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1 Introduction
A financial performance aims to evaluate the return characteristics of a given investment portfolio, and to fulfill the risk and asset allocation constraints provided by investors. The resulting measurement can be used to judge the quality of managerial skillfulness, since fund managers viewed as competitors should be able to process those piece of information not reflected by market prices, then providing an actual value-added service. The balance between reward and risk is condensed into a performance measure as the popular Sharpe ratio (SR), which can be used to rank investment portfolios according to these two characteristics. Other performance measures are alternative to SR, accounting for stylized facts about financial returns such as asymmetrical and fat-tailed distributions. See [4] for a textbook treatment of several performance measures.
The unified mathematical framework of Cherny & Madan (2009) deals with some desirable features many performance measures share. Given a set of (at least) four properties, they identify indices of acceptability
of nonnegative financial performance for a given terminal random cash flow. The basic idea is intimately linked to the well known analysis in Atzner et al. (1999) of coherent risk measure and their acceptance sets, and the subsequent analysis in Carr et al. (2001) of arbitrage pricing and acceptance hedging: There is a continuum of degrees of acceptability of a position, varying with different levels of stressed scenarios supporting positive expectations of cumulative terminal cash flow. Thanks to the duality between coherent risk measures and performance measures, a class of equivalent probability measures, or the corresponding set of their Radon-Nykodim derivatives, give the acceptability of a trade’s cash flow. In other words, an index of acceptability derived from a coherent risk measure must be proportional to the amount of stress tolerated and must yield nonnegative values exceeding floors, so that nonnegative expected cash flows are attained. This fits well the regulatory-capital requirements and the pertaining economic-capital modelling used in practice, for example by banks undertaking ex ante improvements in business-performance tracking through the use of risk adjusted return on capital (RAROC), as the ratio of expected final return to the economic capital measured by a coherent risk measure or by the Value-at-Risk (VaR). Not surprisingly, an acceptability index (AI) may be expressed just in ratio form, as the forerunner static SR.
On the other hand, the industry of fund management claims the use of performance measures such as the Calmar ratio (CR) designed to account for the risk associated to cash flow resulting from the whole investment’s path over a fixed horizon. If this is the case, SR is redefined by set the reward measure in the numerator equal to the expected terminal cash flow, and using the expected maximum drawdown over the horizon as a risk measure in the denominator. The risk-adjusted return then takes into account the future evolution of the market value of the position, not just the terminal one. Albeit this kind of performance measure is widespread among practitioners, it cannot be directly placed into the realm of static AIs studied in [11]. Static risk measures does not embed the cash flow’s path experienced over time. On the other hand, dynamic risk measures have been developed to account for this. First, static risk measures are turned into conditional ones to account for the information available at the risk assessment. Then, on some filtered probability space risk assessment is updated as time elapses and new information arrives, so that a sequence of conditional risk measures depicts a dynamical framework based on different notions of time consistency, see [1]
and the references therein for a detailed review. It is possible to turn things around, and define a coherent monetary risk measure as in Cheridito et al. (2004), yielding a numerical evaluation rather than a sequence of conditional risk measures (random variables). For the special case of a finite sequence of adapted cash flows see
[18, Section 3.2].In the present paper we provide a framework for AIs of performance put in duality with coherent monetary risk measures for bounded càdlàg processes. To recover the information lost in the static setting, one records all possible stressed scenarios during the holding period of a financed position, which is represented by a path rather than a random variable. However, we do not develop a dynamic setup to process information, rather we give a representation based on a static index acting on processes which preserves the main features of an AI: Quasi-concavity, positive homogeneity, monotonicity and Fatou continuity. Eventually, this includes the one-period AI as a particular case. Furthermore, we propose an example of AI for processes related to the one-time step RAROC. Our contribution is similar to that of Bielecki et al. (2014), or the more recent Bielecki et al. (2017), which define coherent risk and AIs for paths in a newly proposed dynamical setting. These authors studied the duality between dynamic coherent risk measures and dynamic coherent acceptability indices where is a set of dates and is a set of (adapted) real-valued stochastic processes modelling cash flows. These authors impose two additional properties to and called independence of the past and dynamic consistency. Nevertheless, this framework cannot deal with performance measures such as the aforementioned CR. Our contribution enable us to overcome this limitation, and to deal with path-transformations like taking the maximum drawdown or the running minimum of a cash flow’s path. Eventually information about time resolution of an investment process (from the spatial viewpoint) is not lost. On the generalization of the AIs to the continuous-time setting see Biagini and Bion-Nadal (2015).
The paper proceeds as follows. Section 2 introduces the essential toolkit for treating some classes of càdlàg processes as models of total cumulative cash flows evolving within a finite horizon, and defines the AIs of performance in this extended framework. The duality concerning such classes are briefly reviewed, together with additional results on their lattice structure used in proving the main representation of this paper. Section 3 is devoted to the generalization of the static AI of [11] from the domain to the collection of bounded càdlàg processes. The new AI is obtained in a straightforward manner, by properly combining the contributions [11, 10]. The multi-period analogue of the static system of supporting kernels is depicted in terms of bi-variate processes reproducing the Radon-Nikodym derivatives corresponding to a static system of kernels. Section 4 deals with the analysis of second order stochastic dominance compatible with the extended AI. Section 5 studies its arbitrage and expectation consistency. Section 6 contains the main example of an AI for processes based on the one-period RAROC, as the ratio of the expected terminal cumulative cash flow to the Average Value-at-Risk of the cash flow’s running minimum. Section 6 provides numerical comparisons of simulated values in order to appreciate the magnitude of AIs and other performance measures. Section 8 contains some concluding remarks.
2 Notations and Preliminary Results
In this paper we model the whole evolution of financial outcomes over a finite time-interval rather than the terminal cumulative cash flow typically handled in performance analysis. Here is the stochastic process modeling the random cash flow resulting from dynamic trading over the investment horizon where We are given a filtered probability space satisfying the usual conditions, i.e., the basis space is complete, the filtration is right-continuous, and the initial information contains all the -null events of Almost surely equal random variables are identified as well as indistinguishable processes on the filtered space, for almost all and all Comparisons among processes are understood in the latter sense. For example, means that is greater than or equal to for all dates and for almost all As usual we set and following [10, 13] we denote
the vector space of (the
-a.s. and for every equivalence classes of) càdlàg processes that are adapted to the filtration. For the characterization we develop in Section 3, the model of a (discounted) cash flow evolving within the horizon is that of a bounded càdlàg process belonging to the stricter class i.e. This is a Banach space equipped with the norm where is the usual norm onDefinition 1.
A performance measure is an AI for processes if it satisfies the following properties:
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Acceptable cash flows at a level form a convex above-level set
In the current context, this is a family describing acceptability for any level The convexity of any is equivalent to the quasi-concavity of namely for any provided are such that and Taking quasi-concavity implies that a diversified position performs better than its components.
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Acceptable cash flows are valued monotonically,
thus is an increasing map and is at least as acceptable as
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The acceptance set is required to be a convex cone, because is not meant to be an investment criterion but rather it measures to what extent moving away from marginal trades supporting the random cash flow results in a new investment direction based on alternative pricing kernels. Hence
i.e. scale invariance is required and the performance of an investment should not depend upon the initial endowment. In other words, is based on a trade in the same direction of and then it has the same level of acceptance.
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The acceptability functional is required to be upper Fatou-continuous for ,
and for every bounded sequence of paths such that converges in probability to i.e. This implies that provided that for every and The bounded sequence can be taken as uniformly.
To obtain the representation results of Section 3, we need the duality relations concerning spaces of càdlàg processes, as well studied more recently in [10] and classically in [13]. We also need the lattice structure of some of such spaces which enforces the involved duality. Recall that a Riesz space is an ordered vector space which is also a lattice, where the norm and the absolute value are in general different. If the norm is in addition monotone in the absolute value of a vector, then it is a lattice norm and completeness entails a Banach lattice. Two special classes of Banach lattices are the AM-spaces and the AL-spaces, based on the following definitions of lattice norms:
-
-norm, whenever implies
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-norm, whenever implies
Thus, a norm complete Riesz space equipped with an -norm is an AM-space, while a norm complete Riesz space equipped with an -norm is an AL-space. A useful result states that a Banach lattice is an AM-space (resp. an AL-space) if and only if its dual is an AL-space (resp. an AM-space), see [2, Chs 8, 9] for more details. The class for generalizes to those càdlàg processes such that This is also a Banach space with norm see Appendix A for a brief review.
Proposition 1.
is a Banach lattice, for Moreover, is an AM-space with order unit.
Proof.
On let us consider the partial ordering for any , and for -almost all Using this partial ordering, if and , which implies that for any , and for -almost all this also implies that the random variables satisfy the inequality
Hence, is a Banach lattice for To show that is an AM space with unit, it suffices to prove that such an order unit is the stochastic process , where -a.s. In order to prove it, we have to show that , where denotes the following order interval of : , with respect to the partial ordering of defined above. The inclusion in , and consequently is obvious. For the opposite inclusion, for any , we may define the stochastic process , such that and moreover . ∎
Another class of Banach lattices related to the geometry of càdlàg processes is for containing the bi-variate processes such that has right-continuous coordinates with finite variation, being predictable with while being optional and purely discontinuous, see Appendix A. We have where is he usual variation of a process. The related positive cone contains those bi-variate processes such that and increasing. The base of this cone is defined as where is the element such that -a.s. The partial ordering implied by on is defined as
and makes a Banach lattices. More generally we have:
Proposition 2.
is a Banach lattice. Moreover, it is an AL-space.
Proof.
The partial ordering defined on , is the following: where
Thus, if this is equivalent to . Thus, is a Banach lattice. This also implies , which means that is an AL-space, since is an AL-space, as well. ∎
For conjugate exponents and the duality between the spaces and plays an important role in our representation of AIs for processes, namely Theorems 1 and 2 in Section 3, as well as other results in Sections 3, 5 and 6. The dual pair is based on the dual pairing defined on see Appendix A. In our main result (Theorem 1) we replace the infimum over classical non-negative expectations with respect to equivalent probability measures (viz. their Radon-Nikodym derivatives), with positive increasing dual processes with unit expected variation. Thus, for every the -increasing family of [11, Theorem 1] is now replaced by an -increasing family where each is a subset of the class defined in [10], containing the bi-variate processes that are in addition nonnegative, increasing and such that see also Appendix A.
3 Basic Representation Result
We give the analogue of [11, Theorem 1], to characterize an AI having a numerical value such that the bounded càdlàg path attains a positive bilinear form (which is the analogue of the expectation in the one-period case), under each bi-variate process from the subset (which is the analogue of the Radon-Nykodim derivative of the absolute continuous probability measure giving the acceptability in the one-period case) corresponding to the level There is a one-parameter class of such sets.
Theorem 1.
A map is an AI for processes if and only if there exists a family of -increasing family such that the representation
| (1) |
holds, with and
Proof.
We claim that defined by (1) satisfies the four properties of an AI. First, we check property (1) of Definition 1 that values only acceptable cash flows evolving during the horizon which belong to a convex level set for any Indeed, assuming that and in addition entail a value of which is then choosing for any bi-variate process we have a value of the linear functional corresponding to each cash flow, which must be Taking a convex combination for we then have and for every in the biggest class we thus have too. This corresponds to the greatest such that which proves the convexity of the level set (or equivalently the quasi-concavity of the index ). For the monotonicity, given two elements of such that the stochastic integral together with the expectation operator defining the bilinear form used in (1) are monotone, then from which property (2) of Definition 1 easily follows. Scale invariance is trivial. To show the upper Fatou-continuity, we first assume for a bounded sequence of elements and some This implies
thus by the Lebesgue’s Dominated Convergence theorem we have
for some Now, for any such any and any we have by construction so that the previous implies too, which is the equivalent formulation of the Fatou property for the AI. ∎
Before completing the proof, we observe that representation (1) of Theorem 1 is equivalently given by
| (2) |
as pointed out in [11]. Indeed, each functional on defined by
is by [10, Corollary 3.5] a coherent risk measure for processes. For passing from to the bigger set the value of will increase to for any Then, the supremum in (1) of Theorem 1 will increase too and obviously the equivalent representation given in (2) holds true. Conversely, for a family of coherent risk measures for processes which is increasing in as a map for a fixed any set corresponding to a risk measure in the family must be bigger anytime increases, due to the representation (2).
Remark 1.
The acceptability set introduced in Definition 1 of Section 2 is equivalently given by
Thus, for every we have a whole family which is clearly decreasing in As a consequence the numerical value of an AI for processes can be recast as
There are several levels at which the performance of a trade can be measured by valuing its riskiness in an acceptable way.
In order to prove the ‘if part’ we need the following characterizations. Based on the AM-AL duality between and we in addition see that for any the coherent risk measure for processes
has the following dual representation:
where each is the polar set of in . Finally, recall that a subset of a vector space is called a wedge if it is convex and it has the property that for any lying in the set we also have that belong to the same set, for every . Putting all things together, we have that proving of Theorem 1 amounts to prove the following:
Theorem 2.
For any AI with the property that every level set of is a wedge, there exists an increasing family of functional sets lying in such that implies and
holds.
Proof.
The level sets of are . For these sets, holds, if . For the equivalent polar sets lying in if this implies Thus, any lies in some This implies that , hence . From the equivalent dual representation of the coherent risk measure , then
and we are done. ∎
The family (viz. system of supporting kernels in [11]) can be characterized as
| (3) |
Then, we have the following maximality property:
Lemma 1.
Proof.
As in the proof of [11, Proposition 1] we use a squeezing argument and show that equation (1) can be split in two opposite inequalities. Let us suppose that defined by (1) and supported by some -increasing family is strictly greater than for any process Then
for some But this implies the existence of which makes negative the bilinear form inside the supremum, contradicting the definition of in (3). To show the reverse inequality, let us suppose that Then, we can find which again makes negative the bilinear form and at the same time makes contradicting the definition (1) of Then,
and is a maximal family. ∎
We state a Lemma which will be useful in the identification of typical AIs for processes, provided that a family is meant to supports as given by Lemma 1. Recall that the space inherits the norm and then it is a Banach space.
Lemma 2.
Proof.
Let be the -increasing family supporting For some take a nonempty -closed and convex set This enable us to find some and such that Thus, by the Hahn-Banach Separation Theorem we further find such that
but this implies which contradicts the representation (3). As a consequence and by the maximality stated in Lemma 1 we are done. ∎
It is worth noting that by choosing and one gets the static AI as in [11].
4 Consistency with Second Order Stochastic Dominance
The consistence of a performance measure for processes with the Second order Stochastic Dominance (SSD), hardly depends upon the definition of SSD itself on the spaces To get the equivalent notion in this space of stochastic processes we require (as in the static setting) that if a trade with random cash flow has given a greater ‘utility’ than another then it should have a higher performance too, Whence, we need to adapt the notion of expected utility in order to characterize this preference relation via SSD.
Given a couple of stochastic cash flows we recall that is the corresponding random variable in and similarly for Therefore, the binary relation defined on by
is the analogue of the SSD in the one-time step setting, where what matter are the terminal cash flows. Here instead, we consider the path-dependency using the running maximum of the reflected (at the origin) processes and As a consequence, we rewrite the above SSD relation as
For the quote on terminal wealth, see [20, p. 671]. In the same paper, we find the characterization of SSD in terms of expected utility. Hence, we define the expected utility on in the following sense:
Definition 2.
For a random cash flow evolving in time the version of expected utility for processes over the horizon is given by
where and is some concave, non-decreasing function.
5 Arbitrage and Expectation Consistency
In Section 3 we introduced the family of coherent risk measures for processes
with the dual representation
where is the polar set of in If we suppose are acceptability subsets of such that , then holds for the equivalent polar sets in The above dual representation equals Moreover, implies . Therefore, given a family of coherent risk measures for processes which is monotone with respect to where contains those bounded càdlàg paths this entails and consequently if then For such AIs arbitrage consistency holds, because
Hence, we proved the following:
Theorem 3.
The AI defined through a family of monotone coherent risk measures for processes in and the order unit 1 of is Arbitrage Consistent with respect to
Now we come to the extension of the expectation consistency stated in [11] for the static case. We again transfer the properties of AIs to the dual system from the dual system
Definition 3.
An AI is called expectation consistent, if and only if then . The functional is the one which corresponds to
Proposition 3.
An AI for processes defined on is expectation consistent if the level set of zero is
Proof.
If the above condition holds, since for every where as usual , we notice that if , this implies ∎
6 An AI for Processes
Assume that describes the continuous-time cumulative random return over a finite horizon, and without loss of generality that the interest rates are zero (avoiding to treat excess returns). We propose to characterize the following AI:
| (4) |
where the denominator represents a coherent risk measure for adapted bounded càdlàg processes, with the convention whenever The above measure is reminiscent of the SR given by
where the denominator is the usual standard deviation of the terminal total cumulative return; the numerator measures the expected reward of the underlying investment just at the horizon. Thus, equation (
4) is a RAROC-type of performance measure provided that the expectation in the numerator is positive. To see why is an AI for processes, we need to find the bi-variate process picked from which is consistent with the representation (1). The right choice is the convex combination:where
In fact, we have the chain of equivalences for
We use Lemma 2 for the closeness feature of the sets supporting this RAROC-type measure. In the convex combination defining the first term projects the whole random return onto the terminal date through the expectation; the second term pertains to the representation of the coherent measure for processes Note that by the assumption
Remark 2.
If one chooses the coherent risk measure for bounded paths
| (5) |
then the above AI can be made operational. Here is the Average Value-at-Risk at the level In fact, a risk measure for bounded processes can be viewed as
the composition of a path-transformation with a one-period risk measure applied to the resulting random variable Obviously, would be a coherent monetary risk measure for bounded processes if and only if is a coherent monetary risk measure for single-period cumulative returns, and transforms the paths of in such a way the properties studied in [10] are preserved. Equivalently, the combined effect of a path-transformation and a static risk measurement is
| (6) |
In the current setting, is the running minimum of It is important to note that other types of path-transformations can be taken into account, see [10, Examples 5.2, 5.5] where and it might be viewed as the continuous-time arithmetic average price of the underlying of an Asian option. Anyway, the acceptability of the proposed stems from the coherence of the static together with the monotonicity of the running minimum (properties (1) and (2) are not destroyed). Moreover, the law invariance of implies that of Clearly, it is also expectation consistent but never arbitrage consistent. Recall that
where as usual the VaR is defined as the negative of the
-quantile of the running minimum’s distribution,
andThe widespread CR
| (7) |
is a classical performance measure depending on the whole investment’s path, but fails to be acceptable as we see below. Here is the drawdown process over of the random return defined as i.e., it is the drop of from its running maximum, while the denominator in equation (7) is the maximum drawdown, i.e. the greatest drop of from its running maximum over the whole horizon (the supremum of reflected at its running supremum). From now on we assume positive performance indices for processes, whenever otherwise the ratio (7) is zero. CR is meant to quantify the expected terminal return of the investment, adjusted by the risk not only at the final date but also including all possible drops from the peaks during the horizon. The easy verification that CR is not an AI for processes is due to the bad behavior of the path-transformation which is not monotone. One can try to replace the expectation (as the one-period risk functional) with the tail-conditional expectation (i.e. the defined on the right tail of the distribution of the maximum drawdown) which is a one-period coherent risk measure, but the lack of monotonicity of the aforementioned destroys acceptability (only convexity is preserved).
Remark 3.
For two bounded càdlàg cash flows such that the concept of law invariant AI developed in [11] in the static case can be translated int the current setting by limiting ourselves to the case of RAROC-type AIs (4). After the corresponding path-transformation is made, the sameness in law of any couple of bounded càdlàg cash flows then translates to thus their transformed paths entail random variables sharing the same probability distribution. We can appeal to
7 Numerical Comparison
To give more insight on the behavior of the performance indices discussed in the previous Section, we present here a simulation exercise based on the following ingredients:
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Generation of two Lévy processes to describe possible patterns of continuously compounded returns over the horizon with 1 year;
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Determination of six empirical distribution functions for the final returns at and the corresponding running minimums and maximum drawdowns within
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Estimation of the sample counterparts
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