picking the right risk-adjusted performance metric

WORKINGPAPER:PickingtheRightRisk-AdjustedPerformanceMetricHIGHLEVELANALYSIS QUANTIK.org

-1-

Thisdocumentmaynotbecopied,publishedorused,inwholeorinpart,forpurposesotherthanexpresslyauthorisedbyQuantik.org.

Investors often rely on Risk-adjusted performance measures such as the Sharpe ratio to chooseappropriate investments and understand past performances. The purpose of this paper is gettingthroughaselectionofindicators(i.e.Calmarratio,Sortinoratio,Omegaratio,etc.)whilestressingoutthemainweaknessesandstrengthsofthosemeasures.

1. Introduction.............................................................................................................................................12. TheVolatility-BasedMetrics.....................................................................................................................2

2.1. Absolute-RiskAdjustedMetrics..................................................................................................................22.1.1. TheSharpeRatio.............................................................................................................................................................2

2.1. Relative-RiskAdjustedMetrics...................................................................................................................22.1.1. TheModigliani-ModiglianiMeasure(“M2”)........................................................................................................22.1.2. TheTreynorRatio..........................................................................................................................................................32.1.3. TheInformationRatio..................................................................................................................................................3

3. HigherandLowerMomentsIndicators.....................................................................................................33.1. TheOmegaRatio.......................................................................................................................................33.2. TheSortinoRatio........................................................................................................................................43.3. TheUpsidePotentialRatio.........................................................................................................................4

1. DrawdownsMeasures..............................................................................................................................41.1. TheCalmarRatio........................................................................................................................................51.2. TheUlcerRatio...........................................................................................................................................5

1. PortfolioValue-at-Risk..............................................................................................................................62. Conclusion................................................................................................................................................63. References................................................................................................................................................6

1. Introduction

Investorsandanalystsinterestedinmeasuringhowgoodisaninvestmentclearlycannotonlyrelyontheabsolutereturnsprovidedbytheasset,orcomparethosereturnstoabenchmark,butmustalsoconsider the Risk they bear holding this investment, i.e. how much returns the investment hasgeneratedrelativetotheamountatRiskithastakenoveraperiodoftime.

There are plenty of alternative Risk-adjusted performance measures, and financial literature haswidelydiscussedthistopic,thepurposeofthispaperisnotgettingthroughallofthem,butthemajorones. Please note that returns, standard deviations, and other metrics can be computed on anyperiod,itisneverthelessadvisedtoannualisefigurestomakethecomparisoneasier.

Onedecidedtostructureandtopresentthoseindicatorsasfollows:

VOLATILITYBASEDMETRICS

AbsoluteRisk-Adjusted

Metrics

RelativeRisk-AdjustedMetrics

SharpeRatio

M2

TreynorRatio

InformationRatio

HIGHERANDLOWER

MOMENTSINDICATORS

OmegaRatio

SortinoRatio

DRAWDOWNMEASURES

CalmarRatio

UlcerRatio

PORTFOLIOVALUE-AT-RISK


-2-


2. TheVolatility-BasedMetrics

Financial analysts frequently use volatility-based metrics to assess portfolio performances.Paradoxically, despite that those measures tend to be very intuitive they remain difficult tointerpret; it can be because of the absence of benchmark (cf. Absolute-Risk Adjusted MetricsSection)ortheproblemtointerprettherelativeperformancesofaportfoliotoabenchmark(cf.Relative-Risk Adjusted Metrics Section). In addition, most volatility-based metrics rely onhistorical data and on the assumption that returns are normally distributed, undermining theimpactofpeak-to-valleyordownsideRisk.

2.1. Absolute-RiskAdjustedMetrics

Those metrics evaluate the portfolio risk-adjusted performances without considering anybenchmark.

2.1.1. TheSharpeRatioIntroducingthenotionofReward-to-VariabilityintheJournalofBusiness(1966),WilliamSharpesetupmilestonesoftheperformanceassessmentofMutualFunds.Theratiomeasurestheexcessreturn of a portfolio to the Risk free rate to the standard deviation of the portfolio. In a Riskadverseframework,investorsarelookingforahighratio.

𝑆ℎ𝑎𝑟𝑝𝑒 𝑅𝑎𝑡𝑖𝑜 =𝑅! − 𝑅!𝜎!

RP=Portfolioreturn(annualised)RF=Riskfreerate(annualised)σP=Standarddeviationofthedailyportfolioreturns

2.1. Relative-RiskAdjustedMetrics

Thosemetricsevaluatetheportfoliorisk-adjustedperformanceswhileconsideringabenchmark.Oneofthemajorweaknessesreliesinthebenchmarkselectionandontheinterpretationofthemetric.Youcaneasilyidentifyifyourportfoliooutperformedorunderperformedtheindex,butyouhavenoclueonhowtheperformancewasachieved.

2.1.1. TheModigliani-ModiglianiMeasure(“M2”)

Modigliani-Modigliani (1997) reconcile the notion of benchmark and Sharpe ratio in a singleformula,andfindthattheportfolioanditsbenchmarkmusthavethesameRisktobecompared.Theideaistoleverordeleveraportfoliosothatitsstandarddeviationisidenticaltothatofthemarketportfolio.TheM2of theportfolio is thereturnof thisportfolio thatcanbecomparedtothemarketreturn.

𝑀! = 𝑅! + 𝑆ℎ𝑎𝑟𝑝𝑒 𝑅𝑎𝑡𝑖𝑜 ∗ (𝜎! − 𝜎!)RP=Portfolioreturn(annualised)σB=StandarddeviationofthedailybenchmarkreturnsσP=Standarddeviationofthedailyportfolioreturns


-3-


2.1.2. TheTreynorRatioThe metric - developed by Treynor (1965) - measures the relationship between the excessreturnof aportfolioon theRisk free rate to theBetaof theportfolio.The ratio isparticularlyadequatetoassessperformancesofextremelydiversifiedportfolios;thisiswhyitislargelyusedbyacademics. Inpractice, investmentanalystscannot ignoretheIdiosyncraticRisk;this iswhytheratioismuchlessusedthantheSharpeone.

𝑇𝑟𝑒𝑦𝑛𝑜𝑟 𝑅𝑎𝑡𝑖𝑜 =𝑅! − 𝑅!𝛽!

RP=Portfolioreturn(annualised)RF=Riskfreerate(annualised)βP=SystemicRisk,Riskthancannoteliminatedbydiversifyingtheportfolio

2.1.3. TheInformationRatio

TheInformationratioindicateshowmuchexcessreturnisgeneratedfromtheamountofexcessRisktakenrelativetothebenchmark.This indicator isappreciatedby investorsthatcaneasilycompare performances of their portfolio to a benchmark and set investment objective to theinvestmentmanager.

𝐼𝑛𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛 𝑅𝑎𝑡𝑖𝑜 =𝑅! − 𝑅!𝜃! − 𝜃!

RP=Portfolioreturn(annualised)RB=Benchmarkreturn(annualised)σP=StandarddeviationofthedailyportfolioreturnsσB=Standarddeviationofthedailybenchmarkreturns

3. HigherandLowerMomentsIndicators

In the above Section, one only considered the first moment (mean of returns), and secondmomentof thedistribution(standarddeviationofreturns);whenactually the thirdandfourthmomentsmust also be considered (i.e. Skewness and Kurtosis, both indicate the shape of thedistribution). Any Risk adverse investor would be interested by (1) high returns, (2) a lowvolatility, (3)more positive returns than negative ones (distribution right skewed), and (4) aleptokurticdistribution(nottoomanyextremereturns,thenthintails).Risk-adjustedmomentsmeasuresrelymostofthetimeonasinglethreshold,aminimalreturnlevelchosenbyinvestors;itisimportanttoprecisethattheselectionofthelattercouldleadtodistortedinterpretationsofthemetric.Inaddition,ifparametersarederivedusingaparametricapproach, one face the sameweaknesses that those specified for volatility basedmetrics (i.e.normalisedreturns,etc.).

3.1. TheOmegaRatio

Shadwick andKeating (2002) elaborate a gain-loss ratio capturing the highermoments in thereturnsdistribution,theOmegaratio.Theauthorssplitthereturnsdistributionaboveandbelowaminimumacceptablereturnfor investorsandassesstheprobabilityofoccurrenceaboveandbelowthepartitioning.Thehigherthemetric,thebettertheinvestment.


-4-


Ω 𝑅𝑎𝑡𝑖𝑜 =1 − 𝐹 𝑅! 𝑑𝑥

!!!"

𝐹 𝑅! 𝑑𝑥!!"!

=𝑈𝑝𝑠𝑖𝑑𝑒 𝑃𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙𝐷𝑜𝑤𝑛𝑠𝑖𝑑𝑒 𝑃𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙

Ri=DailyreturnsRTI=Minimalacceptabledailyreturn[a,b]=Intervalofpossiblereturns

3.2. TheSortinoRatio

One already mentioned that the Sharpe ratio does not tell us anything about the returnsdistribution, is the distribution of returns symmetric around the mean, or mainly driven byreturns above themean (upsidepotential) orbelow themean (downsidepotential).Anaturalextension of the Sharpe ratio is the Sortino indicator only using the downside Risk as adenominatorandreplacingtheRisk-freeratebytheminimalacceptablereturn.

𝑆𝑜𝑟𝑡𝑖𝑛𝑜 𝑅𝑎𝑡𝑖𝑜 =𝑅! − 𝑅!𝜃!

RP=Portfolioreturn(annualised)RT=MinimalacceptableportfolioreturnσD=Standarddeviationofthedailyportfolioreturnsbelowtheminimalacceptablereturn

3.3. TheUpsidePotentialRatio

The Upside Potential ratio - developed by Sortino, Van der Meer and Plantinga (1999) - is ahybrid version between the Omega ratio and the Sortino ratio. The indicator has the upsidepotentialfornumerator(asfortheOmegaratio),andthedownsideRiskfordenominator(asfortheSortinoratio).

𝑈𝑝𝑠𝑖𝑑𝑒 𝑃𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝑅𝑎𝑡𝑖𝑜 =1 − 𝐹 𝑅! 𝑑𝑥

!!!"

𝜃!= 𝑈𝑝𝑠𝑖𝑑𝑒 𝑃𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙𝐷𝑜𝑤𝑛𝑠𝑖𝑑𝑒 𝑅𝑖𝑠𝑘

Ri=DailyreturnsRTI=Minimalacceptabledailyreturn[a,b]=IntervalofpossiblereturnsσD=Standarddeviationofthedailyportfolioreturnsbelowtheminimalacceptablereturn

1. DrawdownsMeasures

Intuitive Risk-adjusted measures may be derived from past drawdowns of the returnsdistribution.Drawdownsconsistof focusingonpeak-to-valleydeclinesduringaspecificperiodof investment; usually the percentage between the peak and the subsequent though. Themaximum drawdown may be defined as the maximum loss from a peak-to-valley over thespecified period. Drawdowns are very sensitive to the frequency of themeasurement intervaland to the length of the sample; assetsmeasured on a daily intervalwill tend to have higherdrawdowns than assets measured on a monthly interval; in addition managers with a largehistoricaldatasetwilltendtohavelargerdrawdowns.


-5-


1.1. TheCalmarRatio

The Calmar ratio developed by Terry Young (1991) is ameasure very frequently used in theHedgeFund industry and in the context of both illiquid assets and lack of historical data. TheCalmar ratio shares the same numerator as the Sharpe ratio, but gets for denominator themaximumdrawdownovertheperiod(i.e.1year).

𝐶𝑎𝑙𝑚𝑎𝑟 𝑅𝑎𝑡𝑖𝑜 =𝑅! − 𝑅!𝑀𝑎𝑥 𝐷

RP=Portfolioreturn(annualised)RF=Riskfreerate(annualised)MaxD=MaximumDrawdownovertheperiod

1.2. TheUlcerRatioPeterG.Martin(1987)developedtheUlcer indexwiththepurposeofnotonlyconsideringthedepthofthepricedeclinebutalsoitsduration.Theindexmeasuresthenegativereturnofeachperiodbelowthepreviouspeakorwatermark.Consequently,deepandlonglastingdrawdownswillhaveasignificantimpactontheratio.

𝑈𝑙𝑐𝑒𝑟 𝑅𝑎𝑡𝑖𝑜 =𝐷!!

𝑛

!

!!!

Di=Drawdownsincethelastpeakintheperiodn=Numberofobservations/daysintheperiod

CumulativeReturns

Time

Drawdown

MaximumDrawdown

Periodi


-6-


1. PortfolioValue-at-Risk

Another approach would consist in assessing the loss of the portfolio using a Value-at-Riskmethod. The different methods to compute the VaR (i.e. Historical, Parametric, Monte-Carlo)have already been developed in another Section of this website (Quantik.org - “DownloadCenter”,“VaRModels(Parametric,Monte-Carlo)”).

2. Conclusion

If the universe of Risk-adjusted performance measures is very large and still continues tobroaden,thispaperwasdedicatedtohighlightthemainindicatorsthatfrequentlyareusedintheindustry. We also expected to stress out the main weaknesses associated to the differentapproaches.

3. References• Goodwin,ThomasH.,1998,“TheInformationRatio”;• KeatingC. and ShadwickW.F., 2002, “AUniversal PerformanceMeasure”, Journal of PerformanceMeasurement,

vol.6,n°3;• Martin P and McCann B, 1989, “The Investor’s Guide to Fidelity Funds: Winning Strategies for Mutual Fund

Investors”;• ModiglianiF.andModiglianiL.,1997,“Risk-AdjustedPerformance”,JournalofPortfolioManagement,pp.45-54;• SharpeW.F,1966,“MutualFundPerformance”,TheJournalofBusiness,39(1),pp.119-138;• SortinoF.A.andVanderMeerR.,1991,“DownsideRisk”,JournalofPortfolioManagement,pp.27-32;• SortinoF.A.,VanderMeerR.andPlantingaA.,1999,“TheDutchTriangle”,JournalofPortfolioManagement,vol.

18,pp.50-59;• TerryW.Young,1991,"CalmarRatio:ASmootherTool",Futuresmagazine;• TreynorJ.L.,1965,“HowtoRateManagementofInvestmentFunds”,HarvardBusinessReview43,pp.63-75.

picking the right risk-adjusted performance metric

Documents