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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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81

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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)


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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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

REFERENCES
Bernardo A and O Ledoit, 2000
Gain, loss and asset pricing
Journal of Political Economy 8(1), pages
144–172
Cascon A, C Keating and W
Shadwick, 2002
The omega function
Working paper, Finance Development
Centre, London
Farinelli S and L Tibiletti, 2002
Sharpe thinking with asymmetrical
preferences
Mimeo
Fishburn P and G Kochenberger,
1979
Concepts, theory, techniques; two-piece
Von Newmann-Morgenstern utility
functions
Decision Sciences 10, pages 503–518
Huang C and R Litzenberger,
1988
Foundations for financial economics
North-Holland

84

RISK JUNE 2004 ● WWW.RISK.NET

Kahneman D and A Tversky, 1979
Prospect theory: an analysis of decision
under risk
Econometrica 47(2), pages 263–292
Keating C and W Shadwick, 2002
A universal performance measure
Journal of Performance Measurement
6(3), pages 59–84
Knight J and S Satchell, 2002
Performance measurement in finance
Butterworth-Heinemann Finance,
Quantitative Finance Series
Malevergne Y and D Sornette,
2003
VAR-efficient portfolios for a class of
super- and sub-exponentially decaying
assets return distributions
Forthcoming in Quantitative Finance
Mittnik S and S Rachev, 1993
Modeling asset returns with alternative
stable distributions
Econometric Reviews 12, pages
261–330

Rabin M and R Thaler, 2001
Anomalies: risk aversion
Journal of Economic Perspectives 15,
pages 219–232
Rom B and K Ferguson, 2001
A software developer’s view: using postmodern portfolio theory to improve
Investment performance measurement
In Managing Downside Risk in Financial
Markets, edited by Sortino and Satchell,
Butterworth Heinemann
Sancetta A and S Satchell, 2002
The hypercube transform
Preprint
Schmidt U, 2003
Reference-dependence in cumulative
prospect theory
Journal of Mathematical Psychology 47,
pages 122–131
Sharpe W, 1966
Mutual funds performance
Journal of Business, January, pages
119–138

Sharpe W, 1994
The Sharpe ratio
Journal of Portfolio Management, fall,
pages 49–58
Sornette D, P Simonetti and J
Andersen, 2000
φq-field theory for portfolio optimization:
“fat-tails” and non-linear correlations
Physics Reports 335, pages 19–92
Sortino F and S Satchell, 2001
Managing downside risk in financial
markets
Butterworth Heinemann
Sortino F and R Van der Meer,
1991
Downside risk
Journal of Portfolio Management, summer, pages 27–31
Tversky A and D Kahneman, 1992
Advances in prospect theory: cumulative
representation of uncertainty
Journal of Risk and Uncertainty 5,
pages 297–323



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