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NoN-Normality of market returNs: a framework for asset allocatioN DecisioN makiNg wiNter 2010

When market stress and dis-ruptions hit portfolios espe-cially hard, it is natural for investors to ask whether

anything could have been done differently. We revisit our tools and strategies looking for ways to improve them. And often, it is in such times of ref lection that we arrive at impor-tant insights and begin to craft effective solu-tions. The turmoil of 2007/2008 is just one among many such financial crises over the past 30 years—rare and unpredictable events that can, and should, challenge conventional ideas about portfolio construction.

Of course, we can never know in advance when such events will occur and how bad they will be. We do know, however, that we empirically observe such events with much greater frequency than current models allow for. So, the proper line of questioning for investors becomes: Can current risk manage-ment frameworks be modified to better cap-ture the long-term, downside risk associated with these rare but dangerous anomalies—i.e., frameworks that take into account more than just one-standard-deviation events—and better ref lect their observed frequency?

The goal of this article is to present such a revised framework.

In broad scope, we believe that two specif ic weaknesses in conventional risk assessment may be contributing to a quantifi-able underestimation of portfolio risk:

• Frameworks assuming “normality”:1 Conventional asset allocation frame-works typically make a range of assump-tions about the “normality” of asset returns, the most problematic of which are that returns are independent from period to period2 and normally dis-tributed. In reality, we can empirically observe that in many cases returns are not independent, and in all cases, they are not normally distributed.

• Inadequate risk measures: If one adopts an asset allocation framework that incor-porates non-normality, then standard deviation becomes ineffective as the primary quantifier of portfolio risk.

Using the latest statistical methods, however, we believe that these shortcomings can be addressed. And based on the results presented here, we argue that a modified risk framework may help investors improve port-folio efficiency and resiliency.

Of course, no single statistical frame-work—no matter how robust—can render portfolios immune to such extreme events. Furthermore, a quantitative framework is only one input into the asset allocation pro-cess and cannot replace the professional skill and judgment necessary to arrive at an appro-priate strategy. The importance of allowing for subjective—and often qualitative—factors in decision making remains. Furthermore,

Non-Normality of Market Returns: A Framework for Asset Allocation Decision MakingAbdullAh Z. Sheikh And hongtAo QiAo

AbdullAh Z. Sheikh

is director of research in the Strategic Investment Advisory Group at J.P. Morgan Asset Management in New York, NY. [email protected]

hongtAo QiAo

is a strategic advisor in the Strategic Investment Advi-sory Group at J.P. Morgan Asset Management in New York, [email protected]


there is always an explicit need to account for the investor’s specific circumstances, including liabilities, when arriving at an appropriate portfolio allocation.

As sudden market disruptions go, the events of the past year—the subprime mortgage crisis and ensuing global financial meltdown—are not without precedent. Just in the last three decades, investors have been faced with a number of financial crises:

• LatinAmericandebtcrisisintheearly1980s• Stockmarketcrashof1987• U.S.savings&loancrisisin1989–1991• WesternEuropeanexchangeratemechanismcrisis

in1992• EastAsianfinancialcrisisof1997• RussiandefaultcrisisandtheLTCMhedgefund

crisisin1998• BurstingofU.S.technologybubblein2000–2001.

While these anomalous events are rare, we observe such extreme “non-normality” in real-world mar-kets more frequently than current risk management approaches allow for. Said another way, we believe that conventionally derived portfolios carry a higher level of downside risk than many investors believe or current portfolio modeling techniques can identify.

The primary reason for this underestimation of risk lies in the conventional approach to applying mean-variance theory, which was pioneered by Harry Markowitzin1952.Traditionalmean-varianceframe-works have become the bedrock of top-down asset allocation decision making. As suggested, a standard assumption in the mean-variance framework, and indeed many other holistic asset allocation frameworks, is that future asset class returns will be independent from period to period and normally distributed.

Despite being widely recognized as overly sim-plistic, suchbroad assumptionsof“normality”haveappeal due to the ease with which they can be imple-mented. To implement an asset allocation framework based on normal asset return distributions, practitioners need only make two assumptions for each asset class (mean and standard deviation) and one assumption for each pair (co-variance). The latter applies because one implicitly assumes the relationship between each pair of asset classes is linear (another problematic issue discussed later in this article).

But what if asset returns are not normallydistributed?

In fact, in the real world we can observe empiri-cally that returns are not normally distributed. This leads us to ask: How would incorporating non-normality affect the strategic asset allocation process?

EMpiRical EvidENcE of NoN-NoRMality iN assEt REtuRNs

Addressing the problem of non-normality entails employing quantitative techniques that are rather more sophisticated than traditional asset allocation techniques. However, our end goal is conceptually very simple: to produce an asset allocation framework that balances future portfolio return expectations with the potential for more robust downside risk management.

First,however,wemusttesttoseewhetherandwhat kinds of non-normality may be present in empiri-cally observed asset returns. In this section, we present the results of such tests, which indicate the presence of several types of non-normality that are typically not allowed for in traditional asset allocation frameworks—to the detriment of risk estimation.

We address three categories of non-normality3 in the following order:

1. Serial correlation in asset returns—this occurs when one period’s return is correlated to the previous period’s return, inducing dependence over time.

2.“Fat” left tails (negative skewness and leptokurtosis)—this occurs when extreme negative returns are observed, with a magnitude and frequency greater thanimpliedbythe“normal”distribution.

3. Correlation breakdown in joint asset class returns—this occurs during periods of high market volatility and is typically not captured by linear correlation matrices.

Throughoutthissection,webaseourtestson15yearsofmonthlyreturndatatoMay31,2009.4 In later sections we incorporate these empirical results into a revised asset allocation framework.

serial correlation of Returns

Foreachassetclass,weformallytestforserialcor-relationbycalculatingwhatiscalledthe“Q-statistic.”

JAI-SHEIKH.indd 10 12/11/09 11:25:58 AM

the JourNal of alterNative iNvestmeNts wiNter 2010

In this test, we are looking to either affirm or reject the “nullhypothesis,”whichassertsthatserialcorrelationdoes not exist in the data.

Forthosefamiliarwiththisstatisticalprocedure,we note that the complete off icial moniker for the “Q-statistic”istheLjung–BoxQ-statistic. It is com-puted as follows:

Q T TT jLB

j

j

k

= +−=

∑( )22

1

τ

Where τj is the j-th serial correlation, k is the number

of lags being tested and T is the number of observations.If the Q-statistic for a given asset class has signifi-

canceata5%confidencelevel(i.e.,ap-valueoflessthan0.05),wecanconcludethatthereissufficientevidenceto reject the null hypothesis of no serial correlation. In this case, we must allow for the effect of serial correlation on future asset class returns. If the p-value is higher than0.05,thenullhypothesisisnotrejectedandweconclude that there is insufficient evidence to reject the null hypothesis of no serial correlation.

InExhibit1weshowtheresultsoftestingforserialcorrelation in the return series of our universe of eight asset classes.

We find statistical evidence of f irst-order serial correlation5 in the returns of several asset classes: Inter-nationalEquity,EmergingMarketsEquity,HedgeFundofFunds,PrivateEquity,andCommodities.

We hypothesize that serial correlation is common in alternative investment strategies—such as Hedge FundofFundsandPrivateEquity—becausetheyoftenhold illiquid and hard-to-price assets. The diff iculty in valuing the underlying assets at regular intervals requires managers or administrators to estimate prices (for example, with reference to the closest marketable security or based on certain economic indicators).

If current asset prices are derived, for example, by updating last month’s asset prices (after allowing for changes in the economic environment since the last valuation), then serial correlation ref lects a gradual rec-ognition of the true underlying value of the asset. Such a valuation methodology would explain our empirical observation of first-order serial correlation in the asset

e x h i b i t 1testing for serial correlation in Monthly asset class Returns

Source: J.P. Morgan Asset Management. For illustrative purposes only.



returns of our alternative asset classes. And a con-sequence of gradual (rather than instantaneous) recognition of the true underlying value of an asset class is that conventional risk estimates derived from a serially correlated return stream will be underestimated.

In the caseof InternationalEquity andEmergingMarketsEquity,thepresenceofserialcorrelation is a very recent phenomenon. In fact, it is a direct consequence of the financial crises experienced globally by investors over 2008 and into2009.

Public equity markets around the world have fallen precipitously over 2008 and into 2009,displayingmonth-upon-monthofnega-tivereturns.InDevelopedInternationalEquity,9ofthelast12monthlyreturnstoOctober2008werenegative.InthecaseofEmergingMarketsEquity,8ofthelast12monthlyreturnswerenegative. This has had the effect of inducing sta-tistically significant serial correlation in monthly public equity market returns for International andEmergingMarketsEquity.

Unfortunately, the presence of serial cor-relation distorts the true risk characteristics of an asset class. In particular, it is incorrect to model future returns assuming independence, if in fact returns are serially correlated (as our statistical tests indicate). If it is not corrected, serial correlation can reduce risk estimates from a time series by inappropriately smoothing asset class volatility.

Evidence of “fat” left tails

Our second form of non-normality relates to observing negative returns in greater magnitude and with a higher probability than implied by the normal dis-tribution. As an example, let us consider U.S. equities.

Exhibit2showsasimplehistogramofmonthlyreturnsforU.S.equitiesoverthelast15years.Itillus-trates that the distribution is clearly not symmetric. In fact it has a very noticeable negative skew (i.e., leans to the right) as well as excess kurtosis (i.e., is more peaked than a normal distribution; it has a kurtosis value of 4.1,whichisabovethe“normal”valueof3.0).Wecanconfirm this intuitive observation with a simple statis-ticaltest:theJarque–Bera( J-B)test6 is a simple test of

non-normality and conclusively rejects the null hypoth-esis of normality in the return distribution.

WhenwerepeattheJ-Btestusingourotherassetclasses, we find that those results, too, reject the null hypothesis. Their returns also exhibit non-normality. We can illustrate the understatement of downside risk by comparing two types of density functions:

1. The density function of the normal distribution.7

2. The kernel (i.e., empirical) density function “implied”byactualhistoricalreturns.

InExhibit3,weshowdensityfunctionresultsforall eight asset classes. Our focus, in particular, is on the left tails of the density function and on whether the normal distribution underestimates the probability of outcomes at these extremes.

These kernel (or empirical) density functions clearly indicate“fat”lefttailsrelativetothoseimpliedbythenormaldistribution.ThetableinExhibit4summarizesthe results of our tests, which identify non-normality in all eight asset classes and fat left tails across all asset classes. The fat left tail phenomenon implies that using a normal distribution to model returns underestimates the frequency and magnitude of downside events.

e x h i b i t 2Histogram of u.s. Equity Returns, 15 years Ending May 2009




e x h i b i t 3density function Results for all Eight asset classes—“fat” left tails in Historical Returns



correlation Breakdown: converging correlations during periods of Market stress

It is a common observation in financial literature that correlations tend to be quite unstable over time. Depending on the historical period, different asset classes can provide varying degrees of diversification. Market eventssuchasthe1987stockmarketcrashandthe1997Asian financial crisis illustrate how contagion—the risk of one market negatively impacting and destabilizing another—can lead to cases of converging correlations.

In this section, we test the hypothesis that correlations and volatility are positively related. In particular, we inves-

tigate whether correlations between asset classes tend to increase during periods of high market volatility and stress, compared to periods of relative calm. If this proves to be the case, it means that the benefits of portfolio diversification may be overestimated by traditional frameworks that rely on linear correlation matrices. In other words, diversifica-tion may not materialize precisely when an investor needs it most—i.e., during periods of high market volatility.

To test our hypothesis,8 we analyze simple Pearson correlations9 over two distinct historical periods—a full 10-year period versus a shorter, high-volatility period of market stress. Our chosen volatility measure is the Chicago BoardOptionsExchangeVolatilityIndex(CBOEVIX).Exhibit5plotsCBOEVIXvaluessinceitsinception.

e x h i b i t 3 (continued)


e x h i b i t 4Empirical Evidence of “fat” left tails in asset Returns




TheCBOEVIX tracks expected equityvolatility over a 30-day period.10 It is widely regarded as one of the most reliable measures of market uncertainty available, and it is traded in realtimeonadailybasis.Weidentify“high-volatility”periodsinrelationtothelong-termVIXaverage.

More specifically, our two test periods are defined as follows:

1. The control period is a full 10-year history, fromOctober1997toSeptember2007,duringwhichtheVIXaveraged20.9%.

2. The period of market stress used for com-parisonis fromAugust1998toSeptember1999,duringwhichtimetheVIXaveraged30.9%—considerablyhigherthanits10-yearaverage. This high-volatility period captured

e x h i b i t 5cBoE viX since inception


e x h i b i t 6correlation Matrix—full period and High volatility period




both the Asian and the Russian f inan-cial crises, which led to signif icant market uncertainty.

Thecorrelationmatrices inExhibit6 indicatethat investing in certain pairs of assets should pro-vide diversif ication benefits—i.e., they should either benegativelyor lowlycorrelated.Forexample, let’sconsidercorrelationsofassetclasseswithU.S.Equi-ties.Basedonhistoricalcorrelationsover theentire10 years, we expect the following asset classes to provide somediversificationbenefitsrelativetoaU.S.Equity-dominated portfolio:

• U.S.Bonds(ρ=–0.21)• REITs(ρ = 0.31)• HedgeFundsofFunds(ρ = 0.46)• PrivateEquity11 (ρ = 0.61)

In times of market stress, however, these correla-tionsbreakdown.PanelBofExhibit6showsasimplecorrelation matrix for the high-volatility period from August1998toSeptember1999.Wehavehighlightedin bold where correlations have converged (i.e., reducing diversif ication) and in italic where correlations have diverged (i.e., increasing diversification).

The results are striking. Nearly all the correlation coefficients increased to some degree, with only four showing a decrease. Our main diversifiers did not pro-vide the diversification benefits anticipated. The correla-tioncoefficientswithU.S.Equitieswereasfollows:

Long-Run During Market Coefficient StressU.S.Bonds –0.21 –0.02REITs 0.31 0.58HedgeFundofFunds 0.46 0.60PrivateEquity 0.61 0.86

Our analysis reaffirms our intuition, suggesting ample evidence of correlation convergence during periods of market stress.

Again, conventional frameworks which assume normality—in this case normal distribution of joint returns—prove inadequate to account for what we observe in real-world markets. And inappropriately assuming linearity of correlations can lead to a signifi-cant underestimation of joint negative returns during a market downturn.

iNcoRpoRatiNg NoN-NoRMality iNto aN assEt allocatioN fRaMEwoRk

Basedonempiricaltesting,wehavedemonstratedthat conventional asset allocation frameworks under-state downside risk because they do not allow for non-normality. However, we believe that achieving optimal portfolio efficiency—based on a more precise estimation of portfolio risk—requires investors to incorporate non-normal return distributions into the asset allocation and portfolio optimization process.

In this section, we offer a statistical methodology for incorporating the three categories of non-normality already discussed. We address them in the same order as they were presented in the previous section:

1. Serial correlation2.“Fat”lefttails(negativeskewnessandleptokurtosis)3. Correlation breakdown (converging correlations).

incorporating the impact of serial correlation

Our empirical work in the last section shows that certainassetclasses—InternationalEquity,EmergingMarketsEquity,HedgeFundofFunds,PrivateEquityand Commodities—test positive for serial correlation. This result implies that, over time, single period returns are inf luenced by, and not independent from, previous periods.First-orderserialcorrelation,ifnotcorrectedfor in the underlying data, will mask true asset class volatility and bias risk estimates downwards, leading to underestimation of asset class risk and, hence, port-folio risk.

Inorderto“adjust”fortheimpactofserialcor-relation on asset returns, we apply a variation of Fisher–Geltner–Webb’s“unsmoothing”methodology,12 wherebywere-calculatean“unsmoothed”returnstreamfrom our original data. The new adjusted return series should be more ref lective of the risk characteristics of the underlying data-generating process. Notably, our new series should have a higher volatility, demonstrate no serial correlation, and have a cross-correlation structure with other asset classes similar to our original data.

Ourprocedure for “unsmoothing” returns (orcorrecting for serial correlation in the underlying data series) is a two-step process:



Step 1: We determine the correlation coefficient at lag one13 (i.e., previous month’s return) for each return series, by running the following regression:

Rt = a + bR

(t–1)

where Rt represents the return at time t.

The regressions indicate estimates shown in Exhibit7forthe“serial”correlationcoefficients,based on monthly returns.

Step 2: We then produce our unsmoothed returnseriesasfollows,basedonthe“serial”cor-relation coefficient already derived:

Rt (corrected) = (R

t–bR

t–1)/(1–b)

After applying the methodology outlined above, we obtain a new data series R

t (corrected)

that should display no serial correlation. Once again we use the Q statistic to test the unsmoothed data for serialcorrelation.TheresultsareshowninExhibit8.

The test indicates that our revised data is no longer serially correlated. More important, our transformations of the underlying data series do not alter the mean return of the series over the time period considered. However, removing the effects of serial correlation does increase thestandarddeviationofreturns,asshowninExhibit9.Asdiscussed,thehighervolatilityofour“unsmoothed”return series is in line with our expectations, as serial correlation masks volatility and understates risk.

Finally,weevaluatetheimpactonassetcorrelationsto assess the viability of our unsmoothed data series—as our unsmoothing procedure should not affect correla-tions. We compare our original correlation matrix using indexdata(Exhibit10,PanelA)withournewcorrelation

matrix derived from unsmoothed data (Exhibit 10,PanelB).Andweseethat,asanticipated,theimpactofunsmoothing on asset correlations is negligible.

incorporating the impact of “fat” left tails into our framework

All of our asset classes show fat left tails, com-pared to the normal distribution. Our approach to dealing with such left-tail risk involves what is called a “semi-parametricapproach”:thatis,insteadofmodelingreturns using a single distribution (as is traditionally done with the normal distribution), we fit the left and right tails separately from the interior of the distribution.

Hence, we segment each return distribution into three parts—the left tail, right tail, and interior. We then“fit”eachsegmentofthedistributionseparately.This approach enables us to more accurately capture the potential distributions associated with the empirical data in each segment of the distribution.

To fit the left and right tails of the distribution we turntoExtremeValueTheory—abodyofworkspecifi-cally designed to look at the probability of low-probability buthigh-riskevents.ThecentralpremisebehindExtremeValueTheoryisthattheprobabilityofobservingextremevalues (such as negative financial market returns) can be modeled using any one of several extreme value distri-butions.14 Our choice of extreme value distribution is the generalized Pareto distribution (GPD).15 We choose the GPD distribution, rather than other extreme value dis-tributions, due to the ease with which it can be adapted to modeling financial market returns.

e x h i b i t 7correlation coefficients at lag one

Notes: Note the coefficient β ref lects the strength of the autocorrelation.aAt the 5% level.


e x h i b i t 8tests for serial correlation on corrected data for up to six lags




The“fitting”processisdoneintwosteps:

1.Fittingtheleft(andright)tailoftheGPDtoourhistorical data. This is done by calibrating the tail of the GPD to extreme values in the sample using the methodofmaximumlikelihood.Extremevaluesare defined by a threshold. This method is com-monlyreferredtoasthe“peaks-over-threshold”methodwithinExtremeValueTheory.

2.Fitting the interiorof adistribution is accom-plished using a non-parametric (kernel or empir-ical) approach.

Hence, in aggregate, we derive a semi-parametric probability density function to describe the data gen-erating process. We refer to this broad approach as a semi-parametric GPD approach. We can test the results of the fitting process by analyzing the cumulative den-sityfunction(CDF)oftheactualdataversusthesemi-parametric GPD approach:

• Whatwearelookingforisabetterfittotheactualdistribution of empirically observed data—as shown in the empirical cumulative density function.

• Whatwe see is that ineachcase (Exhibit11),the GPD distribution is a good fit to real-world data.

• Theimprovedfitisparticularlynoticeableintheleft tail, where the normal distribution under-states risk.

To enhance our conviction in these results, we needtostatisticallytestthis“goodnessoffit”bothforthe whole distribution as well as specifically for the tails.

WeapplytheKolmogorov–Smirnov(KS)16 empirical distribution test under two hypotheses: first, the null hypothesis that returns are normally distributed, and second, under the hypothesis that they are sampled from asemi-parametricGPD.Exhibit12setsoutourresults.We also show the probability that our null hypothesis stands statistical significance.

Our empirical tests confirm our view that the semi-parametric GPD distribution provides a better fit for the historical data—when compared to the normal distribu-tion—for all our asset classes. Given our focus is on left tail risk, however, we can also gauge fit by considering the quantile–quantile(QQ)plotsoftheempiricaldataagainstthose implied by the semi-parametric GPD distribution.17

AsseeninExhibit13,ineachcase,thelowestquin-tiles are a much better fit with the semi-parametric GPD distribution than with the normal distribution, indicating that the semi-parametric GPD distribution more accu-rately captures the frequency of extreme negative events.

incorporating the impact of converging correlations into our framework

One of our empirical findings in the previous sub-section was that correlations converge during periods of high market volatility and stress. Our approach to dealingwiththisphenomenonistoapply“copula”the-ory—a body of work that explicitly looks at the impact of contagion or converging correlations at the total portfolio level. Copulas are mathematical functions that allow us to model the joint distribution of asset returns separately from the marginal (or individual asset class) distributions.Byconsideringjointdistributions,weturnour focus to how asset classes behave together (rather than individually, as we did in the last subsection).

e x h i b i t 9volatilities Before and after “unsmoothing” for serial correlation




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e x h i b i t 1 1fitting the semi-parametric gpd distribution to Historical data



In a traditional framework, joint distributions are captured by simple (Pearson) correlations. Unfortunately, simple correlations assume a linear relationship between individual asset classes, and we have already shown empirically that linearity breaks down at the extremes. Copulas, on the other hand, allow us to differentiate the relationship between asset classes in times of market stress, from more normal times.

In particular, we observe empirically that not only doindividualassetclasseshave“fatter”lefttailsthanallowed in a normal distribution, combinations of asset classes exhibit fat joint left tails—i.e., a higher inci-dence of joint negative returns in times of market stress. Copulas allow us to model both the fat marginal left tails for individual asset classes (using the semi-parametric GPD approach explained earlier), while at the same time

e x h i b i t 1 1 (continued)

e x h i b i t 1 2testing goodness of fit of Normal and semi-parametric gpd distribution





e x h i b i t 1 3QQ plots—Normal distributions versus semi-parametric gpd distributions






modeling more accurately the fat joint left tails ( joint negative returns) as empirically observed.

JustastheapplicationofExtremeValueTheoryinvolves choosing from among several types of extreme value distributions, applying copula theory requires us to choose a copula type. Here we use the t-copula (based on the Student t distribution)18 because it enables us to better capture the effects of converging correlations—i.e., allows fatter tails than the normal distribution.

This approach better ref lects the empirical obser-vations of converging correlations during periods of

high volatility, which tend to be accompanied by nega-tive returns. Specifically, we model joint fat left tails byadjustingthecopula’s“degree-of-freedom”param-eter: a parameter of 30 degrees of freedom is virtually identical toanormaldistribution.By lowering theparameter, we can increase asset class dependence to model observed asset class behavior in times of market stress.

InExhibit14,weillustratetheimpactofmodi-fying the normal marginal distributions by adding a t-copula with a low degree-of-freedom parameter.


Notes: The results are based on applying Extreme Value Theory to the raw “smoothed” data. In our framework—because we require independence of asset returns—we remove the effect of serial correlation before fitting a semi-parametric GPD.




e x h i b i t 1 4Modeling Joint dependence in asset Returns using copulas




In order to show the effect of modeling increased jointdependence,weproduceascatterplotof2,500simulations of U.S. large-cap equity returns and Hedge FundofFundsreturnsfirst,assumingnormalitywithsimple correlations; and second, assuming normality with a Student t copula added (with two degrees of freedom19).

Compared to the traditional multivariate normal framework, our simulations show the impact of increased dependence through a higher joint incidence of negative returns between U.S. large-cap equities andHedgeFundofFunds.This illustratesthephe-nomenon of fat joint left tails we alluded to earlier.

Our simulations also illustrate increased depen-dence of other extreme joint occurrences. In other words, we can see an increased incidence of negative U.S.large-capequityreturnswithpositiveHedgeFundofFundsreturns(thetopleftquadrant)orpositiveU.S.Equity returnswith positiveHedgeFundof Fundsreturns (the top right quadrant). This is visible as the “star”patterninthescatterplot.

This occurs because our Student t distribution is symmetric around the mean (like the normal dis-tribution) and, hence, incorporating fat joint tails increases the probability of observing all extreme joint outcomes (positive and negative)—not just extreme joint negative outcomes. This can be seen as a drawback of our choice of the Student t distribu-tion, as—in reality—correlations do exhibit asym-metric behavior, i.e., the convergence properties are different in extreme positive markets compared to extreme negative markets.

However, our choice of t copula is a vast improve-ment when compared to linear correlation matrices used in traditional models, as the latter completely ignore the effect of correlation convergence. Incor-porating fat joint left tails using copula theory allows us to more robustly model the empirical phenomenon of correlation breakdown during periods of market stress.

EstiMatiNg dowNsidE poRtfolio Risk iN a NoN-NoRMal fRaMEwoRk

Now that we have a statistically solid framework for incorporating non-normality into the asset allocation process, we can apply this framework to a

hypothetical U.S.-domiciled investor. We will assume a well-diversif ied portfolio with an initial value of $1 billion and allocations across our eight major asset classes. Detailed allocations to each asset class are shown inExhibit15.

We can now generate forward-looking projections of our hypothetical portfolio’s real value20 in our newly revisedMonteCarlosimulationframework.Foreachof our 10 forward years, we generate 10,000 simula-tions.21 We then measure downside portfolio risk as the conditionalvalueat riskat the5thpercentileof theportfolio.

Wedefineconditionalvalueatrisk(CVaR95

) as the average (real) portfolio loss (relative to the starting value)intheworst5%ofscenarios,basedonour10,000Monte Carlo simulations. It is simply the average real lossintheworst500(5%of10,000)scenarios(i.e.,theleft tail of the portfolio distribution).

e x h i b i t 1 5Hypothetical portfolio allocation

Note: Sharpe ratio calculated based on expected risk-free return of 4.0% per year as per J.P. Morgan Asset Management Long Term Capital Market Assumptions (please see Appendix B for details).




a Better Risk Quantifier: conditional value at Risk

We believe that empirical evidence suggests an imperative to incorporate var ious types of “non-normality”intotheassetallocationandport-folio modeling process, specif ically to better under-stand and model downside portfolio risk. Yet if we take this step, we have to ask whether or not our conventional risk measure (i.e., standard deviation) is up to the new task.

We would argue that, in a framework based on non-normality, standard deviation may not be investors’ most appropriate measure of portfolio risk because it punishes the desirable upside movements as hard as it punishes the undesirable downside move-ments. This is generally inconsistent with investor

risk preferences—primarily as observed in the field of behavioral f inance.22

Conditionalvalueatrisk(CVaR95

) overcomes many of the drawbacks of standard deviation as a risk measure. Primarily, as it only measures risk on the downside, it captures both the asymmetric risk preferences of investors and the incidence of fat left tails induced by skewed and leptokurtic return distributions.Further,giventhewidespreadusebymajor institutional investors and regulators of its f irst cousin—value at risk—we judge it to be the most appropriate risk measure to incorporate into our framework.

ThecalculationforCVaR95

is illustrated graphi-callyforthecurrentallocationinExhibit16.Exhibit16shows the projected frequency of the portfolio’s real gains (or losses) at the end of 10 years. We calculate

e x h i b i t 1 6Histogram of projected cumulative portfolio gain (loss) at the End of 10 years assuming Non-Normality

Notes: CVaR95

is defined as the average real cumulative loss in the worst 5% or 500 simulations. This is equal to $302 million for the current portfolio.




CVaR95

by taking the average real cumulative loss in theworst5%ofsimulations.

TheCVaR95

of the current allocation, based on our new methodology, is $302 million. In other words, the portfolio can expect to lose (on average) $302 million in theworst5%ofcases(basedonoursimulationresults).This risk is significant. It indicates a real return (i.e., afterallowingforinf lation)of–30.2%ontheportfolioover an extended time horizon—a result our investor is unlikely to be very happy with.

Forthesameportfolio,however,riskcalculationsthat assume normality23wouldresultinaCVaR

95 figure

of$205million.Incorporatingnon-normalityincreasesourestimateofCVaR

95bynearly50%. In absolute

terms,theriskunderestimationis$97millionor9.7%of the portfolio’s initial value.

We should note that while the increased risk associated with modeling non-normality is striking, incremental risk in itself is not necessarily a reason for changing a plan’s asset allocation (unless an absolute thresholdvalueor“riskbudget”hasbeenbreached,anissue we address in more detail in the last section). More specifically, if one assumes an arbitrarily (but equally) higher downside risk for all asset classes, this in itself would not impact the efficiency of the portfolio—i.e., itsefficiencywouldnotchangeatall,albeittheCVaR

95

would now be higher.The reason non-normality can impact asset allo-

cation is that the downside risk associated with dif-ferent asset classes is very different. Most obviously, equity and equity type asset classes entail greater degrees of downside risk than, for example, f ixed income type investments. Hence, because the down-side risk characteristics of different asset classes are different—and cannot be accounted for using tradi-tional modeling techniques or risk measures, such as standard deviation—the eff icient allocations in our CVaR

95 motivated non-normal framework must also

bedifferentfromatraditionalframework.Forthisrea-son—notmerelyduetohigherCVaR

95 f igures—we

believe investors need to quantitatively incorporate the impact of non-normality into the asset allocation process.

In the next section, we consider the impact of non-normality on optimized portfolio solutions for investors with a long-term investment horizon.

iMplicatioNs of NoN-NoRMality oN optiMal poRtfolio solutioNs

In this section we assess the impact of non-normality onportfolioefficiency,usingCVaR

95 as the risk measure

of choice.24

Our optimization process involves minimizing CVaR

95 for a given expected return (or equivalently,

maximizingreturnforagivenCVaR95

). In any case, theportfolioswederivehereare“eff icient”inthatthey offer investors the optimal tradeoff between (downside) risk and reward. The optimization process is based on linear programming techniques25 applied to 10,000 Monte Carlo simulations (generated in the previous section).

Optimizations in general—whether based on modern or, as in our case, post-modern portfolio theory—suffer from certain inherent drawbacks. A par-ticular drawback associated with traditional frameworks is the sheer number of constraints applied to the opti-mization process. Constraints reduce the credibility of the optimization process as, by applying constraints, the practitioner is guiding the model towards a particular solution—rather than allowing the model to calculate one itself independently.

In deriving our optimal portfolios, we do not apply any constraints. Hence, we let the model dictate the direction and magnitude of results.

Weconstructanefficientfrontiermadeof10“effi-cient”portfolios:

• Our f irst portfolio comprises our minimumCVaR

95 portfolio—such that there is no other asset

allocation (derived from our simulations) that min-imizestheportfolio’sCVaR

95 further over 10 years.

It is our minimum risk portfolio.• Ourtenthportfoliocomprisesourhighestreturn

portfolio—such that there is no other asset alloca-tion (derived from in our simulations) that maxi-mizes the return of the portfolio further over 10 years.

• The intermediate eight portfolios are equallydistributed between the first and tenth in terms of expected return, but minimizing CVaR

95

over 10 years at each point along the eff icient frontier.



In aggregate, a plot of the expected compound return(in%peryear)versustheCVaR

95 (in $ mil-

lions) constitutes our eff icient frontier, shown in Exhibit17.

The efficient frontier created using a non-normal framework differs markedly from the equivalent frontier produced from a traditional model26 built on assump-tionsofnormality.Notablyallpointsalongthe“new”eff icient frontier demonstrate—for a given expected return—significantlyhigherrisk,ascapturedbyCVaR

95.

Detailed allocations underlying each efficient frontier areshowninAppendixB.

These results confirm our intuition that capturing the non-normality associated with equity and alternative return distributions should increase the magnitude of CVaR

95 of any portfolio with an allocation to these asset

classes. Crucially, as traditional frameworks fail to cap-ture non-normality, they underestimate the downside risk associated with such portfolios. Hence, the efficient frontier derived from a traditional model is likely to

show a misleading, lower-risk figure for portfolios with a similar expected return.

optimal portfolio solutions for our Hypothetical investor

In this subsection, we ask ourselves how incor-porating non-normality into our return distributions affects our optimal portfolio solutions for our hypo-thetical investor.

Exhibit 18 compares our current hypotheticalinvestor with a portfolio allocation using an identical targetarithmeticreturnof9.3%,butoptimizedusingour revised non-normal framework.27Forillustrativepurposes, we also show the optimized allocation derived from a traditional mean-variance framework with a targetarithmeticreturnof9.3%.

ThefinalcolumninExhibit18showsanopti-mizedportfoliowithin the investor’soriginal “riskbudget.”Wedef ine theriskbudget in termsof theportfolio’sCVaR

95 assuming normality—a f igure of

e x h i b i t 1 7Efficient frontier Based on traditional and Non-Normal frameworks




$205millionwecalculateinthelastsection.Weassumethat this figure constitutes the investor’s original abso-lute tolerance for downside risk and that incorporating non-normality results in a breach of this threshold. The optimized allocation in the final column reduces downside risk to bring it within the investor’s original risk budget.

Our results indicate that the “optimal” port-folio—with a similar target return to the current portfolio—using a traditional mean-variance frame-work actually increases (rather than decreases) risk by 14%relativetothecurrentallocation—asdefinedbyCVaR

95. As a traditional framework minimizes stan-

dard deviation28—which we argue is an inadequate risk measure—it inadvertently exposes our investor to even worse scenarios on the downside than the cur-rent allocation.

However, the more signif icant issue by far—and the biggest drawback of the traditional approach—is that it produces a highly concentrated and imprac-tical asset allocation. This is because in the absence of formal constraints, it over-allocates to particular asset classes (based on small differences in assump-tions). This limits our ability to draw useful insights into the portfolio construction process using such a framework.

On the other hand, our non-normal CVaR95

based framework produces a diversif ied solution with allocations across the asset class spectrum. Despite the fact that our investor already holds quite a diversif ied portfolio, our framework suggests that there is still scope for our investor to improve portfolio eff iciency further.

e x h i b i t 1 8optimization Results

Note: Sharpe ratio calculated assuming risk-free return of 4.0%.




Both“optimal”portfoliosidentifiedbythenon-normal framework improve the expected Sharpe ratio andreducetheCVaR

95 relative to the current allocation.

This signifies that our optimal portfolios are more effi-cient(inSharperatioandCVaR

95 space) than the cur-

rent hypothetical portfolio.The “optimal” portfolio targeting the current

expected return results in a decrease in allocation to fixed income and equities, and an increase in alloca-tion to alternatives. The optimal portfolio targeting the originalriskbudgetof$205millionresultsinhigherallocations to fixed income and alternatives and a lower allocationtoequities(basedonJPMorganLongTermCapital Market Assumptions).

coNclusioNs

Until recently, investors have been constrained in their ability to incorporate non-normality into the assetallocationprocess.Butnow,withtheavailabilityofsophisticated statistical tools, we can meet this challenge. Why should we change?

The most straightforward answer is that this is how the world really works—i.e., we empirically observe non-normality with much greater frequency that cur-rent mean variance frameworks allow for. The more important answer is that ignoring non-normality in equity (and equity-type) return distributions signifi-cantly understates downside portfolio risk—in the worst of the worst cases, potentially posing a solvency risk for the investor.

We also believe that investors need to allow for downside risk in a more robust fashion than standard deviationmeasureshavetraditionallyassumed.ForthisreasonwerecommendCVaR

95. This measure is a better

fit for investors’ asymmetric risk preferences, as well as the fat left tails recognized by non-normal asset alloca-tion frameworks.

Final ly, an asset a l location incorporatingnon-normality has the benefit of reducing the need for external constraints. Investors often impose such constraints in an effort to get normal frameworks to provide non-normal solutions, i.e., to better ref lect the non-normality we see in the real world. A frame-work that builds in non-normality up front, however,

provides a much more direct, efficient, and elegant way of addressing the problem.

Ultimately, we believe the quantitative results illustrate the point best and speak for themselves: incorporating non-normality may reduce the portfo-lio’s volatility, improve its eff iciency (Sharpe ratio), and reduce its risk relative to unpredictable, extreme negative events.

So, we argue for a new asset allocation framework because, beyond its pure statistical merit, there lies a significant, practical benefit for investors: the potential to improve portfolio efficiency and resilience, in light of a clearer understanding of portfolio risk.

A p p e n d i x Ageneralized pareto distribution

ForthoseinterestedinthegeneralizedParetodistribu-tion, we provide supporting equations here.

The probability density function for the generalized Pareto distribution with shape parameter k ≠ 0, scale param-eter σ, and threshold parameter θ, is given by

y f x k kx k

= =

+ −

− −

( | , , )( )σ θ

σθ

σ1

11

1

for θ < x, when k > 0, or for θ < x<–σ/k when k < 0.In the limit for k = 0, the density is for θ < x.

y f xx

= =

− −

( | , , )( )

01σ θσ

θσe

If k = 0 and θ = 0, the generalized Pareto distribution is equivalent to the exponential distribution. If k > 0 and θ = σ, the generalized Pareto distribution is equivalent to the Pareto distribution.

A p p e n d i x bdetailed asset allocations and assumptions

underlying Efficient frontiers



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ENdNotEs

A special thanks to Jianxiong Sun for his significant contribution to development of the original framework and paper. The views expressed in this article are entirely those of the authors and do not necessarily ref lect those of JP Morgan. The authors alone are responsible for any errors and/or omissions. This article is an updated version of a stand-aloneJPMorganpublicationdatedMay2009.

1The normal distribution—recognizable by its bell-shaped probability density function—is a statistical distribu-tion commonly used to model asset class returns in traditional mean-variance optimization frameworks.

2Although“independence”isnotinstrictstatisticaltermsaformof“normality,”weincludeitherebecausetheassumptionof“independence”isoneofthecentraltenetsofconventional asset allocation frameworks built around the conceptof“normality”ofassetreturns.

3There are other forms of non-normality one can observeinassetreturns.Forexample,heteroskedasticityorserial correlation in volatilities (also called volatility clus-tering) is another such form.

4Our sources for these return streams are Datastream, Bloomberg,BarclaysCapital,MSCI,HFRI,FTSE,andDowJones.

5First-orderserialcorrelationimpliesthatthereturnfrom one month is correlated with the return from the pre-vious month.

6TheJarque–Bera testmeasures thedeparture fromnormality of a sample, based on its skewness and kurtosis. TheJ-BteststatisticisdefinedasN/6* (S2+(K–3)2/4) where N is the sample size, S is the skewness, and K is the kurtosis.

7The normal distribution density function is parameter-ized using the maximum likelihood estimation method based on our sample of underlying historical data.

8Note that although we do not test whether the changes in correlations are statistically significant, we do believe the changes illustrate our broad point of correlation convergence during periods of market stress.

9Pearson correlations are used to indicate the strength and direction of a linear correlation between two random variables.

10Moreprecisely,theCBOEVIXrepresentstheimpliedvolatilityofS&P500Indexoptions.TheVIXisquotedinterms of percentage points and translates, roughly, to the expectedmovementintheS&P500overthenext30-dayperiod, on an annualized basis.

11PrivateEquitymaybeseenasareturnenhancerratherthan a diversification play. We include it in our diversification analysis—despite a relatively high correlation—for illustra-tive purposes.e

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12Foramoredetailedtreatment,pleaserefertoFisher,Geltner,andWebb[1994].Also,FisherandGeltner[2000].

13NoteInternationalEquity,EmergingMarketsEquity,HedgeFundofFundsandPrivateEquityreturnstestpositivefor serial correlation at lag one.

14An extreme value distribution is defined as the lim-iting distribution of extreme values sampled from any inde-pendent randomly distributed process.

15See Appendix A for more information about the GPD.16The Kolmogorov-Smirnov test is a goodness of fit test

to test whether a sample comes from a particular distribution. Ournullhypothesisistestedat5%significance.

17QQplotsillustratehowdifferentquantilesofasamplecompare to those implied by the theoretical density function. Alldatapointsbeingonastraight45degreelineindicateaperfect fit of the data with the distribution.

18The Student t distribution comes from the same family of distributions as the normal. A Student t distribu-tion is calibrated on only one parameter, i.e., the degrees of freedom. The higher the degrees of freedom, the closer it is to a normal. At 30 degrees of freedom, the Student t distribu-tion is virtually identical to the normal distribution. Hence, assuming a lower degree of freedom parameter imposes sig-nificant increased dependence between asset classes returns.

19Note our model calibrates the degrees of freedom parameter for the Student t distribution assuming the mar-ginal distributions are semi-parametric GPD. However, to illustrate the effect of adding the copula, we choose a low degree-of-freedom parameter of two and assume the marginal distributions are normal (rather than semi-parametric GPD). Thisallowsustoisolateandillustratetheeffectoffat“ joint”tailsseparatelyfromfat“marginal”tails.

20Our model calculates real portfolio value by dis-counting the nominal portfolio value using projected inf lation. Inf lation itself is projected stochastically in our framework.

21We believe 10,000 simulations should reduce simula-tion error sufficiently to allow us to draw robust inferences from the results.

22A key tenet of behavioral finance is the idea of loss aversion, that is, a tendency of investors to prefer avoiding losses than making gains. This translates to risk preferences that are asymmetric in nature.

23OurestimatesofCVaR95

under a traditional frame-work were derived with an identical asset allocation, except that asset returns were assumed to be individually and jointly normallydistributed.Riskandcorrelationswerederivedbasedonthesame15-yearhistoricalperiodfromJune1994toMay2009.

24It is worth noting that when optimizing under the assumption of normality, the choice of risk measure (whether standarddeviationorCVaR

95) is irrelevant. This is because a

normal distribution is symmetric about its mean—so choosing a different percentile (e.g., two standard deviations)—does not impact the derived solution. Under non-normality, the choice of risk measure is critical to the derived solution.

25Linearprogrammingisrequiredbecauseaswemoveaway from traditional “normal” assumptions, there areno shortcuts to computing total portfolio risk. Individual downside risk measures—such as standard deviation—can no longer be aggregated to the total portfolio level using analytical formulae.

26Our traditional efficient frontier was derived under a Monte Carlo simulation framework with identical constraints, except that asset returns were assumed to be individually and jointly normally distributed.

27All optimizations are based on J.P. Morgan Asset Man-agementLongTermCapitalMarketReturnAssumptions.

28Minimizing standard deviation for a given expected return target is the equivalent of maximizing the Sharpe ratio.

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