.

    It pays to look at the definition carefully. First we require that the strategy should cost nothing to set up. Very often, when we buy an asset and sell another to crystallize the value of an arbitrage strategy, the resulting portfolio will not be self-financing – which means that the proceeds from the sale of the asset we are shorting will not exactly match the cost of the asset we want to buy. This means that we must either borrow or lend some money. Since the interest on the borrowing or lending up to the end of the investment horizon of the strategy must be locked in for the strategy to be an arbitrage (why?), this means that we must buy or sell a discount bond. The price of this discount bond is the ‘balancing item’ that makes the initial set-up cost of the portfolio equal to zero.

  • Part III - The Conditions of No-Arbitrage

  • Riccardo Rebonato
  • Book:
    Bond Pricing and Yield Curve Modeling
    Published online:
    25 May 2018
    Print publication:
    07 June 2018, pp 183-184

  • Summary

    [Ketchup economists] have shown that two quart bottles of ketchup invariably sell for twice as much as one quart bottle except for deviations traceable to transaction costs.

    L. Summers, On Economics and Finance, 1985

    In this part of the book we look at the conditions of no arbitrage from several different angles.

    We start in Chapter 11 with a treatment in a discrete-time, two-state economy. This treatment requires exceedingly little mathematical knowledge. The generality of the result is limited but, hopefully, this is more than compensated by the clarity of intuition that the treatment affords.

    In the following chapter we move to a more traditional derivation of the no-arbitrage conditions, based on the construction of a risk-less portfolio. As we do this, we will be a small step away from deriving the Partial Differential Equation that bond prices (and, indeed, all interest-rate sensitive securities) must obey to prevent arbitrage, but we will not cross that particular bridge until Chapter 16. I prefer not to deal with the Partial Differential Equation at this stage in order to keep the focus strictly on the no-arbitrage conditions, which lead to, but do not require, the derivation of the fundamental Partial Differential Equation. As always, we will eclectically jump from a discrete-time to a continuous-time treatment in chase of the simplest and most transparent way to present the derivations.

    In Chapter 13 we then look at no-arbitrage from a more abstract point of view, by introducing the concept of the stochastic discount factor. This treatment will give a deeper and more general understanding of no-arbitrage; it will enable the reader to understand the contemporary literature on term-structure modelling; and it will allow the weaving together of concepts introduced in different contexts, or in different treatments of the no-arbitrage conditions. For instance, we shall see that the Sharpe Ratio, the market price of risk, the volatility of the stochastic discount factor (and, for those reader who know some stochastic calculus, the volatility of the change-of-measure-related Radon– Nikodim derivative) are all in essence one and the same thing.

  • 3 - Links among Models, Monetary Policy and the Macroeconomy

  • from Part I - The Foundations
  • Riccardo Rebonato
  • Book:
    Bond Pricing and Yield Curve Modeling
    Published online:
    25 May 2018
    Print publication:
    07 June 2018, pp 49-62

  • Summary

    THE PURPOSE OF THIS CHAPTER

    In our treatment we do not provide a macroeconomic or monetary-economics foundation of yield curve modelling. In this sense, our approach is therefore ‘reduced-form’: we acknowledge that deeper drivers (say, expected inflation, or the output gap) than the variables we use (say, the short rate, or the slope of the yield curve) affect the yield curve; but, for a variety of reasons, in our modelling we choose to make use of the latter, less fundamental, variables instead of the fundamental ones.

    Why would we want to do so? Perhaps because they are more easily observable (estimating the slope of the yield curve requires a Bloomberg terminal; assessing the output gap is a tad more difficult). Perhaps because they synthetically embed a lot of information about the more fundamental variables (much as the temperature of a gas ‘contains’ information about the velocities of its molecules). Perhaps because they give rise to a simpler analytical treatment (we will show, for instance, that bond prices are given by expectations of a function of the path of the short rate). More generally, whenever one uses a reduced-form approach, one has a rough idea of the link between the fundamental variables and the phenomenon at hand, but one despairs of one's ability to connect all the modelling dots from the former to the latter. One therefore takes a number of judicious shortcuts through the toughest hairpin turns, hoping that one will still land in the right direction.

    Even if we embrace a high-level, reduced-form approach it is still very useful to have at least an approximate picture of the links between the reduced-form variables that we use and the more fundamental variables that drive the yield curve. Providing this link is the task undertaken in this chapter. As we do so, we also present the first extension of the simple Vasicek model that we will encounter in Chapter 10.

  • 1 - What This Book Is About

  • from Part I - The Foundations
  • Riccardo Rebonato
  • Book:
    Bond Pricing and Yield Curve Modeling
    Published online:
    25 May 2018
    Print publication:
    07 June 2018, pp 3-23

  • Summary

    It is not my intention to detain the reader by expatiating on the variety, or the importance of the subject, which I have undertaken to treat; since the merit of the choice would serve to render the weakness of the execution still more apparent, and still less excusable. But […] it will perhaps be expected that I should explain, in a few words, the nature and limits of my general plan.

    Edward Gibbon, The Decline and Fall of the Roman Empire

    MY GOAL IN WRITING THIS BOOK

    In this book I intend to look at yield-curve modelling from a ‘structural’ perspective. I use the adjective structural in a very specific sense, to refer to those models which are created with the goal of explaining (as opposed to describing) the yield curve. What does ‘explaining’ mean? In the context of this book, I mean accounting for the observed yields by combining the expectations investors form about future rates (and, more generally, the economy) and the compensation they require to bear the risk inherent with holding default-free bonds. (As we shall see later, there is a third ‘building block’, ie, convexity.)

    This provides one level of explanation, but one could go deeper. So, for instance, the degree of compensation investors require in order to bear ‘interestrate risk’ could be derived (‘explained’) in more fundamental terms from the strategy undertaken by a rational, risk-averse investor who is faced with a set of investment opportunities and wants to maximize her utility from consumption in amultiperiod economy. I will sketch with a broad brush the main lines of this fundamental derivation, but will not pursue this line of argument in great detail. The compensation exacted by investors for bearing market risk (the ‘market price of risk’) will instead be empirically related (say, via regressions) either to combinations of past and present bond prices and yields, or to past history and present values of macroeconomic variables.

    Another way to look at what I try to do in this book is to say that I describe the market price of risk in order to explain the yield curve. If one took a more ‘fundamental’ approach, one could try to explain the market price of risk as well, but would still have to describe something more basic, say, the utility function.

  • 34 - Generalizations: The Adrian–Crump–Moench Model

  • from Part VII - What the Models Tell Us
  • Riccardo Rebonato
  • Book:
    Bond Pricing and Yield Curve Modeling
    Published online:
    25 May 2018
    Print publication:
    07 June 2018, pp 663-687

  • Summary

    THE PURPOSE OF THIS CHAPTER

    In this chapter we generalize the treatment in Chapters 32 and 33. As the reader will recall, in those chapters we played match makers, trying to unite the statistical and the no-arbitrage research strands when the state variables were chosen to be (proxies of) the yield principal components.

    The generalization presented in this chapter follows the treatment in Adrian, Crump and Moench (2013), who

  • choose some linear combination of yields as state variables;

  • impose that bonds should be priced as expectations of the state-pricedeflator (doing so ensures absence of arbitrage);

  • enforce the affine assumption for the yields and the market price of risk;

  • express the excess returns in terms of the affinely constrained stateprice deflator;

  • recast the resulting equations in such a form that they lend themselves to an econometric estimation of the parameters of the process for these state variables; and

  • establish a connection between the parameters of the statistical models and the ‘yield coefficients’ of the underlying exponentially affine model.

    • The heart of this procedure is point 5 (the econometric estimation). By carrying out this estimation, the authors are able to determine the state-dependent market price of risk, which constitutes the key ‘structural’ bridge between the real-world and risk-neutral measures in their explanatory view of yield curve dynamics.

    We report and comment on the estimates of the term premia in the closing sections of the chapter, and we revisit the issue of under what conditions expected excess returns are the same as term premia. Finally, we discuss what happens when they are not.

    We stress that, besides being of intrinsic interest, the model by Adrian et al. (2013) can also be seen as a representative instance of similar modelling approaches, such as the one in Ang and Piazzesi (2003).

    THE STRATEGY BEHIND THE ADRIAN–CRUMP–MOENCH MODEL

    One of the most important features of the approach is by Adrian, Crump and Moench (2013) is that a ‘traditional’ affine model has demurely receded in the background.

  • 35 - An Affine, Stochastic-Market-Price-of-Risk Model

  • from Part VII - What the Models Tell Us
  • Riccardo Rebonato
  • Book:
    Bond Pricing and Yield Curve Modeling
    Published online:
    25 May 2018
    Print publication:
    07 June 2018, pp 688-713

  • Summary

    A well-known scientist (some say it was Bertrand Russell) once gave a public lecture on astronomy. He described how the earth orbits around the sun and how the sun, in turn, orbits around the center of a vast collection of stars called our galaxy. At the end of the lecture, a little old lady at the back of the room got up and said: “What you have told us is rubbish. The world is really a flat plate supported on the back of a giant tortoise.” The scientist gave a superior smile before replying, “What is the tortoise standing on?” “You're very clever, young man, very clever,” said the old lady. “But it's turtles all the way down!”

    Stephen Hawking, A Brief History of Time, 1988

    THE PURPOSE OF THIS CHAPTER

    If one uses a structural affine model to describe the yield curve, one has to establish an affine relationship between the state variables and the market price of risk. In non-trivial cases (ie, if the affine relationship is not just a constant), the resulting behaviour for the market price of risk is stochastic – because the state variables are stochastic. However, if the values of the state variables are known (in statistical parlance, if one conditions on a particular set of values for the state variables), there can be one and only one possible value for the market price of risk.

    So, for instance, if one of the state variables is the slope of the yield curve, and, on the basis of our regression studies, we impose a deterministic (affine) relationship between market price of risk and the yield curve slope, then the instantaneous expected excess return will always be exactly the same whenever the slope has a certain value.

    In this chapter we break the deterministic link between the market price of risk and the return-predicting factor. We do so by introducing a model that enforces the regularities uncovered by the empirical investigation on average, but not at each point in time. We argue that this will result in a more realistic decomposition of the observed market yields into an expectation and a riskpremiumcomponent.

Page 3 of 3