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Cutting edge

l

Investment management

Generalising universal

performance measures

Performance and risk measurement are fundamental quantitative activities in finance, and

new ways of measuring them are always of interest. A recently proposed procedure is the

universal performance measure. Theofanis Darsinos and Stephen Satchell show that, in fact,

it is a special case of a class of measures and not necessarily the best measure in this class.

Consultants and fund managers may find these methods appealing, however, as they do

provide a succinct summary of financial performance/risk

W

hen distributions are symmetric and the classical mean-variance

capital asset pricing model (CAPM) is valid, performance measures can be extracted directly from the model. In such a framework, the Sharpe ratio, introduced in Sharpe (1966), is the most popular

and commonly used statistic. It is simply defined as the expected excess

return of an investment w over a benchmark/reference point r (that is,

E[(w – r)]), divided by the corresponding standard deviation. For the classical Sharpe ratio, r is the risk-free rate of return rf. More recently, a generalised version (Sharpe, 1994), called the information ratio, replaced the

risk-free rate with the arbitrary benchmark r. Other measures that evaluate the performance of an investment portfolio relative to a benchmark are

Jensen’s alpha and the Treynor ratio.

However, when returns are asymmetric and mean-variance rules are no

longer efficient, the aforementioned measures cease to capture the essential features of the distribution. While this has been a widely recognised

problem in finance, not least because of the presence of skewness in financial data, remedies have only recently been suggested. In the early

1990s, Sortino & Van der Meer (1991) considered the failings of the Sharpe

ratio and introduced a risk-adjusted performance measure, which came to

be known as the Sortino ratio. The Sortino ratio is the equivalent of the

Sharpe ratio in downside volatility space and thus considers the expected

excess return (w – r) over the benchmark r, this time divided by a measure of downside risk (the second lower partial moment). Even more recently, Keating & Shadwick (2002) and Cascon, Keating & Shadwick (2002),

introduced a new measure known as the omega function, defined as the

ratio of the payout of a ‘virtual’ call option E[max(w – r, 0)] over the payout of a ‘virtual’ put option E([max(r – w, 0)]. They describe it as a universal performance measure, hence the title of this article (more information

on both the omega function and the Sortino ratio are provided below).

The aim of this article is to present a discussion, generalisation and

some new results on the aforementioned functions. We show that the omega

function is a specialisation of a risk measure derived from prospect theory, and as such enjoys certain advantages over performance measures derived from expected utility theory. It is consistent with stochastic dominance

and can be interpreted in the same way as the Sortino ratio or other measures that divide expected excess return by some measure of downside

risk (see, for example, Sortino & Satchell, 2001, and Knight & Satchell,

2002, among others). Although easy to calculate numerically it is also possible to calculate some explicit formulas for certain distributions, and we

do so in the fourth section of this article.

The structure of the article is as follows. In the next section, we provide a discussion of the relevant features of prospect theory to help readers unfamiliar with this subject. We then establish the connection between

prospect theory and the omega function, and provide expressions for general omega functions (generalised universal performance measures). The

80

RISK JUNE 2004 ● WWW.RISK.NET

subsequent section introduces stochastic dominance and deals with dominance results and the consistency of the omega function. We then present

some analytic results and an empirical illustration with the FTSE 100 and

the S&P 500 indexes, and then conclude.

Risk aversion, prospect theory and performance measurement

As pointed out by Rabin & Thaler (2001), the hypothesis of expected utility maximisation has already been sufficiently criticised and often shown

to fail in explaining the behaviour of people acting in the presence of uncertainty. Indeed, it is becoming increasingly accepted that risk aversion

can be better explained by extending the expected utility framework using

prospect theory. Prospect theory (see Kahneman & Tversky, 1979, and

Tversky & Kahneman, 1992) is an approach to decision-making that measures the utility of gain and loss relative to some reference point r. If r

were fixed, it would be no different from the two-piece Von NewmannMorgenstern utility functions advocated by Fishburn & Kochenberger

(1979). How it differs is that r is variable and often dependent on initial

wealth w0. Another difference between prospect theory and expected utility, not discussed further here, is that prospect theoretic decisions can be

based on transformed probabilities f(p) so that unlikely events such as outof-the-money options are deemed more likely than their objective probability p would suggest (the converse applies for more probable events).1

In prospect theory, a full description of preferences involves specifying the

utility function relative to r, V– r > 0. Thus, letting U(⋅) be utility, with w being

(final) wealth and r = r(w0) being the reference point, it is usual to write

the prospect utility function as:

U ( w) = U1 ( w − r ) if w > r

= −U 2 ( r − w) if w ≤ r

(1)

for U1(⋅) and U2(⋅) increasing functions. Sometimes, to emphasise the asymmetry between gains (w – r) and losses (r – w), we write (1) as:

U ( w) = U1 ( w − r ) if w > r

(2)

= − λU 2 ( r − w) if w ≤ r

for λ > 0. Such a specification is called a loss aversion utility function.

Suppose now that for r non-stochastic we consider the function:

E U1 ( w − r ) / w > r

ψ w (r ) =

E U 2 ( r − w) / w ≤ r

1

The characteristic features of cumulative prospect theory are rank-dependence,

reference dependence and sign-dependence. Recently, Schmidt (2003) has discussed

(cumulative) prospect theoretic reference dependence and developed an axiomatisation.

Indeed, to derive the general functional form often complex conditions are required

beyond the standard properties (continuity, weak ordering, stochastic dominance).

Schmidt (2003) uses such concepts as sign-dependent comonotonic trade-off consistency.

The conditions can be less complex if a particular parametric form for utility is desired

Such a function measures the expected utility of gains divided by (minus)

the expected utility of losses and seems a natural metric of performance.

Thus if w1 and w2 were two random prospects and ψw1(r) > ψw2(r) for all r

it would seem plausible that ψ would be a sensible metric to rank alternative investments, since ψw1(r) would be larger if w1 has the larger expected gain and the smaller expected loss, in much the same way as a higher

Sharpe ratio is implied by a larger mean and a smaller standard deviation.

Suppose now that, following numerous authors, we parameterise U1(⋅)

and U2(⋅) by:

U1 ( w − r ) = ( w − r )

U 2 ( r − w) = ( r − w)

n

n

(3)

Ω (r ) =

E max ( w − r , 0)

E max ( r − w, 0)

=

I 2 (r )

I1 ( r )

where:

I1 ( r ) = ∫a F ( w) dw

r

and:

I 2 ( r ) = ∫r (1 − F ( w)) dw

b

a ≤ r ≤ b and D = [a, b] is the domain of the distribution F(⋅). Rewriting

the above as:

I1 ( r ) = ∫a F ( w) dw

then:

r

E ( w − r ) w > r µ +

= n

−

n

E ( r − w) w ≤ r µ n

n

ψ w (r ) ≡ ψ n (r ) =

= wF ( w) a − ∫ wf ( w) dw

a

r

(4)

= rF ( r ) − E w w ≤ r F ( r )

where:

r

µ −n = E ( r − w) w ≤ r =

n

∫ ( r − w)

−∞

n

pdf ( w) dw

prob ( w ≤ r )

= E r − w w ≤ r pr

(5)

denotes the nth conditional lower partial moment and similarly µn+ denotes

the nth conditional upper partial moment and is defined analogously to

equation (5). It is these last definitions that we shall proceed with in further calculations. Consider now:

n

µ nr = E ( w − r )

n

= E ( w − r ) w > r × prob ( w > r )

(6)

and similarly:

I 2 ( r ) = ∫r (1 − F ( w)) dw

b

= E w w > r (1 − F ( r )) − r (1 − F ( r ))

= E w − r w > r (1 − pr )

we see that the omega function is a special case (n = 1) of the ψn(⋅) function of prospect theory (defined in equation (4)):

Ω ( r ) ≡ Ω1 ( r )

+ E ( w − r ) w ≤ r × prob ( w ≤ r )

µ nr =

(1 − pr ) + ( −1)

n

µ n− pr

Ω1 (µ ) = 1

Now from (10) we have:

1 1

1− j

∑ j µ j (µ − r )

pr 1 − pr

j =0

Ω1 ( r ) =

+

−

1 − pr pr

(1 − pr ) µ1

Further calculations follow from the fact that E[w] = µ, since:

n

µ nr = E ( w − r )

n

= E ( w − µ + µ − r )

n n

j

n− j

= E ∑ ( w − µ ) (µ − r )

j =0 j

Now denoting the jth moment about the mean as:

j

µ j = E ( w − µ )

=

µ nr =

n

n

n− j

(9)

j =0

Hence:

n

∑ j µ j (µ − r )

j =0

n− j

(1 − )

pr µ −n

− ( −1)

n

pr

1 − pr

(10)

and:

lim ψ n ( r ) ≡ ψ n (µ ) =

r →µ

µn

(1 − pr ) µ n−

µ−r

= Ω1 ( r ) − 1

µ1− pr

(8)

∑ j µ j (µ − r )

− ( −1)

n

pr

1 − pr

µ−R

+1

µ1− pr

Hence:

(µ1 = 0), we have:

ψ n (r ) =

1 − pr

pr

Also, from (11) it follows that:

µ nr

pr

n

ψ n (r ) =

− ( −1)

−

1 − pr

(1 − pr ) µ n

n

E r − w w ≤ r pr

= ψ1 ( r )

(7)

where pr = prob(w ≤ r). So:

E w − r w > r (1 − pr )

=

n

µ n+

r

(11)

Cascon, Keating & Shadwick (2002) define the omega function (universal performance measure) as:

(12)

We have thus shown that the omega function has a natural performance

measurement interpretation and can actually be interpreted in the same way

as measures that divide expected excess return by some measure of downside risk. It is also interesting to note that in the context of the gain/loss literature, Bernardo & Ledoit (2000) have considered the gain over loss function,

defined as expected positive excess returns divided by the expected negative excess returns under some risk-adjusted probability measure to develop a theory for asset pricing. Their insights can be extended to the omega

function. Furthermore, it is obvious from the above analysis that omega functions of higher order (that is, Ωn(r), n > 1) can also be defined in terms of

the ψn(r) function. Thus for the general omega function we have:

n

Ωn ( r )

(µ − r )

=

pr µ −n

n

+

n

∑ j µ j (µ − r )

j =1

pr µ −n

n− j

− ( −1)

n

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Investment management

E ( X ) = E (Y )

If we wish to have a scale invariant distribution, generalising a Sharpe ratio,

we arrive at the following generalised universal performance measure2:

n

(µ − r ) = n Ω

n

pr µ −n

n

(r ) −

and:

n

∑ j µ j (µ − r )

j =1

pr µ n−

n− j

r

+ ( −1)

n

S ( r ) = ∫ ( FX ( z ) − FY ( z )) dz ≤ 0 ∀r ∈[0,1]

(13)

For example, when n = 2, the omega function Ω2(r) corresponds to the

Sortino ratio3 so that:

(µ − r ) = 2 Ω r − µ 2 + 1

(14)

2( )

2 p µ−

pr µ −2

r 2

But note that, unlike Ω1(r), extra information is needed to calculate formula (13). We therefore lose the one-to-one relationship between the riskadjusted performance measure and Ωn(r), when n > 1.

Since we are generalising the previous risk measures (see equations

(12) and (14)), it is worth noting that the class of risk measures is expanded,

while the class of distributions applicable to our generalisation is reduced.

This is because omega needs only the first moment, Sortino needs the second, while µ+n and µ–n will need the nth. In this sense, our article both generalises and specialises the omega function.4 It should be noted here that

Farinelli & Tibiletti (2002) have also advocated a related variation of the

omega function and provided a similar generalisation to ours, but theirs

differs in that the numerator is an arbitrary upper partial moment while the

denominator is similar.

0

≥ Y then ψ (r) ≥ ψ (r) V

– r ∈ [0, 1]. For proof, see

Proposition 1. If X SSD

1x

1y

the appendix.

We finally consider the case of second-order stochastic monotonic dominance (MSSD). Here, the set of utility functions D*2 is defined as all riskaverse and non-satiable individuals. It is well known (see, for example,

Huang & Litzenberger, 1988, section 2.9) that this is equivalent to E(X) ≥

E(Y) and S(r) ≤ 0, V– r ∈ [0, 1], so that the result follows from a slight adjustment of the previous arguments. Indeed, in this case ψ1x(r) ≥ ψ1y(r) is

an immediate consequence of FSD.

Further refinements for ψnx(r) and ψny(r) can also be proved in terms

of higher-order stochastic dominance. The arguments follow similar lines

to the ones presented above involving successive partial integrations. The

basic relationship stems from the fact that:

r

pr µ −n = ∫ ( r − w) pdf ( w) dw = n! F n ( r )

n

0

where:

r

F n ( r ) = ∫ F n −1 ( s ) ds

0

Dominance results and the ψn(r) function

Dominance is a topic that is little used in practice since the existence of

a dominated asset/portfolio would create arbitrage opportunities. We say

that one asset dominates (D) another if it always outperforms it. Firstorder stochastic dominance (FSD) is a weaker comparison than dominance (D) since for asset A to stochastically dominate asset B, it need not

always outperform it. Instead, the probability of A exceeding any given

level of return should be higher than B (geometrically, this is equivalent

to the cumulative return distribution function (CDF) of A always lying

below (or touching) the CDF of B, but never crossing it. Hence, A always

has a lower probability of poor returns than B. Unlike dominated assets,

stochastically dominated assets may actually exist, although not as optimal holdings for any investors. In a CAPM world, if asset A has a higher

mean and lower risk than asset B, then all mean-variance investors should

prefer A, and so A mean-variance dominates B.

In this section, we discuss the relationship between notions of stochastic

dominance and their implications for generalised universal performance

measures. Let X and Y be the rates of return of two prospects. It is an im≥ Y, that if X is preferred to

mediate consequence of FSD, denoted by X FSD

Y for all individuals with increasing utility functions then FX(r) ≤ FY(r) V– r

∈ [0, 1] (FX(⋅) and FY(⋅) are the distribution functions of X and Y). This is

– ≤

an if-and-only-if relationship. Now FSD implies that E(X) ≥ E(Y) and µ1x

– and µ– are the lower partial moments of the two prospects

µ–1y, where µ1x

1y

as defined for example in equation (5), so that we have ψ1x(r) ≥ ψ1y(r).

We next consider second-degree stochastic dominance (SSD), denoted

≥ Y. This case is not implied by FSD since individuals may have utilby XSSD

ity functions that are not necessarily monotonically increasing. Here we

basically assume that the only information we have about an individual is

that they are risk-averse. We thus assume that rates of return lie on [0, 1]

and that the set of utility functions D2 are those that are risk-averse and

whose first derivatives are continuous except for a countable subset of [1,

≥ Y if and only if:

2]. Let u ∈ D2 be any element of D2; then X SSD

1

1

0

0

∫ u (1 + x) dFX ( x) ≥ ∫ u (1 + y ) dFY ( y )

≥

It is also well known (Huang & Litzenberger, 1988, section 2.5) that X SSD

Y if and only if:

82

RISK JUNE 2004 ● WWW.RISK.NET

for n = 1, 2, ... and:

r

F 0 ( r ) = ∫ pdf ( s ) ds

0

It follows immediately that for D*n = {u; u′(⋅) ≥ 0, u′′(⋅) ≤ 0, u′′′(⋅) ≥ 0, ... ,

un(⋅)(–1)n ≥ 0} (that is, those utility functions whose first n derivatives alternate in sign), it is a consequence of proposition 1 that for two gambles

X and Y:

r

(

)

⌠ F n −1 ( s ) − F n −1 ( s ) ds ≤ 0∀r

Y

⌡ X

0

will imply that prxµ–nx ≤ pryµ–1y and since u′(⋅) ≥ 0 we should also have µX

≥ µY. We have thus proven the following proposition.

Proposition 2. If X n-order stochastically dominates Y (NSD), denoted

≥ Y, then:

X NSD

µX − r

µ −r

≥ Y

−

−

n

n

prx µ nx

pry µ ny

V– r. For proof, see the appendix.

Proposition 2 tells us that any person with u ∈ D*3 will prefer asset X

to asset Y if the Sortino ratio (n = 2) of X is greater than or equal to the

Sortino ratio of Y, for all r. However, anybody with u ∈ D*2 will prefer asset

2

Note here that the comparison of a Sharpe ratio is at a given point r. This is not

equivalent to a comparison of a generalised measure (for example, the omega function)

since such measures are compared for all r

3 The Sortino ratio considers the expected excess return over a target threshold divided by

a measure of downside risk. Mathematically, the measure is usually expressed as:

µ−r

S=

r

2

2

∫ ( r − w) pdf ( w) dw

−∞

where:

2

∫−∞ ( r − w)

r

2

pdf ( w) dw

is known as the second lower partial moment. Following our notation, the equivalent

expression for the Sortino ratio is:

µ−r

2

pr µ −2

Readers unfamiliar with the Sortino ratio can find a good discussion in Rom & Ferguson

(2001)

4 We are grateful to an anonymous referee for drawing our attention to this point

A. Summary statistics of monthly relative

prices/returns (that is, Pt/Pt – 1) for the FTSE

100 and S&P 500 index

X to asset Y if the omega function of X is greater than or equal to the omega

function of Y, for all r. But which of these two is the more plausible?

It seems to us that the Sortino ratio should be preferred for the following reason. If u ∈ D*3 then it is straightforward to prove that u(⋅) must exhibit decreasing absolute risk aversion. Decreasing absolute risk aversion

matters because it implies that as your wealth increases your dollar investment in a risky asset goes up as well. There is a great deal of controversy

about whether this is a sensible requirement for investors. If, however, u

∈ D*2, then an investor can have increasing absolute risk aversion, as in the

case of quadratic utility. Such a person will prefer assets based on higher

omega, notwithstanding his/her tendency to hold less money in risky assets as he/she gets wealthier. This consideration is a bit concerning.

FTSE 100

0.88042

1.08857

1.00355

1.0055

0.0434

–0.587539

0.27487

0.88042

0.91191

1.06463

Minimum

Maximum

Mean

Median (50% quantile)

Standard deviation

Skewness

Excess kurtosis

1% quantile

5% quantile

95% quantile

Note: April 1995–May 2003

Some analytic results

The easiest way to proceed is to estimate Ω1(r) or, indeed, any of the measures by employing historical data. However, for the parametrically inclined reader we provide some analytic results based on a suitable

parameterisation of the relative return distribution.5 A tractable and relevant choice for the distribution of relative prices is the Weibull distribution. For example, Mittnik & Rachev (1993), using the Kolmogorov-Smirnov

test, suggest that the Weibull distribution is the most suitable candidate to

describe S&P 500 daily returns. The Weibull distribution has also recently

been used in finance by Sornette, Simonetti & Andersen (2000), Malevergne & Sornette (2003) and Sancetta & Satchell (2002), among others. One

of its strengths is its ability to model the tails of a distribution and it provides closed-form expressions (in terms of incomplete gamma functions)

for partial moments, and the distribution function moments.

We have shown that:

µ−r

= Ω1 ( r ) − 1

µ1− pr

S&P 500

0.85420

1.09672

1.00789

1.00961

0.0488

–0.5506

–0.0247

0.85420

0.91993

1.07815

1. Omega function (universal performance

measure of order n = 1)

Ω

100

FTSE 100

S&P 500

80

60

40

20

Cross point of FTSE

100 with S&P 500

measure at r = 0.92

r

0.90

0.95

1.00

1.05

1.10

Suppose now that wealth/price/relative price w is governed by a Weibull

distribution with shape parameter a and scale parameter λ. It follows that:

r a

pr = prob ( w ≤ r ) = F ( r ) = 1 − exp −

λ

2. Sortino ratio (universal performance

measure of order n = 2)

(15)

S

and:

f ( w) = pdf ( w) = a

w

λa

a −1

w

exp −

λ

(16)

15

The omega function, Ω1(r), for a Weibull distribution with shape parameter a and scale parameter λ is given by:

Ω1 ( r ) = 1 +

(

)− r

a

r 1 − exp ( − ( λr ) ) − λ Γ (1 + a1 ) − Γ (1 + a1 , r

λ

λΓ 1 +

1

a

10

µ−r

µ 2− pr

(

λΓ 1 +

2

( ( ) ) − 2rλ Γ (1 +

2

r 1 − exp −

r

λ

a

1

a

1

a

a

)

(17)

=

)− r

) − Γ (1 + 1a , λr

a

a

) + λ Γ (1 + ) − Γ (1 +

2

2

a

Cross point of FTSE

100 with S&P 500

measure at r = 0.949

5

a

r

0.90

where Γ(z) is the gamma function evaluated at the value of z. Furthermore,

an analytic expression for the Sortino ratio under Weibull law is:

2

FTSE 100

S&P 500

20

a

2 ra

,

a λa

)

where Γ(c, z) is the incomplete gamma function. Estimation of the Weibull

parameters a and λ can be performed using various analytic methods such

as least squares regression, maximum likelihood or the method of moments.

Empirical calculations. We now provide an empirical illustration of

the theory presented above using the FTSE 100 index and the S&P 500

index as two representative funds to be compared. From the class of generalised universal performance measures discussed above, we concentrate

for simplicity on two measures. First, the generalised universal performance

measure of order n = 1 (that is, (µ – r)/(µ–1 pr) = Ω1(r) – 1); we shall call

0.95

1.00

1.05

1.10

this the omega measure because of the measure’s one-to-one relationship

with the Ω1(r) function). Second, the generalised universal performance

measure of order n = 2, that is:

µ−r

2

µ −2 pr

This is the Sortino ratio/measure. We use monthly closing price data for

the two indexes for the period from April 1995 to May 2003. We then

transform the closing price (level) data to relative prices (or relative returns). This basically means that if Pt denotes the closing price of the index

at day t, we define the relative price (relative return) for day t as xt = Pt /Pt

– 1. In table A, we present summary statistics for the monthly relative

prices/returns for the FTSE 100 and the S&P 500 indexes over the period

5

Proofs of the results in this section are available upon request from the authors

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83

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April 1995 to May 2003. It should be noted that skewness of relative prices

is negative for both indexes.

To calculate the performance measures (µ – r)/(µ–1 pr) and:

µ−r

2

Appendix

≥ Y then S(r) ≤ 0 V

–r ∈ [0, 1] by (b).

Proof of proposition 1. If X SSD

This then implies that:

µ −2 pr

r

∫ ( FX ( z ) − FY ( z )) dz

non-parametrically (empirically) we substitute:

0

N

µ = (1 / N ) ∑ xt

r

t =1

N

0

0

(

)

(

∈[0,1] ⇒ prx r − µ1−x ≤ pry r − µ1−y

t =1

and:

N

r

= prx − pry r − prx µ1−x + pry µ1−y ≤ 0 ∀r

µ1− pr = (1 / N ) ∑ ( r − xt )I t

µ −2 pr

r

= FX ( z ) − FY ( z ) z 0 − ∫ zpdf X ( z ) dz + ∫ zpdfY ( z ) dz

Note that prx = prob(x ≤ r). Similarly defined is pry. Now since E(X)

= E(Y) from (a), the result follows.

Proof of proposition 2. NSD implies that V– u ∈ D*n, E(u(X)) ≥

E(u(Y)) which in turn implies that µX ≥ µY and:

= (1 / N ) ∑ ( r − xt ) I t

2

t =1

where It denotes the indicator function and is defined as It = 1 whenever

xt ≤ r and It = 0 whenever xt > r.

In figure 1, we compare the FTSE 100 with the S&P 500 by means of

their empirical omega measures.6 We see clear evidence that the omega

function for the FTSE 100 dominates that of the S&P 500 in the left tail. For

example, the FTSE 100 should be preferred whenever r < 0.92. The opposite is true when the inequality is reversed, and fund managers with r

> 0.92 would be better off with the S&P 500. Indeed, we would not expect to see complete outperformance by the omega or other measures for

all values of r as this corresponds to strong requirements of stochastic dominance, as proved in propositions 1 and 2. Essentially, this would suggest

that one market, either the US or the UK, is redundant.

In figure 2, we compare the two funds by means of their Sortino measure. The central message is much the same, with the Sortino ratio suggesting that the FTSE 100 should be preferred whenever r < 0.95. All this

may seem inconclusive, since an individual with a low benchmark would

prefer the UK market to the US, while another with a higher benchmark

would prefer the US to the UK. These individuals differ in that the choice

of r reflects where they measure risk from. But this gets to the essential

point we want to emphasise in this article. Comparing an omega function

or similar measure for all benchmarks r is much more stringent than at a

particular point r, as in traditional Sharpe ratio analysis.

)

r

n −1

n −1

∫ ( FX ( s) − FY ( s)) ds ≤ 0

0

the omega function and the Sortino ratio as special cases. These functions

can be motivated by reference-dependent prospect theory and can be demonstrated to be compatible with stochastic dominance. We derive analytic results for these measures based on the Weibull distribution and estimate our

model for both the FTSE 100 and the S&P 500 monthly prices for the past

eight years. Not surprisingly, we fail to find total dominance of one market

over the other (which would deem the dominated market redundant) although we do find evidence that the FTSE 100 dominates the S&P 500 for

large losses based on monthly returns for the past eight years.

Theofanis Darsinos is an associate in the fixed-income and relative value

research group at Deutsche Bank in London. Stephen Satchell is reader

in financial econometrics in the faculty of economics and politics at the

University of Cambridge. The authors would like to thank two anonymous

referees for useful comments that greatly improved the manuscript. The

opinions or recommendations expressed in this article are those of the

authors and are not representative of Deutsche Bank AG as a whole.

Email: theo.darsinos@db.com, steve.satchell@econ.cam.ac.uk

Conclusion

This article presents an analysis of a new family of risk/performance measures that we call generalised universal performance measures. These include

6

Similar results are obtained using the Weibull parameterisation and are available upon

request to the authors

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