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Diversification Returnand Leveraged PortfoliosEDWARD QIAN

EDWARD QIANis the chief investmentofficer in the MultiAsset Group at PanAgoraAsset Management inBoston, [email protected]

Portfolio rebalancing is the simplest

and clearest technique that wi th

few exceptions adds incremental

value to fixed-weighted multi-asset

portfolios. T h e incremental value, often

referred to as diversification return (Willen-

brock [2011] and Booth and Fama [1992]),

is attributed to the fact that the variance of a

fixed-weighted portfolio is in general smaller

than the weighted sum of individual vari-

ances because of portfolio diversification. As

a consequence, the geometric return of the

portfolio is greater than the weighted sum

of the geometric returns of individual assets.

Indeed, the whole is bigger than the sum of

its parts!

Portfolio rebalancing is essential for har-

vesting diversification return. A portfolio com-

posed of a single security (the extreme case of

nondiversification) requires no rebalancing and

hence yields no diversification return. A diver-

sified portfolio, if left alone and not rebalanced,

does not provide diversification return either,

and worse still can become nondiversified over

time as in the case of capitalization-weighted

indices. In other words, diversification and

rebalancing are inseparable. But what is the

underlying dynamic of rebalancing that leads

to a positive diversification return for a typ-

ical portfolio? This question is crucial to an

understanding of the source of diversification

return and in extending the analysis to lever-

aged long—short portfolios.

• It turns out that the underlying portfolio

dynamics of long-only unleveraged portfolios

is mean reverting (i.e., the strategy of selling

winners and buying losers). A simple example

suffices to illustrate the point. Take an example

of a two-asset 50/50 portfolio. Suppose Asset

1 returns 50% and Asset 2 returns 0%. In this

case, the portfolio will drift to 60/40 (75/125

and 50/125) at the end of the period. To rebal-

ance the portfolio to the original 50/50 mix,

we would sell 10% of Asset 1 (the winner) and

buy 10% of Asset 2 (the loser). This mean-

reverting strategy also applies in portfolios

with more than two assets. The weights of

the assets that have positive excess returns

versus the portfolio will drift higher while

the weights of assets that have negative excess

returns versus the portfolio will drift lower.

As a result, rebalancing necessarily requires

selling the former group (the winners) and

buying the latter group (the losers).

W h y could such a mean- rever t ing

strategy generate positive value added for

a long-only portfolio? The reason is that it

takes advantage of randomness or volatility of

asset returns, which is always present regard-

less of how different the average returns of

the individual assets might be. In the case

when all assets have an identical cumulative

return over time, the mean-reverting strategy

would sell assets that happen to overshoot the

long-term average and buy assets that happen

to undershoot their long-term average. As a

14 DIVERSIFICATION RETURN AND LEVERAGED PORTFOLIOS SUMMER 2012

result, the mean-reverting strategy generates positivediversification return versus the weighted sum of theindividual returns.

The connection between portfolio rebalancing andthe mean-reverting strategy is only true for long-onlyunleveraged portfolios. The connection breaks downfor leveraged portfolios that have short positions forwhich portfolio rebalancing would entail the oppositestrategy: trend following. This crucial difference in therebalancing mechanism should raise concerns for inves-tors. Over time, institutional investors have invested inmany alternative strategies that employ a great deal ofleverage, which is often implicit. These strategies includehedge funds, private equity, and real estate. The mostrecent addition is the risk-based risk parity strategy ofQian [2005] that employs explicit leverage to achieverisk balance as well as portfolio risk-return targets.Retail investors, in the past often lacking the tools toleverage, have been increasingly attracted to leveragedexchange-traded funds (ETFs).

What happens to the diversification returns of lev-eraged portfolios? Do they exist? Are they still positive?These are important questions because diversificationreturn can have a material impact on the returns of theseportfolios. This is especially true in a low-return envi-ronment for traditional assets. Although few investorsuse diversification return as the main criterion for port-folio construction, many investors adhere to the practiceof regular rebalancing once the policy portfolios are set.These investors should be made aware of the actions theyneed to take to rebalance and of the associated benefitsand costs. In this article, we study these questions andprovide analytic and empirical answers, some of whichmight seem startling and perhaps counterintuitive.

REBALANCING LEVERAGED PORTFOLIOS

In contrast to the long-only unleveraged case,rebalancing leveraged portfolios involves momentum ortrend following (i.e., buying winners and selling losers).Although this might appear surprising, a simple examplesuffices to illustrate the logic.

Consider again a two-asset 200/100 portfolio with200% long in Asset 1 and 100% short in Asset 2. Sup-pose Asset 1 returns 50% and Asset 2 returns 0%. Atthe end of the period. Asset 1 grows to 300% and Asset2 remains at —100%, so that the net value of the port-

folio doubles. As a result, the portfolio weights shrinkto 150/50 (300/200 = 150% and 100/200 = 50%). Torebalance the portfolio to the original 200/100 targetweights, we would buy an additional 50% of Asset 1(the winner) and short an additional 50% of Asset 2(the loser). It is easy to prove mathematically that whena leveraged portfolio has positive returns, gross leveragedeclines; thus, leverage would have to increase to getback to the original weights. (See Appendix B.)

The opposite is true when a leveraged portfoliosuffers losses. Suppose Asset 1 returns -25% and Asset 2returns 0%. Then Asset 1 declines to 150% and Asset 2stays at -100%, so the net value of the portfolio ishalved. As a result, the portfolio weight is now 300/200(150/50 = 300% and 100/50 = 200%). To rebalance theportfolio to the original 200/100 mix, we would sell100% of Asset 1 (the loser) and buy (or short cover) 100%of Asset 2 (the winner). In general, when a leveragedportfolio has negative returns, gross leverage increases,and the portfolio must be deleveraged to get back to thetarget weights. Viewed in this context, to stop-loss orto deleverage—sometimes hailed as a distinctive invest-ment decision by hedge fund managers—can be thoughtof merely as a mechanical decision to rebalance long—short portfolios at times when fresh capital is not readilyavailable.

Does this trend-following rebalancing strategygenerate diversification return? By the same argumentwe used for a mean-reverting strategy, the intuitiveanswer is that it should generate a negative diversifica-tion return, a fact that we will prove later.

GENERAL FORMULAFOR DIVERSIFICATION RETURNS

Before we show why the diversification return canbe negative for leveraged portfolios, we present the gen-eral formula for diversification return; see Willenbrock[2011] for a derivation. Given a fixed-weight portfoliowith N assets, suppose the weights of the assets are {w ,u>^,- • •,iv^, which can be either positive or negative, andvolatilities of their returns are (a , (5^,---,(5^). Also sup-pose the volatility of the portfolio is G . Then the diver-sification return of the portfolio, which is the portfolio'sgeometric mean^^ in excess of the weighted sum of indi-vidual geometric means {g^, g2,---,gj^, is approximately

SUMMER 2012 THEJOURNAL OF PORTFOLIO MANAGEMENT 15

yr ST 2r, = Ç — > W.Ç. ~ —\ / u'.<J. —O

one-half the weighted sum of individual variances lessthe portfolio variance.

(1)

Writing the portfolio variance in terms of theweighted sum of individual volatilities and correlations,p , we have

(2)

Equation (2) is the general result of diversificationreturns for fixed-weight portfolios, regardless of whetherthey are long-only or long—short. Put differently, thegeometric mean of a fixed-weight portfolio is approxi-mately the weighted average of the geometric mean ofindividual assets plus the diversification return.'

We now explore some special cases to illustratethe intuition behind the diversification return and moreimportantly, the impact of long—short positions andleverage ratios on the diversification returns.

CASE OF ONE RISKY ASSET

The simplest case is one in which portfolios consistof one risky asset and a risk-free asset. When we useexcess return, the case actually redvices to just one assetbecause the risk-free asset has zero excess return andzero excess volatility. For instance, in the two previousexamples, we could have a 50/50 portfolio of the S&P500 Index (Asset 1) and cash (Asset 2) and a leveraged200/100 portfolio of two times the S&P 500 Index(Asset 1) and short 100% cash (Asset 2), respectively.With one risky asset. Equation (2) simplifies to

1 /' 2 \ 2

r, = —\w, -UK a : (3)

Equation (3) is intuitive in several respects. First,the diversification return is zero if Wj = 0 or w = 1 (i.e.,when the portfolio is invested 100% in either the riskyasset or the risk-free asset). No rebalancing occurs andhence no diversification return is earned. Second, thediversification is positive when 0 < u j < 1, in which casethe portfolio is long-only and unleveraged. The diver-

sification becomes negative, however, when W| < 0 orwhen w^>\. In other words, when the portfolio is eithershort the risky asset or is leveraged by shorting cash, thediversification return would be negative. The cause of itsnegativity, as we discussed previously, is that for these twocases, the rebalancing is trend following rather than meanreverting. Lastly, the diversification return is proportionalto the variance of excess returns of the risky assets. Thismakes sense because the higher the variance, the largerthe potential deviation in the portfolio to the originalweights and the greater the shift in the rebalancing.

Exhibit 1 plots the quadratic function of the weightin Equation (3). Not only does it demonstrate the intu-ition, but it also reveals the fact that the maximumdiversification return is achieved when the portfolio is50/50. This is because when the portfolio is evenly splitbetween the two assets—actually, any two assets—thedrift caused by random returns is maximized, and there-fore the rebalance would also have the most benefit.

Applying Equation (3) to inverse and leveragedETFs that are rebalanced daily provides a cautionary talefor investors. To access how large the potential returnslippages or negative diversification returns are, we pro-vide numerical examples in which we assume that therisk index has an annual volatility of 20%, similar tothe annual volatility of the S&P 500 Index. The dailyvolatility is thus about 1.3% (=20%/V250) if we assumedaily returns are independent. Using Equation (3),we obtain the daily return slippage for the followingfive ETFs: -3X, -2X, - IX, 2X, and 3X (shown inExhibit 2). The second row displays the annualized slip-page. The annual slippage of the —3X ETF is a whopping—21.3%, whereas the annual slippage of the 2X ETF is—11.3%. These return slippages create significant returnhurdles for these leveraged and inverse (or both) ETFs.For example, when the index annual return is —5%, anaive investor might expect the super bearish —3X ETFto yield 15%. But, in reality, the likely outcome is —6.3%(=15% — 21.3%). An even more puzzling case could bethat in a year when the index is down, the -3X ETFunderperforms the -2X ETF.

This example of leveraged portfolios of a singlerisky asset seems to indicate that shorting and leveragelead to negative diversification returns. Is this also thecase with multiple-risky-asset portfolios? The answeris, "not necessarily". The reason is that those portfolioshave the benefit of diversification among multiple assets,which the present case lacks."


E X H I B I T 1Quadratic Function (w, -w') That Shows the Relationship between Weight and Diversification Return

150% -100% -50°/ 150% 200% 250%

E X H I B I T 2Daily and Annualized Diversification Returns (Slippages) of Inverseand Leveraged ETFs of an Index with Annual Volatility of 20%

-3X -2X 2X

Daily

Annual

-0.10%

-21.3%

-0.05%

-11.3%

-0.02%

-3.9%

-0.02%

-3.9%

1

2

4i

_2 , _ 2Tj -\-u>^ü^ —I

2 ^ 2

(4)

3X

LONG-ONLY UNLEVERAGEDFIXED-WEIGHT PORTFOLIOS

Before we move on to leveraged multi-asset port-folios, we first provide some analysis of long-only unlev-eraged multi-asset portfolios. For illustration, we usethe simplest case with just two risky assets for which thediversification return is given by

Comparing Equation (4) with Equa-(3), we recognize the first two terms

as the diversification returns for the twoassets individually and the last term repre-sents the diversification between the twoassets. The coefficient of p', is negativebecause the volatilities and the "weights arepositive in the present case. Thus, we con-elude that the diversification return increases

with declining correlation, attaining the maximum whenÇ)\ = -1 and the minimum when p' = 1. Namely,

-0.05%

-11.3%

12

\

2 + " 2( 2

(5)

(6)

Exhibit 3 shows the diversification return of tvi o-asset portfolios with five different correlations. For vola-tility, we assume 20% for Asset 1 (a proxy for equityindices) and 5% for Asset 2 (a proxy for investment-grade

SUMMER 2012 THE JOURNAL OF PORTFOLIO MANAGEMENT 17

E X H I B I T 3Diversification Returns of Two-Asset Long-Only Portfolios vs. Asset Weights under DifferentCorrelation Assumptions

0.80%

0.70%

0.60%

0.50%

0.40%

0.30%

0.20%

0.10%

0.00%

0%

= 0.5

20% 40% 60%

Asset Weight

80% 100%

bond indices). Because the weights varyfrom 0% to 100%, the shapes of the linesresemble those of Exhibit 1 within the samerange. As the correlation declines from 1to —1, the diversification return rises. Themaximum effect is always at 50/50 (i.e., anequal-weighted portfolio). Whereas this istrue for two-asset unleveraged cases, it isnot true for portfolios with three or moreassets. When the two assets are completelycorrelated, the diversification return canstill be as high as 0.28%, or 28 basis points(bps). When they are perfectly negativelycorrelated, the diversification return is0.78%, or 78 bps, for the 50/50 mix.

Shown in Exhibit 4, Panel A, are theresults for a 60/40 portfolio with 60% instocks and 40% in bonds with volatilityat 20% and 5%, respectively. The diver-sification return would be 0.63%, 0.51%,and 0.39% for three different correlations,p = -0.5, 0, 0.5, respectively. Exhibit 4shows that lower correlation leads not only

E X H I B I T 4Diversification Returns and Volatilities of 60/40 Portfoliosand Unleveraged and Leveraged Risk Parity Portfolios

Panel A: Diversification Returns and Volatilities of 60/40 Portfolios

60/40

Diversification return

Portfolio volatility

p = -0.5

0.63% ^

11.1%

p = 0

IK 0.51% j j

12.2%

Panel B: Diversification Returns and Voiatliities of 20/80(Unleveraged Risk Parity) Portfolios

20/80

.Diversification return


Panel C: Diversificationof Risk Parity Portfolios

p = -0.5

0.42%

4.00%

p = 0

0.34% _

5.66%

Returns, Volatilities, and Leverage

p = 0.5

1^0 .39%

13.1%

p = 0.5

0.26%

6.93%

Ratios

Risk Parity p = -0.5 p = 0 p = 0.5Diversification return


Portfolio leverage


to lower portfolio volatility but also to higher diversifi-cation return. There are implications of this diversifica-tion return analysis to asset allocation decisions betweenstocks and bonds. It is known that the correlation betweenstocks and bonds varies greatly when the bonds are ofdifferent credit quality. The correlation is typically nega-tive when the bonds are high-quality government bonds.The correlation is near zero if the bond investment is inaggregate bond indices. The correlation is highly positiveif the bond investment is in corporate bonds and high-yield bonds. In the last case, not only is the benefit ofrisk reduction small, but the high correlation also tendsto decrease the diversification return.

It is worth pointing out that the diversificationreturn is always non-negative for a long-only unlever-aged multi-asset portfolio. This has been assumed true,but has never been proven in the literature until now. Weprovide a mathematical proof in Appendix A. In otherwords, for long-only unleveraged portfolios, diversifica-tion always pays. In a hypothetical world of zero return forall assets, the only positive returns that investors can earnwith fixed-weight portfolios are from rebalancing.

LEVERAGED LONG-ONLY PORTFOLIOS

In a leveraged long-only portfolio, two opposingforces affect its diversification return. On one side,because the portfolio remains long only, the positivediversification effect among the assets continues and isactually magnified by the leverage. On the other side isthe leverage itself, of the whole portfolio, which tendsto generate negative diversification return as we showedearlier. We analyze the two effects and their contribu-tions using the previously derived formula for generalportfolios.

Assuming the leverage ratio of a long-only port-folio to be L, we write

(7)

In Equation (7), the weights are of the risky assets.When L > 1, the portfolio is leveraged by a short positionin cash of size L — 1. We now rescale the weights by theleverage and denote them as w' = w./L. These rescaledweights sum to one (i.e.,^ _ w^. = 1).

We express the diversification return in Equation(2) in terms of the scaled weights, and we have

\

and rewriting Equation (8) as

(8)

(9)

We now recognize that the term inside the paren-theses is the diversification return of the long-onlyportfolio with the rescaled weights. Moreover, thedouble summation in the second term is the varianceof that rescaled portfolio. Therefore, Equation (9) canbe expressed as

(10)

Equation (10) provides an intuitive interpretationof two sources of diversification return of the leveragedportfolio, relative to the diversification return of thescaled unleveraged portfolio. The first source is positive,it is the leverage ratio times the diversification return ofthe rescaled portfolio. The second term is the same formas Equation (3); it is the negative diversification returnof leveraging a single asset, in this case, the rescaledportfolio. It is a function of the leverage ratio and thevolatility of the scaled unleveraged portfolio.

Consider three cases of the leverage ratio. First,when L = 1, the two portfolios are identical, and thediversification remains unchanged. Second, when L > 1,the first term increases the diversification return, but thesecond term is negative. The net result could be eitherpositive or negative, depending on the size of leverage,volatility, and diversification return of the rescaled port-folio. Third, the result is equally applicable when L < 1for a portfolio that is not fully invested. In this case, thefirst term would decrease but would remain positive,whereas the second term is now positive. The portfoliois essentially long only and unleveraged with N risky

SUMMER 2012 THE JOURNAL OF PORTFOLIO MANAGEMENT 19

assets and one risk-free asset; the diversification returnis always positive.

DIVERSIFICATION RETURNOF RISK PARITY PORTFOLIOS

We now apply the results to risk parity portfolioscomposed of the two assets considered before. Becauserisk parity portfolios equalize risk contributions fromstocks and bonds, they typically require leverage inorder to achieve the levels of risk of a traditional 60/40portfolio.

With a volatility assumption of 20% for stocks and5% for bonds, the unleveraged version of the risk parityportfolio would be a 20/80 portfolio (i.e., 20% in stocksand 80% in bonds). Exhibit 4, Panel B, shows the diver-sification returns of the 20/80 portfolios with three dif-ferent correlations. Compared to the 60/40 portfolios,the diversification returns of the unleveraged portfoliosare lower and the portfolio volatilities are much lower.This is not too surprising since the maximum diversifi-cation return is achieved with the 50/50 equal-weightedportfolio. And 60/40 is closer to a 50/50 portfolio thana 20/80 portfolio.

Risk parity portfolios use leverage to balance riskcontribution as well as to target total portfolio risk. Wethus choose leverage ratios on the 20/80 portfolio suchthat the portfolio risks match those of the 60/40 port-folios. Exhibit 4, Panel C, lists the leverage ratios forthree different correlation assumptions and the resultingdiversification returns.

When the correlation is -0.5, the 20/80 portfoliorisk is only 4% and the required leverage is relativelyhigh at 278%. At 278% leverage, the diversificationreturn increases from 0.42% to 0.77%, which is actuallyhigher than that of a 60/40 portfolio, which is 0.63%.With negative correlation between the two assets, thebottom-up mean-reverting strategy dominates thetop-down trend-following strategy. The net result isan increase in diversification even though the leverageratio is rather high.

When the correlation is zero, the required leveragedeclines to 215%. In this scenario, the diversificationreturn stays at 0.34%. The bottom-up and top-downeffects caused by leverage are roughly equal, but thediversification return of 0.34% is still lower than that ofthe 60/40 portfolio at 0.51%.

When the correlation is 0.5, the required leverageis only 190%. In this case, however, the leverage lowersthe diversification return from 0.26% to 0.08% becausethe high correlation between the two assets limits thediversification return from the bottom-up effect.

Comparing the three cases, we conclude that lowcorrelation among the two assets is the key to main-taining a high level of diversification return as the under-lying portfolio is being leveraged. From this perspective,high-quality government bonds again serve as a betterdiversifier than corporate or high-yield bonds.

Exhibit 4, Panel C, also shows that in all three casesthe diversification returns of the leveraged portfoliosremain positive despite leverage. Of course, this can't betrue if the leverage ratios are much higher, especially inthe case of positive correlation. Therefore, it is importantthat the leverage ratio be kept at levels that are consistentwith a positive diversification return.

To more clearly see the impact of both leverageand correlation on the diversification return. Exhibit 5displays the diversification return of the risk parity port-folios under five different correlation assumptions as theleverage ratio is increased from 100% to 300%. The caseof —1 correlation is rather special and unrealistic. Butnevertheless, the diversification return increases linearlywith the leverage because the volatility of the risk parityportfolio is zero. Qian [2011] used this as an example toillustrate how leverage can propel a portfolio that consistsof assets with no returns of their own but has a highSharpe ratio due to diversification to deliver increasedreturns with the same Sharpe ratio. When correlationis —0.5 or 0, the diversification return increases initiallyand then declines when the leverage ratio increases suf-ficiently. When the correlation is positive, the diversifica-tion return declines with increasing leverage but remainspositive when the leverage is modest.

TURNOVER OF PORTFOLIO REBALANCING

Positive diversification returns might seem to bea free lunch, but we must not forget that the analysesand results so far are free of transaction costs. Diversi-fication return is not entirely free because it requirestrading to regularly rebalance portfolios. Even intui-tively, we should know higher levels of diversificationreturn, either positive or negative, must require a higherlevel of turnover. Therefore, it is important to have esti-mates of turnover and net diversification returns based

2 0 DIVERSIFICATION RETURN AND LEVERAGED PORTFOLIOS SUMMER 2012

E X H I B I T 5Diversification Returns of Risk Parity Portfolios as a Functionof Leverage Ratio under Different Correlation Assumptions

2.0%

1.5%

1.0% -

0.5%

0.0%10p%

-0.5%

-1.0%

-1.5%Leverage Ratio

on trading costs. In this section, we provide analyticresults for rebalancing turnover.

Total portfolio turnover is the sum of buying andselling of the individual assets in the portfolio. The amountof trading in individual assets increases with the portfolioweights and return differences between the individualassets and the total portfolio. The higher the weights andthe return differences are, the higher the turnover. Thefollowing formula, derived in Appendix B, provides anestimate of average rebalancing turnover:

o'^ol-2ç p ' P(11)

In Equation (11), u>. is the weight of asset i, G is thereturn volatility of asset i, G is the return volatility ofthe portfolio, and p_, is the return correlation betweenasset i and the portfolio. Thus, the average turnover isproportional to a weighted sum (by absolute weights)of the volatilities of return difference between the indi-vidual assets and the portfolio.

Exhibit 6 plots the average turnover of a two-assetlong-only portfolio as a function of asset weights fordifferent values of correlation. The 50/50 portfolio hasthe maximum turnover. Moreover, turnover increases

as the correlation between the two assetsdecreases. Notice the similarity of Exhibits3 and 6. In Exhibit 6 the diversificationreturn gain is of constant proportion tothe amount of trading. In terms of rela-tive magnitude, the maximum turnoveris about 10% for a correlation o f -1 andthe maximum turnover is about 6% for acorrelation of+1. If we assume a transac-tion cost of 1% per 100% turnover, whichis rather conservative for liquid stocks andbonds, then the transaction costs of theportfolio rebalancing of the 60/40 port-folios are between 6 and 10 bps, whichare much less than the diversificationreturns.

Exhibit 7 plots the rebalancing turn-over of the leveraged risk parity portfo-lios as a function of the leverage ratio fordifferent correlations. For a correlation of- 1 , the turnover (the solid line) increaseslinearly with leverage. But for all othercases, the increase in turnover acceler-

ates as leverage increases; the turning point seems tobe around 200%. This is likely due to the increasingvolatility of the return difference between the individualassets and the overall portfolio as leverage is raised.

Another interesting feature is the reversal of rankorder between turnover and correlation. When leverageis low, roughly below 150%, turnover is lower for caseswith higher correlation. But as leverage increases, turn-over becomes higher for cases with higher correlation.This switch reflects the relative magnitudes of turn-over used for intra-asset rebalancing to turnover usedfor portfolio-level leverage rebalancing. The case witha correlation equal to 1 (the solid line with markers)started as having the lowest turnover when leverage islow, but it has the highest turnover when leverage is closeto 300%. When leverage is low, most of the turnoveris for intra-asset rebalancing, which is lower for highercorrelation cases. When leverage is high, most of theturnover will be used for leverage rebalancing, which ishigher for cases with higher correlation because highercorrelation makes the two assets more likely to move intandem, causing higher fluctuation in leverage.

The turnover results have two important implica-tions to asset selection in risk parity portfolios. First,the combination of low or even negative diversifica-

SUMMER 2012 THE JOURNAL OF PORTFOLIO MANAGEMENT 2 1

E X H I B I T 6Average Turnover of Two-Asset Long-Only Portfolios vs. Asset Weights under Different CorrelationAssumptions

12.00%

10.00%

8.00%

6.00%

4.00%

2.00%

0.00%20% 40% 60%

Asset Weight

80% 100%

E X H I B I T 7Average Turnover of Risk Parity Portfolios as a Function of Leverage Ratio under Different CorrelationAssumptions

40%

100% 150% 200%

Leverage Ratio

250% 300%


tion returns, shown in Exhibit 5, and high turnover,shown in Exhibit 7, spells an undesirable outcome forleveraged portfolios with high cross-asset correlation.The high turnover, of course, would make the realizeddiversification even lower or more negative. Second, thehigher turnover associated with increasing leverage alsosuggests that illiquid assets, such as high-yield bonds thathave mLich higher transaction costs than liquid stocksand investment-grade bonds, should be avoided.

CONCLUSION

This article provides analytical results regarding port-folio rebalancing and the associated diversification returnsfor different kinds of portfolios. The first contribution ofthe article is to relate portfolio rebalancing to underlyingportfolio dynamics. For long-only unleveraged portfolioswe show that rebalancing amounts to a mean-revertingstrategy and that the diversification return is always positivefor multi-asset-class portfolios. But for short (or inverse)and leveraged portfolios, we show that rebalancing at thetop-down level amounts to a trend-following strategy thatdetracts from diversification return.

It is important to note that we are not making asweeping statement about the efficacy of mean-revertingor trend-following strategies commonly employed inactive portfolio management. First, as active strategies,both inean-reverting and trend-following strategies canadd or detract value, albeit over different horizons. Thisis not in conflict with our results because diversificationretLirn is not defined in the traditional sense of activereturn. This is because the "benchmark" return, or theweighted sum of the geometric means as defined inEquation (1), is not normally associated with an actualportfolio.

We highlight the pitfalls of inverse and leveragedETFs based on a single risky asset, for instance, a cap-weighted equity index, which often carries a significantnegative diversification return, or return slippage overtime, against index return multiples.

The second important result that we report in thearticle concerns the diversification returns of risk parityportfolios. We show that the diversification returns ofleveraged long-only portfolios can generally be decom-posed into two parts, both of which are related to ascaled unleveraged portfolio. The first part is the positivediversification return from rebalancing among individualassets at the bottom-up level, which is amplified by

leverage. The second part is the negative diversificationreturn caused by the leverage of the overall portfolio.Our numerical example shows that diversification returnis, in general, positive for leveraged risk parity portfoliosof two assets (stocks and bonds) when the leverage ratiois not too high (under 300%). For portfolios with moreasset classes, the diversification return can be expectedto rise due to the fact that the bottom-up diversificationeffect will be higher with more assets.

Even though diversification return is not an essen-tial part of the traditional asset allocation decision process,it is important to consider it for leveraged portfolios suchas risk parity portfolios, in order to avoid negative diver-sification returns. Our analysis reveals two importantfactors in determining portfolio diversification return.The first is the correlation between different assets. Theanalysis shows that the lower the correlation, the higherthe diversification return. Our stock and bond examplestrongly suggests that the best diversifying assets to stocksare high-quality government bonds due to their negativecorrelation to stocks. In contrast, the worst diversifyingassets to stocks might be high-yield corporate bonds dueto their high correlation to stocks.

The second important factor is turnover in port-folio rebalancing. For long-only unleveraged portfolios,tLirnover tends to be low and declines with increasingasset correlation. But for leveraged long-only portfo-lios, turnover rises quickly with increasing leverage andincreasing asset correlation. This result again suggeststhat in order to reduce turnover and maintain port-folio leverage of leveraged portfolios, such as risk parityportfolios, assets with low to negative correlations toeach other and with high liquidity should be selected,and assets with high correlation to each other and lowliquidity should be avoided.

A P P E N D I X A

THE NON-NEGATIVITYOF DIVERSIFICATION RETURNSFOR LONG-ONLY UNLEVERAGEDPORTFOLIOS

For any long-only unleveraged portfolio, the diversifi-cation return is the lowest if all assets are perfectly correlatedwith one another (i.e., p., = 1). Under such condition, theportfolio's volatility becomes the weighted sum of individualvolatilities.

SUMMER 2012 THE JOURNAL OF PORTFOLIO MANAGEMENT 2 3

Therefore, according to Equation (2), we have a lowerbound of the diversification return.

1r>-\ y.uKü-

(A-1)

To prove that the right-hand side of Equation (A-l) isnon-negative, we make use of the Cauchy—Schwarz inequality,which states that for any real numbers {a., h),i = \,---,N, thefollowing inequality holds:

(A-2)

Let a. = yjw. and ¿, = ^Jw^C5. in Equation (A-2), so thatwe have

It is easy to see that when L> \ and r >0, then L < L' P new

(i.e., positive return of a leveraged portfolio results indeclining leverage, while negative return results in increasingleverage).

The amount of trading required in rebalancing, whetherbuying or selling, is equal to the change of the weight, that is.

- - W . =• • ~ w(r —r ) (B-2)

For assets with long positions, the weights drift higher(lower) if the return is higher (lower) than the portfolio return.For assets wth short positions, the weights drift lower (higher)if the return is higher (lower) than the portfolio return.

The total turnover is the sum of the absolute values ofweight changes.

w. (B-3)

(A-3)To obtain average turnover, we take the expectation of

Equation (B-3). Because the weights are fixed, we have

For an unleveraged portfolio, the sum of weights

equals one, ¿_¡._^w-, =1- Using this fact in Equation (A-3)and squaring both sides proves that the right side of Equation(A-l) is non-negative. Hence, the diversification return of along-only unleveraged portfolio is non-negative.

For leveraged portfolios with negative weights, this proofis not valid. Equality holds for Equation (A-3) only when allvolatilities are the same. Because we have already required thatall correlations must equal one, equal volatihty implies that allassets are the same asset. We thus conclude that the diversifi-cation of a long-only unleveraged portfolio is always positiveexcept in the case of a single-asset portfolio.

A P P E N D I X B

CHANGE IN WEIGHTS AND REBALANCINGTURNOVER ESTIMATE

If the individual asset weight is w., the return is r., andthe portfolio return is r , then the new weight of the asset willbe u'.(i + r)/(l + r). For a long-only leveraged portfolio, thetotal leverage changes to

.(1 +r)> (tf. + w.r. ) =

(B-4)

The return difference between individual assets and theportfolio assuines a normal distribution.

, -|x ,a' + a] -2p, (B-5)

We have assumed |4,. and |X are average returns of theasset i and the portfolio; a. and a are the volatility of the asseti and the portfolio; and p_, is the correlation between the two.When the average returns are equal, the expectation of theabsolute return difference can be derived analytically. Weuse the result as an approximation of the average turnover.We have

avg(TO)^J-^ JO' -i-a~ -2p a G (B-6)

ENDNOTES

'This is an approximation for average returns, whichis accurate only in circumstances with modest ^veights andreturns. Also, in general, the equality is often violated for


cumulative returns over multiple periods even if it holds foraverage returns.

^Skeptics might argue that cap-weighted indices arenot exactly a single risky asset. It is imperative to realizeeven though cap- Areighted indices might consist of hundredsor even thousands of securities, they are not fixed-weightportfolios and hence have no diversification returns sincecap-weighted indices never rebalance, hi fact, this might hethe Achilles' heel of cap-weighted indices, on top of theirrisk concentration. Without regular rebalancing, freqLienthooms and busts of individual stocks, sectors, and countrieswithin the indices simply take investors on wild rides thatgo nowhere over time. Viewed in this light, equal-weightedindices might seem naive to begin with. But regular rebal-ancing at least provides guaranteed positive diversificationreturns over time. This feature goes a long way to explainwhy equal-weight indices often heat cap-weighted indicesin the long run.

REFERENCES

Booth, D., and E. Fama. "Diversification Returns and AssetContribution." Financial Analysts Journal, Vol. 48, No. 3(1992), pp. 26-32.

Qian, E. "Risk Parity and Diversification." TheJournal ofInvesting, Vol. 20, No. 1 (2011), pp. 119-127.

. "Risk Parity Portfolios." Research paper, PanAgoraAsset Management, 2005.

Willenhrock, S. "Diversification Return, Portfolio Rebal-ancing, and the Commodity Return Puzzle." Financial Ana-lysts Journal, Vol. 67, No. 4 (2011), pp. 42-49.

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