Optimal Strategies Under Omega Ratio Carole Bernard∗, Steven Vanduffel†, Jiang Ye
‡
February 14, 2018
Abstract We study portfolio selection under the objective of maximizing the Omega ratio, proposed by Keating and Shadwick (2002) as an alternative to the Sharpe ratio for performance assessment of investment strategies. We show that in a continuous-time setting of the financial market the problem is ill-posed, i.e., maximizing the Omega ratio leads to excessive risk taking. By imposing additional restrictions we show that the Omega ratio maximizing strategy is still very risky and may coincide with the choice made by risk neutral investors. We conclude that caution is needed when using the Omega ratio for making asset allocation decisions.
Keywords: Investment analysis, Optimal portfolio choice, Omega ratio, Ill-posedness, Performance measurement, Risk assessment.
∗
Carole Bernard, Department of Accounting, Law and Finance, Grenoble Ecole de Management (email:
[email protected]). † Corresponding author: Steven Vanduffel, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Bruxelles, Belgium. (email:
[email protected]). ‡ Jiang Ye, PhD candidate at Vrije Universiteit Brussel, Pleinlaan 2, 1050 Bruxelles, Belgium. (email:
[email protected]).
1
1
Introduction
The first quantitative approach to determine an optimal investment portfolio is due to Markowitz (1952). In his framework, investors maximize expected return (reward ) for a given variance (risk ). The Markowitz mean-variance optimal portfolios are located on the efficient frontier, which in its most basic form is a straight line1 with slope that is called the Sharpe ratio (Sharpe (1966)). Specifically, all portfolios on the efficient frontier have maximum Sharpe ratio. Hence, the Sharpe ratio appears as a measure that can be used to assess the performance of investment portfolios. A drawback, however, is that while the Markowitz portfolios are optimal for riskaverse expected utility investors when returns are elliptically distributed, this feature does no longer hold in the presence of asymmetry (skewness). Evidence2 for skewness in returns has been presented in various papers including Arditti and Levy (1975), Eberlein et al. (1998), K¨ uchler et al. (1999), Carr et al. (2002) and Adcock et al. (2015). To account for higher-order information contained in returns, Keating and Shadwick (2002) propose the Omega ratio. In essence, it considers the ratio between the heaviness of the distribution in the upper tail (good outcomes) and the heaviness in the lower tail (bad outcomes). Hence, positive skewness is seen as favorable. Using data on hedge funds, Keating and Shadwick (2002) show that the Omega ratio enables to accommodate the information that is contained in the higher moments of the hedge funds’ return distribution and provides a new perspective on performance assessment of the hedge funds. This feature is also confirmed by Bertrand and Prigent (2011) and Van Dyk et al. (2014) who conclude that the Omega ratio is a useful measure for assessing the performance of investment funds. Therefore, it appears natural to study optimal payoff selection under the objective of maximizing the Omega ratio. In this paper, we study the following problem: does the payoff that maximizes the Omega ratio exist and if so what are its properties? In the literature, there are different approaches to deal with portfolio selection problems. A first approach aims at finding an optimal static portfolio; see for instance Markowitz (1952). There are different variations but the main idea is that at initial time the total budget is allocated to the risky assets and the risk-free asset, and there is no further trading activity (rebalancing) during the investment horizon. In other words, finding the optimal portfolio consists in determining the proportions (weights) that are allocated to the different assets at initial time. In this context, the vector of weights is typically called the portfolio and corresponds to a payoff (terminal wealth) at maturity, which is a linear combination of (terminal values of) the risky assets and the risk-free asset. The “well-posedness” of portfolio selection problems can be ensured by imposing short-selling and borrowing constraints, as these lead to a set of admissible weights that is compact in Rn . Note also that in this instance the problem of maximizing the 1
The efficient frontier takes this form in the presence of an asset earning the risk free rate and when short selling is allowed. 2 For instance, according to Arditti and Levy (1975), multi-period returns may be skewed whilst single-period returns are not. Lau and Wingender (1989) and Fogler and Radcliffe (1974) report however opposite conclusions.
2
Omega ratio can be rewritten as a Linear Programming problem and optimal solutions can be readily determined; see Avouyi-Dovi et al. (2004), Guastaroba et al. (2016), Kane et al. (2009) and Vilkancas (2014). The second strand of literature also splits the initial budget over the risky assets and the risk-free asset, but further trading is allowed in that on a continuous basis assets can be (partially) sold and with the proceeds other assets can be purchased. This approach, labeled as continuous-time portfolio selection, was initiated in Merton (1969, 1971) and has a long tradition in the mathematical finance literature. In this case, optimization can be carried out over a set containing all possible payoffs having as initial cost the available budget. This set of payoffs is typically unbounded and non compact, which can potentially lead to ill-posed optimal portfolio selection problems; see for instance He and Zhou (2011) and Biagini and Pinar (2013). Given the optimal payoff at the end of the investment horizon, the weights that need to be allocated to the different assets at each point in time (thus not only at initial time) can be obtained by using for instance a delta hedging approach (complete market) or, if the optimal payoff cannot be dynamically replicated (which may occur in an incomplete market) an implicit assumption is made that some financial institution is prepared to deliver the desired payoff. This paper belongs to the second strand of literature and our contributions can be summarized as follows: First, we show that in a continuous-time setting of the financial market (standard in mathematical finance literature), the problem of maximizing the Omega ratio is typically ill-posed (Theorem 1 and Theorem 2) and that one can find a payoff with an arbitrarily large Omega ratio. Second, to deal with this issue we impose restrictions on our framework. The basic restriction we impose is that the payoff is bounded between two constant values 0 and M or more generally that it is bounded by two probability distributions F and G. We show that in this setting the optimal payoff appears as a binary (digital) or trinary option. In particular, the optimal payoff is closely linked to the payoff that is obtained by a risk neutral investor, which is at odds with a basic paradigm in finance that states that investors are risk averse. Third, we investigate the Black-Scholes setting in more detail. We derive the payoff with optimal Omega ratio in explicit form and also investigate the case in which possible investment choices are restricted to so-called constant-mix investment strategies,3 which are popular in a real world investment environment. In this case, the Omega ratio maximizing strategy coincides with the riskiest constant-mix strategy that is available. We conclude that investment decisions and performance assessment based on the Omega ratio are subject to criticism and should be carried out with caution. The rest of the paper is organized as follows. Section 2 describes the market and presents the optimal payoff selection problem under the objective of maximizing the Omega ratio. We show that the problem is ill-posed. We then add some constraints to reduce the set of admissible payoffs in Section 3. Specifically, we consider bounded payoffs and derive in explicit form payoffs that maximize the Omega ratio. We find that these payoffs are very aggressive and may not 3 These strategies are characterized by a given proportion invested in risky asset that is kept constant (through continuously trading) during the investment horizon.
3
be appealing to risk averse investors. In Section 4, we further elaborate on the limitations of optimizing the Omega ratio by constructing the utility function such that the optimal payoff of an expected utility maximizer would coincide with the payoff that maximizes the Omega ratio. In Section 5 we illustrate the results of our paper in a Black-Scholes market. Section 6 explores some possible extensions. Specifically, we briefly study optimal payoff selection under so-called Kappa ratios, which were introduced in Kaplan and Knowles (2004) and which incorporate the Omega ratio as a special case.
2
Market Setting
2.1
Pricing
Consider investors with a given finite investment horizon T and no intermediate consumption. We assume a continuous-time financial market that is free of arbitrage, perfectly liquid and frictionless (no transaction costs, no trading constraints). The market contains a risk-free asset, which yields the risk-free rate r > 0 and a risky asset S := (St )06t6T . Let (A, F, P) be the corresponding probability space under the physical probability P, where F denotes the natural filtration generated by the risky asset. Unless otherwise indicated, all expectations are taken with respect to P and are denoted as E[ · ]. Under these assumptions, there exists at least one state-price density process that can be used for arbitrage-free pricing. We assume that market participants agree on one such state-price density process4 , further denoted by ξ := (ξt )06t6T . The initial cost (t = 0) of a terminal payoff XT paid at maturity time T is then given as c (XT ) = E[ξT XT ] = EQ [e−rT XT ],
(1)
where ξT = e−rT dQ dP , and Q is a (risk-neutral) measure. In some specific market settings the state-price density process ξ is path-independent, i.e., ξt (0 6 t 6 T ) solely depends on St and not on the entire asset price history up to time t. This feature of path-independence for the state-price density process holds for instance in the BlackScholes market model and also in the exponential L´evy market model when Esscher pricing is used. In the remainder of the paper, however, we do not make any assumptions regarding the possible path-independence of the state-price density process ξ. In what follows, we will make use of the essential infimum and essential supremum of the state-price process evaluated at maturity time T , which are respectively defined as follows: ξ := essinf ξT := inf {x > 0 | P(ξT 6 x) > 0} , ξ := esssup ξT := sup {x > 0 | P(ξT 6 x) < 1} . 4
Note that in a complete market there is exactly one state-price density process.
4
2.2
Omega ratio
The Sharpe ratio (Sharpe (1966)) has always been a reference in performance measurement. However, it is solely driven by the mean and the variance of the investment returns. In an attempt to account for higher-order characteristics (such as skewness), Keating and Shadwick (2002) propose the Omega ratio. We first formally define the Omega ratio and next aim at finding payoffs that have maximum Omega ratio. In what follows we use the notation (x)+ to denote max(x, 0). Definition 1 (Omega ratio). Let L > 0 be a given reference level. Consider a payoff XT such that both E(XT − L)+ and E(L − XT )+ are finite, and such that P(XT < L) > 0. The Omega ratio of a payoff XT is defined as Ω(XT ) =
E(XT − L)+ , E(L − XT )+
(2)
where L > 0 is a given reference level. The Omega ratio thus measures the ratio between the upper tail heaviness (from L onwards) and the corresponding lower tail heaviness (below the level L). It is clearly a law-invariant objective in that the Omega ratio of a payoff XT only depends on the probability distribution of XT under the physical probability P. In addition, the objective function is consistent with increasing preferences, i.e., Ω(X) > Ω(Y ) if X > Y a.s.
(3)
Therefore, the Omega ratio is consistent with first-order stochastic dominance.5 This result follows from Theorem 1 in Bernard et al. (2015a), which proves that preferences are increasing and law-invariant if and only if they satisfy first-order stochastic dominance. From the identity (XT − L)+ − (L − XT )+ = XT − L, it immediately follows that Ω(XT ) can also be expressed as
Ω(XT ) = 1 + In this expression the ratio
E[XT ]−L E(L−XT )+
E[XT ] − L . E(L − XT )+
(4)
is called the Sharpe Omega ratio (Kazemi et al. (2004)).
In a similar way as for the Sharpe ratio, the Sharpe Omega ratio balances excess return and risk, but the excess return is measured with respect to the threshold L (and not with respect to the risk-free investment ω0 exp(rT )), and risk is measured by the lower tail heaviness (and not the standard deviation). 5 X dominates Y in first-order stochastic dominance if its distribution FX is pointwise smaller than the distribution of Y , i.e., for all x ∈ R, FX (x) 6 FY (x).
5
2.3
Maximization of the Omega ratio is an ill-posed problem
In this paper, we consider the problem of obtaining the payoff that has maximum Omega ratio for a given budget. We assume that there is limited possibility for leverage by limiting the set of admissible payoffs to the ones that are uniformly bounded by below. Without this assumption, arbitrage cannot be ruled out in a continuous-time framework,6 i.e., one makes gains almost surely at zero cost, which implies ill-posedness of the optimal selection problem we consider. We thus impose this constraint, which is natural in a continuous-time setting considered in the mathematical finance literature (see also Delbaen and Schachermayer (2006)). Furthermore, without loss of generality, we assume that the lower bound is zero, i.e., we restrict admissible payoffs to non-negative payoffs. We consider the following optimization problem: Problem 1 (Maximization of the Omega ratio). max
Ω(XT )
(5)
{XT >0 | c(XT )=ω0 }
where ω0 > 0 is the initial budget that is available for investing. We first study the well-posedness of this problem. Specifically, we say that the above Problem 1 is ill-posed if there exists a payoff, which, for the given initial budget ω0 , has an arbitrarily large Omega ratio. First, we deal with the case in which the threshold L is smaller than what can be obtained by solely investing in the risk-free asset (i.e., ω0 erT ). It is intuitive that in this case arbitrarily large Sharpe ratios can be obtained by constructing payoffs that closely follow the payoff obtained by investing in the risk-free asset. Theorem 1 (ill-posed problem). Problem 1 is ill-posed, i.e., there exists a payoff that has an arbitrarily large Omega ratio in the following two cases: 1. ω0 erT > L, 2. ω0 erT = L and ξ = +∞. Proof. If ω0 erT > L, consider XT ≡ ω0 erT . Clearly, c (XT ) = ω0 but Ω(XT ) is not well-defined. By deviating from this payoff, we can easily construct a sequence of payoffs with arbitrarily large Omega ratios. Choose N ∈ N big enough to make sure ( XTN
=
(L − (L −
1 N
< L, N1 < ξ and define the payoff XTN ,
1 N )1ξT >N + aN 1ξT 6N 1 1 + aN 1 1 ξT 6ξ− N N )1ξ>ξT >ξ− N
6
if ξ = +∞, if ξ < +∞,
To see why this is true, the example of so-called doubling strategies is instructive. Under a doubling strategy the investor invests repeatedly in a zero-price security that yields +x or −x. Specifically, the investor doubles the exposure x each date until he wins, after which he stops betting. If the investor can play infinitely many times, he will ultimately make a gain at zero cost and hence there is arbitrage. However, pursuing such doubling strategy is only possible if one allows losses to be unbounded. In this regard, Delbaen and Schachermayer (2006), p.151, conclude that “This type of strategy has to be ruled out: there should be a lower bound on the player’s loss.”
6
in which aN is chosen to satisfy the budget constraint c(XTN ) = ω0 . Because of the budrT
−LQ(ξT =ξ) get constraint, aN > L and limN →+∞ aN = ω0 e Q(ξ > L.7 Furthermore, observe that T N )+aN Q(ξT 6
Q(ξT > N )P(ξT 6 N ) Q(ξT 6 N )P(ξT > N ) erT E[ξT 1ξT >N ] = lim N −→+∞ E[1ξT >N ] = +∞,
Ω(XTN ) =
lim
N −→+∞
where in the last step we use the inequality E[ξT 1ξT >N ] > N E[1ξT >N ] = N P(ξT > N ). Requiring that the threshold L satisfies the condition L > ω0 erT appears very reasonable, but is not enough to avoid ill-posedness of the optimization problem. The following theorem essentially shows that when there are states of the world in which consumption is extremely cheap, it is possible to construct payoffs with arbitrarily high Omega ratio. Theorem 2 (Ill-posed problem). If ξ = 0, then Problem 1 is ill-posed, i.e., there exists a payoff that has an arbitrarily large Omega ratio. Proof. Let ε > 0, define ( XTε := bε 1ξT 6ε =
0
if ξT > ε
bε
if ξT 6 ε
,
in which bε satisfies the budget constraint c (XTε ) = ω0 , i.e., bε =
ω0 erT . Q(ξT 6 ε)
We choose ε sufficiently small such that bε > L. Such ε exists because of the fact that Q(ξT 6 ε) tends to 0 when ε tends to ξ = 0. Then, we have that Ω(XTε ) = 7
(bε − L)P(ξT 6 ε) . LP(ξT > ε)
Q(ξT < N ) = erT E [ξT 1ξT ε) ε−→0 P(ξT > ε) ω0 E[1ξT 6ε ] = lim ε−→0 L E[ξT 1ξT 6ε ] = +∞,
lim Ω(XTε ) =
ε−→0
lim
where we use the inequality E[ξT 1ξT 6ε ] 6 εE[1ξT 6ε ]. Note that an identical proof shows that ω0 erT P(ξT 6 ε) ε−→0 Q(ξT 6 ε) = +∞.
lim E[XTε ] =
ε−→0
lim
In other words, the payoffs that we construct with arbitrarily large values for their Omega ratio also exhibit arbitrarily large values for their expected value. This observation is also consistent with the findings of Caporin et al. (2016) who state that that under the Omega ratio the trade-off between return and risk is mostly guided by the mean return. In order to be as exhaustive as possible on the well-posedness of Problem 1, we end this section with the following two remarks. Remark 1 (Well-posedness). If ξ < +∞, ξ > 0 and ω0 erT 6 L, then Problem 1 is well-posed. Indeed, let A1 := {ω ∈ A|XT (ω) > L}, then it can be readily verified that for any payoff XT having initial cost ω0 , the Omega ratio can be expressed as E [(XT − L)1A1 ] E (L − XT )1A\A1 ξ E (XT − L)(ξ − ξT )1A1 + E (L − XT )(ξT − ξ)1A\A1 + ω0 − Le−rT . = + ξ E (L − XT )1A\A1 ξ
Ω(XT ) =
Hence, ξ Ω(XT ) 6 , ξ and it is not possible to construct a payoff with arbitrarily high Omega ratio. Moreover, in the special case in which ω0 erT = L and A is a finite set, we can also explicitly derive a payoff with maximum Omega ratio. Indeed, in this case ξ := maxj=1,··· ,n ξT (ωj ) = ξT (ωk ) for some ωk , and ξ := minj=1,··· ,n ξT (ωj ) = ξT (ωl ) for some ωl . It can be readily verified that the payoff XT∗
8
defined as L + εl XT∗ (ω) = L L − εk
if ω = ωl if ω ∈ A\{ωk , ωl } if ω = ωk ,
where εl ξT (ωl )P(ωl ) = εk ξT (ωk )P(ωk ), εl > 0 and εk > 0 has an Omega ratio that is equal to ξ ξ,
which is the maximum possible. In fact, when ω0 erT = L the Omega ratio corresponds to
the Gain-Loss ratio, proposed in Bernardo and Ledoit (2000) and further studied in Biagini and Pinar (2013). The conditions ξ < +∞ and ξ > 0 that ensure well-posedness of Problem 1 (when ω0 erT = L) can also be found there. Remark 2 (Summary table). Theorems 1 and 2 as well as Remark 1 make it possible to be conclusive on the well-posedness of Problem 1 in all but one case. We report our findings in Table 1. Note that in this table the cases 2,3,4 and 5 cover situations in which the Omega ratio coincides with the Gain-Loss ratio proposed in Bernardo and Ledoit (2000) (when ω0 erT = L) and allow to conclude that maximization of the Gain-loss ratio is a well-posed problem if and only if ξ < +∞ and ξ > 0, a result that conforms well with the results reported in Bernardo and Ledoit (2000) and in Biagini and Pinar (2013). Case 1
ω0 erT > L
Ill-posed problem
Case 2 ω0 erT = L, ξ = +∞, ξ > 0
Ill-posed problem
Case 3 ω0 erT = L, ξ = +∞, ξ = 0
Ill-posed problem
Case 4 ω0 erT = L, ξ < +∞, ξ > 0 Well-posed problem Case 5 ω0 erT = L, ξ < +∞, ξ = 0
Ill-posed problem
Case 6 ω0 erT < L, ξ < +∞, ξ > 0 Well-posed problem Case 7 ω0 erT < L, ξ < +∞, ξ = 0
Ill-posed problem
Case 8 ω0 erT < L, ξ = +∞, ξ > 0
Undetermined
Case 9 ω0 erT < L, ξ = +∞, ξ = 0
Ill-posed problem
Table 1: Summary table for the ill-posedness of Problem 1
As for the cases 1, 2 and 3, the stated conclusions regarding well-posedness follow from Theorem 1. The conclusions for the cases 5, 7 and 9 follow from Theorem 2, and the conclusions for cases 4 and 6 follow from Remark 1. Only Case 8 stays undetermined. In a continuous-time setting for optimal payoff selection, the conditions ω0 erT 6 L and ξ > 0 are both necessary to obtain well-posedness of Problem 1. The first condition is rather natural, but the second condition is harder to justify. In fact, to the best of our knowledge ξ = 0 holds in all continuous-time models considered in the mathematical finance literature, including for instance the seminal Black-Scholes market and the exponential L´evy market with Esscher pricing; see also Biagini and Pinar (2013) for a similar statement. In summary, the statement 9
that Problem 1 is ill-posed is fairly general. In the next section we impose additional constraints to make the optimization problem meaningful.
3
Payoffs with maximum Omega ratio under additional constraints
From now onwards we will make the assumption that ξT has a density on R+ and that ξ = 0 (see above). Furthermore, we will also assume that the level of the reference level L is sufficiently above the budget ω0 in that we require ω0 erT < L.
(6)
To study payoff selection under the objective of maximizing Omega ratio in a meaningful way we then need to further constrain the set of admissible payoffs. Specifically, we consider the following problem in which we restrict the set of admissible payoffs to be bounded between two constant values:8 Problem 2 (Maximization of Omega ratio for bounded payoffs). max c(XT ) = ω0 0
Ω(XT ),
6 XT 6 M
where ω0 > 0 is the initial budget available for investing and M > L.
9
To solve the above optimization problem, i.e., to determine the payoff with maximum Omega ratio, we make use of results on cost-efficiency (useful for characterizing optimality of payoffs) and on convex order (useful for comparing payoffs with respect to their riskiness). For the ease of presentation, we list hereafter the results we need.
3.1
Results on cost-efficiency and convex order
Cost-efficiency has been introduced in the literature by Dybvig (1988). A payoff is cost-efficient if it is the cheapest payoff that has a given probability distribution under P. Cost-efficient 8
To deal with the ill-posedness of Problem 1, one may consider to restrict the optimization problem to the set of unbounded payoffs having a given expected value rather than to the set of uniformly bounded payoffs as we do. It can be shown that under such mean restriction the optimization problem will be well posed. However, optimal payoffs derived in such setting are not interesting for the investor since for any given payoff we can always construct another payoff that has a higher Omega ratio than the original one, but which moreover also has a higher expected value. The proof for this statement is similar as the proof for the ill-posedness of Problem 1 under the conditions of Theorem 2; see also the comments that follow the proof of Theorem 2. 9 When M 6 L all admissible payoffs have zero Omega ratio and the problem is not interesting .
10
payoffs paid at maturity T must be decreasing (anti-monotonic) in the state price density ξT (Proposition 1 of Bernard et al. (2014)). Hence, the cost-efficient payoff ZT that has a given distribution function F is given as ZT = F −1 (1 − FξT (ξT )).
(7)
It then follows that optimal investment choices resulting from an optimization of an increasing law-invariant objective function (i.e., fully determined by the probability distribution of the portfolio’s terminal payoff) are decreasing in the state-price density ξT . This basic characterization of optimality has been applied in a series of papers that deal with various optimal payoff selection problems of interest; see Proposition 3.1 of Bernard and Vanduffel (2014) as well as Corollary 3.4, Theorem 6.1 and Proposition 6.6 of Bernard et al. (2015b), and Theorem 2.1 and Corollary 2.2 of Bernard et al. (2017). Specifically, the following lemma makes it possible to reduce the set of admissible payoffs to the set that solely contains payoffs that are decreasing in the state-price density. Lemma 1 (Monotonicity in ξT ). For any increasing law-invariant objective function and for any payoff XT with initial price c that is not almost surely decreasing in ξT , there exists a payoff YT with the same initial price c that strictly improves the objective function. Proof. Denote by F the cumulative distribution function (cdf) of XT and denote by ZT the unique cost-efficient payoff with the same cdf F as XT (Proposition 1 of Bernard et al. (2014) when ξT is continuously distributed). As XT is not almost surely decreasing in ξT , P(ZT 6= XT ) > 0 and c(XT ) − c(ZT ) > 0 (by definition of cost-efficiency). Furthermore V (ZT ) = V (XT ) because V (·) is law-invariant. Consider the payoff YT = ZT + (c(XT ) − c(ZT ))erT , we have V (YT ) = V (ZT + (c(XT ) − c(ZT ))erT ) > V (ZT ) = V (XT ), because V is increasing, and the price of YT is exactly c(XT ). The second result that plays a key role in deriving optimal payoffs with maximum Omega ratio concerns convex order, which is a useful device to order payoffs with respect to their variability. We define convex order formally and provide a lemma that shows why this concept is so useful in the study of optimal payoffs. Definition 2 (Convex Order). We say that a payoff XT is smaller in convex order than a payoff YT (denoted by XT 4cx YT ) if and only if for all convex functions v(·) E [v(XT )] 6 E [v(YT )] . Lemma 2 (Convex larger payoffs are cheaper). Assume that XT and YT are decreasing in ξT and that YT 4cx XT .Then c(XT ) 6 c(YT ), 11
with strict inequality10 if FXT 6= FYT . Proof. We have YT 4cx XT if and only if α
Z ψXT (α) := 0
FX−1T (u)du
α
Z 6 0
FY−1 (u)du = ψYT (α) for all 0 6 α 6 1, T
and ψXT (1) = E [XT ] = E [YT ] = ψYT (1) (Theorem 3.A.5 in Shaked and Shanthikumar (2007)). The counter-monotonicity between XT and ξT implies that (XT , ξT ) ∼ (FX−1T (U ), Fξ−1 (1 − U )) T for some U ∼ U(0, 1). Therefore, we get the following expression for E [XT ξT ]: Z
1
E [XT ξT ] = 0
Z
Fξ−1 (1 − u)FX−1T (u)du T
1Z ξ
= ξZ
= 0
Z =
ξT
0
0
Z
0 ξ
106t6F −1 (1−u) dtFX−1T (u)du
1
106u6F¯ξ
T
−1 (t) FXT (u)dudt
ψXT (F¯ξT (t))dt.
0
Here F¯ξT (t) = 1 − FξT (t). The assertions thus follow from the representation Z
ξ
E [YT ξT ] − E [XT ξT ] =
(ψYT − ψXT )(F¯ξT (t))dt.
0
3.2
Payoff with maximum Omega ratio
The following proposition shows that the optimal solution to Problem 2 is a trinary option (i.e., a payoff that takes three possible values). The proof of it builds on the following insight: A given cost-efficient payoff that meets the budget requirement can be improved by concentrating the higher outcomes (those that occur in the range [L, M ]) onto the boundary values L and M in such a way that the mean remains preserved. Doing so increases the risk of the payoff in the sense of convex order and thus makes it cheaper11 . (Lemma 2). The same conclusion applies when concentrating the lower outcomes (occurring in the range [0, L]) onto 0 and L, and the best payoffs are thus those that are concentrated on the values 0, L and M . Proposition 1 (Payoff with maximum Omega ratio). The payoff that solves Problem 2 is unique 10
Since f (x1 , x2 ) = x1 x2 is a supermodular function, the proof of the non-strict inequality also follows as a corollary from Theorem 3.2 in M¨ uller (2003); see also Lemma A.3 in M¨ uller and Scarsini (2001). We thank Alfred M¨ uller for suggesting a proof that yields strict inequality. 11 The fact that when choosing between two payoffs with the same (nominal) mean the more risky one is cheaper (and thus yields a higher expected return) corresponds to economic intuition.
12
and belongs to the set B defined as 0 a,b a,b a,b = ω0 , a > b > 0, XT = B := XT c XT L M
if a 6 ξ T if b6 ξ T < a if ξT < b
.
Proof. From Lemma 1, a candidate payoff n XT is decreasing witho ξT and thus of the form −1 XT = FXT (1 − FξT (ξT )). We define c = inf x|FX−1T (1 − FξT (x)) 6 L and construct payoffs XTb for b ∈ (0, c] as XTb = M 1ξT a > 0, XTa,b = L −1 G (1 − FξT (ξT ))
if ξT ∈ (0, a) ∪ [min(b, cu ), b) if ξT ∈ [max(a, cl ), min(b, cu )) if ξT ∈ [b, ξ) ∪ [a, max(a, cl ))
where cl = inf x|G−1 (1 − FξT (x)) 6 L and cu = sup x|F −1 (1 − FξT (x)) > L . Proof. A candidate payoff XT is cost-efficient and thus writes as XT = FX−1T (1 − FξT (ξT )). We n o define c = inf x|FX−1T (1 − FξT (x)) 6 L and note that c > cl . In a similar way as in the proof of Proposition 1, we can construct XTa , a ∈ (0, c] as XTa = F −1 (1 − FξT (ξT ))10M and G = 1XT >0 , we find that F −1 (p) = M , G−1 (p) = 0, for all 0 < p < 1 and thus obtain that Proposition 1 and Proposition 2 are special cases of Proposition 3 and Proposition 4, respectively.
4
Inferring the corresponding utility function
In this section, we derive the utility function such that an investor who maximizes expected utility of terminal wealth obtains the same payoff as if he is maximizing the Omega ratio. To do so, we use the explicit correspondence between the expected utility setting of Von Neumann and Morgenstern (1947) and any increasing law invariant objective function. Proposition 5 (Inferred utility). The payoff that solves Problem 2 is also a solution to the following maximum expected utility problem ˜ (XT )] E[U
max c (XT ) = ω0 0
(9)
6 XT 6 M
˜ (x) is the following concave piecewise linear utility function in which U
˜ (x) = U
k0 x k1 (x − L) + k0 L k1 (M − L) + k0 L 16
if 0 6 x < L if L 6 x < M , if x > M
(10)
where k0 = Fξ−1 (1 − p0 ), p0 = P(ξT > a), k1 = Fξ−1 (1 − p1 ), p1 = P(ξT > b) in which a and b are T T as defined in Proposition 1 (so that p0 6 p1 and k0 > k1 ). Proof. The optimal payoff in Proposition 1 belongs to the set B and its cdf is of the form p0 F (x) = p1 1
if 0 6 x < L if L 6 x < M ,
(11)
if x > M
in which p0 = P(ξT > a) and p1 = P(ξT > b). Applying Theorem 3 in Section 4.2 (Bernard ˜ . It is written in the form et al. (2015a)), we can get the corresponding utility function U ˜ (x) := U
Z c
x
Fξ−1 (1 − F (y))dy, T
˜ as in (10). in which c is any value such that F (c) > 0. We can choose c = 0 and we obtain U Proposition 5 shows that the optimal payoff obtained by maximizing the Omega ratio on a bounded domain can also be obtained by maximizing a piecewise linear utility over [0, M ] and flat over [M, +∞). In the special case of an optimal payoff that only takes the values 0 and M (as in Proposition 2) the utility function is linear over (0, M ), which is exactly the utility function of a risk neutral investor. This shows again that by optimizing Omega ratio one may not properly account for risk. Remark 4. A series of papers including Pedersen et al. (2002), Zakamouline and Koekebakker (2009) and Zakamouline (2014) study the correspondence between expected utility and performance measures. The basic idea is that a good performance measure needs to satisfy the criterion that it yields a ranking among payoffs that is consistent with expected utility, i.e., the higher the performance measure the higher the utility (Zakamouline (2014)). Specifically, Zakamouline (2014) presents a framework that provides a sound theoretical foundation for performance measures that are based on partial moments of the distribution. In the context of static strategies it is shown that so-called Kappa performance measures (Kaplan and Knowles (2004)) are good performance measures in that they are consistent with (can be rationalized by) investors having utility functions that are of the (fairly general) form of a piecewise linear utility plus a power utility function. In a limiting case, the Kappa measure reduces to the Omega ratio but while the candidate explaining utility function is piecewise linear, the Omega ratio does no longer yield a consistent ranking and can thus not be rationalized in this framework; see Section 2.4 in Zakamouline (2014). In our paper we deal with continuous-time portfolio selection and we rationalize the Omega ratio in that we show that there exists a piece-wise linear utility function (on a bounded domain) that yields the same optimal payoff as when maximizing the Omega ratio. However, the optimal payoff is always risky and in a special case corresponds to the optimal demand of a risk-neutral investor, stretching the rationalization to its limits and confirming that
17
the Omega ratio is to be used with caution.
5
Examples in Black-Scholes Market
Now we give a specific model for the one-dimensional Black-Scholes market in which r is the continuously compounded constant risk-free rate. Here the dynamics of the stock price St under the physical measure P is given by dSt = µdt + σdWt , St where Wt is a standard P-Brownian motion, µ is the drift and σ is the volatility. Let T be the investment horizon. In the Black-Scholes market it is known that ξT = α
ST S0
−β ,
(12)
in which σ2 θ2 θ µ−r θ µ− T − r+ T , β = and θ = . α = exp σ 2 2 σ σ
(13)
We also assume that µ > r. In general, optimal payoffs are decreasing with ξT (see also Lemma 1), but since ξT is decreasing in ST all optimal payoffs can be expressed as increasing functionals in ST . In particular, they are path-independent.14 In Section 5.1, we illustrate how to derive the optimal Omega ratio payoff in the BlackScholes setting. In Section 5.2, we restrict ourselves to constant-mix strategies and derive the optimal constant-mix strategy that maximizes the Omega ratio. Finally, in Section 5.3, we illustrate the implied utility function that provides the same payoff as the one that maximizes the Omega ratio.
5.1
Maximizing the Omega ratio in Black-Scholes Market
From Proposition 1 we know that in a Black-Scholes market the payoff solving Problem 2 has a three-point distribution. The next proposition provides the payoff explicitly. 14
Note that under the assumption µ < r, optimal payoffs will be decreasing in ST . In this case, the optimal payoffs will be similar to the ones we present further in this section. The difference is merely that each time we have inequalities of the form {ST > a0 } this inequality will be reversed, and the values for the threshold a0 will be different. Finally, when µ = r, ξT will no longer be continuously distributed and the results on optimal payoffs do no longer hold. Note that in this case all payoffs with distribution F have the same cost, therefore there is no uniqueness anymore of the optimal payoff. If one payoff is optimal, then all payoffs with the same distribution are also optimal.
18
Proposition 6. In a Black-Scholes market, the payoff solving Problem 2 is given by
XT∗
=
√ If L < M Φ θ T − Φ−1 1 −
ΩL (x) := M −L L ,
0
if ST 6 a0
L
if a0 < ST 6 b0 , b0 > a0 > 0.
M ω0 erT M
if b0 < ST
, then b0 is the unique solution to Ω0L (x) = 0 with AΦ −Y (x) +
Φ Φ−1 −AΦ Y (x) +
C 2
!
C 2
+B −C
,
(14)
√
M −ω0 erT , C =θ L −1 Φ −AΦ(Y (b0 )
T , Y (x) = −Φ−1 (Q(ST > x)) − C2 ; and a0 is the unique solution to Y (x) = + C2 ) + B − C2 . 1 √ √ β erT − θ 2 T + rT Otherwise, a0 = b0 = S0 α exp θ T Φ−1 1 − ω0M .
in which A =
B=
The proof of Proposition 6 is given in Appendix A.1. ∗ Remark 5. We also find that the maximum expected (see also √ return L Proposition 2) an ω0 erT −1 ∗ 1− M admissible payoff can have is given as L = M Φ θ T − Φ . In other words, √ erT is the expected return of the payoff that, among all admissible M Φ θ T − Φ−1 1 − ω0M
payoffs, has maximum expected return. Example 1. In this example we illustrate Proposition 6. The market parameters µ, σ and r as well as the current value of the stock S0 , the initial budget ω0 , the upper bound M and the time horizon T are all displayed in Table 2. Table 2: Parameters of the Black-Scholes market
S0 50
ω0 60
M 100
µ 0.05
r 0.03
σ 0.2
T 1
√ Since L > ω0 erT = 61.8273 must hold and L∗ = M Φ θ T − Φ−1 1 −
ω0 erT M
= 65.5771,
we choose L in the range [62, 70]. When L ∈ [62, 65.5771], we obtain a trinary option in which b0 is obtained as a solution to Ω0L (x) = 0 and a0 solves Y (x) = Φ−1 [−AΦ Y (b0 ) + C2 + B] − C2 . In the left panel of Figure 1 we display a0 and b0 as a function of L. The corresponding (maximum) Omega ratio is displayed in the right panel. When L ∈ [65.5771, 70], we always obtain the same binary option in which a0 = b0 = 47.5525 and its expectation is equal to L∗ = 65.5771. In other words, for an initial investment of 60 we obtain as optimum a payoff that takes the values 0 or 100 and that has an expected return of more than 8 percent, confirming that optimality is driven
19
by the expected return. ( XT∗ =
0
if ST 6 47.5525
100
if ST > 47.5525
.
Figure 1: (Left Panel) Plot of a, b w.r.t. L; (Right Panel) Plot of the Omega ratio w.r.t. L.
70
1.5
b
Omega Ratio
a
65
1.4
60
1.3
55
1.2
50
1.1
45
1
40
0.9
35 62
63
64
65
66
67
68
69
0.8 62
70
63
64
L
5.2
65
66
67
68
69
L
Restriction to set of constant-mix strategies in a Black-Scholes Market
The so-called constant-mix strategy is an investment strategy that consists in keeping the proportion that is invested in the risky asset constant during the entire investment period. To achieve this situation, one trades the risky asset on a continuous basis. When its value has increased more than that of the bank account a fraction of it needs to be sold and when its value has decreased, the risky asset needs to be purchased. It is well-known that its stochastic value Xtλ satisfies dXtλ = ((1 − λ)r + λµ)dt + λσdWt , Xtλ
(15)
in which λ is the proportion of the risky asset S in the portfolio. Assuming (without loss of generality) that the initial budget is taken equal to one, we get the solution of (15) as Xtλ = exp{((1 − λ)r + λµ − 0.5λ2 σ 2 )t + λσWt }.
(16)
The constant-mix strategy is popular in a real-world investment environment and is shown to be optimal for investors who use a Constant Relative Risk Aversion (CRRA) utility function for making investment decisions in the Black-Scholes market. Specifically, the optimal investment for such investor is a constant-mix strategy with a proportion λ that depends on the risk aversion;
20
70
the higher the risk aversion the lower the proportion invested in the risky asset. A special role is played by the constant-mix investment strategy that maximizes expected log utility (or equivalently, that maximizes the expected log-return). This strategy is called the Growth Optimal Portfolio (GOP) and is of particular appeal since on an infinite time horizon it almost surely beats any other strategy. In this section we investigate which constant-mix strategy yields maximum Omega ratio. Unfortunately, the problem is ill-posed in that again, even within the restricted set of strategies, one can create a payoff with an infinitely large Omega ratio. Proposition 7 (Omega ratio of constant-mix strategies). In a Black-Scholes market, the Omega ratio of a constant-mix strategy with parameter λ, XTλ is an increasing function of λ and limλ→+∞ Ω(XTλ ) = +∞. The proof of Proposition 7 is given in Appendix A.2. Our proof shows that the investors would increasingly borrow money to increase the proportion invested in the risky asset, as doing so increases the Omega ratio of the payoffs. If short-selling is not allowed, one would take a 100% position in the risky asset to achieve maximum Omega ratio. In other words, in the context of constant-mix strategies, maximizing the Omega ratio leads to pursuing a strategy that is as aggressive as possible. Hence, using the Omega ratio to measure the performance of investment funds merely amounts to evaluating their riskiness.
5.3
Utility function in the Black-Scholes market
We apply Proposition 5 to determine the implied utility function that an expected utility maximizer should have to find the same payoff as the one that maximizes Omega ratio. Using the specifications of the Black-Scholes market (see Section 5.1) we find that −β a 2T ln α +rT +0.5θ S0 √ p0 = Φ − θ T −β ln α
b S
+rT +0.5θ2 T
0 √ p1 = Φ − θ T √ F −1 (y) = exp Φ−1 (y)θ T − rT − θ2 T ξT 2
.
(17)
Proposition 8. In a Black-Scholes market, the payoff that solves Problem 2 also solves the following maximum expected utility problem max c (XT ) = ω0 0
6 XT 6 M
21
˜ (XT )] E[U
˜ (x) is given as in which U
˜ (x) = U
where k0 = α
6
a0 S0
−β
, k1 = α
b0 S0
k0 x k1 (x − L) + k0 L k1 (M − L) + k0 L −β
if 0 6 x < L if L 6 x < M , if x > M
and where a0 and b0 are defined as in Proposition 6.
Optimal payoff under Kappa ratios
Consider a payoff XT such that both E[(XT − L)+ ] and E[(L − XT )+ ] are finite, and such that P(XT < L) > 0. Kaplan and Knowles (2004) define the Kappa ratio as E[XT ] − L κ(XT ) = q , n E (L − XT )n+ in which L > 0 is a given reference level and n is a strictly positive integer. Specifically, when n = 1 we obtain15 that κ(XT ) = ΩXT − 1 and when n = 2 the Kappa ratio is called the Sortino ratio originally introduced in Sortino and Price (1994). It is of interest to study optimal payoff selection under Kappa ratios. We formulate the following problem. Problem 4 (Maximization of Kappa ratio for bounded payoffs). max c(XT ) = ω0 0
κ(XT ),
6 XT 6 M
where ω0 > 0 is the initial budget that is available for investing and M > L. Theorem 3 (Payoff with maximum Kappa ratio). The payoff that solves Problem 4 is unique and belongs to the set D defined as D := XTa,b
a,b c X = ω0 , a < 0, b > 0, XT∗ = T
15
L−
M q
n−1
0
−b a , n−1 −nL −b > ξT > −b a a , n−1 −nL −b . ξT > a
if ξT 6 aξT +b −n
if if
Note that Omega ratio is also a special case of the Farinelli and Tibiletti ratio (Farinelli and Tibiletti (2008)), which also shows some similarity with the Kappa ratio.
22
.
Proof. To solve Problem 4, we can specify E[XT ] and solve a series of problems of the form (18) κ(XT ).
max c(XT ) = ω0 E[XT ]
(18)
=B
0 6 XT 6 M
To solve (18), we introduce L∗ defined in (8) and discuss two cases. If L∗ > L, the optimal payoff XT∗ of (18) follows E[XT∗ ] > L. Otherwise κ(XT∗ ) 6 0 and there exists XT such that E[XT ] > L (because L∗ > L), so κ(XT ) > 0 which contradicts the optimality of XT∗ . We consider (18) such that E[XT ] = B > L. In this case, maximization of κ(XT ) amounts to minimization of E (L − XT )n+ . For any ω ∈ Ω, consider the following auxiliary problem min {(L − x)n+ − aξT (ω)x − bx}.
M >x>0
When aξT (ω)q+ b > 0, the optimum x∗ = M , when −nLn−1 6 aξT (ω) + b 6 0, the optimum x∗ = L −
n−1
aξT (ω)+b , −n
and when aξT (ω) + b 6 −nLn−1 , the optimum x∗ = 0. The optimal
payoff XTa,b is of the form,
XTa,b =
M L−
q
n−1
aξT +b −n
if
0
if
−b a , n−1 −nL −b > ξT > −b a a , −nLn−1 −b ξT > . a
if ξT 6
(19)
where a < 0, b > 0 is determined by c(XT ) and E[XT ]. The sign of a and b is determined by cost-efficiency theory. Let B goes through the interval (L, L∗ ), the optimal payoff belongs to D. If L∗ 6 L, Maximization of κ(XT ) is equivalent to maximization of E (L − XT )n+ . Since (L − x)n+ is a convex function, we obtain that the optimal payoff XTc is of the form ( XTc
=
M
if ξT 6 c,
0
if ξT > c,
in which c is determined by the budget constraint ω0 . This can be seen as the special condition of (19) when a goes to negative infinity. When L∗ 6 L the optimal payoff is once again a digital option indicating that Kappa ratio maximization is also to be considered with caution.
23
7
Final Remarks
In this paper, we study the optimal portfolio choice for an investor who uses the Omega ratio as a criterion for making investment decisions (see Caporin et al. (2014) for an overview paper on performance measures). In a continuous-time setting of the financial market (standard in the mathematical finance literature), this problem is typically ill-posed and leads to excessive risk taking. If we restrict ourselves to bounded payoffs the problem can be solved but we confirm that the optimal payoff is very risky and is closely linked to the optimum for a risk-neutral expected utility maximizer. The proof of our results makes use of convex order and the theory of cost-efficiency. Our study shows that one needs to be very careful when using the Omega ratio and in particular when maximizing it. It appears intuitive that the risky character of a payoff with maximum Omega ratio is driven by the lack of consistency between Omega ratio and second order stochastic dominance; see Klar and M¨ uller (2017) and Caporin et al. (2016) who also point out the excessive sensitivity of the Omega ratio to the portfolio mean return. Whilst lack of consistency between Omega ratio and second order stochastic dominance certainly contributes to explaining the properties of Omega ratio maximizing payoffs, it is not the only source of explanation – when comparing two payoffs XT and YT one also has to account for their costs c(XT ) resp. c(YT ). In fact, the core insight in this paper is that increasing the risk of a payoff (in the sense of convex order) can be done without negative impact on its Omega ratio nor its expected value, whilst strictly decreasing the cost. In a financial context, the study of consistency between a performance measure (such as the Omega ratio) and a stochastic ordering concept should be based on payoffs that are normalized for cost (i.e., by studying
XT c(XT )
and
YT c(YT ) ).
We leave this topic for further
research. Finally, we note that the Omega ratio (performance measure) is technically closely related to the expectile (risk measure), as shown for instance in Theorem 8 of Bellini et al. (2016); see also Bellini and Di Bernardino (2015) who exploit this connection to provide more intuition on the definition of expectiles. The expectile has appealing properties in that it is the only risk measure that is both elicitable and coherent (Bellini and Bignozzi (2015) and Ziegel (2016)) and has gained some interest in the risk management literature (Acerbi and Szekely (2014), Bellini and Di Bernardino (2017), Cai and Weng (2016), Jakobsons and Vanduffel (2015)). In future research it would be of interest to study the precise connection between the Omega ratio and the expectile and their proper use in risk management and portfolio optimization.
References C. Acerbi and B. Szekely. Back-testing expected shortfall. Risk, page 76, 2014. C. Adcock, M. Eling, and N. Loperfido. Skewed distributions in finance and actuarial science: a review. European Journal of Finance, 21(13-14):1253–1281, 2015.
24
F. D. Arditti and H. Levy. Portfolio efficiency analysis in three moments: the multiperiod case. Journal of Finance, 30(3):797–809, 1975. S. Avouyi-Dovi, S. Morin, and D. Neto. Optimal asset allocation with omega function. Technical paper, Banque de France, 2004. F. Bellini and V. Bignozzi. On elicitable risk measures. Quantitative Finance, 15(5):725–733, 2015. F. Bellini and E. Di Bernardino. Risk management with expectiles. European Journal of Finance, 2015. F. Bellini and E. Di Bernardino. Risk management with expectiles. The European Journal of Finance, 23(6):487–506, 2017. F. Bellini, B. Klar, and A. M¨ uller. Expectiles, omega ratios and stochastic ordering. Methodology and Computing in Applied Probability, forthcoming, doi:10.1007/s11009-016-9527-2, 2016. C. Bernard and S. Vanduffel. Mean–variance optimal portfolios in the presence of a benchmark with applications to fraud detection. European Journal of Operational Research, 234(2):469–480, 2014. C. Bernard, P. P. Boyle, and S. Vanduffel. Explicit representation of cost-efficient strategies. Finance, 35(2):5–55, 2014. C. Bernard, J. S. Chen, and S. Vanduffel. Rationalizing investors choices. Journal of Mathematical Economics, 59:10–23, 2015a. C. Bernard, F. Moraux, L. R¨ uschendorf, and S. Vanduffel. Optimal payoffs under state-dependent preferences. Quantitative Finance, 15(7):1157–1173, 2015b. C. Bernard, D. Cornilly, and S. Vanduffel. Optimal portfolios under a correlation constraint. Quantitative Finance, pages 1–13, 2017. A. E. Bernardo and O. Ledoit. Gain, loss, and asset pricing. Journal of political economy, 108(1):144–172, 2000. P. Bertrand and J.-l. Prigent. Omega performance measure and portfolio insurance. Journal of Banking & Finance, 35(7):1811–1823, 2011. S. Biagini and M. C. Pinar. The best gain-loss ratio is a poor performance measure. SIAM Journal on Financial Mathematics, 4(1):228–242, 2013. J. Cai and C. Weng. Optimal reinsurance with expectile. Scandinavian Actuarial Journal, 2016(7): 624–645, 2016. M. Caporin, G. M. Jannin, F. Lisi, and B. B. Maillet. A survey on the four families of performance measures. Journal of Economic Surveys, 28(5):917–942, 2014. M. Caporin, M. Costola, G. M. Jannin, and B. B. Maillet. On the (ab) use of omega? SSRN Working Paper Series, 2016. P. Carr, H. Geman, D. B. Madan, and M. Yor. The fine structure of asset returns: An empirical investigation. Journal of Business, 75(2):305–333, 2002. F. Delbaen and W. Schachermayer. The mathematics of arbitrage. Springer Science & Business Media, 2006. P. H. Dybvig. Distributional analysis of portfolio choice. Journal of Business, 61(3):369–393, 1988. E. Eberlein, U. Keller, and K. Prause. New insights into smile, mispricing, and value at risk: The hyperbolic model. Journal of Business, 71(3):371–405, 1998. S. Farinelli and L. Tibiletti. Sharpe thinking in asset ranking with one-sided measures. European Journal of Operational Research, 185(3):1542–1547, 2008.
25
H. R. Fogler and R. C. Radcliffe. A note on measurement of skewness. Journal of Financial and Quantitative Analysis, 9(3):485–489, 1974. G. Guastaroba, R. Mansini, W. Ogryczak, and M. G. Speranza. Linear programming models based on omega ratio for the enhanced index tracking problem. European Journal of Operational Research, 251(3):938–956, 2016. X. D. He and X. Y. Zhou. Portfolio choice via quantiles. Mathematical Finance, 21(2):203–231, 2011. E. Jakobsons and S. Vanduffel. Dependence uncertainty bounds for the expectile of a portfolio. Risks, 3 (4):599–623, 2015. S. Kane, M. C. Bartholomew-Biggs, M. Cross, and M. Dewar. Optimizing omega. Journal of Global Optimization, 45(1):153–167, 2009. P. D. Kaplan and J. A. Knowles. Kappa: A generalized downside risk-adjusted performance measure. Journal of Performance Measurement., 8:42–54, 2004. H. Kazemi, T. Schneeweis, and B. Gupta. Omega as a performance measure. Journal of Performance Measurement, 8:16–25, 2004. C. Keating and W. F. Shadwick. An introduction to Omega. AIMA Newsletter, 2002. B. Klar and A. M¨ uller. On consistency of the omega ratio with stochastic dominance rules. Working paper, 2017. U. K¨ uchler, K. Neumann, M. Sørensen, and A. Streller. Stock returns and hyperbolic distributions. Mathematical and Computer Modelling, 29(10):1–15, 1999. H.-S. Lau and J. R. Wingender. The analytics of the intervaling effect on skewness and kurtosis of stock returns. Financial Review, 24(2):215–233, 1989. H. Markowitz. Portfolio selection. Journal of Finance, 7(1):77–91, 1952. R. C. Merton. Lifetime portfolio selection under uncertainty: The continuous-time case. The review of Economics and Statistics, pages 247–257, 1969. R. C. Merton. Optimum consumption and portfolio rules in a continuous-time model. Journal of economic theory, 3(4):373–413, 1971. A. M¨ uller. On the interplay between variability and negative dependence for bivariate distributions. Operations Research Letters, 31(2):90–94, 2003. A. M¨ uller and M. Scarsini. Stochastic comparison of random vectors with a common copula. Mathematics of Operations Research, 26(4):723–740, 2001. C. S. Pedersen, S. E. Satchell, et al. On the foundation of performance measures under asymmetric returns. Quantitative Finance, 2(3):217–223, 2002. M. Shaked and J. G. Shanthikumar. Stochastic orders. Springer Science & Business Media, 2007. W. F. Sharpe. Mutual fund performance. Journal of Business, 39(1):119–138, 1966. F. A. Sortino and L. N. Price. Performance measurement in a downside risk framework. the Journal of Investing, 3(3):59–64, 1994. F. Van Dyk, G. van Vuuren, and A. Heymans. Hedge fund performance evaluation using the sharpe and omega ratios. International Business & Economics Research Journal, 13(3):485, 2014. R. Vilkancas. Characteristics of omega-optimized portfolios at different levels of threshold returns. Business, Management and Education, 12(2):245–265, 2014.
26
J. Von Neumann and O. Morgenstern. Theory of games and economic behavior, 2nd Ed. Princeton University Press, 1947. V. Zakamouline. Portfolio performance evaluation with loss aversion. Quantitative Finance, 14(4):699– 710, 2014. V. Zakamouline and S. Koekebakker. Portfolio performance evaluation with generalized sharpe ratios: Beyond the mean and variance. Journal of Banking & Finance, 33(7):1242–1254, 2009. J. F. Ziegel. Coherence and elicitability. Mathematical Finance, 26(4):901–918, 2016.
A A.1
Proofs Proof of Proposition 6
Proof. Since ξT is decreasing in ST , it follows from Proposition 1 that the optimal payoff XT∗ is of the form
XT∗
=
0
if ST 6 a
L
if a < ST 6 b, b > a > 0,
M
if b < ST
and we only need to specify a and b. The Omega ratio of XT∗ is given as Ω(XT∗ ) =
(M − L)P(ST > b) AP(ST > b) = . LP(ST < a) P(ST < a)
The cost of XT∗ satisfies M Q(ST > b) + LQ(a 6 ST < b) = ω0 erT from which the following relationship between a and b is obtained: Q(ST < a) = B − AQ(ST < b).
(20)
From the the definition of Y (x), −1
Y (a) = Φ
C C [−AΦ Y (b) + + B] − . 2 2
(21)
Using (21), Ω(XT∗ ) can be written as a function of b, ΩL (b) :=
Ω(XT∗ )
=
AΦ −Y (b) + Φ Φ−1 −AΦ Y (b) +
27
C 2
C 2
+B −C
! ,
(22)
where we have also used that16 P(ST > x) = Φ(Φ−1 (Q(ST > x)) + C) = Φ −Y (x) +
C 2
. Since
b > a, ΩL (b) is defined for b ∈ [bmin , +∞) in which bmin
β1 √ √ ω0 erT −1 − 0.5θ T + rT := S0 α exp θ T Φ 1− . M
The derivative of ΩL (b) is given as Ω0L (b) =
φ(−Y (b) + C2 )h(Y (b))Y 0 (b) 2 , Φ Φ−1 −AΦ Y (b) + C2 + B − C
in which C C C −1 + B − Y (b) − C − h(Y (b)) :=A Φ( − Y (b)) exp Φ −AΦ Y (b) + 2 2 2 C + B] − C . AΦ Φ−1 [−AΦ Y (b) + 2 2
∂h(Y (b)) ∂b
The sign of Ω0L (b) is determined by h(Y (b)). By calculating
, we find that h(Y (b)) is
strictly decreasing in b and also that lim h(Y (b)) = −Φ Φ
−1
b→+∞
ω0 erT 1− M
√
−θ T
< 0.
√ It can be verified that h(Y (bmin )) > 0 ⇐⇒ L < M Φ θ T − Φ−1 1 − there is only one root b0 of the equation
Ω0L (b)
ω0 erT M
, in which case
= 0 (or equivalently of the equation h(Y (b)) = 0)
and ΩL (b) attains its maximum value in b0 . From the relationship between a and b, we obtain −1 that a0 is the solution (b0 ) + C2 ) + B − C2 . By contrast, h(Y (bmin )) 6 = Φ rT −AΦ(Y √ to Y (x) e 0 ⇐⇒ L > M Φ θ T − Φ−1 1 − ω0M in which case ΩL (b) is a decreasing function of b and attains its maximum in bmin .
A.2
Proof of Proposition 7
Proof. The Omega ratio of a constant-mix strategy XTλ is given as E[XTλ ] − L +1 E(L − XTλ )+ exp(d3 (λ)) − L = + 1, LΦ(−d2 (λ)) − exp(d3 (λ))Φ(−d1 (λ))
Ω(XTλ ) =
√ where d1 (λ) = − log(L)−((1−λ)r+λµ+0.5λ λσ T 16
Note that under P, ln
ST S0
∼ N ((µ −
2 σ 2 )T
√ , d2 (λ) = d1 (λ)−λσ T and d3 (λ) = ((1−λ)r+λµ)T .
σ2 )T, σ 2 T ) 2
and under Q, ln
28
ST S0
∼ N ((r −
σ2 )T, σ 2 T ). 2
To study the sensitivity of Ω(λ) := Ω(XTλ ) with respect to λ, we calculate the derivative 0
Ω (λ) =
√ √ 2 (λ)) L T exp(d3 (λ))φ(d1 (λ))((µ − r) T Φ(d1 (λ))−Φ(d − σ( exp(dL3 (λ)) − 1)) φ(d1 (λ)) (LΦ(−d2 (λ)) − exp(d3 (λ))Φ(−d1 (λ)))2
.
(23)
Hence, we obtain that Ω0 (λ) > 0 ⇔ G(λ) > 0 (note that L > erT ) in which √ Φ(d1 (λ)) − Φ(d2 (λ)) G(λ) := (µ − r) T −σ φ(d1 (λ))
exp(d3 (λ)) −1 L
(24)
We find then that the derivative of G(λ) is given as √ 0 φ(d2 (λ)) d1 (λ)(Φ(d1 (λ)) − Φ(d2 (λ))) G (λ) = (µ − r) T d1 (λ) 1 − + φ(d1 (λ)) φ(d1 (λ)) √ √ log(L) − rT φ(d1 (λ)) − φ(d2 (λ)) + d1 (λ)(Φ(d1 (λ)) − Φ(d2 (λ))) √ , + 0.5σ T = (µ − r) T φ(d1 (λ)) λ2 σ T 0
and thus G0 (λ) > 0 ⇔ φ(d1 (λ)) − φ(d2 (λ)) + d1 (λ)(Φ(d1 (λ)) − Φ(d2 (λ))) > 0. Define H(x) := φ(x) − φ(x − A) + x(Φ(x) − Φ(x − A)), where A is a positive constant. Since Φ(x) is a logconcave function it holds that the increasing function φ(x) + xΦ(x) is also non-negative. This √ implies that H(x) > 0. Choosing x = d1 (λ) and A = λσ T , we find G0 (λ) > 0 and G(λ) is thus an increasing function in λ, λ > 0. Since limλ→0+ G(λ) = σ(1 −
erT L
) > 0, it follows that
G(λ) > 0 when λ > 0 and thus also that Ω(λ) is an increasing function of λ, λ > 0. Finally, limλ→+∞ (Ω(XTλ )) = +∞ because its denominator is upper bounded by L and its numerator converges to +∞, which gives a proof of the second statement of the proposition.
29
