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9054 results in Finance and accountancy

Page 129 of 453

  • 23 - Excess Returns: Setting the Scene

  • from Part VI - Excess Returns
  • Riccardo Rebonato
  • Book:
    Bond Pricing and Yield Curve Modeling
    Published online:
    25 May 2018
    Print publication:
    07 June 2018, pp 415-430

  • 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.

  • 20 - The Value of Convexity

  • from Part V - The Value of Convexity
  • Riccardo Rebonato
  • Book:
    Bond Pricing and Yield Curve Modeling
    Published online:
    25 May 2018
    Print publication:
    07 June 2018, pp 351-370

  • 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.

  • 5 - The Risk Factors in Action

  • 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 81-97

  • 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.

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

  • from Part VI - Excess Returns
  • Riccardo Rebonato
  • Book:
    Bond Pricing and Yield Curve Modeling
    Published online:
    25 May 2018
    Print publication:
    07 June 2018, pp 431-448

  • 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

  • prove that if ‘forward come true’ then the invest-long–fund-short strategy will, on average, reap no profits;

  • derive an expression for the difference between the market yields and the ‘actuarially expected’ yields;

  • spell out the precise link between excess returns, the market price of risk and risk premia;

  • discuss again why looking at the ‘yields-do-not-move’ assumption is interesting and important; and

  • revisit the distinction between the real-world and the subjective probabilities (this is a topic to which we will return in greater detail in Chapter 28).

    • What do we mean by ‘actuarially expected yields’? We mean that we take the expectation over (minus) the exponential of the paths of the short rate in the real-world measure. This gives us the ‘virtual’ bond prices that investors would demand if they did not care for risk. From these ‘virtual’ actuarial bond prices (ie, the bond prices that would prevail if investors did not care about risk) we derive ‘virtual’ actuarial yields, which we can compare with the market (Q-measure) yields.

    Understanding of all this will put us in good stead to discuss the literature that we present in the remaining chapters of this part of the book.

    DECOMPOSING AND INTERPRETING THE APPROXIMATE EXCESS RETURNS

    The ‘Carry’ Description

    If we discretize the results obtained at the end of the previous chapter we see that the realized (approximate) excess return can be decomposed as follows:

    Looking at the last line, we find two familiar terms.

    The first term is the change between the time-t n-year yield and the value of the (n - 1)-year yield at time-(t + 1), all multiplied by the (n - 1) duration. It is the roll-down term.

  • 29 - The Spanning Problem Revisited

  • from Part VI - Excess Returns
  • Riccardo Rebonato
  • Book:
    Bond Pricing and Yield Curve Modeling
    Published online:
    25 May 2018
    Print publication:
    07 June 2018, pp 547-556

  • 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.

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

  • from Part II - The Building Blocks: A First Look
  • Riccardo Rebonato
  • Book:
    Bond Pricing and Yield Curve Modeling
    Published online:
    25 May 2018
    Print publication:
    07 June 2018, pp 160-182

  • 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.

  • Part V - The Value of Convexity

  • Riccardo Rebonato
  • Book:
    Bond Pricing and Yield Curve Modeling
    Published online:
    25 May 2018
    Print publication:
    07 June 2018, pp 349-350

  • 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.

  • Part VI - Excess Returns

  • Riccardo Rebonato
  • Book:
    Bond Pricing and Yield Curve Modeling
    Published online:
    25 May 2018
    Print publication:
    07 June 2018, pp 413-414

  • 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.

  • 26 - Excess Returns: The Recent Literature – I

  • from Part VI - Excess Returns
  • Riccardo Rebonato
  • Book:
    Bond Pricing and Yield Curve Modeling
    Published online:
    25 May 2018
    Print publication:
    07 June 2018, pp 473-496

  • 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.

  • 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.

Page 129 of 453