the myth of diversification_jpm.2009.36.1

26 THE MYTH OF DIVERSIFICATION FALL 2009

Perhaps the most universally acceptedprecept of prudent investing is to diver-sify, yet this precept grossly oversim-plifies the challenge of portfolio

construction. Consider a typical investor whorelies on domestic equities to drive portfoliogrowth. This investor will seek to diversify thisexposure by including assets that have low cor-relations with domestic equities. Yet the corre-lations, as typically measured over the full sampleof returns, often belie an asset’s diversificationproperties in market environments when diver-sification is most needed, for example, whendomestic equities perform poorly. Moreover,upside diversification is undesirable; investorsshould seek unification on the upside. Ideally, theassets chosen to complement a portfolio’s mainengine of growth should diversify this asset whenit performs poorly and move in tandem with itwhen it performs well.

In this article, we will first describe themathematics of conditional correlations assumingreturns are normally distributed. Then, we willpresent empirical results across a wide variety ofassets, which reveal that, unlike the theoreticalprofiles, empirical correlations are significantlyasymmetric. We are not the first to uncover cor-relation asymmetries, but our empirical investi-gation updates prior research and extends theanalysis to a much broader set of assets. Finally,we will show that a portfolio construction tech-nique called full-scale optimization producesportfolios in which the component assets exhibitrelatively lower correlations on the downside

and higher correlations on the upside than mean-variance optimization.

CORRELATION MATHEMATICS

It has been widely observed that corre-lations estimated from subsamples comprisingvolatile or negative returns differ from corre-lations estimated from the full sample ofreturns.1 These differences do not necessarilyimply that returns are non-normal or gener-ated by more than a single regime. Consider ajoint normal distribution with equal means of0%, equal volatilities of 15%, and an uncondi-tional (full-sample) correlation of 50%. Supposewe condition correlations on both assets, asillustrated in Exhibit 1, mathematically,

(1)

where the variables x and y are observed valuesfor each asset, ρ(θ ) is the conditional correla-tion, and θ is the threshold. Longin and Solnik[2001] labeled these conditional correlationsexceedance correlations.

Exhibit 2 shows the exceedance correla-tion profile for x and y. We generated this pro-file using the closed-form solution presented inthe appendix. The profile’s peak shows thatwhen we truncate the sample to include onlyobservations when both x and y return apositive value (or when they both return anegative value), the correlation decreases from

ρ θθ θ θθ θ

( ), | ,

, | ,=

> > >< <

corr( ) if

corr(

x y x y

x y x y

0

)) if θ <⎧⎨⎩ 0

The Myth of DiversificationDAVID B. CHUA, MARK KRITZMAN, AND SÉBASTIEN PAGE

DAVID B. CHUA

is an assistant vice president at State Street Associatesin Cambridge, [email protected]

MARK KRITZMAN

is president & CEO at Windham Capital Management in Cambridge, [email protected]

SÉBASTIEN PAGE

is a senior managingdirector at State StreetAssociates in Cambridge,[email protected]

JPM-CHUA KRITZMAN:Layout 1 10/19/09 5:35 PM Page 26

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50% to 27%. Moving toward the tails of the distribution,the correlation decreases further. For example, if we focuson observations when x and y both return –15% or less,the correlation decreases to 18%. This effect, which iscalled the conditioning bias, may lead us to concludefalsely that diversification increases during extreme marketconditions.

To avoid the conditioning bias in our empiricalanalysis, we compared empirical correlation profiles withthose expected from the normal distribution. Our goal wasto determine whether correlations shift because returns aregenerated from more than a single distribution or if theydiffer simply as an artifact of correlation mathematics.

EMPIRICAL CORRELATION PROFILES

Prior studies have shown that exposure to differentcountry equity markets offers less diversification in down

markets than in up markets.2 The same istrue for global industry returns (Ferreira andGama [2004]); individual stock returns (Ang,Chen, and Xing [2002], Ang and Chen[2002], and Hong, Tu, and Zhou [2007]);hedge fund returns (Van Royen [2002b]);and international bond market returns (Cap-piello, Engle, and Sheppard [2006]). The onlyconditional correlations that seem toimprove diversification when it is mostneeded are the correlations across assetclasses. Kritzman, Lowry, and Van Royen[2001] found that asset class correlationswithin countries decrease during periods of

market turbulence.3 Similarly, Gulko [2002] found thatstocks and bonds decouple during market crashes.

We extended the investigation of correlation asym-metries to a comprehensive dataset comprising equity, style,size, hedge fund, and fixed-income index returns. Exhibit 3shows the complete list of the data series we investigated.For most asset classes, we generated correlation profiles withthe U.S. equity market because it is the main engine of growthfor most institutional portfolios. We also generated correla-tion profiles for large versus small stocks, value versus growthstocks, and various combinations of fixed-income assets.

To account for differences in volatilities, we stan-dardized each time series as follows: (x – mean)/standarddeviation. We then used the closed-form solution out-lined in the appendix to calculate the correspondingnormal correlation profiles.

Exhibit 4 shows the correlation profile between theU.S. (Russell 3000) and MSCI World Ex-U.S. equity mar-kets, as well as the corresponding correlations we obtainedby partitioning a bivariate normal distribution with the samemeans, volatilities, and unconditional correlation. Observedcorrelations are higher than normal correlations on thedownside and lower on the upside; in other words, interna-tional diversification works during good times—when it isnot needed—and disappears during down markets. Whenboth markets are up by more than one standard deviation,the correlation between them is –17%. When both marketsare down more than one standard deviation, the correlationbetween them is +76%. And in times of extreme crisis, whenboth markets are down by more than two standard devia-tions, the correlation rises to +93% compared to +14% forthe corresponding bivariate normal distribution.

We investigated the pervasiveness of correlationasymmetry across several asset classes. For each correlation

FALL 2009 THE JOURNAL OF PORTFOLIO MANAGEMENT 27

E X H I B I T 1Stylized Illustrations of Exceedance Correlations

E X H I B I T 2Correlation Profile for Bivariate NormalDistribution

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profile, we calculate the average differencebetween observed and normal exceedance cor-relations.

We calculate these average differencesfor up and down markets, as follows:

(2)

The variables μdn and μup are theaverage differences for down and up mar-kets, respectively; is the observedexceedance correlation at threshold θi; andρ(θi) is the corresponding normal correla-tion. For up and down markets, we use nthresholds (θi > 0 and θi < 0, respectively)equally spaced by intervals of 0.1 standarddeviations.4 Exhibit 5 summarizes our resultsin detail and Exhibit 6 shows the corre-sponding ranking of correlation asymme-tries defined as μdn – μup. We find thatcorrelation asymmetries prevail across avariety of indices. The correlation profilefor equity-market-neutral hedge funds raisesquestions about such claims of market neu-trality. Only a few asset classes offer desirable

downside diversification, or decoupling. And unfor-tunately, most of these asset classes—MBS, highyield, and credit—failed to diversify each otherduring the recent subprime and credit-crunchcrisis of 2007–2008.

The last column of Exhibit 5 shows the per-centage of paths—from a block bootstrap of all pos-sible 10-year paths—for which asymmetry (μdn – μup)has the same sign as the full-sample result shown inthe third column. We find a reasonable degree ofpersistence in the directionality of correlation asym-metries, which suggests historical correlation pro-files should be useful in constructing portfolios.

PORTFOLIO CONSTRUCTIONWITH ASYMMETRIC CORRELATIONS

How should we construct portfolios if down-side correlations are higher than upside correla-tions? Research on conditional correlations has led

μ ρ θ ρ θ θ

μ ρ θ

dn i ii

n

i

up i

n

n

= − ∀ <

=

=∑1

0

11

[ ( ) ( )]

[ ( )

�

� −− ∀ >=∑ ρ θ θ( )]ii

n

i1

0

�ρ θ( )i


E X H I B I T 3Data Sources

E X H I B I T 4Correlation Profile between U.S. and World Ex-U.S., January1979–February 2008


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to several innovations in portfolio construction. In general,conditional correlations lead to more conservative portfo-lios than unconditional correlations. For example, Camp-bell, Koedijk, and Kofman [2002] presented two efficientfrontiers for the allocation between domestic stocks (S&P500) and international stocks (FTSE 100). The first fron-tier uses the full-sample historical returns, volatilities, andcorrelation; the second substitutes the downside correla-tion for the full-sample correlation. Campbell, Koedijk, andKofman found that for the same level of risk the down-side-sensitive allocation to international stocks is 6% lowerand cash increases from 0% to 10.5%. We present similar evi-dence of these shifts by deriving optimal allocations basedon downside, upside, and full-sample correlations.

Exhibit 7 shows expectations and optimal portfo-lios for U.S. equities, World Ex-U.S. equities, and cash.Downside correlations lead to an increase in cash allo-cation from 3% to 9% and a higher allocation to WorldEx-U.S. equities, while upside correlations lead to an all-equity portfolio with a higher allocation to the U.S. market.

Exhibit 7 also shows the impact of correlations onexpected utility. For example, the portfolio constructed ondownside correlations—holding everything else constant—has higher utility (0.0633) than the portfolios constructedon unconditional correlations (0.0630) and upside corre-lations (0.0627), if the downside correlations are realized.

Another approach for addressing correlation asym-metry is to dynamically change our correlation assumptions.


E X H I B I T 5Average Excess Correlations versus Normal Distribution (%)


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Previous research suggests that regime-switching modelsare ideally suited to handle correlation asymmetries insample.5 From a practitioner’s perspective, the questionremains whether regime shifts are predictable. For example,Gulko [2002] suggested a mean-variance regime-switchingmodel that calls for the investor to switch to an all-bondportfolio for one month following a one-day crash. Unfor-tunately, the superiority of this strategy is unclear—the

author’s model relies on six events and assumes that bondsoutperform stocks for one month after a market crash.

Our goal was not to develop active trading strate-gies, thus we did not try to predict regime shifts. Instead,we focused on strategic asset allocation with the argu-ment that portfolios should be modeled after airplanes,which means they should be able to withstand turbulencewhenever it arises, because it is usually unpredictable.


E X H I B I T 6Correlation Asymmetries (μdn – μup) Ranked


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Strategic investors, such as pension plans, are just like air-line pilots—their goal is not to predict the unpredictable,but their portfolios should weather the storms.

FULL-SCALE OPTIMIZATION

Our approach, called full-scale optimization (Cremers,Kritzman, and Page [2005] and Adler and Kritzman [2007]),identifies portfolios that are more resilient to turbulencebecause they have better correlation profiles. It does not seekto exploit correlation asymmetries directly, but instead max-imizes expected utility. By so doing, our approach constructsportfolios in which assets diversify each other more on thedownside and move together more on the upside than port-folios derived from mean-variance analysis.

In contrast to mean-variance analysis, whichassumes that returns are normally distributed or thatinvestors have quadratic utility, full-scale optimizationidentifies the optimal portfolio given any set of returndistributions and any description of investor preferences.It therefore yields the truly optimal portfolio in sample,whereas mean-variance analysis provides an approxi-mation to the in-sample truth.

We apply mean-variance and full-scale optimizationto identify optimal portfolios assuming a loss-averse investorwith a kinked utility function. This utility function changesabruptly at a particular wealth or return level and is rele-vant for investors who are concerned with breaching athreshold. Consider, for example, a situation in which an

investor requires a minimum level of wealth tomaintain a certain standard of living. Theinvestor’s lifestyle might change drastically ifthe fund penetrates this threshold. Or theinvestor may be faced with insolvency fol-lowing a large negative return, or a particulardecline in wealth may breach a covenant on aloan. In these and similar situations, a kinkedutility function is more likely to describe aninvestor’s attitude toward risk than a utilityfunction that changes smoothly. The kinkedutility function is defined as

(3)

where θ indicates the location of the kink andυ the steepness of the loss aversion slope.

In Exhibit 8, we illustrate the full-scaleoptimization process with a sample of stock and bondreturns. We compute the portfolio return, x, each period asRS × WS + RB × WB, where RS and RB equal the stock andbond returns, respectively, and WS and WB equal the stockand bond weights, respectively. We use Equation 3 to com-pute utility in each period, with θ = –3% and υ = 3.

We then shift the stock and bond weights until we findthe combination that maximizes expected utility, which forthis example equals a 48.28% allocation to stocks and a51.72% allocation to bonds. The expected utility of theportfolio equals 0.991456. This approach implicitly takesinto account all features of the empirical sample, includingpossible skewness, kurtosis, and any other peculiarities ofthe distribution, such as correlation asymmetries.

To minimize the probability of breaching the threshold,full-scale optimization avoids assets that exhibit high down-side correlation, all else being equal. Exhibit 9 shows anexample using fictional distributions generated by MonteCarlo simulation. It shows an optimization between a fic-tional equity portfolio and a fictional market neutral hedgefund. Mean-variance is oblivious to the hedge fund’s highlyundesirable correlation profile because it focuses on thefull-sample correlation of 0% as a proxy for the hedge fund’sdiversification potential. It invests half the portfolio in thehedge fund and, as a consequence, doubles the portfolioexposure to losses greater than –10%. In contrast, full-scaleoptimization chooses not to invest in the hedge fund at all.6

U xx x

x x( )

ln( ),

( ) ln( ),=

+ ≥× − + + <

⎧⎨⎩

1

1

if

if

θυ θ θ θ


E X H I B I T 7Mean-Variance Optimization with Conditional Correlations


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Using empirical data, we identified optimal countryportfolios by allocating across the U.S., U.K., France,Germany, and Japan, and we also optimized across theentire sample of 20 asset classes presented in Exhibit 3.7

In each case, we evaluated the mean-variance efficientportfolio with the same expected return as the trueutility-maximizing portfolio. For purposes of illustration,we set the kink (θ ) equal to a one-month return of –4%.

Before we optimized, we scaled each of the monthlyreturns by a constant in order to produce means that con-form to the implied returns of equally weighted portfo-lios. This adjustment does not affect our comparisons

because we apply it to both the mean-variance and full-scale optimizations; hence, the differences we find arisesolely from the higher moments of the distributions andnot their means.

We measure the degree of correlation asymmetry(ξ ) in each of the optimal portfolios as

(4)

where wi is asset i’s weight in the portfolio, is theweighted average of correlations between asset i and theother assets in the portfolio when the portfolio is down,and is the weighted average of correlations betweenasset i and the other assets in the portfolio when the port-folio is up.

Exhibit 10 shows the correlation asymmetry forcountry allocation portfolios using data from January 1970to February 2008. It reveals the following:

1. Downside correlations are significantly higher thanupside correlations.

2. Full-scale optimization provides more downsidediversification and less upside diversification thanmean-variance optimization.

3. Full-scale optimization reduces correlation asymmetryby more than half compared to mean-varianceoptimization.

ρiup

ρidn

ξ ρ ρ= −∑ ∑w wii

idn

ii

iup


E X H I B I T 8Full-Scale Optimization

E X H I B I T 9Impact of Correlation Asymmetries on Full-Scale and Mean-Variance Portfolios

E X H I B I T 1 0Weighted-Average Correlations for Country Allocation, January 1970–February 2008


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Exhibit 11 shows the utility gain obtained by movingfrom the mean-variance to the full-scale optimal portfolio,as well as the turnover required to do so. Also, it shows resultsfor subsamples and for the multi-asset optimization. In allcases, full-scale optimization improves correlation asym-metry when compared to mean-variance optimization. Insome cases, full-scale optimization turns correlation asym-metry from positive to negative, which means the portfoliohas lower downside correlations than upside correlations.Utility gains are highest for the multi-asset optimization.When hedge funds are included, the gain reaches 254.11%.

Note that utility gains are not perfectly correlatedwith improvements in correlation asymmetry—in somecases, large improvements in correlation asymmetry leadto a relatively small utility gain. For example, looking atcountry allocation, full-scale optimization improves cor-relation asymmetry by a greater amount in the 1970–1979subsample than in the 1980–1989 subsample (16.10%versus 5.27%), while the improvement in utility is lower(9.96% versus 23.40%). This result occurs because thekinked utility function puts a greater premium on down-side diversification than upside unification. Also, when wetake into account other features of the distribution, allasymmetry improvements are not created equal in termsof expected utility. For example, Statman and Scheid [2008]found that return gaps provide a better definition of diver-sification because they include volatility. Full-scale opti-mization addresses this issue by using the entire return

distribution. It is even possible to observe deteriorationin correlation asymmetry associated with an increase inutility. But overall, our results show that to the extent thatcorrelations are an important driver of utility—as opposedto volatilities, or returns, or other features of the joint dis-tribution—an improvement in correlation asymmetry willlead to an improvement in utility.

In general, the stability of our results is not surprisinggiven the reasonable level of stability shown in the under-lying correlation profiles (Exhibit 5). Although our maingoal is to solve the problem of utility maximization insample, Adler and Kritzman [2007] provided a robustdemonstration that full-scale optimization outperformsmean-variance optimization out of sample. Our findingshelp explain their results. We suggest that a significantportion of this out-of-sample outperformance comes fromimprovements in correlation profiles. In other words, con-ditional correlations matter—and mean-variance opti-mization fails to take them into consideration.

CONCLUSION

We measured conditional correlations to assess theextent to which assets provide diversification in down mar-kets and allow for unification during up markets. We firstderived conditional correlations analytically under theassumption that returns are jointly normally distributedin order to measure the theoretical bias we should expect


E X H I B I T 1 1Full-Scale vs. Mean-Variance Optimization and Correlation Asymmetries (%)

Note: *Sample includes hedge funds.

Proxy for U.S. Equities is MSCI USA.


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from conditional correlations. We then measured conditionalcorrelations from empirical returns, which revealed that cor-relation asymmetry is prevalent across a wide range of assetpairs. Finally, we turned to portfolio construction. Weshowed that conventional approaches to portfolio con-struction ignore correlation asymmetry, while full-scaleoptimization, which directly maximizes expected utilityover a sample of returns, generates portfolios with moredownside diversification and upside unification than alter-native approaches to portfolio formation.

A P P E N D I X

Conditional Correlation for Bivariate NormalDistributions

Let X = (x, y) ~ N(0, Σ) be a bivariate normal randomvariable, where x and y have unit variances and unconditionalcorrelation ρ. Then the correlation of x and y conditional onx < h and y < k is given by

The variances and covariance of these conditionalrandom variables can be written in terms of the momentsmij = E[xiy j|x < h, y < k] as

Let L(h, k) denote the cumulative density of our bivariatenormal distribution,

As shown in Ang and Chen [2002], the first and secondmoments can be expressed as

L h k m h k k h

L h k m k h

( , ) ( , ; ) ( , ; )

( , ) ( ,10

01

= +=

ψ ρ ρψ ρψ ;; ) ( , ; )

( , ) ( , ) ( , ; )

ρ ρψ ρχ ρ ρ χ+

= − −

h k

L h k m L h k k h202 (( , ; )

( , ) ( , ) ( , ; ) ( , ;

h k

L h k m L h k h k k h

ρχ ρ ρ χ ρ02

2= − − ))

( , ) ( , ) ( , ; ) ( ,and L h k m L h k h h k k k h11 = + +ρ ρ ψ ρ ρ ψ ;; )

( , ; )

ρρ− Λ h k

L h kx xy y

dx( , ) exp( )

=−

− − +−

⎛⎝⎜

⎞⎠⎟

1

2 1

2

2 12

2 2

2π ρ

ρρ

ddyhk

−∞−∞∫∫

var( | , )

var( | , )

x x h y k m m

y x h y k m m

< < = −

< < = −20 10

2

02 0012

11 10 01and cov( , | , )x y x h y k m m m< < = −

corr( , | , )cov( , | , )

var( |x y x h y k

x y x h y k

x x h< < = < <

< ,, ) var( | , )y k y x h y k< < <

where ψ(.), Λ(.), and χ(.) are given by

and φ(.) and Φ(.) are the PDF and CDF, respectively, of the univariate standard normal distribution.

ENDNOTES

1See, for example, Longin and Solnik [2001], Kritzman,Lowry, and Van Royen [2001], and Van Royen [2002b].

2See, for example, Ang and Bekaert [2002], Kritzman,Lowry, and Van Royen [2001], Baele [2003], and Van Royen[2002a] on regime shifts; Van Royen [2002a] and Hyde, Bredin,and Nguyen [2007] on financial contagion; and Ang and Bekaert[2002], Longin and Solnik [2001], Butler and Joaquin [2002],Campbell, Koedijk, and Kofman [2002], Cappiello, Engle, andSheppard [2006], and Hyde, Bredin, and Nugyen [2007] oncorrelation asymmetries.

3Kritzman, Lowry, and Van Royen [2001] condition ona statistical measure of market turbulence rather than returnthresholds.

4Exceedance correlations are computed for any thresholdwith more than three events.

5See, for example, Ramchand and Susmel [1998], Angand Bekaert [2002], and Ang and Chen [2002].

6Note that full-scale optimization will not always pro-duce the most concentrated portfolio. It might invest a signif-icant proportion of a portfolio in an asset with high full-samplecorrelation, but negative downside correlation, while mean-variance might ignore the asset altogether.

7We exclude the World Ex-U.S. asset class because it isredundant with the country choices.

REFERENCES

Adler, T., and M. Kritzman. “Mean-Variance versus Full-ScaleOptimisation: In and Out of Sample.” Journal of Asset Manage-ment, Vol. 7, No. 5 (2007), pp. 302–311.

Ang, A., and G. Bekaert. “International Asset Allocation withRegime Shifts.” Review of Financial Studies, Vol. 15, No. 4 (2002),pp. 1137–1187.

ψ ρ φ ρρ

ρ

( , ; ) ( )

( , ; )

h k hk h

h k

= − −

−

⎛

⎝⎜⎜

⎞

⎠⎟⎟

= −−

Φ

Λ

1

1

2

ρρπ

φρ

ρ

χ ρ ψ

2 2 2

22

2

1

h hk k

h k k k

− +

−

⎛

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To order reprints of this article, please contact Dewey Palmieri [email protected] or 212-224-3675.



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