NEW MEASURES FOR PERFORMANCE EVALUATION Alexander ...

NEW MEASURES FOR PERFORMANCE EVALUATION Alexander Cherny∗ ∗

Dilip Madan∗∗

Department of Probability Theory

Faculty of Mechanics and Mathematics Moscow State University 119992 Moscow Russia E-mail: [email protected] ∗∗

Robert H. Smith School of Business Van Munching Hall University of Maryland College Park, MD. 20742, USA E-mail: [email protected]

1

NEW MEASURES FOR PERFORMANCE EVALUATION Abstract This paper characterizes performance measures satisfying a set of proposed axioms. We develop four new measures consistent with the axioms and show that they improve on the economic properties of the Sharpe Ratio and the Gain-Loss Ratio. In our treatment, the performance measures, or the indices of acceptability, are linked to positive expectations resulting from a stressed sampling of the cash flow distribution. Theoretically, it is shown that the level of acceptability varies directly with the amount of stress tolerated. In an empirical application, the performance measures are applied to cash flows generated by writing options on the SPX and the FTSE. This exercise reveals that acceptability levels are U -shaped in the strike direction.

An earlier version of this article was entitled “On measuring the degree of market efficiency”. We are grateful to two anonymous referees for a very careful reading of the manuscript and a number of valuable remarks and suggestions that enabled us to improve the quality of the paper. We thank the participants of the Global Derivatives Conference Paris, Second AMaMeF Conference, Stanford-Tsukuba Workshop, Third Annual Meeting of CARISMA London, and Hedge Fund Replication Forum New York, seminar participants at the universities of Columbia, Cornell, Florida, New York, North Carolina, Maryland, Princeton, and Stanford, as well as our colleagues from Bloomberg and Derivatives Technologies Foundation Amsterdam for helpful comments and discussions. Special thanks are to Gurdip Bakshi for very valuable comments and suggestions.

2

Introduction This paper develops a new collection of performance measures by first formulating a set of axioms that such measures should satisfy. Measures satisfying these axioms are termed indices of acceptability, and our representation result provides a complete characterization of such indices. The index of acceptability is a nonnegative real number, and associated with each level of the index is a collection of terminal cash flows seen as random variables that are acceptable at this level. In a multiperiod context all trading strategies are taken to be self-financed accessing a terminal cash flow at zero cost. There is an underlying event space and the terminal cash flow associated with an investment or trading strategy is seen as a random variable in some probability space. In our introductory discussions, we fix the event space to be finite, so that the space of random variables is finite-dimensional. For any level of acceptability the nonnegative random variables accessed at zero cost represent arbitrages and they are acceptable at all levels. The cash flows acceptable at a given level therefore contain the nonnegative random variables or the positive orthant in the space of random variables. More generally the cash flows acceptable at a level are, as a consequence of the axioms, some convex cone larger than the positive orthant. This means that convex linear combinations of random variables acceptable at a level are also acceptable at the same level, as are positive scalar multiples. Our sets of acceptable cash flows are therefore in the spirit of Artzner, Delbaen, Eber, and Heath (1999) and Carr, Geman, and Madan (2000). We show that associated with every cone of acceptability is a supporting set of pricing kernels. Cash flows are acceptable at a level only if they have a positive expectation under every pricing kernel in the associated supporting set. Furthermore, the higher the level of acceptability, the larger is the supporting set of pricing kernels and correspondingly, the smaller is the cone of acceptability. As the level of acceptability tends to infinity, the cone of acceptability contracts to the nonnegative random variables or arbitrages. The widest possible cone of acceptability is attained when the supporting set of pricing kernels has only one element. In this case we have for acceptability the half space of random variables with a positive expectation under the single supporting pricing kernel. A byproduct of defining the levels of acceptability lies in measuring the efficiency of markets. We propose that an economy that merely eliminates arbitrages has a zero level of efficiency. In somewhat more efficient economies one would expect that trades with very high levels of acceptability are also eliminated. In doing so, we are taking a specific view 3

of efficiency, namely that of the exclusion of good trading opportunities, and recognize that there are other dimensions to the concept of efficiency, for example, informational efficiency. There is a large empirical literature testing market efficiency via the validity of an asset pricing model (see, for example, Fama and French (1992), (1996), Campbell, Lo, MacKinlay, and Whitelaw (1997)). The desirable trades are then seen as those with a high alpha or positive excess return. There is here a positive excess return from the perspective of a single pricing kernel and from the viewpoint of acceptability considered here, this is weak support for a good trade or an acceptable cash flow as the cone of acceptability is now among the widest possible cones. The results of this paper should be of interest to those seeking good trading opportunities and using for this purpose either alphas or Sharpe Ratios. From yet another perspective (Jacod and Shiryaev (1998)) we see that market efficiency in the sense of arbitrage exclusion occurs just if zero is a possible weighted average outcome for the set of possible asset price moves and some weights. When zero is such an average outcome, one may use the associated weights to construct a measure under which we have zero excess returns. Describing this measure may be arduous. In this sense, the mere exclusion of arbitrages is a rather weak hypothesis that is satisfied in all liquid markets when all trades are exposed to a positive loss probability. The key question is then not that of assessing arbitrage exclusion, but rather that of understanding better what other non-arbitrage trades are also excluded in markets with a higher efficiency level. In order to formalize operational and tractable indices of acceptability, we restrict attention to defining acceptability in terms of the probability law or distribution function of the cash flow. Such cones of acceptability are termed law invariant as all random variables with the same probability law have the same level of acceptability. It turns out that a random cash flow outcome will have a high acceptability level if its distribution function withstands high levels of stress or equivalently a stressed sampling has a positive expectation. The level of acceptability is then proportional to the level of stress. An illustrative example is provided by constructing the expectation of the minimum of n independent draws from the distribution of the cash flow to establish, by the positivity of such a stressed expectation, the cash flow acceptability at level n. The paper provides a number of other ways to stress a distribution function and it describes the associated supporting pricing kernels. The first law invariant measure that one is accustomed to consider in connection with 4

good trades is the Sharpe Ratio (Sharpe (1964)).1 But as pointed out by Bernardo and Ledoit (2000), the Sharpe Ratio measure does not really respect arbitrage and can be zero for an arbitrage when we have a positive cash flow with a finite mean and an infinite variance. This led Bernardo and Ledoit (2000) to propose the Gain-Loss Ratio as a measure of performance. This ratio goes to infinity for arbitrages, and we show later that it is an index of acceptability, in that it is consistent with the axioms we introduce here for such a measure. A limitation of the Gain-Loss Ratio is that it treats small losses and large losses symmetrically. To rectify this situation we consider the tilt coefficient that essentially computes the highest level of risk aversion for exponential utility such that the marginal utility weighted mean with exponential marginal utility is still positive. Our motivation for incorporating such an exaggeration of losses relative to a simultaneous deflation of gains is rooted in economic principles advocating weightings proportional to marginal utility. Our analysis however reveals that the tilt coefficient does not yield a convex set of acceptable cash flows. We thus recognize that the tilt coefficient can be regarded as a tentative approach to using risk aversion as a measure of acceptability, but it fails to satisfy the axioms for an index of acceptability introduced here. The indices of acceptability proposed here may also be differentiated from more classical performance measures that are typically constructed as ratios of reward to risk (see, e.g., Biglova, Ortobelli, Rachev, and Stoyanov (2004) and Rockafellar and Uryasev (2002)). In contrast, our indices may be viewed as analogs of the tilt coefficient that are consistent with the axiomatic structure for acceptable cash flows introduced in Artzner, Delbaen, Eber, and Heath (1999) and further developed here. Our measures are thereby more closely related to the intuitions embedded in classical economics. We provide four new explicit examples of law invariant indices of acceptability: AIM IN , AIM AX, AIM AXM IN , and AIM IN M AX (“AI” stands for “acceptability index”) based on four different types of stressed sampling. • For AIM IN one constructs a stressed sample on forming the expectation of the minimum of several draws from the cash flow distribution. • For AIM AX one constructs a distribution from which one draws numerous times and takes the maximum to get the cash flow distribution being evaluated. • AIM IN M AX and AIM AXM IN combine the above procedures.

5

For each of these measures we describe the set of supporting pricing kernels. The last two measures share the property with classical state-price densities of having densities that tend to infinity or zero as losses tend to negative infinity or gains tend to infinity. We report the results of an empirical evaluation of various performance measures, including the classical ones and the new ones, for simple strategies of writing call and put European options on the SP X and the F T SE, holding them to maturity, and paying out the required cash flows. It is observed that the level of acceptability is U -shaped in the strike direction. These results are broadly consistent with similar observations in the literature, for example, Jackwerth and Rubinstein (1996). The outline of the paper is as follows. Section 1 presents the axioms for the indices of acceptability as well as some additional desirable properties. The characterization theorem is presented in Section 2. Section 3 considers examples of popular measures as well as new measures motivated by our axioms. The main task of this section is to compare a number of examples to determine performance measures with the best economic properties. The results of our empirical study are presented in Section 4. Section 5 concludes. All the proofs are given in the Appendix.

1

Axioms for Measures of Performance

We propose in this section a set of axioms that a measure of trading performance should satisfy. For this purpose the financial outcomes of trading are modeled by zero-cost terminal cash flows seen as random variables on a probability space (Ω, F , P ). Even though the underlying market structure may involve dynamic trading in liquid assets over a number of periods, we take all trades to be financed to a terminal date with intermediate returns likewise banked to the same date. We define our performance measures on this space of random variables, following in this regard Artzner, Delbaen, Eber, and Heath (1999). Noticeably we exclude arbitrary lotteries that combine random variables into hypothetical gambles that may not be available as the outcomes of realistic trading strategies. With a view to avoiding technicalities associated with the finiteness of moments, we restrict attention to the class of bounded random variables given by L∞ = L∞ (Ω, F , P ). By a measure of performance we will mean a map α from L∞ to the extended positive half-line [0, ∞]. For a random variable X ∈ L∞ meaning the financed or banked terminal cash flow from a trading strategy (thus, we assume the presence of the risk-free rate), α(X) measures the performance or quality of X. 6

The acceptability indices will be designed to convey an understanding of the set of potential personalized pricing kernels that see the trade as one of positive marginal value. We shall ensure that higher levels of an index are associated with a larger pool of consenting kernels. In this connection we observe that many situations in investment allocation and risk management involve a representative action where the decision maker is not acting on personal account. In such circumstances trades that enhance a specific choice criterion like a single expected utility can be problematic. The subjective probability distribution may be at odds with wider opinion, for example. The difficulty arises if and when the trade is subsequently called into question and it then has to be defended before these contrary views. From such a perspective an arbitrage is a safe transaction as it has the consent of all parties. For these reasons we connect high acceptability levels with an ever increasing collection of pricing kernels consenting to the trade. It is precisely these considerations that motivate our detailed study of the structure of supporting kernels, which will be performed in Section 2. We shall consider eight properties that a measure of performance might satisfy, and they are discussed in turn in short subsections of this section. The first four of these properties will define what we term to be an acceptability index. The remaining properties are additional ones enabling us to make further comparisons between acceptability indices.

1.1

Quasi-Concavity

For an performance measure α, it is natural to define the set Ax of trades acceptable at a level x as those with performance above x: Ax = {X : α(X) ≥ x} ,

x ∈ R+ .

The quasi-concavity property requires these sets to be convex. Combined with the scale invariance property described below, this means that Ax are convex cones. Acceptable opportunities have already been characterized in Artzner, Delbaen, Eber, and Heath (1999) and Carr, Geman, and Madan (2001) as convex cones or sets containing the positive orthant. We now merely introduce the idea of levels of acceptability indexed by the real number x, but keep the basic property of convexity for all the levels. Thus, the quasi-concavity condition requires the function α to satisfy the property if α(X) ≥ x and α(Y ) ≥ x, then α(λX + (1 − λ)Y ) ≥ x for any λ ∈ [0, 1].

7

1.2

Monotonicity

A basic property of acceptability is the general preference for more over less, or formally monotonicity. Technically this is the condition that if X is acceptable at a level and Y dominates X as a random variable, then Y is acceptable at the same level. The monotonicity property requires this condition to be satisfied for all the levels, i.e. if X ≤ Y a.s., then α(X) ≤ α(Y ).

1.3

Scale Invariance

There is some debate about whether sets of acceptability should be convex cones and hence scale invariant or whether they should just be convex sets. The axiomatization in Artzner, Delbaen, Eber, and Heath (1999) opted for a conic structure, while Carr, Geman, and Madan (2001), F¨ollmer and Schied (2002), and Frittelli and Rosazza Gianin (2002) argued for convex sets. When designing or evaluating trades, we may focus attention on two separate considerations involved, the direction of the trade and its size or scale. We note that a number of issues affect the scale of trades and include the level of personal risk aversion, the depth of the market, and the resulting impact of the trade on the terms of trade, or the wealth or borrowing ability of the individual trading. Some of these considerations are market based, while others are personalized. The direction of trade is presumably a relatively more objective consideration with the size then left to more personal matters. Additionally many of the examples of early measures of performance like the Sharpe Ratio or the Gain-Loss Ratio that we seek to improve, are scale invariant. Scale invariance requires that the level of acceptability of X does not change under scaling, i.e. α(λX) = α(X) for λ > 0. This formulation is also in keeping with the motivating intuition behind the performance measures. A performance measure is not a choice criterion but merely an indicator of the size of support provided to a marginal trade in the direction X. We recognize that once we move away from marginal trades, even personalized pricing moves away from linear functionals and pricing kernels. As we focus our attention on the size and structure of the set of pricing kernels supporting the trade direction, it is natural to consider cones of acceptable cash flows with their natural emphasis on selecting directions.

8

1.4

Fatou Property

In order to obtain the characterization theorem, we will need a certain continuity property, which we will call the Fatou property. It requires that if (Xn ) is a sequence of random variables such that |Xn | ≤ 1, α(Xn ) ≥ x, and Xn converge to X in probability, then α(X) ≥ x.

1.5

Law Invariance

This property requires that when two cash flows have the same probability distribution, they should have the same level of performance. Formally we write law

if X = Y, then α(X) = α(Y ), law

where X = Y means that X and Y have the same probability distribution. As a consequence we shall be able to construct the performance level of the cash flow from just the knowledge of its distribution function. Practically this is very useful. Let us also remark that the performance measures like the Sharpe Ratio or the Gain-Loss Ratio satisfy this property. We recognize that law invariance is a strong assumption that ignores dependencies between the cash flow and personal investor situations. For example, two cash flows can have the same distribution but be very differently correlated with marginal utility leading to a natural preference of one over the other. As such, a high value of the performance measure is a first order favorable result supporting a trade, and one may wish to investigate further whether there are mitigating circumstances with respect to relevant codependencies.

1.6

Consistency with Second Order Stochastic Dominance

From the viewpoint of preference theory, it is reasonable to require that if one trade is preferred to another by any market participant, then it should have a higher performance. If the participants’ preferences are described by expected utility, then we get the property if Y second order stochastically dominates X, then α(X) ≤ α(Y ), where second order stochastic dominance means that E[U (X)] ≤ E[U (Y )] for any increasing concave function U .

9

1.7

Arbitrage Consistency

An arbitrage in our setting is a positive random variable X with P (X > 0) > 0. As arbitrages are universally acceptable, it is desirable that the level of performance for such outcomes be set at infinity, and so for arbitrage consistency we require that X ≥ 0 a.s. if and only if α(X) = ∞. We have skipped here the condition P (X > 0) > 0 because formally α(0) = ∞ for any performance measure satisfying monotonicity, scale invariance, and the Fatou property that is moreover unbounded above (indeed, by monotonicity property combined with scale invariance and unboundedness, we see that α(1) = ∞; then, by scale invariance, α(ε) = ∞ for any ε > 0; finally, by the Fatou property, α(0) = ∞).

1.8

Expectation Consistency

Arbitrage consistency deals with the high values of the index. In contrast, expectation consistency deals with its low values and requires that if E[X] < 0, then α(X) = 0; if E[X] > 0, then α(X) > 0.

We shall consider a variety of candidate measures of performance and study their properties in Section 3. We comment here on the relationship between acceptability indices as we have defined them and individual objective functions like expected utility. The latter seek to determine a preference level of an accessed cash flow. They measure how good a trade is from the perspective of a particular market participant. Expected utility is clearly not scale invariant, and so it is different from our acceptability indices. Apart from these formal distinctions, we note the differences that stem from the different objectives being pursued. Expected utility measures preference levels. In particular, it ranks in preference two positive cash flows. An acceptability index is designed to measure the extent of support for a marginal trade in the specified direction. In the presence of arbitrage consistency, two positive cash flows both have full support for trade in their respective directions, and so they have the same infinite level of acceptability. Acceptability is not a choice criterion, but it may serve as a risk management tool or a constraint in selecting among investments, especially in the contexts of a representative action. 10

2

Characterization of Performance Measures

In this section we present a number of equivalent descriptions for acceptability indices. They are naturally related to coherent risk measures, and a summary of results related to risk measures permits the presentation of a number of economically important parallels. First, we have the set of supporting kernels that are to be used in establishing acceptability at any level. This set might usefully be pared down to what are called the extreme measures that generate all the other measures in the set of supporting kernels on taking convex combinations. Finally, we have the convex cones of acceptability that contain arbitrages and help us visualize how far one has gone from arbitrage by opening up the cone towards the acceptance of certain loss situations. We therefore provide a brief overview of coherent risks in Subsection 2.1 and pass on to acceptability indices in Subsection 2.2, where we present a number of important parallels between the two subsections.

2.1

Coherent Risk Measures

The theory of acceptability indices we propose here builds on the theory of coherent risk measures and, more particularly, on the associated acceptability sets studied in Artzner, Delbaen, Eber, and Heath (1999) and Carr, Geman, and Madan (2001). According to the basic representation theorem proved by Artzner, Delbaen, Eber, and Heath (1999) for a finite Ω and by Delbaen (2002) in the general case, any coherent risk measure admits a representation of the form ρ(X) = − inf E Q [X] Q∈D

(1)

with a certain set D of probability measures absolutely continuous with respect to P . A trade X is acceptable if it has negative risk, i.e. ρ(X) ≤ 0. This occurs just if all measures in the supporting set D evaluate X at a nonnegative expectation. From the risk perspective, the measures in the set D are called generalized scenarios by Artzner, Delbaen, Eber, and Heath (1999). Focusing more on acceptability, Carr, Geman, and Madan (2001) used the term test measures. For acceptability, the candidates for such test measures should be potential pricing kernels of market participants in no-trade economies or economies with no liquid assets. Since we are not specific about the exact market structure and are not specifying the set of liquid assets available, we work with this wider collection of personalized densities available in the no-trade situation. The supporting set D defining acceptability or equivalently a coherent risk measure 11

through (1) is not unique (for example, if it is not convex, then D and its convex combinations define the same ρ). However, there obviously exists the largest such set given by D = {Q ∈ P : E Q [X] ≥ −ρ(X) ∀X ∈ L∞ }, where P denotes the set of probability measures absolutely continuous with respect to P . The set D is called the set of supporting kernels of ρ and it is of primary importance in applications of coherent risks to pricing. For example, the fundamental theorem of asset pricing provided by Carr, Geman, and Madan (2001) relates the absence of strictly acceptable opportunities to the existence of a risk-neutral measure in this set. Another important object associated with a set of kernels supporting a cone of acceptability is as follows. The set of extreme measures corresponding to a random variable X is defined as the set of supporting kernels Q, at which the minimum of expectations E Q [X] is attained. We denote it by Q∗ (X). In typical situations (see examples in Section 3), Q∗ (X) is a singleton; the corresponding measure will be again denoted by Q∗ (X), with a slight abuse of notation. The understanding of the extreme measures essentially synthesizes the set of supporting kernels as they may all be obtained from the set of extreme measures by taking convex combinations. Another important feature of extreme measures is seen from the following relation (see Cherny (2007, Th. 2.16)): if the extreme measure Q∗ (X) exists and is unique, then, for any Y ∈ L∞ , the marginal change of risk of X in the direction Y is the negative of the expected value of Y under the measure Q∗ (X): lim ε−1 [ρ(X + εY ) − ρ(X)] = −E Q

∗ (X)

ε↓0

[Y ].

(2)

This relation suggests that Q∗ (X) might be considered as the coherent analog of the personalized pricing measure for an agent employing the risk measure ρ and possessing a portfolio producing the cash flow X. Finally, we introduce the acceptability set associated with a coherent risk measure: A = {X ∈ L∞ : ρ(X) ≤ 0}. This is the set of positions that have a positive expectation under each measure from the set of supporting kernels, i.e. the positions supported by all those measures. The acceptability set is a convex cone containing the positive orthant L∞ + . It is clear that ρ(X) = inf{m ∈ R : X + m ∈ A}, i.e. ρ(X) is the smallest amount of money that should be added to X such that the sum is acceptable (i.e. belongs to the acceptability set). 12

2.2

Acceptability Indices

Recall that an acceptability index is a map satisfying quasi-concavity, monotonicity, scale invariance, and the Fatou property. We first establish the basic representation theorem for these maps. It states that any acceptability index is linked to an increasing one-parameter family of sets of probability measures so that the value α(X) is the largest level x such that X has a positive expectation under each measure from the set corresponding to the level x. In other words, the acceptability at level x means that all measures from the corresponding family value X positively. The index α(X) is then the highest level of acceptability attained. Theorem 1 A map α : L∞ → [0, ∞] unbounded above is an acceptability index if and only if there exists a family of subsets (Dx )x∈R+ of P such that Dx ⊆ Dy for x ≤ y and n o α(X) = sup x ∈ R+ : inf E Q [X] ≥ 0 , (3) Q∈Dx

where inf ∅ = ∞ and sup ∅ = 0. The above representation shows that acceptability indices are tightly linked with coherent risk measures. Namely, each functional ρx (X) = − inf E Q [X], Q∈Dx

x ∈ R+

is a coherent risk measure. Thus, if α is an acceptability index, then it can be represented as α(X) = sup{x ∈ R+ : ρx (X) ≤ 0}

(4)

with a family (ρx )x∈R+ of coherent risks increasing in x (i.e. the map x 7→ ρx (X) is increasing for any X ∈ L∞ ). Conversely, if (ρx )x∈R+ is such a family, then the family of their sets of supporting kernels (Dx )x∈R+ is clearly increasing in x. As a conclusion, we get that α is an acceptability index if and only if there exists an increasing one-parameter family of coherent risks with the property: α(X) is the largest level x such that X is acceptable to the level-x risk measure. For acceptability indices, the analog of the set of supporting kernels is the system of supporting kernels provided by the following lemma. Proposition 2 Let α be an acceptability index. Then there exists the maximal system (Dx )x∈R+ , for which (3) is true, i.e. for any other system (Dx0 )x∈R+ satisfying (3), we have Dx0 ⊆ Dx for any x ∈ R+ . It is given by Dx = {Q ∈ P : E Q [X] ≥ 0 for any X such that α(X) > x}, 13

x ∈ R+ .

(5)

Finding the system of supporting kernels of an acceptability index is important in understanding the structural relationship between different indices. It is also useful in making judgements on the suitability of an index for a proposed application. The richness of the set of trades acceptable at a given level is closely related to the diversity of measures in the system of supporting kernels. There should be some diversity to induce a proper structure, but yet one does not wish to have so much diversity that we start entertaining completely orthogonal measures that have no common basis. The lemma below will be extensively used in Section 3 to identify systems of supporting kernels for various acceptability indices. By the L1 -closedness of a subset of P we mean the L1 -closedness of the set of its Radon-Nikodym derivatives with respect to P . Lemma 3 Let (Dx )x∈R+ be a family of convex L1 -closed subsets of P satisfying the condition Dx = ∩y>x Dy for any x ∈ R+ . Define α by (3). Then (Dx )x∈R+ is the system of supporting kernels of α. The substitute of an extreme measure for an acceptability index is the extreme system (Q∗x (X))x∈R+ , where Q∗x (X) is the set of elements of Dx (here (Dx )x∈R+ is the system of supporting kernels), at which the minimum of expectations E Q [X] is attained. In typical situations (see examples of Section 3), Q∗x (X) is a singleton; we will denote the corresponding measure again by Q∗x (X). Note that the knowledge of Q∗x (X) for all X provides us with the knowledge of the risk measure ρx . Namely, ρx (X) is the negative of the expectation of X with respect to the extreme measure: ∗

ρx (X) = −E Qx (X) [X],

x ∈ R+ .

Of particular interest is the measure Q∗ (X) = Q∗α(X) (X). Indeed, if the x 7→ ρx (Y ) is continuous in x for any Y ∈ L∞ , then, as follows from (2), EQ

∗ (X)

[Y ] > 0 =⇒ ∃δ > 0 : ∀ε ∈ (0, δ), α(X + εY ) > α(X),

EQ

∗ (X)

[Y ] < 0 =⇒ ∃δ > 0 : ∀ε ∈ (0, δ), α(X + εY ) < α(X).

In other words, for a portfolio X, a marginal trade in the direction Y increases the value of the index if and only if Y has a positive expectation under Q∗ (X). Thus, Q∗ (X) appears as the coherent analog of the personalized pricing measure for an agent employing the index α and possessing the position X. Finally, let us provide another look at acceptability indices. The acceptability system associated with an index α is defined as Ax = {X ∈ L∞ : ρx (X) ≤ 0}, 14

x ∈ R+ .

This is a family of convex cones of random variables containing the positive orthant L ∞ + and decreasing in x. The value α(X) is then the largest number x such that X belongs to the level-x acceptability set: α(X) = sup{x ∈ R+ : X ∈ Ax }. Thus, for a risk measure, all the positions are split in two classes: acceptable and not acceptable. In contrast, for an acceptability index we have a whole continuum of degrees of acceptability defined by the system (Ax )x∈R+ , and the index measures the degree of acceptability of a trade. To provide a visual illustration of the relation between coherent risks, acceptability indices, acceptability sets, and acceptability systems, we present in Figure 1 an example for the case when Ω consists of two points ω1 , ω2 . Then any random variable X is represented as a point (X(ω1 ), X(ω2 )) on the plane. The left-hand graph illustrates the acceptability cone A of a coherent risk measure ρ. The right-hand graph illustrates the flow of acceptability cones Ax of an acceptability index α. A

6

A x∗

Ax

A∞ 6

A0

X



X

-

α(X) = x∗

ρ(X)

Figure 1. (a) Acceptability cones associated with coherent risks. (b) Acceptability cones associated with acceptability indices.

To conclude this section, let us remark that Theorem 1 provides a description of indices satisfying the first four properties of the previous section. The law invariance and the second order monotonicity for acceptability indices are studied in Subsection 3.6, where it is shown that they are equivalent one to the other, and the description of corresponding indices is provided. As for the remaining two properties of the previous section, it can be S shown that an acceptability index is arbitrage consistent if and only if the closure of x Dx

coincides with P, where (Dx )x∈R+ is the system of supporting kernels; the expectation consistency for an acceptability index is equivalent to the property D0 = {P }. 15

3

Performance Measures and Acceptability Indices: Examples

We consider in this section a variety of performance measures. For each of them, we study the eight properties introduced in Section 1. Moreover, for each measure, which is an acceptability index, we identify the structure of the system of supporting kernels and the form of extreme measures. This is needed, first, for understanding the width of the set of these kernels and, second, is important for applications to pricing and hedging in incomplete markets, as the subsequent research shows. The structure of extreme measures also allows us to compare different examples based on economic principles. In the following, we will identify measures from P typically denoted by Q with their densities with respect to P typically denoted by Z.

3.1

Sharpe Ratio SR(X)

The first measure we consider is the Sharpe Ratio of the cash flow X defined by the ratio of the mean to the standard deviation σ(X):  E[X]    σ(X) SR(X) =   0

if E[X] > 0, otherwise

(we exclude the negative values since all our performance measures are positive). This measure is clearly quasi-concave, scale invariant, law invariant, and expectation consistent. It also has the Fatou property. It is well known, however, that the Sharpe Ratio does not satisfy monotonicity property and hence is not an acceptability index. To observe the violation it is sufficient to compare the tosses of fair coins with outcomes a1 , a2 and b1 , b2 , respectively, where 0 < a1 < a2 < b1 < b2 with a1 , a2 very close to each other so that the first coin has a small standard deviation that inflates its Sharpe Ratio above that of the second coin. The same example shows that the Sharpe Ratio is not consistent with second order stochastic dominance. It is not consistent with arbitrage either as was noted explicitly in an example by Bernardo and Ledoit (2000) taking a positive random variable with an infinite variance to produce a positive cash flow with a zero Sharpe Ratio. They proposed the Gain-Loss Ratio as a performance measure free of this drawback.

16

3.2

Gain-Loss Ratio GLR(X)

The Gain-Loss Ratio is defined as the ratio of the mean to the expectation of the negative tail:

 E[X]    E[X − ] GLR(X) =   0

if E[X] > 0, otherwise,

where X − = max{−X, 0}. This measure is clearly monotone, scale and law invariant, arbitrage and expectation consistent. It also satisfies the Fatou property, and we observe below that it satisfies quasi-concavity. Hence, it is an acceptability index. To observe that it satisfies quasi-concavity, suppose that GLR(X) ≥ x and GLR(Y ) ≥ x, where x > 0. We then have equivalently that E[X] ≥ xE[X − ] and E[Y ] ≥ xE[Y − ]. By the convexity of the function x− , we get   xE (λX + (1 − λ)Y )− ≤ x(λE[X − ] + (1 − λ)E[Y − ]) ≤ E[λX + (1 − λ)Y ], and so GLR(λX + (1 − λ)Y ) ≥ x. The Gain-Loss Ratio is also consistent with second order stochastic dominance. To observe this property, consider X, Y such that Y second order stochastically dominates X. Suppose GLR(X) = x > 0. Then E[X] = xE[X − ]. As the function −x− is increasing and concave, we deduce that E[Y − ] ≤ E[X − ]. Then E[Y ] ≥ E[X] = xE[X − ] ≥ xE[Y − ] or equivalently GLR(Y ) ≥ x. The system of supporting kernels and the extreme measures associated with GLR are provided by the proposition below. We denote by qλ (X) the (right) λ-quantile of X. We will say that X has a continuous distribution if its distribution function is continuous. Proposition 4 (i) The system of supporting kernels of GLR is given by Dx = {c(1 + Y ) : c ∈ R+ , 0 ≤ Y ≤ x, E[c(1 + Y )] = 1},

x ∈ R+ .

(ii) Let X have a continuous distribution and x ∈ R+ . Then the function Z λ ϕ(λ) = qλ (X) + xλqλ (X) − x qs (X)ds, λ ∈ (0, 1) 0

is strictly increasing and ϕ(0+) < E[X] < ϕ(1−). Let λ∗ = sup{λ : ϕ(λ) ≤ E[X]}. Then the extreme measure Q∗x (X) is unique and is given by 1 + xI(X ≤ qλ∗ (X)) dQ∗x (X) = . dP 1 + xλ∗ 17

Although the Gain-Loss Ratio satisfies all the properties discussed in Section 1, it has an essential drawback from the economic viewpoint. It is that the extreme measures Q ∗x have bounded densities, which means that large losses are exaggerated up to a finite level. On the other hand, it is reasonable economically if large losses were exaggerated up to infinity. Note in this respect that the personalized pricing density cU 0 (W ) of an agent using a utility (i.e. concave increasing) function U and possessing a wealth W is unbounded provided that W is unbounded below and U 0 (−∞) = +∞. Thus, the consent from the densities supporting the Gain-Loss Ratio may be questionable. These considerations lead us to consider a performance measure more closely related to such economic theoretic measure changes.

3.3

Tilt Coefficient T C(X)

The tilt coefficient of a cash flow X may be thought of as the highest level of absolute risk aversion for exponential utility such that the cash flow is still attractive to such a utility at the margin:    T C(X) = sup λ ∈ R+ : E Xe−λX ≥ 0 ,

  where sup ∅ = 0. We may view the expectation E Xe−λX as the expected marginal utility weighted cash flow with the exponential utility function and the risk aversion

coefficient λ. If T C(X) is positive, then all the risk aversions below T C(X) approve trading in the marginal direction X. It is easy to see that T C is monotone, law invariant and has the Fatou property. It is arbitrage and expectation consistent. However, it is not quasi-concave or scale invariant, and hence, it is not an acceptability index. To observe the absence of quasi-concavity, we may find a number a > 0 such that the function xe−x is convex on [a, ∞). We then choose a set A with 0 < P (A) < 1 and define random variables X, Y such that X, Y ∈ [a, ∞) a.e. on A and X 6= Y on A. On the complement of A we take X = Y = c, where c is the constant such   that E[Xe−X ] = E Y e−Y = 0. We have thus organized that T C(X) = T C(Y ) = 1.

However, when we consider the average, we have   X +Y Xe−X + Y e−Y X +Y exp − < 2 2 2
and so T C X+Y < 1. 2

a.e. on A,

With respect to scale invariance we note that T C(θX) = T C(X)/θ, and so one may 18

consider a revised measure in the form T C(X)E[X] when E[X] > 0 with the value 0 if E[X] ≤ 0. We would then preserve scale invariance but not quasi-concavity. The tilt coefficient also fails to be consistent with second order stochastic dominance. To observe this violation consider X as above with T C(X) = 1, but now define Y to equal X on the complement of A and set it equal to b = E[XIA ]/P (A) on A. Then Y   dominates X in second order, but due to the convexity of xe−x on A, we have E Y e−Y <   E Xe−X = 0. Hence, T C(Y ) < 1. We now present a counterpart of the Gain-Loss Ratio, which overcomes the problem

of having bounded extreme densities and is an acceptability index.

3.4

Coherent Risk-Adjusted Return on Capital RAROC(X)

The Coherent Risk-Adjusted Return on Capital is defined as the ratio of the mean to the risk, which is measured by a coherent risk measure ρ:  E[X]   if E[X] > 0,  ρ(X) RAROC(X) =   0 otherwise.

We use the convention RAROC(X) = +∞ if ρ(X) ≤ 0. It is assumed that P belongs to the set of supporting kernels D of ρ. In particular, this property is automatically satisfied if ρ is law invariant, as can easily be derived from the result of Kusuoka (2001). Obviously, RAROC satisfies all the properties of an acceptability index. It is law invariant if and only if ρ is law invariant. By Theorem 5 given in Subsection 3.6, these properties are further equivalent to consistency with second order stochastic dominance. Clearly, RAROC is expectation consistent. However, RAROC has an essential drawback: it is not arbitrage consistent as can be noted by considering X with P (X < 0) > 0 and ρ(X) < 0. The system of supporting kernels has the form Dx =

1 x {P } + D, 1+x 1+x

x ∈ R+ .

Indeed, for x > 0, we have RAROC(X) ≥ x ⇐⇒ E[X] ≥ −x inf E Q [X] Q∈D

1 x ⇐⇒ E[X] + inf E Q [X] ≥ 0 1+x 1 + x Q∈D ⇐⇒ inf E Q [X] ≥ 0 Q∈Dx

19

(in order to check the first equivalence, one should consider the cases inf Q∈D E Q [X] < 0 and inf Q∈D E Q [X] ≥ 0 separately; in the first case the equivalence is obvious, while in the second one we have RAROC(X) = ∞ according to our agreement, so that the inequality on the left is satisfied, and E[X] ≥ 0 according to the inclusion P ∈ D, so that the inequality on the right is also satisfied). Thus, (Dx )x∈R+ defines RAROC. By Lemma 3, it is the system of supporting kernels. If inf Q∈D E Q [X] is attained at a unique measure Q∗ (X) (this is typically the case as seen from the considerations of the next subsections), then, clearly, the set of extreme measures for X consists of a unique measure Q∗x (X) =

x 1 P+ Q∗ (X), 1+x 1+x

x ∈ R+ .

Thus, RAROC has the drawback of lacking arbitrage consistency, and we still do not have a performance measure having all the desirable properties. But now we come to the new performance measures that are introduced in this paper. The first one is the Tail VAR acceptability index. It has the same characteristics as the Gain-Loss Ratio but in fact serves as an intermediate step for introducing indices with the best properties.

3.5

TVAR Acceptability Index AIT (X)

A basic coherent risk measure is the Tail Value at Risk defined as T VARλ (X) = − inf E Q [X], Q∈Tλ

where λ is a parameter from (0, 1] and Tλ is the set of probability measures absolutely continuous with respect to P such that dQ/dP ≤ λ−1 . If X has a continuous distribution, then (see F¨ollmer and Schied (2004, Remark 4.48)) the above infimum is attained at the measure Q∗ (X) with dQ∗ (X) = λ−1 I(X ≤ qλ (X)). dP

(6)

Then T VARλ (X) = −E[X | X ≤ qλ (X)], i.e. T VAR appears as the negative of the expectation of the cash flow conditioned on the event that it falls below its λ-quantile. In particular, this motivates the term Tail Value at Risk. For more information on this risk measure, we refer to F¨ollmer and Schied (2004, Section 4.4) and Rockafellar and Uryasev (2002).

20

An important feature of T VAR is that it is not a single risk measure but rather a one-parameter family of risk measures decreasing in λ. Recalling representation (4), we may now define the TVAR acceptability index  AIT (X) = sup x ∈ R+ : T VAR

1 1+x

(X) ≤ 0 .

If X has a continuous distribution, then we can provide a more direct representation: AIT (X) = inf{λ ∈ (0, 1] : E[X | X ≤ qλ (X)] ≥ 0}


−1

− 1.

This representation is very clear intuitively: AIT (X) = λ−1 ∗ − 1, where λ∗ is the smallest value λ such that the conditional mean of X over its λ-tail is positive; the higher the quality of X is, the smaller is the value λ∗ , and the higher is the value AIT (X). The law invariance of AIT is inherited from the same property of T VAR; the latter property is obvious if X has a continuous distribution and is seen from F¨ollmer and Schied (2002, Lemma 4.46) in the general case. Consistency with second order stochastic dominance is also inherited from the same property of T VAR: T VARλ (X) ≥ T VARλ (Y ) whenever Y second order stochastically dominates X; see F¨ollmer and Schied (2004, Corollary 4.59). AIT is moreover arbitrage and expectation consistent, as follows from the relations lim T VARλ (X) = − essinf X(ω) = − sup{c ∈ R : X ≥ c a.s.}, λ↓0

ω

lim T VARλ (X) = −E[X]. λ↑1

Clearly, AIT is represented by the system Dx = {Z : 0 ≤ Z ≤ x + 1, E[Z] = 1},

x ∈ R+ ,

and, by Lemma 3, it is the system of supporting kernels. For X with a continuous distribution, the extreme measures have densities given by
dQ∗x (X) = (x + 1)I X ≤ q 1 (X) , 1+x dP

x ∈ R+ ,

as seen from (6). These measures are even more extreme than those supporting GLR. Here we ignore gains altogether employing measures that are zero for gains and are uniform with respect to the size of losses. From the perspectives of economic considerations, these are unreasonable measures and probably more so than those associated with GLR. But we are now ready to introduce our main example, which would possess all the desirable properties. 21

3.6

WVAR Acceptability Indices AIW (X)

A generalization of T VAR is the Weighted Value at Risk defined as the mixture of T VAR λ with different risk levels λ by a probability measure µ on (0, 1]: Z W VARµ (X) = T VARλ (X)µ(dλ). (0,1]

One can check that this is a coherent risk measure. Before introducing the associated acceptability index we describe this risk measure in an alternative way that readily yields the tractable examples we later construct. For this purpose we introduce the function Z yZ Ψµ (y) = 0

λ−1 µ(dλ)dz,

y ∈ [0, 1].

(z,1]

In fact, we have in µ ↔ Ψµ a one-to-one correspondence between probability measures µ on (0, 1] and concave distortions, i.e. increasing concave continuous functions Ψ : [0, 1] → [0, 1] such that Ψ(0) = 0, Ψ(1) = 1. Indeed, the differentiation of Ψµ establishes that it is monotone and concave. Changing the order of integration shows that Ψµ (1) = 1, while clearly Ψµ (0) = 0. The inverse map is given by µ(dy) = −yΨ00µ (dy), where Ψ00µ is the second derivative in the sense of distributions, i.e. it is the probability measure on (0, 1] given by Ψ00µ ((a, b]) = (Ψµ )0+ (b) − (Ψµ )0+ (a), where (Ψµ )0+ is the right-hand derivative of Ψµ with the convention (Ψµ )0+ (1) = 0. According to F¨ollmer and Schied (2004, Theorem 4.64), Z W VARµ (X) = − yd(Ψµ (FX (y))),

(7)

R

where FX is the distribution function of X. Given the importance of this result for the development here, we briefly present the required connection: Z W VARµ (X) = T VARλ (X)µ(dλ) (0,1] Z Z −1 =− λ ydFX (y)µ(dλ) (0,1] (−∞,qλ (X)) Z Z =− y λ−1 µ(dλ)dFX (y) R (FX (y),1] Z = − yd(Ψµ (FX (y))). R

Note that the right-hand side of equality (7) is just the negative of the expectation of a random variable having Ψµ (FX ) as its distribution function. The above representation of W VAR is very convenient in constructing particular representatives of this 22

class, and below we employ it to construct new risk measures M IN VAR, M AXVAR, M AXM IN VAR, and M IN M AXVAR. It is also at the core of defining the acceptability index associated with W VAR — the AIW index. Moreover, the above representation provides a straightforward formula for the numerical evaluation of W VAR. Namely, if x1 , . . . , xN are historic or Monte Carlo realizations of some financial variable and X has the corresponding empirical distribution (i.e. the uniform distribution on the set {x1 , . . . , xN }), then W VARµ (X) = −

 n − 1   n − Ψµ , x(n) Ψµ N N n=1

N X

(8)

where x(1) , . . . , x(N ) are the values x1 , . . . , xN in the increasing order. Thus, the numerical evaluation of W VARµ (X) consists in ordering the realizations of X and summing them up with weights determined by µ through the function Ψµ . According to Cherny (2006), the set of supporting kernels of W VARµ is given by Dµ = {Z : Z ≥ 0, E[Z] = 1, and E[(Z − y)+ ] ≤ Φµ (y) ∀y ∈ R+ },

(9)

where Φµ is the convex conjugate of Ψµ : Φµ (y) = sup (Ψµ (z) − yz),

y ∈ R+ .

z∈[0,1]

For X with a continuous distribution, the minimum of expectations E Q [X] over Q ∈ Dµ is attained at the unique measure Q∗ (X) given by dQ∗ (X) = Ψ0µ (FX (X)); dP

(10)

see F¨ollmer and Schied (2004, Corollary 4.74) (the derivative Ψ0µ is defined at all points, except for a countable set, but this is sufficient to determine Ψ0µ (FX (X)) as FX (X) has the uniform distribution). For more information on W VAR, we refer to Acerbi (2002), Cherny (2006), F¨ollmer and Schied (2004, Section 4.6). To provide a visual illustration of the extreme measure densities, we plot in Figure 2 the function Ψ0 ◦ FX appearing in the expression for the extreme density. The distortion Ψ in Figure 2.a corresponds to the AIM IN index introduced in Subsection 3.7; the one in Figure 2.b corresponds to the AIM AX index of Subsection 3.8. The random variable X is taken to be standard normal (although we are working on L∞ , coherent risks can be extended to a wider space). It is seen from (7) that W VARµ ≤ W VARµe if and only if Ψµ ≤ Ψµe . Keeping this

observation in mind and using (7) once more, we can define the WVAR acceptability 23

3 25 2.5 20 2 15 1.5

10 1

5

0.5

–3

–2

–1

0

2

1

3

–3

–2

–1

2

1

x

3

x

Figure 2. (a) Extreme measure densities for Ψ(x) = 1 − (1 − x) 3 . (b) Extreme measure densities for Ψ(x) = x 1/3 .

index by

n

AIW (X) = sup x ∈ R+ :

Z

yd(Ψx(FX (y))) ≥ 0 R

o

(we set sup ∅ = 0), where (Ψx )x∈R+ is a family of concave distortions on [0, 1] increasing pointwise in x. Thus, we distort the distribution function of X more and more severely and look for the largest stress level such that the expectation of X under the corresponding distortion remains positive. Let us remark that T VARλ is a particular case of W VAR with µ equal to the delta mass concentrated at the point λ, and, for this µ, we have Ψµ (y) = λ−1 y ∧ 1. Thus, AIT is a particular case of AIW with Ψx (y) = (1 + x)y ∧ 1. The numerical calculation of AIW can be performed via (8). The fact that AIW is an acceptability index follows from representation (4). It is law invariant and consistent with second order stochastic dominance, as follows from the same properties of W VAR (the second property for W VAR means that W VARµ (X) ≥ W VARµ (Y ) whenever Y second order stochastically dominates X), which, in turn, are inherited from the same properties of T VAR. It is not hard to see that AIW is arbitrage consistent if and only if Ψx (y) tends pointwise to 1 on (0, 1] as x → ∞; AIW is expectation consistent if and only if Ψx (y) tends pointwise to y as x ↓ 0. In order to identify the system of supporting kernels, introduce the right modification Ψ+ x = limε↓0 Ψx+ε and define the dual functions Φx (y) = sup (Ψ+ x (z) − yz),

x, y ∈ R+ .

z∈[0,1]

It is then clear from (9) that the system Dx = {Z : Z ≥ 0, E[Z] = 1, and E[(Z − y)+ ] ≤ Φx (y) ∀y ∈ R+ },

24

x ∈ R+

defines AIW . Obviously, Φx = limε↓0 Φx+ε , so that Dx = (Dx )x∈R+ is the system of supporting kernels.

T

ε>0

Dx+ε . By Lemma 3,

According to (10), for X with a continuous distribution, the extreme measures have densities given by dQ∗x (X) 0 = (Ψ+ x ) (FX (X)), dP

x ∈ R+ ,

where the differential is taken in y. In the theory of coherent risks, the two basic characterization results are the representation of general risk measures given by Artzner, Delbaen, Eber, and Heath (1999) and the representation of law invariant risk measures given by Kusuoka (2001). Theorem 1 is the analog of the former one for indices. The theorem below is the analog of the latter one. It shows that AIW indices serve as basic building blocks for constructing all the law invariant acceptability indices. It also establishes the equivalence between law invariance and consistency with second order stochastic dominance. Moreover, in this theorem, we provide another description of law invariance based on the notion of dilatation monotonicity, which was introduced for coherent risks by Leitner (2004) and is also motivated by the notion of factor risks introduced by Cherny and Madan (2006). We will assume that the probability space is atomless. One of equivalent descriptions of this condition is as follows: the probability space supports a random variable with a continuous distribution. Theorem 5 For an acceptability index α on an atomless probability space, the following conditions are equivalent: (a) α is law invariant; (b) α is consistent with second order stochastic dominance; (c) α is dilatation monotone, i.e. for any X ∈ L∞ and any sub-σ-field G ⊆ F , we have α(E[X | G]) ≥ α(X); (d) the system of supporting kernels (Dx )x∈R+ of α is law invariant, i.e. for any x ∈ R+ law

and any Z = Z 0 , we have Z 0 ∈ Dx whenever Z ∈ Dx ; (e) there exists a family (αγ )γ∈Γ of AIW indices such that α(X) = inf αγ (X), γ∈Γ

X ∈ L∞ .

The class AIW is quite wide. For the practical applications, one needs some representatives of this class. Below we introduce four its representatives and compare their properties. We employ the notation ρx (X) = −

Z

yd(Ψx(FX (y))), R

25

x ∈ R+

for the stream of coherent risks associated with the index.

3.7

MINVAR Acceptability Index AIM IN (X)

A particularly attractive and very intuitive concave distortion is given by the function Ψx (y) = 1 − (1 − y)x+1 ,

x ∈ R+ , y ∈ [0, 1].

(11)

For an integer x, we have ρx (X) = −E[Y ], where law

Y = min{X1 , . . . , Xx+1 } and X1 , . . . , Xx+1 are independent draws of X. We call the risk measure ρx M IN VAR and denote the associated acceptability index as AIM IN . It is then the largest number x such that the expectation of the minimum of x + 1 draws from the cash flow distribution is still positive. By working with real values of x in distortion (11), we allow for a real number of draws; it is analogous to the extension of the factorial function to the positive half-line as the Gamma function. The convex dual of the concave distortion Ψx has the form   y 1+ x1   if 0 ≤ y ≤ x + 1, 1 − y + x  x+1 Φx (y) =   0 if y ≥ x + 1.

The densities from Dx therefore have an upper bound of x + 1 on the amount by which large losses may be exaggerated. For X with a continuous distribution, the extreme measure densities are given by dQ∗x (X) = (x + 1)(1 − FX (X))x , dP

x ∈ R+ .

A potential drawback of AIM IN is that this density tends to a finite value x + 1 at negative infinity. As we shall see, the values of x attained are not very large, and hence, the maximal weight on large losses is small. Therefore, consent by such densities may not be economically relevant. This leads us to introduce another representative of the AIW class.

3.8

MAXVAR Acceptability Index AIM AX(X)

Another concave distortion is 1

Ψx (y) = y x+1 ,

x ∈ R+ , y ∈ [0, 1]. 26

For an integer x, we have ρx (X) = −E[Y ], where Y is a random variable with the property law

max{Y1 , . . . , Yx+1 } = X, Y1 , . . . , Yx+1 being independent draws of Y . We call the risk measure ρx (X) M AXVAR and denote the associated acceptability index as AIM AX. Let us remark that this class of distortions is known in the insurance literature as the proportional hazards transform; see Wang (1995). The convex dual of Ψx has the form    1 − y Φx (y) =  x1  1 x    x + 1 (x + 1)y

if y ≤

1 , x+1

if y ≥

1 . x+1

The densities from Dx allow for arbitrary large levels by which losses may be exaggerated as Φx (y) is never zero. For X with a continuous distribution, the extreme measure densities are given by x dQ∗x (X) 1 = (FX (X))− x+1 , dP x+1

x ∈ R+ .

This density tends to infinity at negative infinity but has another potential drawback: it tends to a strictly positive value 1/(x + 1) at positive infinity. This corresponds to an asymptotically linear weighting for large gains. But let us note that the classical personalized pricing densities cU 0 (W ) tend to zero for large values of W provided that W is unbounded above and U 0 (+∞) = 0. Below we give two examples of AIW indices, whose extreme densities are unbounded above and also tend to zero at positive infinity.

3.9

MAXMINVAR Acceptability Index AIM AXM IN (X)

Combining M IN VAR and M AXVAR, we consider the distortion 1

Ψx (y) = (1 − (1 − y)x+1 ) x+1 ,

x ∈ R+ , y ∈ [0, 1].

For an integer x, we have ρx (X) = −E[Y ], where Y is a random variable with the property: law

max{Y1 , . . . , Yx+1 } = min{X1 , . . . , Xx+1 }, X1 , . . . , Xx+1 being independent draws of X and Y1 , . . . , Yx+1 being independent draws of Y . We call the risk measure ρx M AXM IN VAR and denote the associated acceptability

27

index as AIM AXM IN in recognition of the fact that we construct the worst case scenario first using a M IN VAR perspective followed by a M AXVAR perspective. In this case the convex dual Φx (y) of Ψx does not have a closed form. However, it may be computed, and we present in Figure 3 its graph for a sample of x values. We observe that this dual function goes to one as y tends to zero and tends to zero as y goes to infinity. 1

0.9

0.8

0.7 x=2.5 phi(y)

0.6

0.5

x=2.0

0.4

0.3

0.2

0.1

x=1.5

0

1

2

3

4

5 y

6

7

8

9

10

Figure 3. The function Φx (y) for a sample of x values for M AXM IN VAR

For X with a continuous distribution, the extreme measure densities are given by x dQ∗x (X) = (1 − FX (X))x (1 − (1 − FX (X))x+1 )− x+1 , dP

x ∈ R+ .

In particular, this density tends to infinity at negative infinity and to zero at positive infinity.

3.10

MINMAXVAR Acceptability Index AIM IN M AX(X)

Another way to combine M IN VAR and M AXVAR is to consider 1

Ψx (y) = 1 − 1 − y x+1


x+1

,

x ∈ R+ , y ∈ [0, 1].

For an integer x, we have ρx (X) = −E[Y ], where Y is a random variable with the property: law

Y = min{Z1 , . . . , Zx+1 }, law

max{Z1 , . . . , Zx+1 } = X, Z1 , . . . , Zx+1 being independent draws of Z. We call the risk measure ρx M IN M AXVAR and denote the associated acceptability index as AIM IN M AX in recognition of the fact 28

that we construct the worst case scenario first using a M AXVAR perspective followed by a M IN VAR perspective. The convex dual in this case also lacks a closed form, and we plot in Figure 4 the computed dual for a sample of x values. 1

0.9

x=2.5

0.8 x=2.0

phi(y)

0.7

0.6

0.5

0.4

0.3

0.2

x=1.5

0

1

2

3

4

5 y

6

7

8

9

10

Figure 4. The function Φx (y) for a sample of x values for M IN M AXVAR

For X with a continuous distribution, the extreme measure densities are given by 1
x x dQ∗x (X) = 1 − FX (X) x+1 FX (X)− x+1 , dP

x ∈ R+ .

In particular, this density tends to infinity at negative infinity and to zero at positive infinity.

4

Testing Performance Measures

We evaluate the performance measures described above for some simple strategies. This is useful for a variety of purposes. First, we get an idea about the numerical magnitudes of measures and the types of values one may expect to see for them. We also develop some understanding of the relative values of the various measures as they are all computed for the same cash flow streams. Furthermore, as many of the measures represent levels of law invariant acceptability, they may also serve as pricing devices under the physical measure. For example, if we are to quote on put options on the net asset value of a fund of hedge funds and we have access to just the time series of net asset values, then we may work out the price that attains a particular level of acceptability. From such a perspective the acceptability level attained in liquid option markets is of interest. We could then price put options on the fund of hedge funds asset value to attain a level of 29

acceptability comparable to that observed in the exchange traded option markets. Let us remark that applications of acceptability indices to pricing and hedging are a subject of ongoing research. It is also instructive to have an assessment of these measures with a view to testing for good trading strategies or detecting market anomalies. As mentioned in the introduction, much of the literature focuses on high alpha strategies that evaluate a positive expected return under the single measure given by the asset pricing model in place. The set of good trades is then a half space containing the positive orthant. Were the cash flows evaluated for performance using our indices, they had to be positively evaluated by every measure in the set of supporting kernels. The cones defined by our acceptable sets are much smaller than a half space, and they all contain the positive orthant. We present in two subsections the data used and the results obtained.

4.1

Data

We focus attention on the performance measures for cash flows related to writing options, holding the position unhedged and paying out the necessary cash flow at the option maturity. One expects these cash flows to have a positive mean for the writer as the contracts represent a sale of insurance. We studied the measures for these cash flows in two major world indices, these are options on the S&P 500 index and options on the F T SE. In each case we considered 7 strike buckets and 4 maturity buckets and wrote all out of the money options that were puts for strikes below the spot and calls for strikes above the spot. The options were held to maturity and the cash flow paid out. As a result we had 28 cash flows for each strike/maturity bucket, where the number of trades in each bucket was around 4500 for the period beginning in December of 2000 till December of 2005.

4.2

Results for Cash Flows on Option Trades

For each of the series of 28 cash flows we compute eight measures of trade performance. These are the Sharpe Ratio, the tilt coefficient, the Gain-Loss Ratio, the RAROCX10, where the risk is measured by the expectation of the minimum of 10 draws from the distribution. Finally, we evaluate the four new acceptability indices AIM IN , AIM AX, AIM AXM IN , and AIM IN M AX. The results are presented in two tables, one for the SP X and the other for the F T SE. Each table contains eight subtables with 7 rows for 30

the strike buckets and 4 columns for the maturity buckets. The eight subtables display the eight performance measures. These have been replaced by zero in the table to have the results in line with the ones for acceptability indices, which are zero for all trades with negative expectation. The T C are generally the smallest followed by AIM IN M AX, AIM AXM IN , AIM AX, RAROCX10, SR, AIM IN , and GLR. For the SP X all Sharpe ratios are positive, but for the F T SE at the money option sales and put options in the .5 − .75 maturity range they have negative expected value. For the SP X we observe that AIM IN , AIM AX, AIM AXM IN , and AIM IN M AX are generally below 1 with the exception of short maturity deep out of the money calls. All the measures display a U -shaped structure with respect to strike at each maturity. The sharpness of the U -shape on the downside first decreases with maturity but then tends to rise. On the upside the U -shape flattens with respect to maturity. The upside calls generally have a sharper lift than the downside puts. For the F T SE short maturity calls do attain AIM IN above 1. The U -shape in the strike direction is maintained for all maturities. The sharpness of the U -shape with respect to maturity varies. The upside calls have a higher level of acceptability than comparable downside puts. The values of AIM IN M AX, AIM AXM IN are comparable to each other and are below the values for AIM IN , AIM AX, and this is to be expected as the measures now form worst cases in two ways inclusive of both the M IN VAR and M AXVAR strategies. The order (M IN M AX or M AXM IN ) does not seem to matter much, at least, for the cash flows associated with long option positions held to maturity.

5

Conclusion

We treat the absence of arbitrage as a zero level of efficiency and develop measures of efficiency, via the ability of trading strategies to approximate arbitrages. The degree of efficiency is measured by the level of acceptability of a cash flow and arbitrages have an infinite level of acceptability. More generally, the level of acceptability rises with the size of the associated set of measures that evaluate the trade positively. We axiomatize the notion of an acceptability index as a new measure of the degree of acceptability of cash flows. The level of acceptability of trades or investments could be of interest to agents acting on behalf of rational investors. The higher the level of acceptability, the greater is 31

the set of consenting measures and the more likely that it is viewed positively by investors who are not at hand at the decision making point. It should be mentioned that acceptability indices are not aimed at measuring preferences. Indeed, the value of an index is constant on any ray of the form (λX)λ>0 , while this is of course not the case for a preference measure. In the market, where each trade can be scaled by any number λ > 0, an acceptability index might be used to determine the direction of the optimal trade, but it tells nothing about the optimal size. In contrast, a preference ordering determines both the direction and the size of an optimal trade. Three important objects associated with the index are the associated sequence of coherent risks, the class of consenting kernels, and the flow of extreme measures. Actually, the most important of these three is the set of extreme measures as their knowledge immediately yields the associated risk measures, and also extreme measures synthesize the class of consenting measures associated with the index. Moreover, the knowledge of extreme measures allows us to compare different representatives of acceptability indices to determine ones with the best economic properties. The classical performance measures such as the Sharpe Ratio and the tilt coefficient fail to satisfy the axioms defining an acceptability index. The Gain-Loss Ratio is an acceptability index and it satisfies additional desirable properties but has an essential economic drawback: the associated extreme measures are bounded meaning that large losses are exaggerated up to a finite level. This can be overcome by introducing the Coherent Risk-Adjusted Return on Capital, but it has another drawback of lacking arbitrage consistency. However, the characterization of acceptability indices through families of coherent risk measures opens the way for constructing new indices associated with the known representatives of coherent risks. Thus, we construct the T VAR and the W VAR acceptability indices. The first one has the same drawback as the Gain-Loss Ratio of having bounded extreme densities. But the second one is a whole class of acceptability indices, which includes representatives having all the desirable properties. We consider four representatives of this class: AIM IN , AIM AX, AIM AXM IN , and AIM IN M AX. The extreme measure densities for AIM IN have a potential drawback of being bounded. Those for AIM AX also have a drawback of tending to a finite value at positive infinity, but this drawback might be questionable. As for the extreme densities of AIM AXM IN and AIM IN M AX, they exhibit the desirable behavior of tending to infinity at negative infinity and to zero at positive infinity. Needless to say, these indices also satisfy all the desirable properties outlined in Section 1 of the paper. Thus, we are led to rec32

ommend AIM AX, AIM AXM IN , and AIM IN M AX as promising new measures for performance evaluation. Computations of all measures are illustrated for simple strategies of writing options, holding them to maturity unhedged and paying out the required cash flows. The underliers used are the SP X and the F T SE. It is observed that all measures have a U -shape in the strike direction for all maturities. Furthermore, the SP X has a higher level of acceptability on the downside, while the F T SE has a higher acceptability on the upside.

33

Figure 1. (a) Acceptability cones associated with coherent risks. (b) Acceptability cones associated with acceptability indices.

Figure 2. (a) Extreme measure densities for Ψ(x) = 1 − (1 − x) 3 . (b) Extreme measure densities for Ψ(x) = x 1/3 .

Figure 3. The function Φx (y) for a sample of x values for M AXM IN VAR

Figure 4. The function Φx (y) for a sample of x values for M IN M AXVAR

34

Appendix Proof of Theorem 1. Let α be given by (3). Let us check that it satisfies the four properties of acceptability indices. If X, Y are such that α(X) ≥ x and α(Y ) ≥ x, then, for any y < x and any Q ∈ Dy , we have E Q [X] ≥ 0 and E Q [Y ] ≥ 0. Clearly, the same is true for the convex combination of X and Y , which proves the quasi-concavity. Monotonicity and scale invariance are clear. To check the Fatou property, take |Xn | ≤ 1 with α(Xn ) ≥ x such that Xn converge to X in probability. Then, for any y < x, any Q ∈ Dy , and any n, we have E Q [Xn ] ≥ 0, which obviously implies that E Q [X] ≥ 0. Hence, α(X) ≥ x. Let now α be an acceptability index unbounded above. For x ∈ R+ , consider Ax = {X ∈ L∞ : α(X) ≥ x},

(12)

ux (X) = sup{m ∈ R : X − m ∈ Ax }.

(13)

Assume that Ax is a proper subset of L∞ . In this case Ax does not contain negative constants. Indeed, if it contained a negative constant, then, by the monotonicity and the scale invariance of α, it would be the whole L∞ . Furthermore, Ax contains positive constants since α(a) = ∞ for any a > 0 due to the monotonicity and the scale invariance of α. Moreover, we have Y ∈ Ax whenever X ∈ Ax and Y ≥ X a.s. From these considerations it follows that ux takes finite values on L∞ . Let us check that ux has the properties: (i) ux (λX + (1 − λ)Y ) ≥ λux (X) + (1 − λ)ux (Y ) for λ ∈ [0, 1]; (ii) if X ≤ Y a.s., then ux (X) ≤ ux (Y ); (iii) ux (λX) = λux (X) for λ ≥ 0; (iv) if |Xn | ≤ 1 and Xn converge to X in probability, then ux (X) ≥ lim supn ux (Xn ). The first of these properties is implied by the convexity of Ax , which, in turn, is guaranteed by the quasi-concavity of α. The second property is an obvious corollary of the monotonicity of α. The third property for λ > 0 follows from the conic structure of Ax , which is ensured by the scale invariance of α. To check (iii) for λ = 0, note that α(0) = ∞, as explained in Subsection 1.7. This implies that ux (0) ≥ 0. As Ax does not contain negative constants, we have ux (0) = 0, so that (iii) is verified for λ = 0. Finally, to check the fourth property, fix v < lim supn ux (Xn ). Then, after passing on to a subsequence, we have Xn − v ∈ Ax for any n, i.e. α(Xn − v) ≥ x. By the Fatou property of α, this implies that α(X − v) ≥ x, i.e. ux (X) ≥ v. As v is arbitrary, we get ux (X) ≥ lim supn ux (Xn ). 35

Properties (i)–(iv) show that ux is the negative of a coherent risk measure. Let Dx denote its set of supporting kernels. For x such that Ax = L∞ , we take Dx = ∅, so that ux (X) = ∞ = inf Q∈Dx E Q [X]. Then ux (X) = inf Q∈Dx E Q [X] for any x ∈ R+ . As Ax decrease in x, the functionals ux also decrease in x, which means that the sets Dx increase in x. Now, it follows from the equivalence α(X) ≥ x ⇔ ux (X) ≥ 0 that (3) holds. Proof of Proposition 2. Let α be given by (3) with some system (Dx0 )x∈R+ . The family (Dx )x∈R+ is increasing in x. For any X ∈ L∞ , we have the inequality n

Q

o

α(X) ≤ sup x ∈ R+ : inf E [X] ≥ 0 . Q∈Dx

(14)

Indeed, otherwise there exists x0 such that α(X) > x0 , while the above supremum is less than x0 . But then there exists Q ∈ Dx0 such that E Q [X] < 0, which contradicts the definition of Dx0 . Furthermore, Dx0 ⊆ Dx since otherwise we can find Q ∈ Dx0 such that E Q [X] < 0 and α(X) > x, which contradicts the definition of α. Thus, we get the reverse inequality in (14) and, at the same time, the maximality of (Dx )x∈R+ . ex )x∈R+ be the system of supporting kernels of α. Suppose Proof of Lemma 3. Let (D

ex for some x. Then we can find y > x and Q0 ∈ D ex such that Dx is strictly contained in D that Q0 ∈ / Dy . By the Hahn-Banach theorem, there exists X ∈ L∞ such that E Q0 [X] < 0 < inf E Q [X]. Q∈Dx

But then α(X) ≥ y > x, which contradicts (5). Proof of Proposition 4. (i) We have inf E[ZX] ≥ 0 ⇐⇒

Z∈Dx

inf E[(1 + Ye )X] ≥ 0,

0≤Ye ≤x

x ∈ R+ .

Furthermore, it is clear that the latter infimum is attained at Ye = xI(X ≤ 0) and is equal

to E[X] − xE[X − ]. It follows that

inf E[ZX] ≥ 0 ⇐⇒ E[X] ≥ xE[X − ] ⇐⇒ GLR(X) ≥ x,

Z∈Dx

x ∈ R+ .

Thus, (Dx )x∈R+ represents GLR. By Lemma 3, it is the system of supporting kernels. (ii) Fix x ∈ R+ . For 0 < λ1 < λ2 < 1, we have ϕ(λ2 ) − ϕ(λ1 ) ≥ qλ2 (X) − qλ1 (X) + xλ2 qλ2 (X) − xλ1 qλ1 (X) − x(λ2 − λ1 )qλ2 (X) = qλ2 (X) − qλ1 (X) + xλ1 (qλ2 (X) − qλ1 (X)) > 0. 36

Furthermore, ϕ(0+) = lim qλ (X) = essinf X(ω) < E[X], ω

λ↓0

ϕ(1−) = (1 + x) lim qλ (X) − xE[X] = (1 + x) esssup X(ω) − xE[X] > E[X]. λ↑1

ω

For each λ ∈ [0, 1], the density Zλ = belongs to Dx and

1 + xI(X ≤ qλ (X)) 1 + xλ

Rλ E[X] + x 0 qs (X)ds E[Zλ X] = . 1 + xλ

The function f (λ) = E[X] + x

Z

λ

qs (X)ds,

λ ∈ [0, 1]

0

is convex. Thus, for the value a0 = sup E[Zλ X] = max{a ∈ R : a + axλ ≤ f (λ) ∀λ ∈ [0, 1]}, λ∈[0,1]

there exists a unique λ0 ∈ (0, 1) such that a0 + a0 xλ0 = f (λ0 ). It satisfies the conditions xqλ−0 (X) = f 0 (λ0 −) ≤ a0 x ≤ f 0 (λ0 +) = xqλ0 (X), where qλ− (X) denotes the left quantile. It follows that qλ−0 (X) ≤ a0 ≤ qλ0 (X), and hence, Z λ0 − − qλ0 (X) + qλ0 (X)xλ0 ≤ E[X] + x qs (X)ds ≤ qλ0 (X) + qλ0 (X)xλ0 . 0

This means that λ0 = λ∗ . As a result, we get that the maximum of expectations E[Zλ X] over λ ∈ [0, 1] is attained at the unique point λ∗ . Take now an arbitrary Z = c(1+Y ) ∈ Dx and set λ = (cx)−1 −x−1 . Then Zλ = c(1+Ye )

with Ye = xI(X ≤ qλ (X)). Let us prove that E[XY ] ≥ E[X Ye ]. As E[Y ] = E[Ye ], we can shift X in such a way that qλ (X) = 0. Then

E[X(Y − Ye )] = E[X(Y − x)I(X < 0)] + E[XY I(X > 0)] ≥ 0

and moreover the inequality is strict if Y 6= Ye . This shows that the extreme measure of X can only be of the form Zλ with some λ ∈ [0, 1]. Thus, the proposition is proved.

Proof of Theorem 5. (a)=⇒(d) It follows from the Hardy-Littlewood inequality (see F¨ollmer and Schied (2004, Theorem A.24)) that, for a given X ∈ L∞ and a given measure Q, Q

0

inf E [X ] = 0

X ∼X

Z

1

qs (X)q1−s (Q)ds, 0

37

(15)

where qs (Q) means the (right) λ-quantile of Q and the notation X 0 ∼ X means that X 0 and X have the same law. Combining this with (5), we get the law invariance of Dx . (d)=⇒(a) Another application of the Hardy-Littlewood inequality yields Z 1 0 inf E[Z X] = qs (X)q1−s (Z)ds. 0 Z ∼Z

0

From this it is clear that each ux is law invariant. As shown in the proof of Theorem 1, α(X) = sup{x ∈ R+ : ux (X) ≥ 0}, so that the law invariance holds for α as well. (a)=⇒(e) Let ux be given by (12).

According to Kusuoka’s theorem (see

Kusuoka (2001) or F¨ollmer and Schied (2004, Theorem 4.57)), for any x ∈ R+ , there exists a set M0x of probability measures on (0, 1] such that ux (X) = − sup W VARµ (X), µ∈M0x

(if ux = ∞, then M0x = ∅). Let Mx be the largest set for which this representation is true, so that Mx ⊆ My for x ≤ y. Define Γ = {(z, µ) : z ∈ R+ , µ ∈ Mz } and   y if x < z, (z,µ) Ψx (y) =  Ψµ (y) if x ≥ z.

Then

ux (X) = − sup W VARµ (X) = − µ∈Mx

sup z≤x, µ∈Mz

W VARµ (X) = inf

γ∈Γ

Z

yd(Ψγx (FX (y))), R

so that α(X) = sup{x ∈ R+ : ux (X) ≥ 0} Z n o = inf sup x ∈ R+ : yd(Ψγx (FX (y))) ≥ 0 , γ∈Γ

X ∈ L∞ .

R

(e)=⇒(b) This implication follows from the consistency of AIW with second order stochastic dominance. (b)=⇒(c) To prove this implication, it is sufficient to notice that E[X | G] second order stochastically dominates X. (c)=⇒(a) Let ux be given by (12).

Then, for any x > 0, ux has the dilata-

tion monotonicity property: for any X ∈ L∞ and any sub-σ-field G ⊆ F , we have ux (E[X | G])) ≥ ux (X). According to Cherny and Grigoriev (2006), ux is law invariant. Hence, the same is true for α(X) = sup{x ∈ R+ : ux (X) ≥ 0}.

38

References [1] Acerbi, C., 2002, “Spectral Measures of Risk: a Coherent Representation of Subjective Risk Aversion,” Journal of Banking and Finance, 26, 1505–1518. [2] Artzner, P., et al., 1999, “Coherent Measures of Risk,” Mathematical Finance, 9, 203–228. [3] Bernardo, A. and O. Ledoit, 2000, “Gain, Loss, and Asset Pricing,” Journal of Political Economy, 108, 144–172. [4] Biglova, A., et al., 2004, “Different Approaches to Risk Estimation in Portfolio Theory,” Journal of Portfolio Management, 31, 103–112. [5] Campbell, J.Y., et. al., 1997, The Econometrics of Financial Markets, Princeton University Press, Princeton, NJ. [6] Carr, P., H. Geman, and D. Madan, 2001, “Pricing and Hedging in Incomplete Markets,” Journal of Financial Economics, 62, 131–167. [7] Cherny, A., 2006, “Weighted VAR and its Properties,” Finance and Stochastics, 10, 367–393. [8] Cherny, A., 2007, “Pricing with Coherent Risk,” Theory of Probability and Its Applications, 52, 506–540. [9] Cherny, A. and P. Grigoriev, 2006, “Dilatation Monotone Risk Measures are Law Invariant,” Finance and Stochastics, 11, 291–298. [10] Cherny, A. and D. Madan, 2006, “Coherent Measurement of Factor Risks,” Working paper available at http://www.ssrn.com. [11] Cochrane, John H. and Jesus Sa´a-Requejo, 2000, “Beyond Arbitrage: ‘Good Deal’ Asset Price Bounds in Incomplete Markets,” Journal of Political Economy, 108, 79– 119. [12] Delbaen, F., 2002, “Coherent Risk Measures on General Probability Spaces,” in K. Sandmann and P. Sch¨onbucher (eds.), Advances in Finance and Stochastics: Essays in Honor of Dieter Sondermann, Springer, Berlin, 1–37.

39

[13] Fama, E.F. and K.R. French, 1992, “The Cross Section of Expected Stock Returns,” Journal of Finance, 47, 427–465. [14] Fama, E.F. and K.R. French, 1996, “Multifactor explanations of Asset Pricing Anomalies,” Journal of Finance, 51, 55–84. [15] F¨ollmer, H. and A. Schied, 2002, “Convex Measures of Risk and Trading Constraints,” Finance and Stochastics, 6, 429–447. [16] F¨ollmer, H. and A. Schied, 2004, Stochastic Finance: An Introduction in Discrete Time. 2nd Edition, Walter de Gruyter. [17] Frittelli, M. and E. Rosazza Gianin, 2002, “Putting Order in Risk Measures,” Journal of Banking and Finance, 26, 1473–1486. [18] Hansen, L. and R. Jagannathan, 1991, “Implications of Security Market Data for Models of Dynamic Economies,” Journal of Political Economy, 99, 225–262. [19] Jackwerth, J. and M. Rubinstein, 1996, “Recovering probability distributions from option prices,” Journal of Finance, 51, 1611–1631. [20] Jacod, J. and A. Shiryaev, 1998, “Local Martingales and the Fundamental Asset Pricing Theorems in the Discrete-Time Case,” Finance and Stochastics, 3, 259–273. [21] Kusuoka, S., 2001, “On Law Invariant Coherent Risk Measures,” Advances in Mathematical Economics, 3, 83–95. [22] Leitner, J., 2004, “Balayage Monotonous Risk Measures,” International Journal of Theoretical and Applied Finance, 7, 887–900. [23] Rockafellar, R.T. and S. Uryasev, 2002, “Conditional Value at Risk for General Loss Distributions,” Journal of Banking and Finance, 26, 1443–1471. [24] Sharpe, W., 1964, “Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk,” Journal of Finance, 19, 425–442. [25] Wang, S., 1995, “Insurance Pricing and Increased Limits Ratemarking by Proportional Hazards Transforms,” Insurance: Mathematics and Economics, 17, 43–54.

40

1

A high Sharpe Ratio has traditionally been taken as an indicator of a good deal (see,

for example, Hansen and Jagannathan, (1991), Cochrane and Saa-Requejo, (2000))

41

Table 1. Each subtable shows empirically estimated values of the corresponding performance measure for strategies of writing out of the money options on the SP X index, holding them up to maturity and paying out the required cash flow at maturity. Different columns of the subtable correspond to different maturities measured as a fraction of a year. Different rows of the subtable correspond to different values of the moneyness. Strike Ranges

Maturity Range 0-.25

.25-.5

.5-.75

Maturity Range .75-1.0

0-.25

.25-.5

.5-.75

.75-1.0

SR

TC

.85-.9

0.3166 0.1790 0.2136 0.4585

0.0520 0.0247 0.0345 0.1346

.9-.95

0.1970 0.1234 0.1748 0.2802

0.0266 0.0128 0.0246 0.0600

.95-1.0

0.1281 0.0741 0.0857 0.1043

0.0132 0.0050 0.0067 0.0100

1.0-1.05

0.0697 0.0203 0.0127 0.0566

0.0044 0.0004 0.0002 0.0031

1.05-1.1

0.1761 0.2732 0.2879 0.2347

0.0284 0.0683 0.0763 0.0520

1.1-1.15

0.3544 0.3423 0.4538 0.3667

0.0846 0.0939 0.1536 0.1084

1.15-1.2

0.5336 0.4849 0.4950 0.4409

0.1262 0.1393 0.1574 0.1366

GLR

RAROCX10

.85-.9

2.9789 0.8292 0.8997 2.0282

0.3328 0.1073 0.1244 0.2954

.9-.95

1.2354 0.4962 0.6635 0.9792

0.1437 0.0682 0.0961 0.1596

.95-1.0

0.6234 0.2528 0.2768 0.2921

0.0773 0.0380 0.0439 0.0551

1.0-1.05

0.2498 0.0578 0.0345 0.1444

0.0361 0.0100 0.0063 0.0302

1.05-1.1

0.5665 0.9487 1.0264 0.7847

0.1070 0.1861 0.2036 0.1612

1.1-1.15

2.1914 1.5448 2.1740 1.4472

0.2734 0.2344 0.3170 0.2389

1.15-1.2

6.8759 3.8486 2.7580 2.0353

0.7889 0.4555 0.3674 0.2935

AIMIN

AIMAX

.85-.9

0.9115 0.3461 0.4710 1.1245

0.3397 0.1667 0.2059 0.4718

.9-.95

0.4708 0.2762 0.2889 0.4581

0.2003 0.1306 0.1341 0.2304

.95-1.0

0.2508 0.0763 0.1169 0.1884

0.1169 0.0398 0.0593 0.1075

1.0-1.05

0.0644 0.0077 0.0216 0.0245

0.0348 0.0044 0.0123 0.0152

1.05-1.1

0.2461 0.3635 0.4193 0.2764

0.1823 0.2791 0.3238 0.2254

1.1-1.15

0.6849 0.4636 0.8665 0.6080

0.3699 0.3050 0.4894 0.3805

1.15-1.2

1.2659 1.1474 0.9899 0.8882

0.5485 0.5218 0.4718 0.4546

AIMAXMIN

AIMINMAX

.85-.9

0.2249 0.1076 0.1359 0.2994

0.2145 0.1049 0.1315 0.2780

.9-.95

0.1330 0.0856 0.0885 0.1458

0.1291 0.0838 0.0866 0.1404

.95-.1.0

0.0772 0.0259 0.0388 0.0669

0.0759 0.0257 0.0384 0.0657

1.0-1.05

0.0224 0.0028 0.0078 0.0093

0.0223 0.0028 0.0078 0.0093

1.05-1.1

0.1010 0.1500 0.1727 0.1193

0.0983 0.1440 0.1649 0.1155

1.1-1.15

0.2207 0.1721 0.2823 0.2170

0.2095 0.1648 0.2629 0.2050

1.15-1.2

0.3362 0.3166 0.2871 0.2726

0.3127 0.2944 0.2678 0.2546

42

Table 2. Each subtable shows empirically estimated values of the corresponding performance measure for strategies of writing out of the money options on the F T SE index, holding them up to maturity and paying out the required cash flow at maturity. Different columns of the subtable correspond to different maturities measured as a fraction of a year. Different rows of the subtable correspond to different values of the moneyness. Strike Ranges

Maturity Range 0-.25

.25-.5

.5-.75

Maturity Range .75-1.0

0-.25

.25-.5

.5-.75

.75-1.0

SR

TC

.85-.9

0.2505 0.0650 0.0000 0.0988

0.0396 0.0038 0.0000 0.0090

.9-.95

0.1103 0.0000 0.0000 0.0000

0.0100 0.0000 0.0000 0.0000

.95-1.0

0.0462 0.0000 0.0000 0.0000

0.0020 0.0000 0.0000 0.0000

1.0-1.05

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

1.05-1.1

0.5915 0.4471 0.4948 0.4469

0.2599 0.1775 0.2244 0.1822

1.1-1.15

0.6761 0.6869 0.7332 0.5958

0.2326 0.3341 0.4149 0.2851

1.15-1.2

0.5219 0.7185 1.0434 0.8054

0.1167 0.2588 0.5540 0.4188

GLR

RAROCX10

.85-.9

1.8315 0.2645 0.0000 0.2802

0.2089 0.0356 0.0000 0.0528

.9-.95

0.5638 0.0000 0.0000 0.0000

0.0686 0.0000 0.0000 0.0000

.95-1.0

0.1802 0.0000 0.0000 0.0000

0.0245 0.0000 0.0000 0.0000

1.0-1.05

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

1.05-1.1

3.2350 1.9633 2.0273 1.8022

0.4959 0.3532 0.4291 0.3623

1.1-1.15

9.1863 5.7715 4.2449 2.7973

1.0758 0.7685 0.6863 0.4782

1.15-1.2

17.4153 23.4815 16.0821 5.1221

2.0014 2.8906 2.3423 0.7614

AIMIN

AIMAX

.85-.9

0.4579 0.0269 0.0000 0.1385

0.2314 0.0157 0.0000 0.0849

.9-.95

0.1657 0.0000 0.0000 0.0000

0.0867 0.0000 0.0000 0.0000

.95-1.0

0.0526 0.0000 0.0000 0.0000

0.0289 0.0000 0.0000 0.0000

1.0-1.05

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

1.05-1.1

1.1565 0.6869 0.8379 0.7031

0.6760 0.5104 0.6533 0.5422

1.1-1.15

1.7939 1.3843 1.5108 1.0274

0.7746 0.7789 0.9108 0.6721

1.15-1.2

2.1375 2.8520 3.2708 1.8586

0.9083 1.1088 1.2811 0.9130

AIMAXMIN

AIMINMAX

.85-.9

0.1447 0.0099 0.0000 0.0517

0.1401 0.0098 0.0000 0.0509

.9-.95

0.0556 0.0000 0.0000 0.0000

0.0549 0.0000 0.0000 0.0000

.95-1.0

0.0185 0.0000 0.0000 0.0000

0.0184 0.0000 0.0000 0.0000

1.0-1.05

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

1.05-1.1

0.3770 0.2675 0.3313 0.2805

0.3429 0.2492 0.3028 0.2599

1.1-1.15

0.4557 0.4271 0.4886 0.3619

0.4120 0.3852 0.4314 0.3288

1.15-1.2

0.5162 0.6262 0.7293 0.5220

0.4655 0.5498 0.6187 0.4591

43

A

6

A x∗

Ax

6 A∞

A0

X



X

-

α(X) = x∗

ρ(X)

Figure 1. (a) Acceptability cones associated with coherent risks. (b) Acceptability cones associated with acceptability indices.

44

3 25 2.5 20 2 15 1.5

10 1

5

0.5

–3

–2

–1

0

2

1

3

–3

–2

–1

2

1

x

x

Figure 2. (a) Extreme measure densities for Ψ(x) = 1 − (1 − x) 3 . (b) Extreme measure densities for Ψ(x) = x 1/3 .

45

3

1

0.9

0.8

0.7 x=2.5 phi(y)

0.6

0.5

x=2.0

0.4

0.3

0.2

0.1

x=1.5

0

1

2

3

4

5 y

6

7

8

9

10

Figure 3. The function Φx (y) for a sample of x values for M AXM IN VAR

46

1

0.9

x=2.5

0.8 x=2.0

phi(y)

0.7

0.6

0.5

0.4

0.3

0.2

x=1.5

0

1

2

3

4

5 y

6

7

8

9

10

Figure 4. The function Φx (y) for a sample of x values for M IN M AXVAR

47