sesión 4. articulo omega

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Omega 32 (2004) 273–284www.elsevier.com/locate/dsw

An analysis of Spanish investment fund performance:some considerations concerning Sharpe’s ratio

Luis Ferruz Agudo∗, Jos*e Luis Sarto MarzalFaculty of Economics and Business Studies, University of Zaragoza, Gran V��a 2, 50005-Zaragoza, Spain

Received 20 November 2002; accepted 25 November 2003

Abstract

This paper concentrates on the 2nancial analysis of investment performance taking Sharpe’s ratio as a basic point ofreference, as well as giving further consideration to the use of this performance measure as an approximation to a utility index.

We also propose certain changes to Sharpe’s ratio which would, on the one hand, avoid the appearance of inconsistentassessments and, on the other, provide an approach to the use of Sharpe’s performance measure as a utility index. All of themeasures involved in this study have been applied to a sample of Spanish investment funds.? 2004 Elsevier Ltd. All rights reserved.

Keywords: Return; Risk; Fund management performance; Investment funds; Utility indices

1. Introduction

The object of this paper is to analyse Sharpe’s ratio [1–4]not only from the point of view of its usual application as aperformance measure for 2nancial investments but also byway of an approximation to a utility index, representing thesatisfaction obtained by an investor from such investments.

Sharpe’s ratio may be considered as the 2rst measureto combine the two key attributes of 2nancial investments:risks and returns. This risk/return context itself representsa continuation of the conceptual framework developed byMarkowitz [5].

The portfolio selection model designed by Markowitz[5–7] in the context of Portfolio Theory, the inclusion ofrisk-free assets by Tobin [8] and the contributions madeby Sharpe himself [9,2–4] laid the foundations for the cre-ation of the Capital Asset PricingModel (CAPM) developedby Sharpe [1] and described by Fama [10] as the Sharpe–Lintner–Black model.

∗ Corresponding author. Tel.: +34-976-762-494; fax: +34-976-761-791.

E-mail addresses: [email protected] (L. Ferruz Agudo),[email protected] (J.L. Sarto Marzal).

Building on the foundations of Portfolio Theory and themarket equilibrium model, Sharpe [1], Treynor [11] andJensen [12] succeeded in weaving together the strands of riskand returns to establish the 2rst indices for the measurementof portfolio management performance.

Subsequent research related with investment performancehas produced certain criticisms of the CAPM, including thework of scholars such as Roll–Ross [13] and Leland [14].These failures or inconsistencies in the CAPM would aFectthose performance indices using beta as the systematic riskindicator.

Modern 2nancial literature rarely if ever fails to refer tothe CAPM or apply a single factor model, although Carhart’s[15] four factor model is also commonly used. This modelincludes the three factor model created by Fama and French[16] and Jegadeesh and Titman’s [17] “momentum factor”,as Khorama [18] explains in a recent paper.

Other signi2cant lines of research into portfolio manage-ment performance include:

• The work of scholars such as Modigliani and Modigliani[19], who analyse risk-adjusted returns as a measure ofperformance. The performance measures derived fromthis work are in line with those drawn from Sharpe’s ratio.

0305-0483/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.omega.2003.11.006

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274 L. Ferruz Agudo, J.L. Sarto Marzal / Omega 32 (2004) 273–284

• Other authors have tried to break investment perfor-mance down into constituent factors such as portfoliomanagement style, synchronisation with the market andinvestment securities selection. The work of Grinblatt andTitman [20–22], Sharpe [2], Rubio [23], Ferson andSchadt [24], Daniel et al. [25], Christopherson et al. [26],Becker et al. [27] and Basarrate and Rubio [28] stands outin this 2eld. Recent empirical work includes a study byKothari and Warner [29], which concentrates especiallyon what has come to be termed “style characteristics”and “abnormal performance”.

• A third line of research looks at persistence with the ob-jective of predicting future fund performance on the ba-sis of historical data. These issues have been addressedby Grinblatt and Titman [21], Malkiel [30], Elton et al.[31], Carhart [15], Ribeiro et al. [32], Jain and Wu [33],Argawal and Yaik [34], Casarin et al. [35], Hallahan andFaF [36], Droms and Walker [37] and Davis [38] amongothers.

As we have already mentioned, this paper takes Sharpe’sratio as the starting point for performance measurement,since it continues to be a valid reference for the task, ashas been made clear in a number of recent papers, includ-ing work by Stutzer [39] and Muralidhar [40]. Moreover,Sharpe’s ratio does not require the validation or veri2cationof any equilibrium model for 2nancial assets.

A further objective is to tie our analysis in with a mini-mum axiomatic and conceptual framework related with util-ity theory in the presence of risk.

Against this background, Section 2 contains a 2nancialanalysis of the functioning of Sharpe’s ratio and proposescertain small changes, while respecting the essence of theoriginal. Section 3 provides an empirical study of a set ofSpanish investment funds. The paper concludes with a re-view of our 2ndings from the study.

2. Sharpe’s ratio in connection with utility theory in thepresence of risk

2.1. Re6ections on Sharpe’s ratio

Sharpe’s ratio can be used to measure portfolio perfor-mance without the need to validate or verify any prior model,in contrast to Treynor’s and Jensen’s indices, which assumethe validity of the CAPM and, therefore, presuppose opti-mum diversi2cation of the portfolios analysed.

Sharpe’s ratio is expressed as follows:

Sp =Ep − Rf

�p; (1)

where Ep is the average return on a portfolio, p, Rf is theaverage return on a risk-free asset and �p is the standarddeviation in the return on the portfolio, p.

Consequently, this performance measure considers agiven combination for the expected return on a portfolioand the total associated risk.

If Sharpe’s ratio, or any other performance measure forthat matter, is to be considered as an approximation to autility index, it is necessary 2rst to propose a minimum, ob-jective conceptual framework for its application. It seemsreasonable to extend Sharpe’s performance measure to aninvestor utility index in order to derive a complementaryapplication which allows the generalisation of a minimumobjective conceptual framework, on the one hand, and theinclusion of a degree of discretion with regard to the sub-jective perception of risk on the other.

Starting from the framework created by Markowitz [5],we propose six logical/2nancial postulates that take into ac-count basic aspects of Portfolio Theory and Utility Theoryin the presence of risk. These postulates are as follows:

1. Utility or satisfaction depend on risk and returns:

U = f(Ep; �p); (2)

where U is utility, Ep is the expected return on p and�p is the portfolio risk measured in terms of standarddeviation.

2. Utility increases in line with returns if risk remains con-stant:

�U�Ep

¿ 0: (3)

3. Utility decreases as risk increases if returns remains con-stant:

�U��p

¡ 0: (4)

4. There is a positive premium on returns at higher levels ofrisk. As risk increases it must be traded oF against risingreturns:

dEp

d�p¿ 0: (5)

This postulate is a consequence of or inference fromthe preceding two.

5. Marginal returns rise strictly in line with risk. Where riskincreases, the related increment in returns is more thanproportional:

d2Ep

d(�p)2¿ 0: (6)

The combination of this and the fourth postulate impliesa 2nancial risk-return 2eld formed by increasing, convex in-diFerence curves. The degree of risk aversion is representedby the convexity of the indiFerence curve, which is equiva-lent to the demand for higher premiums to trade the accep-tance of higher risk oF against an increase in returns.

6. Positive marginal utility decreases strictly in line withreturns.

L. Ferruz Agudo, J.L. Sarto Marzal / Omega 32 (2004) 273–284 275

Declining marginal utility in the presence of wealth is agenerally accepted principle of utility theory. In this context,however, it would seem appropriate to modify the principlesuch that marginal utility declines with returns, expressedanalytically as follows:

�2U (Ep; �p)�(Ep)2

¡ 0: (7)

This postulate may be relaxed to allow a more generalposition in which marginal utility is decreasing or constantwith returns. Analytically:

�2U (Ep; �p)�(Ep)2

6 0: (8)

Sharpe’s ratio is shown to function appropriately in thecontext described by applying these six postulates, exceptin two cases:

• In accordance with postulate 3, Ep¿Rf must hold forSharpe’s ratio to function properly with regard to risk.Though this is a requirement of long-term 2nancial logic,it may not actually be the case in certain short-term cir-cumstances in the 2nancial markets.

• Contrary to postulate 5, Sharpe’s ratio does not considerscaled increases in returns in the presence of rising levelsof risk. Thus, the indiFerence functions are straight.

2.2. Alternative variation of Sharpe’s ratio when Ep¡Rf

As explained above, when Ep¡Rf Sharpe’s ratio doesnot function properly because postulate 3 does not hold.

One proposal to correct eventual inconsistencies inSharpe’s ratio due to this phenomenon would be to treat thepremium on returns as relative rather than absolute.

In principle, this might be regarded as a mere algebraicsolution. However, it may also be considered meaningful in2nancial terms for the following reasons:

• The treatment of the premium on returns as relative is notwithout precedent in Financial Analysis. For example, thestraightforward use of Net Present Value is not suQcientand therefore it is necessary to establish the Cost-Bene2trelationship in the analysis of alternative investments withdiFering initial outlays.

• In dynamic studies that take the evolution of portfolio per-formance over time into account, the return on risk-freeassets, considered initially as a constant, becomes a sig-ni2cant and volatile variable. As a result, it is more ap-propriate to treat the return on risk free assets in the samemanner as the total risk inherent in the portfolios analysed.

The proposed performance measure would thus be ex-pressed as follows:

Sp(1) =Ep=Rf

�p: (9)

This index resolves possible failings in the original in thepresence of variations in the level of risk.

It should be noted that the performance rankings obtainedfor a given set of portfolios would vary upon applying theconventional expression of Sharpe’s ratio and the proposedvariant. This is because the expressions themselves diFerand, hence, they throw up diverging values.

On the basis of expressions (1) and (9):Sp¿Sp(1) provided that Ep − Rf ¿Ep=Rf .That is, if

Ep¿R2f

Rf − 1: (10)

It is also necessary to establish which of the two indicespenalises risk more or, to put it another way, which expres-sion is more sensitive to increases in the level of risk. Thisimplies consideration of the 2rst partial derivative for eachindex with respect to risk. Thus

�Sp��p

=−Ep − Rf

�2p; (11)

�Sp(1)��p

=−Ep=Rf

�2p(12)

Thus, expression (10) must hold for the 2rst expressionto take a higher negative value than the second (i.e. for theconventional Sharpe’s ratio to penalise performance morethan the alternative).

2.3. Alternatives to Sharpe’s ratio to take into accountscaled increases in returns in the presence of rising levelsof risk

An alternative to resolve the second question mentionedabove, which is arises from postulate 5, might be to mea-sure risk in terms of variance rather than standard deviation.Analytically

Sp(2) =Ep=Rf

�2p: (13)

This index does consider the existence of scaled increasesin returns in the presence of rising levels of risk, and ittherefore represents an approximation to quadratic utilityfunctions. In this case, postulate 5 would indeed hold.

This Sp(2) index should, however, only be used if thereturns on some or all of the portfolios included in the sampleare lower than the return on the risk-free asset employed.

If all of the portfolios in the sample perform in accordancewith the dictates of normal long-term 2nancial logic (whichis to say that Ep¿Rf holds for each portfolio) on the basis ofSharpe’s ratio and using the variance to evaluate increasingaversion in the presence of risking levels of risk, a 2rstalternative index would be

Sp(3) =Ep − Rf

�2p: (14)


The Sp(3) index operates correctly within the frameworkconsidered provided that the return on the portfolio exceedsthat of the risk free asset considered.

As in Section 2.2, the use of the Sp(3) index gives riseto diFering performance rankings for the portfolios fromthose obtained using Sharpe’s original ratio. The reasons aresimilar to those explained for Sp(2).

On the basis of expressions (1) and (14) it appears thatSp = Sp(3)�p.

In which case, the expression Sp¿Sp(3) holds providedthat �p¿ 1.

Following the same pattern of analysis as in Section 2.2,the 2rst derivative for each index needs to be analysed toestablish which is more sensitive to increases in the level ofrisk. Analytically

�Sp��p

=−Ep − Rf

�2p; (15)

�Sp(3)��p

=−2Ep − Rf

�3p: (16)

Thus

�Sp��p

=�Sp(3)��p

�p=2: (17)

Consequently, if �p¿ 2, Sharpe’s ratio is more sensitiveto variations in the level of risk than the alternative Sp(3).If, however, �p¡ 2, the Sp(3) index will impose a greaterpenalty on any eventual increase in the risk inherent in aportfolio.

It should be noted that �p is normally greater than 2, atleast for equity investments. This analysis therefore con2rmsthe intuitive positions that risk is more heavily penalisedif we consider the variance and that indices related withSharpe’s ratio using the variance provide a better 2t for morerisk averse decision-makers.

Given the similarity of the expressions involved, theseconclusions are perfectly applicable to the Sp(1) and Sp(2)indices, since the former is more sensitive to variations inthe level of risk when �p¿ 2, but otherwise Sp(2) penalisesrisk more heavily.

3. Empirical application

The following pages analyse the performance of a set ofSpanish investment funds contained in two data bases. The2rst of these data bases comprises 91 2xed-income fundsfor the period from January 1993 until December 2000, andthe second is formed by 40 equity funds for the period fromJanuary 1995 until December 2000.

In both cases, quarterly returns obtained from each port-folio are analysed within the relevant time-frame. The aver-age quarterly return obtained on 3-month Spanish treasury

bill repos in the 2rst time-frame was 1.46%, while in thesecond it was 1.24%. These 2nancial assets have been usedin this study as the risk-free asset for the analysis of thefunds included in the data bases.

Tables 1a and b respectively show the average quar-terly returns on the 2xed-income and equity portfoliosanalysed, as well as the total levels of risk inherent ineach.

Of the 91 investment funds analysed, 41 were foundto have generated average quarterly returns that werelower than that of the risk-free assets considered. Noneof the 40 equity funds included in the sample showed anegative premium on returns compared to the risk-freeasset.

Tables 2a and b show the performance rankings generatedby the conventional Sharpe’s ratio for the 2xed-income andequity funds, respectively. The ranking presented in Table2b is consistent, since none of the equity funds showed anegative premium on returns. In Table 2a, however, certaininconsistencies are apparent in the ranking.

Foncaixa Ahorro 8 is an example of such inconsistencies,being the last placed in the ranking despite an average returnof 1.12% and a risk factor of 0.76. Other funds, such asBBV Renta Fija Corta 1 and BI Eurobonos, obtained lowerreturns at higher levels of risk but are nonetheless rankedabove Foncaixa Ahorro 8.

These inconsistencies can be resolved by applying alter-native Sp(1), as shown in Table 3a. Table 3b shows a rank-ing of equity funds obtained on the same basis, although theapplication of the alternative index is not essential in thiscase.

Table 4, which reSects the results of applying alternativeSp(2) to the sample of 2xed-income funds, allows for adegree of risk aversion on the part of the rational 2nancialinvestor. It also oFers a consistent ranking of the funds.

The Sp(2) index could also be applied to the equity funds,although preference has been given to alternative Sp(3),since the average returns on these portfolios were in no caselower than the risk-free asset. Table 5 shows the results ofapplying the Sp(3) index.

Both Tables 4 and 5 provide consistent rankings as a resultof the application of performance indices that are suitable foruse as an approximation to the utility obtained by investorsfrom these portfolios.

Upon reviewing all of the performance rankings gener-ated, it becomes clear how the various indices applied giverise to diFerences in the evaluation of each portfolio form-ing part of the data bases. These diFering rankings are theresult of changes in the treatment of risk and returns, the twokey elements of portfolio analysis, depending on the indexapplied.

The subjective perception of risk incorporated through theuse of performance measures as approximations to utilityindices may thus validate a range of performance indices,provided that certain minimum postulates of 2nancial logicare respected.


Table 1Fixed-income investment funds and equity funds comprising the data base and quarterly returns as well as the level of risk inherent in each

Investment fund Ep �p Investment fund Ep �p

(a) Fixed-income investment funds from January 1993 to December 20001 AB AHORRO 1.83 1.86 47 FONCAIXA AHORRO 2 1.23 0.812 AB FONDO 1.94 1.85 48 FONCAIXA AHORRO 4 1.13 0.773 AB FT 1.61 1.76 49 FONCAIXA AHORRO 7 1.62 0.844 AC DEUDA FT 1.58 1.65 50 FONCAIXA AHORRO 8 1.12 0.765 BANESDEUDA FT 1.77 1.89 51 FONCAIXA AHORRO 9 1.14 0.796 BANIF RENTA FIJA 1.76 1.59 52 FONDACOFAR 1.29 0.817 BANKPYME FT 1.82 1.96 53 FONDICAJA 1.33 1.218 BANKPYME MULTIVALOR 1.34 1.20 54 FONDMAPFRE RENTA 1.56 1.549 BASKEFOND 1.59 1.38 55 FONDOATLANTICO 1.54 1.2610 BBVA DEUDA FT 1.84 1.93 56 FONLAIETANA 1.48 0.9111 BBVA HORIZONTE 1.83 1.75 57 FONMARCH 1.57 1.5912 BBVA RENTA FIJA CORTO 1 1.11 0.87 58 FONSEGUR 1.57 1.4713 BBVA RENTA FIJA CORTO 3 1.31 0.88 59 FONSNOSTRO 1.47 1.3014 BBVA RENTA FIJA CORTO 5 1.30 0.90 60 FONTARRACO 1.18 1.2315 BBVA RENTA FIJA CORTO 7 1.41 1.74 61 FONVALOR 1.31 1.0516 BBVA RENTA FIJA LARGO 3 1.48 1.60 62 HERRERO RENTA FIJA 1.51 1.5317 BCH BONOS FT 1.74 1.72 63 IBERAGENTES AHORRO 1.86 2.5818 BCH RENTA FIJA 1 1.86 1.76 64 IBERAGENTES FT 1.89 2.0219 BCH RENTA FIJA 3 1.56 1.66 65 IBERCAJA AHORRO 1.48 1.4920 BETA DEUDA FT 1.80 2.17 66 INVERFONDO 1.29 1.1621 BETA RENTA 1.73 1.87 67 INVERMADRID FT 1.64 1.5322 BI EUROBONOS 0.95 1.66 68 INVERMONTE 1.25 1.0123 BK FONDO FIJO 1.85 1.96 69 INVER-RIOJA FONDO 1.26 0.8424 BM FT 2.13 2.50 70 IURISFOND 1.62 0.7625 BSN RENTA FIJA 1.84 2.42 71 KUTXAINVER 1.43 1.8126 CAIXA GALICIA INV 1.54 2.78 72 LLOYDS FONDO 1 1.48 1.3727 CAJA BURGOS RENTA 1.41 1.03 73 MAPFRE FT 1.36 0.9328 CAJA MURCIA 1.15 1.18 74 MUTUAFONDO 2.11 1.7829 CAJA SEGOVIA RENTA 1.26 0.99 75 NOVOCAJAS 1.28 1.1730 CAM BONOS 1 1.63 1.70 76 P& G CRECIMIENTO 1.28 0.8931 CANTABRIA DINERO 1.28 0.88 77 RENTA 4 AHORRO 1.72 1.5732 CANTABRIA MONETARIO 1.15 0.92 78 RENTCAJAS 1.39 1.1833 CITIFONDO PREMIUM 1.38 0.83 79 RENTMADRID 1.24 0.9334 CITIFONDO RF 1.41 1.59 80 SABADELL BONOS EURO 1.86 1.3135 CUENTAFONDO RENTA 2.01 2.35 81 SABADELL INTERES EURO 1 1.54 0.9736 DB INVEST 1.42 1.31 82 SABADELL INTERES EURO 2 1.53 0.9737 DB INVEST II 1.35 1.27 83 SABADELL INTERES EURO 3 1.75 0.9638 EDM AHORRO 1.40 1.72 84 SANTANDER AHORRO 1.13 0.7339 EUROVALOR RF 1.65 1.76 85 SEGURFONDO 2.01 1.9140 FG TESORERIA 1.36 0.70 86 SOLBANK INTERES EURO 1.53 1.3941 FIBANC FT 1.93 1.75 87 TOP RENTA 1.35 1.7342 FIBANC RENTA 2.01 1.79 88 UNIFOND EURORENTA 1.16 1.8343 FONBANESTO 1.51 1.43 89 URQUIJO RENTA 1.60 1.7044 FONBILBAO FT 2.32 2.89 90 URQUIJO RENTA 2 1.60 1.6945 FONCAIXA AHORRO 10 1.24 0.82 91 ZARAGOZANO RF 1.31 0.9946 FONCAIXA AHORRO 11 1.42 0.83

(b) Equity funds from January 1995 to December 20001 AB BOLSA 5.02 13.21 21 FONBILBAO ACCIONES 5.01 11.532 ARGENTARIA BOLSA 5.66 14.89 22 FONBOLSA 4.39 13.173 BANKPYME SWISS 5.20 8.75 23 FONCAIXA BOLSA 5 6.17 11.934 BBVA BOLSA 2 4.21 13.53 24 FONDBARCLAYS 2 5.39 13.595 BBVA EUROPA BLUE CHIPS 2 6.35 12.76 25 FONJALON ACCIONES 4.58 10.906 BBVA EUROPA CRECIMIENTO 1 4.81 14.03 26 FONJALON II 3.62 8.517 BBVA MIX 60 A 3.25 8.24 27 IBERAGENTES BOLSA 4.93 13.328 BCH ACCIONES 4.82 12.03 28 IBERCAJA BOLSA 4.47 12.32


Table 1 (continued)

Investment fund Ep �p Investment fund Ep �p

9 BETA CRECIMIENTO 4.66 10.82 29 INDEXBOLSA 4.72 12.7310 BK FONDO 5.14 11.61 30 INDOSUEZ BOLSA 3.63 10.0211 BM-DINERBOLSA 5.74 15.53 31 INVERBAN FONBOLSA 4.60 10.9512 BNP BOLSA 5.13 13.48 32 MADRID BOLSA 4.81 13.2013 BSN ACCIONES 5.11 11.99 33 MERCHFONDO 6.39 15.0814 CITIFONDO RV 5.12 13.40 34 METAVALOR 3.51 9.6415 DB ACCIONES 5.66 13.47 35 PLUSCARTERA 4.46 10.9516 DB MIXTA II 4.16 10.31 36 SANT EUROACCIONES 4.78 11.1217 EUROFONDO 2.70 10.58 37 SANTANDER ACCIONES 4.47 10.1518 EUROVALOR BOLSA 4.70 12.42 38 URQUIJO CRECIMIENTO 4.08 10.6619 FG ACCIONES 4.67 12.36 39 URQUIJO GLOBAL 3.12 7.9020 FIBANC CRECIMIENTO 3.71 8.42 40 URQUIJO INDICE 4.29 11.42

Table 2Application of Sharpe’s original ratio to the data base of 2xed-income funds and equity funds

Investment fund Sp Investment fund Sp

(a) Fixed-income funds1 MUTUAFONDO 0.364 47 FONLAIETANA 0.0152 FIBANC RENTA 0.305 48 IBERCAJA AHORRO 0.0123 SABADELL BONOS EURO 0.302 49 BBVA RENTA FIJA LARGO 3 0.0104 SABADELL INTERES EURO 3 0.297 50 LLOYDS FONDO 1 0.0095 FONBILBAO FT 0.297 51 FONSNOSTRO 0.0086 SEGURFONDO 0.283 52 KUTXAINVER −0.0167 FIBANC FT 0.267 53 CITIFONDO RF −0.0318 BM FT 0.265 54 BBVA RENTA FIJA CORTO 7 −0.0339 AB FONDO 0.257 55 DB INVEST −0.03410 CUENTAFONDO RENTA 0.231 56 EDM AHORRO −0.03511 BCH RENTA FIJA 1 0.222 57 FONCAIXA AHORRO 11 −0.05112 IBERAGENTES FT 0.211 58 CAJA BURGOS RENTA −0.05613 BBVA HORIZONTE 0.211 59 RENTCAJAS −0.06314 IURISFOND 0.203 60 TOP RENTA −0.06615 AB AHORRO 0.199 61 DB INVEST II −0.09216 BK FONDO FIJO 0.198 62 CITIFONDO PREMIUM −0.09517 BBVA DEUDA FT 0.194 63 BANKPYME MULTIVALOR −0.10318 BANIF RENTA FIJA 0.187 64 FONDICAJA −0.11219 FONCAIXA AHORRO 7 0.186 65 MAPFRE FT −0.11420 BANKPYME FT 0.183 66 FG TESORERIA −0.14221 RENTA 4 AHORRO 0.162 67 FONVALOR −0.14522 BCH BONOS FT 0.162 68 INVERFONDO −0.14623 BANESDEUDA FT 0.160 69 ZARAGOZANO RF −0.15324 BSN RENTA FIJA 0.156 70 NOVOCAJAS −0.16125 BETA DEUDA FT 0.155 71 UNIFOND EURORENTA −0.16626 IBERAGENTES AHORRO 0.152 72 BBVA RENTA FIJA CORTO 3 −0.17127 BETA RENTA 0.139 73 BBVA RENTA FIJA CORTO 5 −0.17928 INVERMADRID FT 0.119 74 CAJA SEGOVIA RENTA −0.20829 EUROVALOR RF 0.107 75 INVERMONTE −0.20830 CAM BONOS 1 0.100 76 CANTABRIA DINERO −0.20931 BASKEFOND 0.095 77 P& G CRECIMIENTO −0.21132 AB FT 0.083 78 FONDACOFAR −0.22033 URQUIJO RENTA 2 0.081 79 FONTARRACO −0.23534 URQUIJO RENTA 0.080 80 INVER-RIOJA FONDO −0.23835 SABADELL INTERES EURO 1 0.075 81 RENTMADRID −0.24036 SABADELL INTERES EURO 2 0.070 82 CAJA MURCIA −0.26337 FONSEGUR 0.070 83 FONCAIXA AHORRO 10 −0.269


Table 2 (continued)

Investment fund Sp Investment fund Sp

38 AC DEUDA FT 0.070 84 FONCAIXA AHORRO 2 −0.28239 FONDMAPFRE RENTA 0.065 85 BI EUROBONOS −0.30840 FONMARCH 0.064 86 CANTABRIA MONETARIO −0.33641 BCH RENTA FIJA 3 0.060 87 BBVA RENTA FIJA CORTO 1 −0.40342 FONDOATLANTICO 0.058 88 FONCAIXA AHORRO 9 −0.40543 SOLBANK INTERES EURO 0.049 89 FONCAIXA AHORRO 4 −0.42644 HERRERO RENTA FIJA 0.031 90 SANTANDER AHORRO −0.45745 FONBANESTO 0.029 91 FONCAIXA AHORRO 8 −0.45946 CAIXA GALICIA INV 0.028

(b) Equity funds1 BANKPYME SWISS 0.453 21 BNP BOLSA 0.2882 FONCAIXA BOLSA 5 0.413 22 AB BOLSA 0.2863 BBVA EUROPA BLUE CHIPS 2 0.400 23 DB MIXTA II 0.2844 MERCHFONDO 0.342 24 FONJALON II 0.2805 BK FONDO 0.336 25 EUROVALOR BOLSA 0.2796 DB ACCIONES 0.328 26 FG ACCIONES 0.2777 FONBILBAO ACCIONES 0.327 27 IBERAGENTES BOLSA 0.2778 BSN ACCIONES 0.323 28 INDEXBOLSA 0.2749 SANT EUROACCIONES 0.319 29 MADRID BOLSA 0.27110 SANTANDER ACCIONES 0.318 30 URQUIJO INDICE 0.26711 BETA CRECIMIENTO 0.316 31 URQUIJO CRECIMIENTO 0.26612 FONJALON ACCIONES 0.307 32 IBERCAJA BOLSA 0.26213 INVERBAN FONBOLSA 0.307 33 BBVA EUROPA CRECIMIENTO 1 0.25514 FONDBARCLAYS 2 0.305 34 BBVA MIX 60 A 0.24415 BCH ACCIONES 0.298 35 INDOSUEZ BOLSA 0.23916 ARGENTARIA BOLSA 0.297 36 FONBOLSA 0.23917 FIBANC CRECIMIENTO 0.294 37 URQUIJO GLOBAL 0.23818 PLUSCARTERA 0.294 38 METAVALOR 0.23619 BM-DINERBOLSA 0.290 39 BBVA BOLSA 2 0.22020 CITIFONDO RV 0.290 40 EUROFONDO 0.138

Table 3Application of index Sp(1) to the 2xed-income funds and equity funds

Investment fund Sp(1) Investment fund Sp(1)

(a) Fixed-income funds1 IURISFOND 1.460 47 LLOYDS FONDO 1 0.7352 FG TESORERIA 1.336 48 INVERMADRID FT 0.7353 FONCAIXA AHORRO 7 1.311 49 DB INVEST II 0.7264 SABADELL INTERES EURO 3 1.242 50 FONSEGUR 0.7265 FONCAIXA AHORRO 11 1.171 51 BCH RENTA FIJA 1 0.7196 CITIFONDO PREMIUM 1.140 52 FONBANESTO 0.7187 FONLAIETANA 1.106 53 AB FONDO 0.7168 SABADELL INTERES EURO 2 1.083 54 SEGURFONDO 0.7169 FONDACOFAR 1.083 55 BBVA HORIZONTE 0.71410 SABADELL INTERES EURO 1 1.082 56 FONDMAPFRE RENTA 0.69311 SANTANDER AHORRO 1.055 57 BCH BONOS FT 0.69312 FONCAIXA AHORRO 2 1.035 58 IBERCAJA AHORRO 0.68013 FONCAIXA AHORRO 10 1.033 59 HERRERO RENTA FIJA 0.67514 INVER-RIOJA FONDO 1.024 60 AB AHORRO 0.67515 BBVA RENTA FIJA CORTO 3 1.024 61 FONMARCH 0.67216 FONCAIXA AHORRO 8 1.009 62 CAJA MURCIA 0.67017 FONCAIXA AHORRO 4 1.003 63 CAM BONOS 1 0.65618 MAPFRE FT 0.994 64 FONTARRACO 0.65619 FONCAIXA AHORRO 9 0.991 65 AC DEUDA FT 0.652


Table 3 (continued)


20 CANTABRIA DINERO 0.989 66 BBVA DEUDA FT 0.65021 BBVA RENTA FIJA CORTO 5 0.988 67 URQUIJO RENTA 2 0.64622 P& G CRECIMIENTO 0.980 68 BK FONDO FIJO 0.64523 SABADELL BONOS EURO 0.970 69 BCH RENTA FIJA 3 0.64424 CAJA BURGOS RENTA 0.930 70 URQUIJO RENTA 0.64325 RENTMADRID 0.913 71 EUROVALOR RF 0.64226 ZARAGOZANO RF 0.906 72 IBERAGENTES FT 0.64027 CAJA SEGOVIA RENTA 0.870 73 BANESDEUDA FT 0.63928 BBVA RENTA FIJA CORTO 1 0.869 74 BANKPYME FT 0.63529 CANTABRIA MONETARIO 0.859 75 BBVA RENTA FIJA LARGO 3 0.63330 INVERMONTE 0.850 76 BETA RENTA 0.62931 FONVALOR 0.849 77 AB FT 0.62332 FONDOATLANTICO 0.833 78 CITIFONDO RF 0.60933 MUTUAFONDO 0.810 79 CUENTAFONDO RENTA 0.58334 RENTCAJAS 0.801 80 BM FT 0.58035 BASKEFOND 0.792 81 BETA DEUDA FT 0.56736 FONSNOSTRO 0.773 82 EDM AHORRO 0.55937 FIBANC RENTA 0.769 83 BBVA RENTA FIJA CORTO 7 0.55138 BANKPYME MULTIVALOR 0.760 84 FONBILBAO FT 0.54939 INVERFONDO 0.760 85 KUTXAINVER 0.54340 FIBANC FT 0.755 86 TOP RENTA 0.53441 BANIF RENTA FIJA 0.755 87 BSN RENTA FIJA 0.51942 SOLBANK INTERES EURO 0.752 88 IBERAGENTES AHORRO 0.49143 FONDICAJA 0.750 89 UNIFOND EURORENTA 0.43344 RENTA 4 AHORRO 0.748 90 BI EUROBONOS 0.39145 NOVOCAJAS 0.746 91 CAIXA GALICIA INV 0.37946 DB INVEST 0.740

(b) Equity funds1 BANKPYME SWISS 0.480 21 BBVA MIX 60 A 0.3192 FONCAIXA BOLSA 5 0.418 22 URQUIJO CRECIMIENTO 0.3093 BBVA EUROPA BLUE CHIPS 2 0.402 23 CITIFONDO RV 0.3084 BK FONDO 0.357 24 ARGENTARIA BOLSA 0.3075 FIBANC CRECIMIENTO 0.356 25 BNP BOLSA 0.3076 SANTANDER ACCIONES 0.355 26 AB BOLSA 0.3077 FONBILBAO ACCIONES 0.351 27 EUROVALOR BOLSA 0.3058 BETA CRECIMIENTO 0.348 28 FG ACCIONES 0.3059 SANT EUROACCIONES 0.347 29 URQUIJO INDICE 0.30310 BSN ACCIONES 0.344 30 INDEXBOLSA 0.30011 FONJALON II 0.343 31 IBERAGENTES BOLSA 0.29912 MERCHFONDO 0.342 32 BM-DINERBOLSA 0.29813 FONJALON ACCIONES 0.340 33 MADRID BOLSA 0.29414 INVERBAN FONBOLSA 0.339 34 METAVALOR 0.29415 DB ACCIONES 0.339 35 IBERCAJA BOLSA 0.29316 PLUSCARTERA 0.329 36 INDOSUEZ BOLSA 0.29317 DB MIXTA II 0.326 37 BBVA EUROPA CRECIMIENTO 1 0.27718 BCH ACCIONES 0.323 38 FONBOLSA 0.26919 FONDBARCLAYS 2 0.320 39 BBVA BOLSA 2 0.25120 URQUIJO GLOBAL 0.319 40 EUROFONDO 0.206

The level of similarity between the various rankings canbe calculated by applying Spearman’s correlation coeQ-cient, which is expressed as follows:

rs = 1− 6∑d2i

N (N 2 − 1); (18)

where N is the number of funds in each sample, and di isthe diFerence in the position occupied by the fund, i, in eachranking.

This empirical study considers two diFerent data bases,and Spearman’s coeQcient has been applied separately toeach. Table 6 shows the results of the correlation in the


Table 4Application of index Sp(2) to the 2xed-income funds


1 IURISFOND 1.929 47 FONSEGUR 0.4932 FG TESORERIA 1.914 48 INVERMADRID FT 0.4803 FONCAIXA AHORRO 7 1.551 49 RENTA 4 AHORRO 0.4764 SANTANDER AHORRO 1.443 50 BANIF RENTA FIJA 0.4745 FONCAIXA AHORRO 11 1.411 51 IBERCAJA AHORRO 0.4576 CITIFONDO PREMIUM 1.373 52 MUTUAFONDO 0.4567 FONDACOFAR 1.335 53 FONDMAPFRE RENTA 0.4508 FONCAIXA AHORRO 8 1.334 54 HERRERO RENTA FIJA 0.4429 FONCAIXA AHORRO 4 1.297 55 FIBANC FT 0.43210 SABADELL INTERES EURO 3 1.291 56 FIBANC RENTA 0.43011 FONCAIXA AHORRO 2 1.270 57 FONMARCH 0.42112 FONCAIXA AHORRO 10 1.258 58 BCH RENTA FIJA 1 0.40713 FONCAIXA AHORRO 9 1.257 59 BBVA HORIZONTE 0.40714 INVER-RIOJA FONDO 1.215 60 BCH BONOS FT 0.40315 FONLAIETANA 1.211 61 BBVA RENTA FIJA LARGO 3 0.39716 BBVA RENTA FIJA CORTO 3 1.168 62 AC DEUDA FT 0.39517 SABADELL INTERES EURO 2 1.121 63 BCH RENTA FIJA 3 0.38918 CANTABRIA DINERO 1.119 64 AB FONDO 0.38719 SABADELL INTERES EURO 1 1.114 65 CAM BONOS 1 0.38620 P& G CRECIMIENTO 1.101 66 CITIFONDO RF 0.38421 BBVA RENTA FIJA CORTO 5 1.097 67 URQUIJO RENTA 2 0.38222 MAPFRE FT 1.066 68 URQUIJO RENTA 0.37823 BBVA RENTA FIJA CORTO 1 0.996 69 SEGURFONDO 0.37424 RENTMADRID 0.983 70 EUROVALOR RF 0.36525 CANTABRIA MONETARIO 0.935 71 AB AHORRO 0.36426 ZARAGOZANO RF 0.915 72 AB FT 0.35327 CAJA BURGOS RENTA 0.900 73 BANESDEUDA FT 0.33828 CAJA SEGOVIA RENTA 0.881 74 BBVA DEUDA FT 0.33729 INVERMONTE 0.843 75 BETA RENTA 0.33530 FONVALOR 0.805 76 BK FONDO FIJO 0.32931 SABADELL BONOS EURO 0.740 77 EDM AHORRO 0.32632 RENTCAJAS 0.676 78 BANKPYME FT 0.32433 FONDOATLANTICO 0.660 79 IBERAGENTES FT 0.31834 INVERFONDO 0.654 80 BBVA RENTA FIJA CORTO 7 0.31635 NOVOCAJAS 0.638 81 TOP RENTA 0.30936 BANKPYME MULTIVALOR 0.632 82 KUTXAINVER 0.30137 FONDICAJA 0.620 83 BETA DEUDA FT 0.26238 FONSNOSTRO 0.593 84 CUENTAFONDO RENTA 0.24839 BASKEFOND 0.575 85 UNIFOND EURORENTA 0.23740 DB INVEST II 0.573 86 BI EUROBONOS 0.23541 CAJA MURCIA 0.569 87 BM FT 0.23242 DB INVEST 0.565 88 BSN RENTA FIJA 0.21443 SOLBANK INTERES EURO 0.540 89 IBERAGENTES AHORRO 0.19044 LLOYDS FONDO 1 0.536 90 FONBILBAO FT 0.19045 FONTARRACO 0.535 91 CAIXA GALICIA INV 0.13646 FONBANESTO 0.502

ranking of 2xed-income funds, while Table 7 reSects thelevels of similarity in the ranking of equity funds.

The lack of correlation between Sharpe’s ratio and thealternative Sp(1) in Table 6 is immediately striking. This isdue, in the 2rst place, to the fact that it is a non-homogeneuscomparison, since Sharpe’s original ratio ranks the fundsinconsistently. Furthermore, in the relevant time-frame the

return on the risk-free asset was 1.46%, and expression (10)tells us that where Ep¿ 4:63% Sharpe’s ratio will penalisehigher levels of risk in the portfolios more heavily than thealternative Sp(1).

Nevertheless, none of the portfolios analysed has achievedsuch a high average return, and indeed all of the valuesobtained are signi2cantly lower. Consequently, there is a


Table 5Application of index Sp(3) to the equity funds


1 BANKPYME SWISS 0.052 21 DB ACCIONES 0.0242 FIBANC CRECIMIENTO 0.035 22 INDOSUEZ BOLSA 0.0243 FONCAIXA BOLSA 5 0.035 23 URQUIJO INDICE 0.0234 FONJALON II 0.033 24 MERCHFONDO 0.0235 BBVA EUROPA BLUE CHIPS 2 0.031 25 FONDBARCLAYS 2 0.0226 SANTANDER ACCIONES 0.031 26 EUROVALOR BOLSA 0.0227 URQUIJO GLOBAL 0.030 27 FG ACCIONES 0.0228 BBVA MIX 60 A 0.030 28 AB BOLSA 0.0229 BETA CRECIMIENTO 0.029 29 CITIFONDO RV 0.02210 BK FONDO 0.029 30 INDEXBOLSA 0.02211 SANT EUROACCIONES 0.029 31 BNP BOLSA 0.02112 FONBILBAO ACCIONES 0.028 32 IBERCAJA BOLSA 0.02113 FONJALON ACCIONES 0.028 33 IBERAGENTES BOLSA 0.02114 INVERBAN FONBOLSA 0.028 34 MADRID BOLSA 0.02015 DB MIXTA II 0.028 35 ARGENTARIA BOLSA 0.02016 BSN ACCIONES 0.027 36 BM-DINERBOLSA 0.01917 PLUSCARTERA 0.027 37 FONBOLSA 0.01818 URQUIJO CRECIMIENTO 0.025 38 BBVA EUROPA CRECIMIENTO 1 0.01819 BCH ACCIONES 0.025 39 BBVA BOLSA 2 0.01620 METAVALOR 0.024 40 EUROFONDO 0.013

Table 6Spearman’s correlation coeQcient based on the rankings for indicesSp, Sp(1), Sp(2) and Sp(3) applied to 2xed-income funds

Sp Sp(1) Sp(2) Sp(3)

Sp 1 −0.2965 −0.4674 0.9872Sp(1) 1 0.9669 −0.2705Sp(2) 1 −0.4356Sp(3) 1

Table 7Spearman’s correlation coeQcient based on the rankings for indicesSp, Sp(1), Sp(2) and Sp(3) applied to equity funds


Sp 1 0.8623 0.3764 0.5456Sp(1) 1 0.7482 0.8687Sp(2) 1 0.9687Sp(3) 1

wide gulf between the treatment of risk in Sp and Sp(1) forthe portfolios comprising the 2rst data base, resulting in acorrelation coeQcient of −0:2965.

The same reasoning is applicable to the correlation, alsonegative, between the rankings generated by the Sp(2) andSp(3) indices.

The high level of correlation oFered by indices Sp andSp(3) and by Sp(1) and Sp(2) is also interesting, bearingin mind that the diFerence between each pair of measuresrefers to the inclusion of the variance of the returns on theportfolios rather than their standard deviation. Furthermore,the results oFered by Sp and Sp(3) are inconsistent, whileSp(1) and Sp(2) operate correctly.

Finally, on the basis of expression (17), when �p¿ 2 theSp and Sp(1) indices are more sensitive to variations in thelevel of risk in the portfolios than either Sp(3) or Sp(2).For the funds comprising this data base, the levels of

risk in certain portfolios are close to 2, both at the top andthe bottom of the ranking. This explains the high levels ofcorrelation between the pairs of indices.

To complete this review of correlations between the ap-plication of pairs of indices, it should be mentioned thatthe correlation is negative for the pair Sp and Sp(2) and forthe pair Sp(1) and Sp(3), because of the considerable diFer-ences between the expressions, aside from the fact that Spand Sp(3) are inconsistent.The results obtained from the application of Spearman’s

correlation coeQcient to the second data base, shown inTable 7, diFer considerably from those reSected in Table 6.

Firstly, the correlation between Sp and Sp(1), and betweenSp(2) and Sp(3), is very high in contrast to the results for the2xed-income funds. In this time-frame, the average return onthe risk-free asset was 1.24% and, consequently, expression(10) establishes a threshold level of 6.41% for returns onthe portfolios.


Table 8Study of partial derivatives permitting analysis of the conventional Sharpe’s ratio and the alternatives proposed as approximation to utilityindicators in the presence of risk


�Ip=�Ep + + + +�2Ip=�(Ep)2 =0 =0 =0 =0�Ip=��p −∗ − − −∗�Ep=��p + + + +�2Ep=�(�p)2 =0 =0 + +

∗if Ep¿Rf

On the basis of the results shown in Table 1b, the av-erage returns obtained on the equity portfolios were gener-ally lower than, but reasonably close to, this threshold. Thismeans that the sensitivity of all of the indices to risk tendsto be similar and, therefore, the rankings generated from theapplication of each measure have a high correlation.

Secondly, the correlations between the rankings generatedby indices Sp, Sp(3), Sp(1) and Sp(2) are noticeably lowerthan those obtained for the 2rst data base (Table 6).

In this sense, the key reference value was level 2 of thestandard deviation of the portfolios. In the second data base,which comprises equity funds, risk levels are signi2cantlyhigher than 2, as shown in Table 1b. As a result, sensitiv-ity to risk diFers markedly between the pairs of indices. Inparticular Sp(2) and Sp(3) are less sensitive in these circum-stances and penalise risk to a lesser degree than Sp(1) andSp, respectively.

4. Conclusions

The measurement of portfolio performance usingSharpe’s ratio may give rise to inconsistent rankings wherethe average returns on the portfolios considered are lowerthan the average returns on the risk-free asset taken as areference.

More generally, and taking this anomaly into considera-tion, where Sharpe’s ratio is used as an approximation to autility index, it can be seen, among other matters, that it doesnot take strict risk aversion on the part of a rational 2nancialinvestor into account within a normative conceptual frame-work. This needs to be taken into account from the point ofview of the utility or satisfaction obtained by the individualinvestor from the investments in the portfolios analysed.

The proposed series of postulates, which take into accountbasic aspects of utility theory in the presence of risk andthe basic 2nancial logic of Portfolio Theory, allows detailedanalysis of Sharpe’s ratio, bringing out its strengths but alsohighlighting the sources of possible 2nancial anomalies (e.g.where Ep¡Rf ) and the lack of a strict treatment of riskaversion. On this basis, alternative indices belonging to thesame family as Sharpe’s can be generated.

Such alternative indices have been applied together withSharpe’s ratio to a data base containing the returns on Span-ish investment funds. The application of each index givesrise to diFering performance rankings for the portfolios anal-ysed.

Not all these rankings can be considered as valid, with itbeing necessary to exclude classi2cations with inconsistentindexes when Ep¡Rf . In the remaining indexes coherent,albeit diFerent, rankings are generated although, in general,with high correlations. These diFerences are explained by adistinct treatment given to the risk-return combination, witha greater or lower compensation being required for returnwhen there is an increase in risk.

Further research will be required to 2ne tune the con-ceptual framework and operational approach. Among othermatters, the relative premium appears to make little diFer-ence to the rankings, while the measurement of risk usingthe variance may have profound eFects, especially in equityportfolios, where it introduces a bias towards more conser-vatively managed funds.

Acknowledgements

The authors would like to express their thanks to theSpanish Directorate General for Higher Education for theaward of Project PB97-1003, to the Regional Government ofAragon for the award of Project P06/97, to Ibercaja for theaward of Project 268-96 and to the University of Zaragozafor the award of funding through Research Projects 268-77,268-84, and 268-93.

The authors also express their thanks to comments andsuggestions of the anonymous referees.

Any possible errors contained in this paper are the exclu-sive responsibility of the authors.

Appendix

Details regarding the study of partial derivatives permit-ting analysis of the conventional Sharpe’s ratio and the al-ternatives proposed as approximations to utility indicatorsin the presence of risk are given in Table 8.


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