Crisis-Robust Bond Portfolios

Crisis-RobustBond Portfolios

Marie BrièreAriane Szafarz

WORKING PAPER CAAM STRATEGY

N°1 ! April 2008

Crisis-Robust Bond Portfolios

Dr. Marie BrièreHead of Fixed Income, Forex and Volatility Strategy

Crédit Agricole Asset Management

Prof. Ariane SzafarzDirector of the Centre Emile Bernheim

Solvay Business SchoolUniversité Libre de Bruxelles

Dépôt légal : avril 2008ISSN : en cours

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ABOUT THE AUTHORS

Marie Brière is Head of Fixed Income, Forex and Volatility Strategyat Crédit Agricole Asset Management, associate researcher with theCentre Emile Bernheim at Université Libre de Bruxelles and affiliateprofessor at CERAM Business School. She holds a PhD in Economicsand is a graduate of the ENSAE school of economics statistics andfinance. She also teaches empirical finance at Université de ParisIX Dauphine and Paris II Assas. She is the author of a book on theanomalies in the formation of interest rates as well as various scientificarticles published in books, academic and practitioners’ journals.

Ariane Szafarz, PhD in Science, teaches finance and mathematicsat the Université Libre de Bruxelles (ULB), where she is Director ofthe Centre Emile Bernheim, of the PhD programme in ManagementSciences (Solvay Business School), and the European Master inMicrofinance. She was visit ing professor at Université de Lil le II,Université Catholique de Louvain and the Luxembourg School ofFinance. She is the author of six books (published, among others,by Elsevier, Economica and Pearson Education) and many scientificart icles published in books and international academic journalsincluding Econometric Theory, European Economic Review, Journalof Empirical Finance, Oxford Economic Papers, International Journalof Business, Economics Letters.

The authors thank Michel Aglietta, Jerome Barkate, Pascal Blanqué,Tony Bulger, Gabrielle Demange,Christian Gourieroux, Sylvie de Laguiche,

Sandrine Lardic, Bertrand Maillet, Thierry Michel, Valérie Mignon,Kim Oosterlinck, Carlos Pardo, Marco Piersimoni, Benjamin Sahel,

Ombretta Signori, for usefull comments and suggestions.

Comments can be sent to [email protected] or [email protected].

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INTRODUCTION

A financial crisis is typically associated with a rise in the volatility of mostassets. Moreover, if the crisis is “contagious”, things become even worse forinvestors because corre lat ions among asset returns also increase anddiversif ication becomes less eff icient than during quiet periods. As mostidentified financial crises have been positively tested for contagion (Loretanand English (2000), Hartmann et al. (2001), Bekaert et al. (2005)), this problemwarrants close attention.

To reduce investors’ excessive exposure to crisis effects (Chow et al.(1999)), this study builds “crisis-robust” portfolios, i.e. those exhibiting theleast change in volatil ity during crises. Such portfolios enable investors tominimize as much as possible the perverse effects of volatility caused by acrisis. Holding a crisis-robust portfolio is an alternative to a regime-switchingasset allocation (Ang and Bekaert (2002, 2004)), which implies recognizingearly signs of a crisis and switching regimes appropriately1. In this respect,a crisis-robust portfolio is less demanding; and therefore safer, for the investor.

As far as possible, crisis-robust portfolios are supposed to eliminate theneed to rebalance asset allocations according to the climate in the financialmarkets. Choosing a crisis-robust portfolio not only reduces transaction costs;it means that investors do not have to worry constantly about changing theirallocations. Furthermore, it is diff icult to adapt portfolios to the changingvolatil ity/correlation picture in a timely manner because most crises occursuddenly.

In practice, crisis-robust portfolios can help asset managers respect arisk threshold fixed by their clients or publicly announced along with a fund'sobjectives. Indeed, the risk thresholds produced by our model ensure thatcrisis periods interfere as little as possible with a portfolio's composition andrisk. This is an innovation, since funds guaranteeing a maximal risk leveltypically evaluate that risk using time intervals that are long enough to bedominated by non-crisis subperiods.

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Although general, the concept of cr is is-robust portfol io is especial lyappealing when applied to markets that exhibit the mixed effects of criseson volatilities and correlations. Typically, these markets are characterized bya so-called flight-to-quality (FTQ) effect (i.e. all volatilities increase but thecorrelations between the safest and the riskiest assets decrease). Empiricalwork has detected this effect between bonds and stocks and among alsobonds (Bekaert et al. (2005), Berben and Jansen (2005)), while stocks seemto be subject only to contagion (correlations rise).

Thus, contrary to the stock market, the bond market offers an FTQ trade-off during crises that can be exploited to keep portfolios as crisis-insensitiveas possible, the optimum being the crisis-robust portfolio.

One might think that crisis-robust bond portfolios are composed solely ofthe highest quality securities, i.e. developed countries' sovereign bonds andhighly rated corporate bonds, because the volatility of investment grade bondsis low, even during periods of turmoil. However, FTQ effect undermines thisargument. The fact that investors shift from risky to safe securities duringcrises (Beber et al. (2006)) causes a decline in the correlations between riskyand safe assets during such periods (Stivers and Sun (2002), Baur and Lucey(2006)). This is good news for diversification benefits.

Regarding bond portfolios, crises tend to increase volatilities and decreasecorrelations between highly rated and speculative bonds. Therefore, thanksto more effective diversification, a bond portfolio including some high yieldbonds –but not too many, for volatility reasons– may withstand crises betterthan a 100%-safe portfolio. The empirical part of this study is devoted toassessing the extent of this phenomenon, that is, to measuring the fractionof risky assets devoted to optimally “robustifying” the portfolio against financialturmoil.

Regarding the consequences of crises, therefore, bondholders are betterplaced than stockholders, who are subject to harmful contagion. Indeed,bondholders benefit from an FTQ-linked risk trade-off. Crisis-conscious bondportfolio managers should therefore take this into account. The crisis-robust


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approach is one way to fully exploit this trade-off in order to keep portfoliosas crisis-insensitive as possible.

In the empirical part of this study, we have identified seven financial crisesbetween 1998 and 2007 on the basis of the literature. The full dataset iscomposed of weekly returns of government (GVT) and corporate bond indicesin the U.S. and in the Eurozone. Corporate bonds are split into investmentgrade (IG) and high yield (HY) categories. We therefore deal with six differentassets. First, we show that the FTQ has a key effect in the bond class. Indeed,the two correlation matrices (U.S., Eurozone) show that diversification benefitsrise during troubled times in four cases out of six. Accordingly, in these cases,the crisis-robust portfolio includes the lowest-grade asset, confirming theimpact of the volatility trade-off.

The study is organized as follows. Section I defines crisis-robust portfoliosin a general framework and furnishes their analytical expression in a simpletwo-asset sett ing. Section II presents the original bond-market dataset,explains how crisis periods have been delineated, and documents the presenceof FTQ. Section III applies the crisis-robust concept to this dataset. SectionIV concludes.

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I. CRISIS-ROBUST PORTFOLIOS

Because financial markets can be unpredictable, both in crisis periodsand in quiet periods, the crisis-robust portfolio is defined as the one minimizingthe volatility ratio between the two types of period.

Consider the following model. There are two possible regimes: the crisisregime C and the quiet regime Q, but the probabilities of these regimes areunknown. A crisis is therefore seen as a totally unpredictable event that affectsthe markets.

The market includes n risky assets. The stochastic return vectoris taken randomly from either the crisis-period multivariate distribution withcovariance matrix or the quiet time distribution with covariance matrix .Cr ises happen exogenously2 but investors are "cr is is-conscious" (Koleet al. (2005)), i.e. they know that crises are possible. However, they have noprior information on the probability of occurrence. A rationale for the impossibilityof inferring the probability of a next-day crisis could stem from the fact thatcrises are highly erratic, so the probability is unknown and, in most cases,unknowable.

Let , be the composition of portfolio P made up of

n assets, where denotes the proportion of asset i in P. The variance of this

portfolio depends on the regime:

(2.1)

during crises

during quiet times

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The crisis-robust portfolio P* is defined as the one minimizing the varianceratio between the two regimes:

(2.2)

Its composition, denoted by , with , is thus suchthat:

(2.3)

With a crisis-robust portfolio, the investor minimizes the need to rebalancethe asset allocation. Thus, a crisis-robust portfolio reduces transaction costs,all other things being equal. However, the benefits associated with crisis-robustness are likely related to the style of the portfolio manager. For activeinvestors who react even to minor signals, looking for a crisis-robust portfoliomakes litt le sense since they adapt to any changes they perceive on themarket. Conversely, passive investors with a diversified portfolio should bekeenly interested in finding a way to avoid –or at least weaken as much aspossible– the impact of a financial crisis on their wealth.

The optimization in (2.2)-(2.3) may be performed on any set of financialassets provided that the two-regime setting in (2.1) is assumed. Numericalsoftware may then be used to find the appropriate asset allocation. However,to determine the robust portfolio analytically in a simple case, we will assumefrom now on that the market is composed of two assets having volatilities

during quiet times, and during crises. The correlationcoefficient between the returns of the two assets is during quiet times,and during crises. We also introduce parameter representing the ratioof these correlations:

(2.4)

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The position of with respect to unity is left unconstrained. It may eitherbe larger than one in case of contagion or smaller than one when FTQ arises.Therefore, the chosen parameterization will make it easier to conduct furtherdiscussions involving FTQ-specific issues.

To obtain tractable analytical results we assume that the crisis has thesame relative impact on both assets' volatilities, that is, their crisis-to-quiettime volatility ratios are equal. We consequently introduce parameter forthis common ratio. This parameter takes values larger than unity since crisesare associated with a rise in volatility:

(2.5)

Short positions are excluded. The two-regime variance (2.1) of portfolio

P including a proportion of asset 1 and, consequently, proportion

of asset 2, may be expressed as follows:

(2.6)

The first order condition for minimizing the ratio leads to:

(2.7)

I f , meaning that the correlat ion changes during crises, then thecondition simplifies to:

(2.8)


or equivalently:

(2.9)

so that the unique portfolio allocation fulfilling the first-order condition associatedwith the crisis-robust criterion is given by3 with:

(2.10)

However, as il lustrated by Figures 1 and 2 for several point values of theparameters in question, this critical point may well correspond to a maximalvalue of the volatil ity ratio as well as to a minimal value depending on thesign of coefficient defined as:

(2.11)

The sign of indicates whether the two assets are subject to contagion

or to FTQ . In the first case, the critical point corresponds to a maximal

value and is therefore useless. Conversely, when the crises are associated

with a decorrelation of the assets, then the critical point provides the optimal

composition for a crisis-robust portfolio. Proposition 1 summarizes this result.

Proposition 1: I f the market is made up of two assets such that

and , then the ratio reaches an interior minimum

for P = P* composed of of asset i .

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Figure 1Variance in crisis / variance in quiet period

The variance ratio is calculated as a function of

for


Figure 2Variance in crisis / variance in quiet period

The variance ratio is calculated as a function of

for

Portfolio P* is thus the crisis-robust portfolio. Its existence is ensuredthanks to the downward shift in the correlation, which compensates, at leastto some extent, for the rise in the volatilities of the two asset returns. In thecase of contagion, such a compensation effect is impossible and the minimalratio is obtained for corner solutions, i.e. portfolios made up of a single asset.

What is the efficiency of the crisis-robustness criterion? In other words,

how close to one is the optimal variance ratio ? Replacing and

by their expressions and computing both variances yields:

(2.12)

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The next proposition follows from this equation.

Proposition 2: If the market is made up of two assets such that

and , then the minimal ratio is equal to 1 if and

only if .

When FTQ is present, the correlation spread between crisis and quietperiod, , is negative and creates a volatility trade-off. Proposition2 exhibits the threshold to be reached by this correlation spread in order toallow for full compensation, making the volatility of the crisis-robust portfoliototally insensitive to crises. Indeed, we have:

(2.13)

By definition, and , then is always negative. However,in practice, the correlation gap is generally insufficient ( ) to lead tothe existence of a portfolio that is totally insensitive to crises4. Consequently,crisis-robust portfolios should be seen as second-best choices.

On Figure 1, all chosen values of are such that . Therefore,none of the represented curves reaches the level of one for the volatility ratiocorresponding to a fully crisis-insensitive portfolio. Note that the result is notinfluenced by the quiet period volatilities of the two assets but only by thecrisis-to-quiet period ratio .

Conversely, on Figure 2, for , the curve presents a minimumvalue with a unit ratio, while for the volatility ratio can be lower thanone. This is an unrealistic situation to be avoided, making the volatility duringcrisis periods smaller than in quiet times.


II. DATA, DESCRIPTIVE STATISTICS

AND CORRELATION MATRICES

Government and Corporate Bonds

The period under study stretches from July 1998 to May 2007. The datasetis composed of weekly returns of the sovereign and corporate bond indicesof two geographic zones: U.S. and Eurozone. Corporate bonds are split intoIG and HY.

GVT bonds are represented by the 10-year benchmark indices suppliedby Datastream5. These indices, which include coupon returns, are usuallybased on a single bellwether, namely the last bond issued by the country'sTreasury in a given maturity. Factors such as liquidity, issue size and couponsare also taken into account when choosing the index component(s). Forcorporate bonds, two categories of index were used: IG, with ratings betweenAAA and BBB-, and HY, rated from BB+ to CCC. These indices excludeconvertible bonds and include coupon returns. Data are sourced from MerrillLynch (i.e., bids quoted by traders at the ML desk) at the market close6.

Figures 3 presents GVT bond prices, Figure 4 draws all corporate bondprices. HY bond indices decreased sharply between 2000 and 2002 in theaftermath of the stock market collapse. Indeed, HY default rates reachedhistorical peaks at that time. Starting from 1.6% in 1998, they reached 5.1%in 2000 and finally 12.8% in 2002 (an historical high on the period 1971-2006). The HY market also recovered very rapidly after 2002. In 2003, thedefault rate declined to 4.7%, causing a complete change in the HY performancetrend at that date. The particular episode of 2000-2002 explains why globally,for the period 1998-2006 under review, the HY bond index underperformsIG bond returns, showing a negative risk premium to less risky assets.

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Figure 3Government bond pricesCumulative weekly returns in local currency, July 1998–May 2007


Figure 4Corporate bond pricesCumulative weekly returns in local currency, July 1998–May 2007

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Crises studied

Our definit ion of "crisis" is broad. It encompasses five types of event:currencies, sovereign debt, bond or equity crash, corporate bankruptcies(Enron, WorldCom), and crises of confidence (9/11). We have deliberatelyomitted crises of a purely banking nature and economic cr ises such asrecessions or oi l shocks. The real d i ff icu l ty l ies in establ ish ing precisetimeframes for the crises we have selected.

Table I shows the start and end dates of the crises studied in this article.They have been chosen careful ly on the basis of previous research (seeAppendix). Admittedly, while the onset of a crisis is usually easy to identify,the end date is much harder to pinpoint. This awkward problem is highlightedby the Asian crisis studied by several authors.

Table ICrises used in this study


Descriptive statistics, correlations and volatilities

Table II provides descriptive statistics on returns (in local currencies) forall assets in the database, for the whole period (July 1998 - August 2006).All returns have been tested for stationarity (but the results are not reportedhere). The annualized mean return of GVT bonds is around 4.8%. IG corporatebonds exhibit somewhat larger returns with a risk premium between 108 and164 bp while their volatility does not show higher values.

Surprisingly, the mean returns of HY bonds are lower than those of IGbonds, and in the case of Europe, even lower than the return on GVT bonds.The most probable explanation for this, as already mentioned, is the presencein the sample period of a huge breakdown (2000-2002) on the HY bondmarket, leading to a sharp increase in the default rate.

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Table IIDescriptive statistics for returns: All bonds, whole periodWeekly returns in local currency, July 1998–May 2007

*Germany for GVT bond indices.

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Tables III and IV give the descriptive statistics for crises and quiet periods,respectively. Logically, as testified by their average returns, corporate bondssuffer much more from crises than do GVT assets. In fact, HY bond returnsexhibit negative mean returns during crises.

As crises are the exception rather than the rule, the quiet period statisticsgenerally resemble the whole period data. Nevertheless, the anomaly regardingthe risk premia over the whole period (i.e. riskier bonds having smaller meanreturns) disappears in the quiet-period dataset (Table IV), testifying to theimpact of crises on this stylized fact.


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Table IIIDescriptive statistics for returns: All bonds, Crisis periodWeekly returns in local currency, July 1998–May 2007


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Table IVDescriptive statistics for returns: All bonds, quiet periodWeekly returns in local currency, July 1998–May 2007



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Tables V and VI provide the correlation matrices for, respectively, the wholeperiod, the crises, and the quiet period. In all cases, sovereign and IG bondsexhibit posit ive correlations. The same is true within the HY bond class.However, correlations between HY bonds and the other categories take valuesbetween 0.61% and 29.97% during quiet periods. Most of the time they dropsharply during crises, reaching values comprised between -6.15% and 34.59%.The FTQ effect during crises is clearly observed among bonds.

Table VCorrelation matrix: U.S. bondsWeekly returns in local currency, July 1998–May 2007

Panel A: Whole period

Panel B: Crisis period

Panel C: Quiet period

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Table VICorrelation matrix: E.U. bondsWeekly returns in local currency, July 1998–May 2007

Panel A: Whole period

Panel B: Crisis period

Panel C: Quiet period


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Finally, as the theory has shown (see Propositions 1 and 2 in Section 2),correlation spreads and volatility ratios are key elements in determining crisis-robust portfolios. This information is provided in Tables VII to IX, respectively,thus helping to measure the effects of crises on volatilities and diversificationbenefits.

Table VIICorrelation spread matrices*, U.S. bondsWeekly returns in local currency, July 1998–May 2007

*Differences between correlations in crisis and correlations during quiet periods.The cells in grey correspond to the presence of FTQ (correlation decrease).

Table VIIICorrelation spread matrices*, E.U. bondsWeekly returns in local currency, July 1998–May 2007

*Differences between correlations in crisis and correlations during quiet periods.The cells in grey correspond to the presence of FTQ (correlation decrease).

Table IXVolatility ratios*Weekly returns in local currency, July 1998–May 2007

*Crisis volatility over quiet-period volatility.

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Tables VII and VIII show that FTQ is observed in four out of six cases,meaning that crises lead to lower correlations. For instance, on the U.S.market during turmoil, the correlation between sovereign and IG corporatebonds declined more than 8%, from 93.27% to 86.59%. Likewise, the declinein correlation between sovereign and HY bonds reaches 11.14% while intra-corporate bonds (IG and HY) show some contagion (+4.62%). This couldmean that, during crises, U.S. bondholders consider public debt as the onlyreally safe asset. Be that as it may, Tables VII and VIII testify to the simultaneousoccurrence of contagion and FTQ in the bond market, thus confirming therelevance of our two-correlation-regime approach.

Table IX shows that volatility ratios in our dataset all lie between 1.46 (forGVT bonds in Europe) and 2.26 (for HY bonds in the U.S.). Not surprisingly,the ratios take the largest values in the HY bond category. The two-assettheoretical model in Section 2 assumes a constant ratio. While this hypothesisis clearly not met for every pair of assets, it seems still realistic for U.S. andE.U. sovereign and IG bonds at least. Crisis-robust portfolios can howeverbe determined empirically in all cases. The drawback of not respecting theconstant volat i l i ty rat io hypothesis is that neither the existence nor theuniqueness of a crisis-robust portfolio is guaranteed by the theory.


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III. DETERMINING CRISIS-ROBUST BOND PORTFOLIOS

For each geographical zone, we determine the crisis-robust portfolio madeup of two or three assets. In the two-asset case, only one parameter is requiredfor indicating the portfolio composition. Consequently, it is possible to representthe portfolio volatilities in both regimes, as well as their ratios, as functionsof the proportion of an asset in the portfolio.

Take for example the case of U.S. GVT and HY bonds. Figure 5 showsthat both volatility curves are U-shaped and thus reach a minimal risk valuefor a certain proportion of HY bonds in the portfolio (69.91% during calmperiods, 50.14% during crisis periods). The volatility ratio is trickier7 but stillreaches a minimal value at an interior point of the 0%-100% interval. Precisely,the crisis-robust portfolio has a volatility ratio of 1.55 and includes 12.68%of HY bonds. Consequently, HY bonds are indeed present in the crisis-robustportfolio. However, this portfolio is not fully crisis-insensitive since the minimalratio remains above the unit value.

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Figure 5Volatility during the quiet period and during a crisis, and volatility ratio:U.S. portfolios (GVT and HY bonds)


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More general ly, Table X offers the results for the various possibi l i t iesof two –or three– asset U.S. and E.U. portfolios. In each case, we determinethe crisis-robust bond portfolio from the standpoint of a domestic investor.

Table X indicates (left column) the assets allowed in the portfolio, then thepresence or absence of the FTQ effect. The latter piece of information isderived from Tables VII or VIII depending on the geographical zone in question.In accordance with the theory, the two cases of no-FTQ effect correspondto corner solutions: 100% IG bonds in IG-HY U.S. portfol ios, 100% GVTbonds in GVT-IG E.U. portfolios.

Table XComposition of the crisis-robust portfolios

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When the FTQ effect is present, the U.S. and E.U. portfolios differ fromone another by the respective proportions of the riskiest asset required toreach crisis-robustness. While European investors find their optimum withsmall proportions (2.8%, 4.9%), U.S. investors need much larger proportions(12.7%, or even 59.3%). Moreover, the optimal ratio is more advantageousfor Europeans, as it is (slightly) closer to unity, the best ratio being 1.45 forthe E.U. and 1.52 for the U.S.

The difference between the U.S. and Euro portfolio composition is probablyattributable to the 2000-2002 “anomaly” mentioned earlier. In fact, in the U.S.during quiet times, IG bond volati l ity appears lower than sovereign bondvolatility (0.77 < 0.94, see Table IV). According to Proposition 1, the share ofasset 1 volatility in the sum gives the proportion of asset 2 in the crisis-robustportfolio. Even if the results in Table X do not stricto sensu obey the theory(because the volatility ratio is not constant), Proposition 1 is still a good proxyfor the composition of the crisis-robust portfolio. The same “anomaly” occursin Europe, too, but only in the GVT-IG case, where the FTQ effect does notappear. Conversely, the HY European bonds exhibit greater volatility than dothe two other classes. As these cases correspond to the occurrence of theFTQ effect, only relatively small amounts of HY bonds are to be included incrisis-robust portfolios.

Note also that, in both the U.S. and the Eurozone, putting the three assetstogether does not make the optimal portfolio more crisis-robust. Since thetheory is restricted to the two-asset case, we cannot infer the generality ofthis observation. But, on the basis of this stylized fact, we conjecture thatonly a few securities exhibiting FTQ effects would be required to (partially)hedge a portfolio against financial turmoil.


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IV. CONCLUSION

Several past events have pointed to the necessity for investors to takeinto account the possibility that financial crises may occur. That said, eachcrisis has its own pattern: some start slowly, others abruptly; some are short,others last longer; some hit specific assets, others are general. Therefore,waiting for a crisis to erupt in order to rebalance a portfolio is hazardous. Sothis study offers the concept of a crisis-robust portfolio, that is, a portfolioexhibiting the lowest volatility ratio between turmoil and quiet periods. Ourwork highlights the somewhat paradoxical result that introducing risky assetsinto a portfolio can “hedge” against a rise in volatility during crises. This resultis established analytically as well as empirically. In a two-asset framework,the presence of the riskiest asset in the crisis-robust portfolio is solely linkedto the occurrence of a flight-to-quality (FTQ) effect, namely an asset decorrelationduring crises.

The empirical application provided here is rather simple, aimed at illustratingthe fact that crisis-robust portfolios can include a substantial fraction of riskyassets when FTQ is observed instead of contagion. In this respect, the bondmarket offers a natural arena for examples. Other cases of FTQ have alsobeen identified, in particular in the bond-stock correlation (Hartmann et al.(2001), Baur and Lucey (2006)). Therefore, a natural extension of this workwould be to test crisis robustness in the case of larger portfolios that alsoinclude stocks. Indeed, contagion and its unpleasant consequences forportfolio management have been now extensively described. But surprisingly,little attention has been paid to the benefits that can be obtained by overweightingassets whose correlations decrease during crises. Actually, acknowledgingthe FTQ effect helps reduce the perceived “drama” surrounding financialcrises. More empirics are obviously needed to gauge the size of FTQ on globalfinancial markets.

Crisis-robustness is, however, a debatable criterion for portfolio management,for several reasons. First, in this framework, the investor does not choosethe level of risk as such. As safer assets tend to be less influenced by crises

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than highly risky assets, a crisis-robust portfolio will be low-risk overall, butnot necessarily the one with the minimal risk over the whole period or undera given regime. In this respect, the investor’s classic level of risk aversion isreplaced by an aversion to higher volatility during crises.

Second, crisis-robustness concerns volatility only. The fact that returnsdecrease during crises is not taken into account. Further research couldinvestigate the distorsions of the risk-return relationship during crises andcheck whether the volatility gains associated with crisis-robust portfolios areadversely affected by return costs. The intuition is that the result would dependon the level of investors’ crisis-awareness. In markets where portfolios aregenerally composed in a single regime setting, crisis-robust portfolios offerfree hedging, whereas in two-regime markets, recognized by investors andintegrated in their optimization along the lines of our model, asset pricingwould incorporate the pricing of the hedge.

Third, crisis-robustness is formulated here in relative terms, as a ratiobetween the volatilities in the two regimes. Other formulations involving, forexample, the frequency of a crisis or some kind of weighted average volatilitylevel, could lead to better criteria. A ratio has been chosen here for simplicitybut our main goal was to advocate looking for a crisis-conscious investmentrather than having to scrutinize the market each morning and decide whetheror not to change the portfolio's composition.

Going further on this issue would require a more general theoret icalframework for investors’ fears and attitudes regarding crises. In this respect,behavioral finance (Malkiel et al. (2005), Bourachnikova (2007)) has shownthat people tend to overestimate the probability of detrimental but rare events.By not introducing the probability of a crisis occurrence into our model, weprovide a theoretical approach compatible with any view (rational or not) aboutthe possibility of a crisis. Moreover, constraints could be added to the proposedoptimization, such as a maximal level of volatility in either of the given regime,or even in both.


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Finally, the model in this study is static since no composition update isconsidered. This is of course a major limitation, because extensive evidenceof h igh ly unstab le cor re la t ions can be found in econometr ic papers(Engle (2002), Goetzmann et al. (2005)). Further research could thereforeconsider a dynamic counterpart, including for instance a two-regime randomcorrelation coefficient.

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NOTES

1 “Because the regime is not observable, the estimation involves inferringfrom the data which regime prevai ls at each point in t ime” (Ang andBeckaert, 2004, p.89).

2 In this respect, the applications of the model are widely open as no crisisspecification is needed. Indeed, different portfolio management styles maybe sensitive to different crisis types.

3 Other solutions may exist if the no-short selling condition is waived.

4 However, in theory, it can happen that two interior portfolios are crises-insensitive because the correlation gap is so huge that it overcompensatethe variance growth. In such a situation, as a matter of fact, the variance-ratio minimization is inappropriate due to the fact that the denominatormay exceed the numerator.

5 For Eurozone, the German bond index.

6 The indices have minor differences. For investment grade indices, weselected a maturity of 7 to 10 years. However, for HY indices, maturitywas not proposed as a selection parameter, so there are small differencesin duration. Furthermore, the geographical indices for both IG and HYsecurities have median ratings that may vary slightly. Moreover, the numberof index components varies from one country to another.

7 If the assumptions made in the theoretical part were fully satisfied, thiscurve would also exhibit a U-shape. This is not the case since the GVTbond volat i l i ty rat io is 1.56 whi le the HY one is much larger (2.26) .Nevertheless, a numerical optimisation can always be performed.

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APPENDIX: CRISES SELECTED FOR STUDY

In th is s tudy, we examine f ive types of cr is is : (1 ) cur rency cr ises,(2) sovereign debt crises, (3) crises triggered by an equity or bond crash,(4) corporate bankruptcies or loss of confidence (e.g. the collapse of Enron),and (5) crises of confidence arising from severe external events (e.g. 9/11).

• Currency crisis

Brazil 1999

Dungey et al. (2002) say that the crisis began on January 13, 1999 withthe devaluation of the real. It is hard to establish an end date because nolandmark events occurred. However, the crisis is generally referred to asthe "January 1999 Brazilian crisis". We have therefore taken the final dateto be the end of January 1999.

• Sovereign debt crises

Russia 1998

The Russian crisis began on August 17, 1998, when the country defaultedon its debt, and continued until September of that year, when anothercrisis was triggered by the collapse of the hedge fund LTCM. We havetherefore considered these two crises jointly, setting the end date for bothat the end of the LTCM crisis.

Argentina 2001

The crisis began on the November 1, 2001 when Argentina announced adebt restructuring plan. On December 5, the IMF refused to release fundsto help the country, and the Argentine president was forced to resign onDecember 20. On December 23, 2001, the country announced that it was

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in default. For investors, the announcement marked the end of the crisis,and emerging spreads began to narrow (BIS, March 2002).

• Crashes

e-crash 2000

Triggered by the crash in tech stocks, the equity meltdown began onMarch 28, 2000. We have dated the end of the crisis to April 14, 2000,when prices stopped falling. Thereafter, the market entered a period ofstagnation.

• Corporate bankruptcies and crises of confidence

LTCM 1998

The hedge fund Long Term Capital Management (LCTM) collapsed onSeptember 23, 1998. Dungey et al. (2002) consider that the crisis endedwhen the U.S. Federal Reserve decided to cut interest rates in order tocontain the fallout. The Fed's decision was taken unexpectedly betweentwo FOMC meetings on October 15, 1998.

Enron 2001

The onset of the crisis can be dated to November 28, 2001, when Moody'sInvestor Services decided to downgrade Enron, taking it from IG to HY. Althoughit was Moody's decision that sparked the mood of wariness which spread toall financial markets, signs that Enron was in trouble had emerged much earlier.On October 16, 2001, the company lowered its earnings guidance (BIS (March2002)), and on November 8, it announced a retroactive adjustment to all itsresults since 1997. Enron filed for bankruptcy on December 2. It is extremelydifficult to set a precise end date, and we consider that the crisis lastedthroughout December.


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WorldCom 2002

The crisis related to the bankruptcy of WorldCom began on June 25, 2002,when the company revealed accounting inaccuracies concealing lossesof $3.8 bi l l ion in 2001 and 2002; it also announced 17,000 job cuts,equivalent to 20% of the workforce. WorldCom filed for bankruptcy onJuly 11, and its share fell 80% over the next four months. Once again itis very hard to establish an end date because the loss of confidence wasexacerbated by fears relating to terrorist attacks in May and June 2002and to political tensions between India and Pakistan. According to theBIS (September 2002), the most significant crisis-related market movementsoccurred between July 10 and 23. We therefore consider that the crisislasted until end-July 2002.

• Other crisis of confidence

9/11

The terrorist attacks on the USA on September 11, 2001 sparked a crisisof confidence across markets worldwide. It is hard to say precisely whenthe crisis ended, but we have considered that it lasted for the whole ofSeptember.

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“Written by Crédit Agricole Asset Management in collaboration with CentreEmile Bernheim (Solvay Business School, Université Libre de Bruxelles).

Crédit Agricole Asset Management is a French joint stock company (sociétéanonyme) with a registered capital of 546,162,915 euros.

An investment management company approved by the French Securities Authority(Autorité des Marchés Financiers - “AMF”) under No. GP04000036. Registeredoffice: 90, boulevard Pasteur 75015 Paris-France. 437 574 452 RCS Paris.

In each country where they carry on investment business, Crédit Agricole AssetManagement and its affiliates are regulated by the local regulatory authority.This information contained herein is not intended for distribution to, or use by,any person or entity in any country or jurisdiction where to do so would becontrary to law or regulation or which would subject Crédit Agricole AssetManagement or its affiliates to any registration requirements in these jurisdictions.The information contained herein is produced for information purposes only andshall not be considered as an investment advice nor the sole basis for theevaluation of any CAAM’s product. Any data provided herein is based onassumptions and parameters that reflect our good faith judgment or selectionand therefore no guarantee is given as to the accuracy, completeness orreasonableness of any such data. No examination has been executed by uswith respect to the financial, legal, tax, regulatory –and any related aspectsthereof– situation of any addressee of the information herein”.

Investors tend to gravitate towards low-risk securities such as governmentbonds during financial crises. Typically, stock market volatility increasesduring such crises, a situation usually compounded by closer correlationbetween assets. This contagion makes diversification less efficient.For investors, the combination of widespread increases in volatility andcorrelation is pernicious because it forces them to cope with a dual in-crease in risk unless they modify their portfolios. In such cases, they maybe tempted to sell off their most volatile securities and increasethe proportion of the safest investments.The key question here is to detect precisely when a crisis occurs in orderto switch portfolios in time. Because the onset can be sudden, investorsmay miss the optimal reaction date and bear the brunt of the initial shocks.Therefore, with a dual regime, portfolio managers have to monitor thesituation constantly.

In this study, we develop an alternative concept - the crisis-robustportfolio - that consists in determining the optimal asset composition forminimizing the volatility ratio between crises and quiet periods. In otherwords, it means holding a portfolio that is as insensitive as possible toregime switches. In practice, a crisis-robust portfolio enables assetmanagers to respect the volatility thresholds they have promised their clients.

One might think that crisis-robust bond portfolios are composed solelyof the highest quality securities, i.e. developed countries' sovereignbonds, because their volatility is low, even during periods of turmoil. How-ever, a flight-to-quality effect undermines this argument. The decline in thecorrelations between risky and safe assets during such periodsis good news for diversification. Therefore, thanks to more effective diver-sification, a fixed income portfolio including some corporate bonds (in-vestment grade or high yield, depending on the country) may withstandcrises better than a 100% - safe portfolio.

Marie Brière, PhD in Economics, is Head of Fixed Income, Forex and Volatil-ity Strategy at Crédit Agricole Asset Management, associate researcherwith the Centre Emile Bernheim at Université Libre de Bruxelles (ULB) andaffiliate professor at CERAM Business School.

Professor Ariane Szafarz, PhD in Science, teaches finance and mathematicsat the Université Libre de Bruxelles (ULB). She is Director of the CentreEmile Bernheim, the PhD programme in Management Sciences (SolvayBusiness School), and the European Master in Microfinance.


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Crisis-Robust Bond Portfolios