Properties of an ideal risk measureBy: Peter UrbaniRisk like beauty is largely in the eye of the beholder.Although we can probably all agree that risk hassomething to do with the possibility of loss, either inrelative- (keeping up with the Jones’s) or absolute-(keeping above the breadline) terms, there are nowa multitude of possible risk measures available toconfuse you.These range from Standard Deviation, Semi-variance, Downside Deviation, Beta, Sharpe Ratio,Sortino Ratio, Treynor Ratio, Jensen’s measure, M2,LPM, Tracking Error, Information Ratio, Value atRisk, Expected Shortfall, Shortfall Probability,Extreme Tail Loss, Expected Regret, MaximumDrawdown the Kappa function and now the Omegafunction to name but a few. Given this plethora ofoptions which measure is the best? It is also clear that we perceive risk as being relative to something in this case A is perceived toIn order to answer this question we must go back to be less risky than B but risk can be measuredfirst principles and ask what exactly it is we want relative to a number of possible benchmarks.from a risk measure and what the ideal properties of These include:such a measure should be. Only then can we makean informed comparison of different available In the case of a pension fund the value of themeasures. So what are they? funds future liabilities.According to the available academic literature a risk For those who abhor losses, relative to zero.measure should have the following properties: For anyone trying to preserve their wealth relative1) It should be Asymmetric to inflation.2) Relative to one or more benchmarks Relative to a default no-risk investment of having3) Investor-specific cash in the bank.4) Multidimensional Relative to some peer group or benchmark5) Complete in a specific sense6) Numerically positive Relative to our budgeted or target rate of return7) Non-linear For a sector or index fund the relevant sector or index.I shall endeavour to explain some of these conceptsin plain English so that you can decide for Some of these risk benchmarks could also beyourselves which properties you agree or disagree viewed as performance benchmarks except thatwith. falling below them is not simply disappointing but positively undesirable.Asymmetry of risk deals largely with how weperceive risk. Given two potential investments Since investors have different liability profiles andmarked A and B in the following example, most or objectives and may use different riskpeople would intuitively feel that B is riskier than A. benchmarks it is clear that the ideal risk measureThis is because although they both have the same needs to be flexible enough to be both investormean expected return of 10%, B has twice as much specific and accommodate multiple benchmarksvariability as A as denoted by its standard deviation hence multidimensional.of 10% Having justified the first four desirable propertiesB also appears to have more periods when its of a risk measure I will address the last three,returns are below those of A and also when they are Completeness, positivity and non-linearity by wayless than zero. The fact that this disquiets us of examples of those risk measures which fail tosuggests that we are more concerned about the satisfy these requirements. One of the morepotential downside of an investment than its upside attractive risk measures is the probability ofhence our response to risk is asymmetric and so shortfall. Clearly this is a number we are inshould the ideal risk-measure be. general interested in. Unfortunately the probability
of shortfall measure is not ideal because it is not One of the most widely used measures of risk,complete. Standard Deviation or Volatility is not really a measure of risk at all but rather a measure ofIf we consider the case of an investor, who is uncertainty. It is also particularly poorly suited forconcerned about losing capital relative to an use as the ideal risk measure for the followingimportant benchmark and is confronted with two reasons.hypothetical investment possibilities, E and F. Bothhave an expected return of zero relative to the If we consider investments A, C, and D in which bothbenchmark and both have a probability of shortfall of C and D have the same standard deviation as one50%, but are they equally risky? another (10%) whilst A has a standard deviation of 5%. Using standard deviation as your sole measureIf an investors only measure or risk is the shortfall of risk you would be indifferent between C and D.probability then he/she will be indifferent between E But this is clearly wrong since D has an averageand F. However we can see that F has a greater expected return of -10% versus C’s +10%. Manypotential downside and that everywhere in the people object to standard deviation as a riskshaded area also a greater probability of realising measure because it gives equal weight to deviationsthat downside than E. Thus the shortfall probability above the mean and deviations below the mean,measure, although interesting, does not address the whereas investors are likely to be more worriedissue of how severe an event may be. It is thus about “downside deviation” than “upside deviation.”insufficient and incomplete.Similarly if we now use maximum shortfall as theonly measure of risk using example F and G we cansee although both have the same probability ofshortfall of 50% and the same maximum shortfall of-30% it is not clear which is riskier because themaximum shortfall measure alone says nothingabout the size of the typical shortfall. Twoinvestments may have the same worst outcome butone may have many large losses and the other onlya few. Information about the end point of the lowertail of a distribution says little about the distributionoverall. Moreover, we typically have only a few datapoints with which to work in the tail making themaximum shortfall measure both numerically-ill-conditioned and incomplete as a risk measure. Another problem with using standard deviation as a risk measure is that it is not sensitive to order. In the below examples you can see that A and C have the same standard deviation and mean.
However C is clearly riskier than A, having lost 38% bad as losing half of your money. I don’t think so,”of its value from its peak to trough during the he says. “It’s at least 10 times as bad.”hypothetical period shown. Markets that look, or feel, Why VaR is not a coherent measure of riskvolatile often feel that way because of a distinctorder of prices or returns: an order that involveschoppy movements with frequent reversals. This 1) Subadditivitykind of “order dependent volatility” is not captured by For all random losses X and Ythe technical definition of standard deviation, since p(X)+p(Y) > p(X+Y)standard deviation is not sensitive to order. Thispoint has direct application to hedge fund investing, Scenario p(X) p(Y) p(X+Y)since many hedge fund managers employ tradingstrategies whose success or failure will be related 1 0 0 0not to the volatility of markets but to the path that 2 0 0 0markets follow 3 0 0 0 4 0 0 0 5 0 0 0Value at Risk has garnered widespread acceptance 6 0 0 0in recent years as the new measure of risk. Despitethis widespread use it is also not complete in that it 7 0 0 0is not mathematically coherent. In order to be 8 0 0 0mathematically coherent a risk measure must satisfy 9 100 0 100the conditions of: 10 0 100 1001.) Subadditivity2.) Monotonicity VaR @ 85% 0 0 1003.) Positive Homogeneity and; 0 + 0 is not > 1004.) Translation invariance. 2) MonotonicityWithout going into detail on these, suffice it to say If X < Y for each scenario thenthat Value at Risk fails to satisfy both theSubadditivity and Monotonicity conditions. This has p(X) < p(Y)two consequences. The first is that the sum of theparts may be less than that of the whole and Scenario X Ysecondly that the graphing of value at risk as a 1 1.00 5function of returns, as in the mean / value at risk 2 2.00 5frontier, may not result in a neat convex function.This makes finding the optimal point difficult using 3 3.00 5conventional methods. 4 4.00 5 5 5.00 5Fortunately there is one measure of risk, closely 6 5.00 5related to value at risk that is both mathematically 7 4.00 5coherent and complete. That is the ExpectedShortfall measure aka. the conditional Value at Risk 8 3.00 5or Extreme Tail Loss. 9 2.00 5 10 1.00 5What the Expected Shortfall measure provides is a E( r ) 3.00 5.00probability weighted average of the expected losses SD 1.41 0.00in excess of the value at risk. Hence it is the averageof the tail losses conditional on the value at risk VaR 5.83 5.00being exceeded. As a risk measure, Expected E( r ) + 2 x SD 5.83 is not < 5shortfall captures the whole of the downside portionof the relative probability density function and iscomplete. However, its one remaining failing is that 3) Positive Homegenietyit, like value at risk, is a linear measure of risk.The non-linearity of risk is closely related to investor For all L > 0 and random losses Xpsychology and utility. More people insure their p(L X) = L p(X)homes than their pets despite the fact that thepossibility of losing a pet is significantly higher than 4) Translation Invariancelosing a home. This reflects the non-linearity of howwe perceive risk. We perceive a low probability ofexperiencing a large loss as being far worse than a For all random losses X and constant ahigh probability of experiencing a small loss. Frank p(X+a ) = p(X)+aSortino says “VaR. It’s simply a linear measure ofrisk. It says that losing all your money is twice as
Sortino strongly holds the view that it is downside References:risk that is most important. According to this view,the most relevant returns are returns below the A unified approach to upside and downside returnsmean, or below zero, or below some other “target” or – Leslie A Balzar (2001)“benchmark” return.This has lead to a proliferation of measures of Expected Shortfall, a natural coherent alternative“downside risk”: semi-variance, shortfall probability, to Value at Risk - Acerbi and Tasche (2001)the Sortino ratio, etc. Ignoring the specificadvantages and disadvantages of each individual Coherent measures of risk - Artzner and Delbaencandidate to represent “the true nature of risk,” we (1999)would offer two general observations:Frequency vs. Amplitude. The idea of risk as“expected pain” combines two elements: the Peter Urbani, is Head of KnowRisk Consulting. Helikelihood of pain, and the level of pain. The was previously Head of Investment Strategy formeasures described above (with the exception of Fairheads Asset Managers and prior to that SeniorExpected shortfall) focus on one or the other of Portfolio Manager for Commercial Unionthese elements, but not both. Semi-variance (and its Investment Management.descendant, the Sortino ratio) focuses on the size ofthe negative surprises, but ignores the probability of He can be reached on (073) 234 -3274those surprises. Shortfall probability focuses on thelikelihood of falling below a target return, but ignoresthe potential size of the shortfall. If I were forced topick a single quantitative measure of risk, it wouldoffer the concept of “expected return below thetarget,” defined as the sum of the probability-weighted below-target returns. This measure isessentially the area under the probability curve thatlies to the left of the target return level. (Note thatthis definition is broad enough to cover both normaland non-normal distributions.)Other generalisations of this such as the LPMnmeasure and the Omega function which capturesthe full distribution are also available.The final criteria is given as numerical positivityalthough this is more a desirable than essentialrequirement. Personally I prefer to show losspercentages such as value at risk in negative termssince this is more intuitive, but it is more common toshow them +ve because of the widespread use ofquadratic penalties in scientific optimisation.Past performance does not guarantee anythingregarding future performance, and past risk does notguarantee anything regarding future risk. This is trueeven when the historical record is long enough tosatisfy normal criteria of statistical significance. Theproblem is that, just as a performance record isgetting long enough to have statistical significance, itmay no longer have investment significance.because the people and the organization may havechanged. Investors should thus use the full toolboxof available risk measures but not loose sight of thewood for the trees.
Useful Calculations in Excel A B Normal distribution (Prob. density) 0.45 1 Mean 13.13 13.13 2 Std Dev 17.87 0.40 3 CL 0.95 0.35 4 HPR 1 0.30 5 MAR 5.00 0.25 6 0.20 -16.26 7 Normal VaR -16.26 0.15 8 Expected Shortfall -23.73 -23.73 9 Downside Deviation 13.52 0.10 10 Below MAR Deviation 8.53 0.05 11 Shortfall probability 32.46% 0.00 12 Upside Potential 42.52 -60 -40 -20 0 20 40 60 80 13 Average Shortfall -3.79 Normal Probability density Mean is 13.13 14 Upside Potential Ratio 131 Selected probability 5.0% VaR @ 95.0% CL is -16.26 15 Regret 2.24 ETL @ 95.0% CL is -23.73 ABS @ 32.5% CL is 5.00Normal VaR=-(-B1*B4-(NORMSINV(1-B3))*B2*SQRT(B4))Expected Shortfall=-(-B1*B4+(NORMDIST(NORMSINV(1-B3),0,1,FALSE)/NORMDIST(NORMSINV(1-B3),0,1,TRUE))*B2*SQRT(B4))Downside Deviation=SQRT((((NORMDIST(0,B1,B2,TRUE))*(B2^2+B1^2))-((B2^2*NORMDIST(0,B1,B2,FALSE))*B1))/NORMDIST(0,B1,B2,TRUE))Below MAR Deviation=SQRT(((B5-B1)^2+B2^2)*NORMDIST(B5,B1,B2,TRUE)+(B5-B1)*NORMDIST(B5,B1,B2,FALSE)*B2*B2)Shortfall probability=NORMDIST(B5,B1,B2,TRUE)Upside Potential=-(-B1*B4-(NORMSINV(B3))*B2*SQRT(B4))Average Shortfall=-((B5-B1)*NORMDIST(B5,B1,B2,TRUE)+NORMDIST(B5,B1,B2,FALSE)*B2*B2)Upside Potential Ratio=B12/B11Regret=((NORMDIST(B5,B1,B2,TRUE)*(B2^2+B1^2-2*B5*B1+B5^2))-((NORMDIST(B5,B1,B2,FALSE)*B2^2)*(B1-B5)))/NORMDIST(B5,B2,B3,TRUE)/100