diversification and systemic risk

Journal of Banking & Finance 46 (2014) 85–106

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Journal of Banking & Finance

journal homepage: www.elsevier .com/locate / jbf

Diversification and systemic risk

⇑ Tel.: +33 631189878.E-mail address: raffestl@tcd.ie

1 Schematically, this approach allows us to map the individually riskeffect and the contagion externality into two distinct components of systethe former affects in the marginal distribution of each element on theinvestors wealth, while the latter is embedded within the dependencebetween all N investors.

Louis Raffestin ⇑LAREFI, Montesquieu-Bordeaux IV University, Avenue Léon Duguit, 33608 Pessac, France

a r t i c l e i n f o

Article history:Received 8 August 2013Accepted 6 May 2014Available online 27 May 2014

JEL classification:G110G120C650

Keywords:Systemic riskDiversificationCirculant networkCirculant matrixFire sales

a b s t r a c t

Portfolio diversification makes investors individually safer but creates connections between themthrough common asset holdings. Such connections create ‘‘endogenous covariances’’ between assetsand investors, and enhance systemic risk by propagating shocks swiftly through the system. We providea theoretical model in which shocks spread through constrained selling from N diversified portfolioinvestors in a network of asset holdings with home bias, and study the desirability of diversificationby comparing the multivariate distribution of implied losses for every level of diversification. Theremay be a region on the parameter set for which the propagation effect dominates the individually saferone. We derive analytically the general element of the covariance between two assets i and j. We findagents may minimize their exposure to endogenous risk by spreading their wealth across more and moredistant assets. The resulting network enhances systemic stability.

reducing

1. Introduction

When thinking about financial markets and systemic risk, onecan find it useful to consider a group of climbers roped togetherat the top of a cliff. Each climber individually favors being ropedas it lowers his chances of falling off, yet one climber slippingnow threatens the stability of his neighbors. The effect of beingroped on the probability that many or all climbers fall is thus a pri-ori ambiguous.

Prior to the 2008 credit crunch, both market participants andregulatory instances seemed to favor the roped equilibrium,implicitly assuming that individual soundness leads to systemicsoundness. Yet the swiftness with which risk spread throughoutthe market, largely unanticipated, led to suspicions that ‘‘connec-tions’’ may in fact carry systemic risk through the system. BaselIII will apply an extra capital ratio requirement of up to 2.5% to wellconnected establishments. Measures of systemic importancethat account for externalities, such as Covar (Adrian andBrunnermeier, 2011), or Shapley values (Tarashev et al., 2010)have recently gained in popularity.

This change in focus has implications on the desirability of port-folio diversification from society’s perspective. Indeed while the

individually risk reducing effect of diversification has been knownsince Markowitz (1952), diversification also forms connectionsbetween investors through common asset holdings, identified asa major vehicle of contagion in the presence of fire sales (Shleiferand Vishny, 2012). The goal of this paper is to take a first steptowards quantifying this contagion externality and comparing itto the individual risk-reducing effect, in order to get a primaryassessment of diversification’s total impact on systemic risk, aswell as the factors on which this impact may depend.

We do so by setting a model in which investors react to stochas-tic shocks on their wealth by selling/buying assets, which furtherimpact asset prices and wealth, and so on. This occurs in a ‘‘homebiased’’ network, in which investors acquire assets which have lowinformational distance with those they already own. The modelgenerates a normal multivariate distribution1 of investors wealth.Systemic risk is then studied through the probability that a numberg of investors fall below a given bankruptcy threshold K, for differentlevels of diversification. We then discuss the desirability of eachlevel of diversification by attaching to each probability a cost thatgrows exponentially with g.

mic risk:vector ofstructure

86 L. Raffestin / Journal of Banking & Finance 46 (2014) 85–106

We find intermediate numbers of bankruptcies are less likelywith a high degree of diversification, but the probability that manyor all investors fail simultaneously is larger. In a context of highselling constraint and weak demand, the probability of ‘‘extremefailure’’ outcomes becomes non-trivial, so that no or little diversi-fication may become optimal for society. We then introduce atouch of heuristics in the model by allowing agents on the demandside to grow more risk averse in a high variance environment. Wefind this strongly enhances the desirability of higher levels ofdiversification through a more indirect channel: spreading shocksacross assets lowers the chances of triggering a panic. In this con-text, intermediate levels of diversification appear particularlyharmful as they provide linkages through which shocks mayspread without going far enough in minimizing individual riskand the likelihood that a panic occurs.

Our set-up also allows us to derive an analytical expression forthe covariance between assets/investors i and j. In this case covar-iances depend negatively on the distance between i and j in thefinancial network and on the number of assets in the economy.We show that an investor may minimize the variance of his port-folio by seeking assets which are far away from his in the financialnetwork. We discuss the systemic implications of the resulting‘‘optimal’’ network in which all investors are no longer biased,and that of a ‘‘wider’’ network, i.e., one in which there are moreassets. Moving to an optimal network is generally desirable for sys-temic risk, while moving to a wider one is unambiguously so. Amix of more optimality and width is particularly efficient, notablyfor intermediate levels of diversification which previously were themost dangerous option from a systemic perspective.

Previous work on the link between diversification and systemichas been split between three strands of literature which haverarely crossed.

The first concerns small-scale2 models of financial contagion.Such models highlight different channels through which commonasset holding may lead to fire sales, which may in turn degenerateinto systemic events. Schinasi and Smith (2000) for instance showthat routine portfolio rebalancing brings contagion. The scope isincreased when agents are subject to wealth effects as in Kyle andXiong (2001). Goldstein and Pauzner (2004) point out that fire salesmay also result from strategic risk.3

The second strand of literature is based on statistical analysis.One method, taken by Shaffer (1994) or Wagner (2010), is to com-pare the properties of a fully diversified situation to a fully undi-versified one. Both authors show that the risk that all investorsfail simultaneously is necessarily higher in the fully diversifiedsituation. Our study confirms this fact but also goes more in depthby considering any level of diversification and any number of fail-ures. A second approach deals with ‘‘fat tails’’, which may mitigatethe strength of the variance reducing effect, as showed bySamuelson (1967). In particular, Ibragimov et al. (2011) use anindicator of the tail behavior of returns to define a diversificationthreshold. They find that on a given parameter range there canexist a wedge between investors interests and society ones.

2 The models usually feature only 2 assets, and diversification is defined as howevenly an investor spreads his wealth across both. A notable exception is provided byLagunoff and Schreft (1999) who try moving to a 3 assets case. They find that thescope for contagion is decreased.

3 Others authors relate diversification to different amplification mechanisms,which deserve to be mentioned although they will not feature as such in this paper.Market distress may lead to an even wider collapse if it turns into liquidity distress:Adrian and Shin (2010) shows that leverage is negatively correlated to the marketvalue of assets. Allen et al. (2010) find that when investors need to roll debt over,being connected brings a negative reputation externality. We also leave out apotential impact of diversification on ‘‘banks not doing their homework’’. Forinstance, Jiao et al. (2013) show that diversification may reduce the heterogeneity ofinvestors beliefs, Calvo and Mendoza (2000) argue that diversification lowers theincentive for investors to acquire information about securities before selling.

The link between these first two strands is the correlation struc-ture between asset prices. In any contagion model correlationsbetween asset returns appear endogenously as an output, as eventwo securities which are ‘‘fundamentally’’ independent becomelinked through the investors who hold them. In statistical analysishigh correlations are an input, as they may cause ‘‘fat tailed’’portfolios.

The last strand of literature related to this work deals with net-work analysis of financial stability. In particular, this paper may berelated to previous work on the ‘‘robust yet fragile’’ feature of thefinancial system, as in Nier et al. (2007) or Amini et al. (2012).

The broad method of this paper is to bridge these threeapproaches, by proceeding in three steps:

(1) setting up a large scale contagion model. N constrained ‘‘port-folio investors’’ who hold from 1 to N assets are forced to sellto ‘‘convergence traders’’ in response to negative shocks ontheir wealth. This lowers prices, further tightening the con-straint, and so on. Mathematically this translates into a lin-ear system of N recurrence equations of price returns, inwhich the amount of recurrence will depend on the weightof the constraint on investors and the discount on the sales.

(2) within a specified network. We set two characteristics forinvestors: they are home biased, and all have the same pat-tern of asset holdings. The latter leads the network andmatrix of asset holdings and to be circulant. This feature iswhat allows us to solve the system, obtaining analyticalexpressions for the covariances between assets and inves-tors that depend on distance.4

(3) in order to study systemic risk through statistical analysis andanalytical expression of the covariances. Statistical analysis isrun through the multivariate distribution of portfolio losses,which gives us the likelihood that any number of investorsgbetween 0 and N fail, for a given level of diversification n.

To the best of our knowledge two recent papers have used asimilar approach in terms of obtaining endogenous covariancesfrom a theoretical model. Danielsson et al. (2011a,b) define a mul-tivariate model which produces a matrix of covariance of unre-stricted dimensions. The correlations obtained have a‘‘fundamental’’ and an ‘‘endogenous risk’’ component, whereendogenous risk is the risk resulting from ‘‘the actions of marketparticipants which are hard wired in the system’’. Cont andWagalath (2012) specify a similar but more aggregated model,which they calibrate to estimate the realized covariance matrixduring well-known fire sales episodes, such as the aftermath ofthe collapse of Lehman brothers.

Our study places itself within this endogenous risk approach,but differs from these papers in that it puts the network of assetholding at the center of the analysis. In spirit, both papers focuson explaining the pattern of prices correlation during crisis epi-sodes, while our interest lies primarily with the desirability ofdiversification from a systemic perspective.

Finally, the paper shares some of its conclusions with Caccioliet al. (2012) who find that there might be a window on the diver-sification spectrum for which systemic risk may become signifi-cant. Caccioli et al. study the asymptotic properties of thefinancial system as a function of the average number of connec-tions of a node in an otherwise random network. Our model ismicro founded, and specifies a given network based on economicevidence. This gives the model a greater granularity. In particularwe show that different micro foundations lead to different

4 Circulant matrices are a useful tool, and a side goal of the paper is to contribute towidening their use and understanding beyond the fields of pure network analysis andsignal theory in which they are more common.

Fig. 1. Moving from a n = 3 to an n = 5 network when n = 9.

L. Raffestin / Journal of Banking & Finance 46 (2014) 85–106 87

conclusions on systemic risk, even within a network whose generalcharacteristics remain unchanged.

In short, the model is somewhat ‘‘hybrid’’ between micro-founded and network approaches, which allows it to remain bothtractable and general, and provide a wide oversight of the factorswhich may impact on the link between diversification and sys-temic risk.

Section 2 presents the baseline model, and how it brings recur-rence in asset prices. Section 3 presents the distributions of inves-tor wealth for the baseline model, and that with possible panics. Insection 4 we go further by drawing and discussing the realizedcovariances between assets and investors, and studying the impactof a wider and/or a home bias free network on systemic risk.Section 5 concludes.

2. Set-up

The model presented below generates a linear system whichwill allow us to draw N-dimensional covariance matrices betweenassets. Vectors are indicated with bold characters.

2.1. The market

The investment period starts at t = 0 and finishes at t = T, whereeach period t # t þ 1 may be seen as ‘‘a day’’ on the markets, andTis a large but finite number. Financial markets are composed ofN risky assets, which are best seen as stocks. N is finite to keepthe individual benefits from diversification within bounds, so thatmarkets are incomplete. Asset prices have three sources ofmovement:

DPt ¼ DPFt þMRtþ1 þ e�tþ1

where DPt is the actual price evolution between t and t + 1, DPFt the

fundamental price evolution. MRtþ1 is the mean reversion vector, ande�tþ1 is the investor-driven deviations from fundamental value. Allelements are N � 1 vectors. Intuitively, these dynamics imply thatthe behavior of investors may induce actual prices, which aredefined in equilibrium, to deviate from their fundamental value

by an amount e�tþ1 . Such deviations however tend to die out inthe medium/long run, through MRtþ1.

The fundamental price/value of each stock is defined as the dis-counted value of its future dividends. We assume dividends D fol-low an arithmetic Brownian motion, i.e., for stock i at time t:Di;tþ1 ¼ Di;t þ uD

i þ eDi;tþ1. The Gordon growth model yields:

pFi;t ¼

Di;tk þ cst, where k is an exogenous discount factor suited to

the riskiness of the cash-flows, and cst a constant. ConsequentlypF

i;t also follows an arithmetic Brownian motion, which we note:

pFi;tþ1 ¼ pF

i;t þ ui þ eFi;tþ1

where ui is the fundamental drift, and eFi;tþ1 is a normally distributed

shock, N � ð0;r2F Þ. Both are linear functions of the underlying divi-

dend drift and shock: ui ¼uD

ik and eF

i;tþ1 ¼eD

i;tþ1k .

We assume that the fundamental drift is the same for all assets,and that fundamental shocks are independent across asset andtime. Mathematically, ui ¼ u and vector eF

t , the N � 1 vector of gen-eral element eF

i;t , is defined by N � ð0;RF ¼ diagðr2F ÞÞ. These

assumptions will simplify presentation and calculus, but also showhow correlations between assets may arise from the actions ofmarket participants, even starting from a situation in which theyare fundamentally independent. The vector of fundamental pricesis written:

DPFt ¼ uþ eF

The mean reversion component ensures that prices do not go toofar away from their fundamental value. In our model mean rever-sion will be driven by an exogenous supply of ‘‘new’’ investors ateach period t, who will demand asset i if pi;t � pF

i;t < 0, and short sellit if pi;t � pF

i;t > 0. We set this amount of ‘‘new’’ investors to be quitelow, so that prices only converge to their fundamental value in themedium/long run. Mathematically:

MRtþ1 ¼ �kðPt � PFt Þ

where k, the speed of mean reversion, is fairly low.Substituting for DPF

t and MRtþ1 we may then rewrite the pricevector as:

DPtþ1 ¼ u� kðPt � PFt Þ þ eF

tþ1 þ e�tþ1 ð1Þ

Fig. 2. Sequence of events following a shock.

5 More on this assumption in Section 2.4.

e�tþ1, the investor driven deviations vector, will be the focal pointof this paper. It gives the movement in prices resulting from theactions/constraints of investors that may cause prices to deviatefrom their fundamental value. Its exact dynamics will be derivedin the following section, but it is useful to give the outline in words.

e�tþ1 is drawn from the interactions of two types of agents:‘‘portfolio investors’’, whose strategy is based on fundamentaldrifts, and ‘‘convergence traders’’ who exploit the short-term devi-ations from fundamental value. We may also refer to them as longterm and mid term investors due to their investment horizon,which we note respectively LT and MT. LT investors face a con-straint which forces them to sell in response to negative shockson their portfolio. MT investors buy those asset at a discount,which leads prices to fall below their fundamental value. Thisbrings LT investors to sell further, and so on.

The full specialization between long-term and mid-term inves-tors results from an ex-ante arbitrage by both agents. Convergencetraders specialize on spotting and exploiting deviations of pricesfrom their fundamental value, while portfolio investors have amore passive approach. Formally, we assume MT investors pay afixed cost �1 to be able to successfully measure Pt � PF

t , while LTinvestors pay �2 to correctly estimate the unconditional momentsof vector DPtþ1. Such a market segmentation is close to that ofGraham who differentiates an ‘‘active or enterprising approach toinvesting’’ from a ‘‘passive or defensive strategy that takes littletime or effort but requires an almost ascetic detachment fromthe alluring hullabaloo of the market’’ (Graham and Zweig, 2003,p. 101). One may also view LT investors as mutual funds or banks,while MT investors are more speculative agents following arbitragestrategies, such as hedges funds.

Long-term investors hold a diversified portfolio, withn 2j ½1;N� j the level of diversification. The level of diversificationthus impacts the system in two ways. First, the total shock onthe portfolio is likely to be lower when n is large, so that the wealthshocks should be lower. Second, a given shock will trigger sales onmany assets when n is large, so that LT investors and assetsbecome more correlated. Initially each long-term investor I isendowed with ‘‘his’’ asset i, i.e., investor 1 holds asset 1, so thatthey are N representative long term investors.

Note that in what follows we nearly only refer to constrainedselling by portfolio investors, because we are primarily interested

in fire sales. However the model applies equivalently to a situationin which portfolio investors are buyers, as their constraint getslooser following a positive shock on their wealth.5 Convergencetraders are segmented as in Merton (1987).

We introduce some notation: for a given investor I, qi;I repre-sents the actual quantity of asset i, q�i;I the desired one,qI ¼ Ri¼N

i¼1 qi;I is his total investment in all risky assets. qi ¼PI¼N

I¼1 qi;I

is the total quantity of asset i across investors.

2.2. Investors

Both mid-term and long-term investors have CARA utility, withrisk aversion of 1

smtand 1

sltrespectively. Risk aversion is similar

across investors of the same type, and each agent is a price-taker.

2.2.1. Mid-term investorsThey hold assets during tH periods, the time it takes for inves-

tor-driven deviations shocks to return to 0. They pay a fixed cost�1 to monitor those deviations at every t. They are free of regula-tion and have ‘‘deep pockets’’, so that they may hold exactly thequantities they desire: q�i;t ¼ qi;t . As they are segmented each inves-tor operates in one market only, so that we drop the vector nota-tion for the time being. Mathematically each I solves:

Max E �e�w

I;tþtHsmt

� �

u=c wI;tþtH ¼ wI;t þ q�i;I;tðpi;tþtH � pi;tÞ � tH�1

where wI;t is the wealth of investor I at time t. Using the momentgenerating function yields the well-known solution:

qmti;I;t ¼ smt

Eðpi;tþtH � pi;tÞEðr2

tþtH Þ

where r2tþtH is the variance of asset price i at horizon t þ tH.

When tH is large, all trading shocks, whether they have occurredor are expected to, will vanish through mean-reversion. Setting afairly large value for tH we thus have:

Fig. 4. Extreme bankruptcies odds, r/h = 0.6.

Eðpi;tþtH �pi;tÞ¼ EXj¼tþtH�1

¼ E utHþXj¼tþtH

j¼tþ1

MRi;jþXj¼tþtH

j¼tþ1

eFi;jþ

Xj¼tþtH

j¼tþ1

e�i;j

!¼ tHEðuÞþEðpF

i;t�pi;tÞ

As mentioned, convergence traders successfully monitor devia-tions from fundamentals, so that EðpF

i;t � pi;tÞ ¼ pFi;t � pi;t .

First differencing we obtain the demand/supply shock for agiven convergence trader between t and t + 1:

Dqmti;I;t ¼ smt

ðpFi;tþ1 � pi;tþ1Þ � ðpF

i;t � pi;tÞEðr2

tþtH Þ

Rearranging and setting to m the number of convergence trad-ers operating on each market, we obtain the vector of demand/sup-ply shifts from convergence traders between t and t + 1:

) DQ mtt ¼ �hðDPt � DPF

t Þ ð2Þ

where h ¼ msmtEðr2

tþtHÞ represents the total increase in the quantity

demanded in response to a rise in expected returns, and thus indi-cates the strength of the demand. If the distance between actual andfundamental prices remains constant between t and t + 1, thedemand by MT investors will be null. If DPt > DPF

t they will sell, ifDPt < DPF

t they will buy.Note also that starting from a situation in which actual prices

are equal to fundamental prices, there is no mean reversion, andportfolio investors are not constrained and thus do not supply/demand any assets. Market equilibrium then implies

Fig. 3. Distribution of number of bankrup

�hðDPt � DPFt Þ ¼ 0() DPt ¼ DPF

t . Therefore in a ‘‘business asusual’’ scenario in which the constraint of portfolio investors doesnot bind, actual prices should be equal to fundamental ones at anytime t.

2.2.2. Long-term investors2.2.2.1. Their maximization problem. Their time horizon is noted t}.Formally:

tcies in stable regime with r/h = 0.75.

Fig. 7. Desirability of diversification, r/h = 0.6.

MaxE �e�w

I;tþt}slt

� �u=c wI;tþt} ¼wI;tþQ �|t ðPtþt} �PtÞ� t}�2

EðPtþt} �PtÞ¼ E Ptþ t}uþXj¼tþt}

j¼tþ1

MRjþXj¼tþt}

j¼tþ1

eFtþ1þ

Xj¼tþt}

j¼tþ1

e�tþ1�Pt

¼ Eðt}uÞþkEXj¼tþt}

j¼tþ1

ðPFj �PjÞ

using Eq. (1) and the fact that EðeFtþ1Þ ¼ 0 . Since LT investors do not

pay the cost �1 and do not observe the deviations of prices fromtheir fundamental value, they can only estimate it with its long-term value EðPF

j � PjÞ ¼ 0. However as they incur the cost �2 theysuccessfully estimate the fundamental price drift u and covariance,noted Rt} . Formally this means EðPtþt} � PtÞ ¼ t}u and EðRt} Þ ¼ Rt} .Therefore:

Q �ltt ¼slt t}uRt}

Two key elements:

– Since the unconditional moments of Ptþt} � Pt are constant, thedesired quantities will be constant.

– As we shall verify later in equilibrium, the matrix Rt} is sym-metric. This implies that LT investors optimally want to holdan equal share of all assets, so that:

qi;I;t

qI;t¼ 1

nð3Þ

Note that optimally portfolio investors want to hold as manyassets as possible to maximize the benefits of diversification. Thelevel of diversification n is thus equal to the maximum numberof markets they have access to.

Fig. 10. Extreme bankruptcies odds, low panic.

2.2.2.2. Their Constraint. We assume a linear correspondencebetween the total amount of risky assets that an investor may holdat t + 1 and his value-at-risk at t.

qI;tþ1 6 rVaRI;t

where parameter r mitigates the strength of the selling in responseto a change in the VaR. In words, when the constraint binds, achange in the VaR at time t will lead investors to sell/buy risky posi-tions between t and t + 1 to comply with the constraint at t þ 1.Note that a more usual form would be PtQ t 6 rVaRI;t , i.e., the VaRleads investors to manage their monetary exposures rather thantheir quantities solely, we pick this form for technical reasons.6

Each investor’s VaR is given by:

VaRI;t ¼ EtðPtþt} Þ>Glt

I;t þ cstffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðGlt

I;tÞ>Rt}Glt

qwhere Glt

I;t is the vector summarizing the proportions of each assetin total quantity invested by investor I. Using (3) and the fact thatEðPtþt} Þ ¼ Pt þ t}u for LT investors, we have

EtðPtþt} Þ>Glt

I;t ¼ an

Pipi;I;t þ t}u

� �=n

� �, where

Pipi;I;t represents

the sum of the prices of assets that feature in I’s portfolio. First dif-ferencing, we obtain the selling of a given asset i by investor I thatoperates on his constraint:

) Dqlti;I;t ¼

Dpi;I;t�1 ð4Þ

The message conveyed by this equation is simple: when pricesincrease, investors are more wealthy and thus may invest more atthe following period, when prices fall they must lower their expo-sure to risky assets. As long as the constraint does not bind the sup-ply/demand by portfolio investors will be null, and prices willsimply track fundamental values. The rest of the paper studies asituation in which the constraint binds for all LT investors.

Note that VaR constraints as such usually concern banks, so thatone may argue the model applies primarily to markets dominatedby banks, such as ABS. Yet trust funds may also be subject to anexplicit constraint by their beneficiaries. What’s more, in practice

6 More on this choice in Section 2.4.

fire sales may also be driven by increased risk aversion, higheremphasis on the short run, etc. The constraint chosen is thus onlya tractable vehicle for such sales, but captures a wider phenome-non. Therefore we avoid referring to LT investors as ‘‘banks’’ only,and believe our results apply to markets in general.

2.3. Network formation and matrix form

2.3.1. NetworkThe previous section showed how each representative investor I

behaves. To study the implications of such behavior on each pricewe must specify a pattern of asset holdings. Fig. 1 summarizes thenetwork formation: each holding of asset i by investor J is a con-nection between J and the investor I who initially held the asset.Investors are nodes and asset holdings are the connectionsbetween them. The numbers on each link thus represent the assetsthat I and J have in common, indicating how closely related theyare.

As the degree of diversification n increases, each I acquires theasset that is the closest to his right. Then Ik holds only asset k ifn ¼ 1, assets k and k + 1 if n ¼ 2, etc. In general Ik holds assets kto k0, where k0 � kþ n� 1 [N].

The same information may be expressed in matrix form by let-ting each row represent an investor I and each column an asset i,and setting 1 if I holds i, 0 otherwise. For instance if N = 5 andn = 3:

1 1 1 0 00 1 1 1 00 0 1 1 11 0 0 1 11 1 0 0 1

0BBBBBB@

1CCCCCCA

I1 holds assets 1;2;3 so that a1;1 ¼ a1;2 ¼ a1;3 ¼ 1.This network has two important features:

–As each investor chooses to purchase the asset of his nearest‘‘neighbor’’ in the financial network, he exhibits some degree

Fig. 11. Extreme bankruptcies odds, moderate panic.

Fig. 12. Extreme bankruptcies odds, high panic.

of home bias. Home bias refers to the tendency of investors tofavor assets with the lowest distance to those they alreadyown. It is a well documented phenomenon in the literature oninternational diversification. ‘‘Distance’’ between two assets iand j may reflect informational asymmetries betweenthem, the transaction cost of acquiring one when possessingthe other, the fact the investors who hold i are not familiar withj, etc. Here we take no stance on what factor underlie ‘‘dis-

tance’’, seeing it as a combination of heuristic and rationalfactors.

– All investors have a similar pattern of asset holdings, so that the

Fig. 13. Desirability of diversification, high panic.

Fig. 14. Covariances as a function of n for all values of j k� j j.

network in which investor 1 holds assets 1,4,5, is circulant ifinvestor 2 holds 2,5,6, and so on. As we shall see in Section 4,circulant matrices have powerful properties, which will allowus to diagonalize them easily to solve the model and provideanalytical expressions for the covariance between asset i and j.

2.3.2. Matrix formFrom Section 2.2.2 we know that the proportion of asset i in

total quantity is 1/n if i features in the portfolio, and 0 otherwise.The average return on investor I ’s portfolio writes:

aI;iDpi;t=n

And the selling constraint (4) may be re-expressed as:

Dqi;I;t ¼r

aI;iDpi;t�1

i.e., the amount of each asset sold by investor I is a linear function ofthe product of each line I of A by the price vector.

Fig. 15. Covariances as a function of n for all values of j k� j j, panic.

Fig. 16. Covariances between j and k ¼ jþ 2 as a function of N, for all values of n.

Fig. 17. Covariances between j and k ¼ jþ 2 as a function of N for all values of n, panic.

The vector of general element Dqi;I;t summarizing the unit sell-ing by each long-term investor I as a response to a wealth shock,may then be expressed as:

Dqltt ¼

rn2 ADPt�1

What’s more, just as each row of A tells us what assets a giveninvestor holds, each row of A| indicates what investors hold a givenasset i. Thus we may express DQ lt

t , the vector whose general ele-ment Dqi;t is the total selling of asset i between t þ 1 and t, as:

DQ ltt ¼ A|Dqlt

) DQ ltt ¼

rn2 A|ADPt�1 ð5Þ

Let us now take the market clearing conditionDQ lt

t þ DQ mtt þ DQ mr

t ¼ 0, where the terms on the right-hand sideare the demand/supply shocks stemming from respectively portfo-lio investors, convergence traders, and the mean reverting exoge-nous supply of investors. Substituting:

Fig. 18. Covariances j and k as a function of j j� k j, for values of n.

Fig. 19. Home biased versus non-biased investor.

Fig. 20. Bankruptcies odds, comparison home biased/optimal networks.

�hðDPt � DPFt Þ þ DQ mr

n2 A|ADPt�1 ¼ 0

) DPt ¼ uþ DQ mrt

hþ r=h

� �A|ADPt�1 þ eF

tþ1 ð6Þ

The end product of the model is thus a stochastic recurrencesystem that maps how price shocks spread through time andassets. The asset price dynamics we fed into LT and MT investorsmaximization problems are verified in equilibrium.

Summary of the frame of the model:

– portfolio investor of type I holds assets i = I to i = I + n-1– when faced with a negative shock I sells an equal quantity of every

asset, where the amount sold depends on r, the weight of constrainton the investor

– convergence traders buy these assets with a discount that dependson h, the strength of demand

– the market clearing condition then yields DPt ¼ uþ DQ mrt

h þr=hn2

A|ADPt�1 þ eF

2.4. Vector of total change in investors wealth

As mentioned, this paper aims at studying the link between sys-temic risk and diversification by comparing the likelihood thatg 2j ½1;N� j investors go bankrupt. This requires that we get thefirst two moments of the vector of total change in investors wealthduring the crisis. In order to so while conveying maximum eco-nomic intuition and minimizing quantitative heavy lifting, wemake two simplifying assumptions.

First, we set the mean-reversion and fundamental drift to 0, forthree reasons: (a) our time unit is a ‘‘day’’ in the market, so thatthese fundamental long-term trends should be negligible anyway,(b) they would overload the equations as a recurrence system witha drift is slightly more demanding and (c) without changing theinsights of our model, since the variance structure would not bechanged by the inclusion of u, and very marginally by MR.

Second, we will only study how stochastic shocks on assets att = 0 spread through time. We thus set all shocks to 0 past t = 0for all securities, letting total losses depend on the realization ofthe stochastic vector DP0. This leads to the following form for assetprices:

DP0 ¼ eF1

DPt ¼r=hn2

� �A|ADPt�1 ð7Þ

) DPt ¼r=hn2

� �A|A

� �t

DeF1 ð8Þ

Fig. 21. Impact of increasing N on systemic risk.

Fig. 22. Bankruptcies odds, comparison home biased/optimal networks.

7 See Appendix B.

which notably implies EðDPtÞ ¼ r=hn2

h itEðeF

1Þ ¼ 0. Fig. 2 gives achronological description of how a vector of shock at t = 0 propagatesthrough the simplified system:

We label as ‘‘change in the wealth of investor i’’ the evolution ofthe average price of the assets in i’s portfolio throughout the per-iod. Since each row i of matrix A represents the holdings of investorI, this average price shift at period t is It ¼ 1

n ADPt, and the totalprice return over the period, on which we shall focus, is:

I ¼Xt¼þ1t¼0

DIt ¼1n

AXt¼þ1t¼0

ðDPtÞ ¼1n

AXt¼þ1t¼0

� �A|A

� �t

Its expected value is null, and its variance is given by:RI ¼ Eð½I� EðIÞ�½I� EðIÞ�|Þ ¼ EðI� I|Þ, which yields:

RI ¼r2

n2 AXt¼þ1t¼0

� �A|A

� �t" #

AXt¼þ1t¼0

� �A|A

� �t" #

since EðDeF1DeF|

1 Þ ¼ diagðr2F Þ as the fundamental shocks are inde-

pendent across assets. Let us note that:

– I is a linear combination of the initial normally independent dis-tributed shocks on assets, so it is normally distributed.

– We have implicitly assumed that the system is not explosive,i.e., that the eigenvalues of r=h

A|A are below one. We later

prove7 that this boils down to imposing the condition thatr < h. The increase in demand following a drop in prices mustbe superior to that of constrained supply, otherwise prices areever-falling.

2.5. Assumptions

At this point, it is appropriate to discuss 4 assumptions under-lying the model:

(1) Stock prices are normally distributed. The relevance of thispostulate has already been discussed extensively(Andersen et al., 2001): while admittedly a poor descriptionof reality, its merit is to be easy to manipulate, and also inour case to allow for the study of the multidimensional den-sity function. We have chosen normality over log-normality,as the logic of the model dictates we work with price evolu-tion rather than returns.Nonetheless, one of the most common criticism attached tonormal price returns is that they are incapable of capturingthe actual thickness of the tails. Our model shows that undera particular set of circumstances the system endogenouslygenerates a dependence structure between assets whichmake such ‘‘fat tails’’ appear.

(2) The model is symmetric. We assume that negative shocksbring sales but equivalently positive shocks bring purchases.Symmetry comes partly from normality, but it also carriesthe implicit assumption that constrained investors may not

Fig. 23. Desirability, optimal and wider network.

Fig. 24. Lower K and higher rF , both panic and non panic case.

escape the constrained region even when their wealthincreases, that is the desired quantity of assets is alwaysbelow that allowed by the constraint. In reality we mayexpect some investors not to buy assets in response to alooser constraint if they are happy with their holdings.This assumption is purely to simplify analysis. One couldinclude asymmetry in the model fairly easily, for instanceby specifying a proportion of investors who escape the con-straint binding region for each positive price shift. Nonethe-less, since we are only interested in what happens in the left-tail, a choice has been made not to burden the model withparameters that affect the right-tail of the distribution.

(3) Parameters are constant through time. So far, agents stick withtheir assessment of asset moments, and never question thisassessment. In particular, convergence traders alwaysprovide demand with the same discount h, regardless of

the current market conditions. In practice increased riskaversion, or higher emphasis on the short-term may hamperdemand during crises. We account for this in Section 3 byallowing risk aversion to rise in response to high sellingmovements.

(4) The constraint relates linearly quantities and VaR. As stated inSection 2.2.2, a more common form relates monetary hold-ings and VaR. This form also has the advantage of being ini-tial price fall.riven by theory as it is derived from therequirement that capital should exceed the VaR. However,as mentioned such a form does not allow to derive a marketequilibrium, and we are forced to relate solely quantity tothe VaR.

Though this choice clearly comes from necessity, it may bemore justifiable on normative grounds. In reality the link between

shocks on portfolios and selling decisions depends upon many fac-tors, and its exact form is hard to know, so that a linear relationshipmay not be ruled out. For instance looking at the charts providedby Adrian and Shin (2010), a linear form between leverage andasset returns during crisis appears to be an acceptable fit.

3. Multivariate distribution of investor wealth

Having established that total changes in investors wealth areinterdependent and follow a multivariate normal distribution, wenow:

– set a maximum loss before bankruptcy K for each investor– use normality to draw the odds that g 2j ½1;N� j investor fail– study how these odds change with n for different values of

parameters r2F , K, N and r=h

– weight each odd with a cost that is an exponential function ofthe number of failures to discuss the desirability of each n

Then redo these steps in a set-up where investors may panic(Section 3.2).

8 There has been empirical evidence on the presence of fire sales (see for instanceCoval and Stafford (2007)), but it is hard to use it for calibration as researchers canonly conjecture that a given sale has been made out of necessity.

3.1. Baseline model

3.1.1. Choice of parametersBesides n, results will depend on 4 parameters: the fundamen-

tal variance r2F , the maximum loss K that investors can incur before

bankruptcy, the number of assets N, and finally the conditions onthe markets r/h. These two first have a similar and straightforwardimpact in our model: a fundamentally riskier portfolio or a lowerdefault threshold both make each investor marginally riskier. Thusfor conciseness we keep parameters r2

F and K constant throughoutthe paper, relegating simulations with alternative values inAppendix A. For parameters N and r/h, we will present simulationsfor different values.

We choose our baseline parameter set to fit the state of thefinancial markets coming into the 2008 credit crunch. We normal-ize prices and initial quantities at t = 0 to 1.

To estimate the fundamental variance we compute the averagedaily variance of an asset belonging to the S&P 500, from the firstof July 1997 to the first of July 2007. This yields r2

F ¼ 0:000616.The S&P was chosen because it is designed to provide a broaderdescription of the investment opportunities than its counterparts,and equities are the asset class which suits our model the most. Withrespect to the period chosen, in our model noise is only fundamentalin BAU so that the window does not include the subprime crisis.Nevertheless we include the internet bubble, which we consideredas an increased volatility episode rather than a systemic event.

An investor goes bankrupt when his losses exceed his capital K.We use data of the World Bank to set K = 0.08, that is 8% of normal-ized assets. This value stands between the reported capital ratios ofTier 1 capital for US banks of 9.1% in 2008, and that of Europeanones, usually around 5%. We have somewhat arbitrarily chosen tobe on the high side, to reflect the higher weight of the US markets.

N represents the number of assets in our model but in essencedescribes more the number of asset classes. Anecdotal evidencefrom investment funds implies a number of assets between 5and 15. Furthermore, seeing N as the maximum diversificationlevel, we may follow a classic paper from Evans and Archer(1968), who estimate that diversification is no longer profitablepast 10 securities, a belief shared amongst practitioners. We thusset N = 10 for now, but will try rising it to 20 in Section 4, as inCont and Wagalath (2012).

With respect to market conditions r/h, we study 3 scenarios:

– a ‘‘mild’’ one in which the constrained agents are forced to sellassets in fairly small quantities when a shock hits, and thedemand by active investors for such assets is strong. r=h ¼ 0:6

– a ‘‘windy’’ scenario in which the quantities sold by constrainedagents and the demand by convergence traders are both moder-ate. r=h ¼ 0:75

– a ‘‘storm’’ scenario in which the pro-cyclical effect of the VaRconstraint is large, and demand response is weak. r=h ¼ 0:9

Studying 3 scenarios results partly from the lack of data on firesales by investors,8 but is mostly interesting as part of our analysis.

3.1.2. Distribution of total number of bankruptcyFig. 3 summarizes our approach of systemic risk in the interme-

diate scenario r=h ¼ 0:75. Each possible number of investor fail-ures, from 0 to N = 10, has a probability given by the multivariatecumulative normal distribution, and we study how this probabilityvaries with the level of diversification. The dual impact of diversi-fication appears clearly. As n rises investors become more andmore dependent, outcomes in which some investors fail but othersurvive become less likely. In the total diversification case, only the‘‘all survive’’ and the ‘‘all fail’’ outcomes are possible. Due to theindividually risk-reducing impact of diversification, the ‘‘all sur-vive’’ equilibrium increases faster in likelihood, yet we also observea gradual detachment from 0 of the likelihood that every investor fail,reaching a non-trivial 0.5% for n = 10. In this case the overall desir-ability is thus ambiguous, and will crucially depend on how painfulmass failure is to the entire economy.

Figs. 4–6 zoom in on number of failures large enough to consti-tute a systemic event, for all values of r/h.

In Fig. 4 we see that any number of failures above 6 has a nearzero likelihood. For low levels of diversification this results mainlyfrom the higher independence between investors, for high levels ofdiversification it comes from the fact that with r/h = 0.6 shocksquickly die out, so that the contagion externality is limited. InFig. 5 the scope for contagion is increased as we move tor=h ¼ 0:75. In the extreme r/h = 0.9 case, shocks transmit almostfully across investors, so that the probability of mass failureincreases drastically. In particular the odds attached to the ‘‘all-fail’’ outcome reaches up to 15% when n = N. In this case promotinga lower level of diversification amongst investors seems sensible.

3.1.3. WelfareWe weight the probability attached to a given number of fail-

ures against the cost attached to it.If the cost to society increased in a linear fashion with the num-

ber of failures, diversification would unambiguously be desirablefrom society’s perspective. Yet there are many theoretical reasonsfor which this cost may in fact grow exponentially with the num-bers of defaults. In particular, the well-identified channels for con-tagion may be enhanced by bankruptcies: we expect survivinginvestors to become much more risk averse, reputation risk tosky-rock, the liquidity constraint to tighten, etc. Fiordelisi andMarques-Ibanez (2013) for instance show empirically that bankfailures have an extra impact on asset prices. Bernanke (1983) alsopoints out that financial bankruptcies have a more than propor-tional effect on the real economy, through decreased money supplyand increased cost of financial intermediation.

Perhaps due to this high variety of channels and non-linearity, itis hard to estimate the exact cost to society of financial bankrupt-cies. Authors who want to model this cost have used differentmathematical artifices. For instance Ibragimov et al. (2011) define

9 Using the conditional variance decomposition formula:Rt ¼ ½Rt=8Dqi :j Dqi j< k�Þ þ E DPt=8Dqi :j Dqi j< k�ð Þð Þ2�Pð8Dqi :j Dqi j< k�Þ

þ ½ðRt=9Dqi :j Dqi j> k�Þ þ E DPt=9Dqi :j Dqi j> k�ð ÞÞ2�Pð8Dqi :j Dqi j> k�Þ� EðDPtÞ½ �2

where EðDPtÞ ¼ 0 and EðDPt=8Dqi : Dqi < k�Þ ¼ E DPt=9Dqi :j Dð qi j> k�ÞÞ2 ¼ 0, as thedistribution of DQ is symmetric.

10 These values may appear low but one should remember our time unit is a ‘‘day’’.

a time to recovery for the system, which depends of the number ofdefaults. We simply specify the following cost function for society:

CðgÞ ¼ ebg

where g, the number of failures, is a random variable, and b miti-gates the severity of the increase in the cost to society of an addi-tional failure. We show results for values of b 2�0;0:72�. We pickthis interval because it contains the interesting cases. b ¼ 0:72 rep-resents a very high exponentiality of mass failures, as it implies anaverage cost of a single bankruptcy that is 65 times larger wheng ¼ N than it is when g ¼ 1.

The expected cost to society reads:

EðCÞ ¼Xg¼N

PðgÞCðgÞ

where the probabilities Pðg ¼ iÞ were obtained in the previous sec-tion. The Figs. 7–9 show the log of this expected cost for our threevalues of r/h, across b and n. The dashed and solid lines track thelevel of diversification for which the expected cost to society for agiven ðr=h;bÞ couple is the lowest, or highest respectively.

Unsurprisingly, a higher b unambiguously works against higherlevels of diversification, in which mass failure is more likely.Graphically this is shown by the left turns of the dashed line plot-ting the optimal diversification level.

The impression from the previous section remains. In ther/h = 0.6 case the contagion externality is too modest for diversifi-cation not to be desirable. However with large value of b the opti-mal level of diversification goes surprisingly low, reaching n = 5. Inr/h = 0.75 the same logic applies, leading the optimal level to n = 3for a high b. In words this means the cost of an increase in thedependence between investors is still acceptable in exchange forthe huge private benefits of moving from n = 1 to n = 2 and n = 2to n = 3. The contagion externality is larger, leading n = N tobecome the least desirable level for b > 0:7. In the last r/h = 0.9 sit-uation, the scope for contagion is so high that no individual riskreduction justifies it past b ¼ 0:4. The perfectly diversified situa-tion becomes the worst possible situation rapidly, when b > 0:2.

b ¼ 0:7 implies that the ratio of unit of failure in the ‘‘all fail’’over that in the ‘‘one only’’ case is about 54, while b ¼ 0:4 impliesa ratio of 3.65. As mentioned this measure is based more on intu-ition than evidence, we leave it to the reader to assess where itstrue value would stand. In any case its seems fair to say that, ina situation in which financial shocks propagates linearly, there exista reasonable set of parameters in which any level of diversification inthe economy is dominated by a situation in which investors trade onlytheir own assets, and complete diversification is generally not the opti-mal level for society.

3.2. With possible panic

3.2.1. Change in frameworkRemember the demand response to a deviation from funda-

mentals by convergence traders is given by h ¼ msmtEðr2

tþt� Þ. We now

allow the strength of this demand response to change in the faceof extreme selling movement on the markets. In this set-up thiscould happen because MT investors review their estimation ofthe variance Eðr2

tþt� Þ in periods of higher volatility, or because theirrisk aversion 1=smt rises, as they begin to doubt their short-termability to absorb escalating losses. Somewhat arbitrarily we decideto attribute the rise in r/h to risk aversion when sales go beyond acertain threshold k�. The system now writes:

DPt ¼r=h�

n2 A|A� �t

eF1 if 9Dqi; j Dqi j> k�

DPt ¼r=hn2 A|A

� �t

eF1 if 8Dqi; j Dqi j6 k�

where h� is the discount prevailing in a situation of ‘‘panic’’. There-fore in this framework investors ‘‘panic’’ when constrained sales getpassed k� on a given market. It only takes one extreme movementon one market to trigger panic. Both experience from 2008 and the-ory may justify this form: increased risk aversion is highly conta-gious so that investors operating on the troubled market may leadothers to grow more risk averse, accrued counterpart uncertaintyis higher when one asset falls drastically than when all fall moder-ately, margin calls may be triggered when losses exceed a certainlevel, etc.

We note a the probability that one or more market reaches thesales threshold. It is given by the multivariate cumulative normaldistribution U since sales depend linearly on the price shocks att = 0, which are normally distributed. We thus have a no-panicprobability of 1� a ¼ probð8Dqi; j Dqi j6 k�Þ ¼ Uðk�; . . . ; k�Þ, whereU represents the cumulative normal multivariate distribution.The expected value and variance of the sales vector at t = 0 arerespectively 0 and RDQ ¼ r2

n2 A|A� �2.

Two things should be noted here: first this technique is consis-tent with our set-up since the sales are the largest at t = 0. Thismeans the threshold is either immediately or never reached inour framework, but in both cases r=h� of r=h remains constantafterward. In other words the initial sales shock ‘‘sets the tone’’for the rest of the crisis episode. Second, the expected value ofDPt remains zero regardless of the regime, so that the variancewith panic is given9 by Rt ¼ aRP

t þ ð1� aÞRNPt , where subscripts P

and NP refer to the ‘‘panic’’ and ‘‘no panic’’ cases respectively. Simi-larly the variances for investors over time is simply the weightedaverage of covariances matrices in the panic and no panic cases:

RI ¼ aRPI þ ð1� aÞRNP

I ð9Þ

3.2.2. Distribution of investors bankruptcyIn the last section we have seen r/h = 0.6 implies a very limited

scope for contagion, r/h = 0.9 a large one. These values are thus nat-ural candidates to estimate the no panic and panic case respec-tively. Regarding the panic threshold, similar to last section wetry three different values for the tendency for short-term investorsto panic:

– ‘‘easy panic’’: panic occurs, and thus r/h moves to 0.9, whensales on a given market exceed 0.005.

– ‘‘moderate panic’’: panic occurs when sales on a given marketexceed 0.01.

– ‘‘difficult panic’’: panic occurs when sales on a given marketexceed 0.05.

Normalizing r = 1, these values represent 0.5%, 1% and 5% of nor-malized total quantity of each asset.10 Figs. 10–12 summarize ourfindings.

In Fig. 10, the extreme selling required to trigger panic is veryunlikely to happen when n > 2, so that the figure resembles thenon-panic r/h = 0.6 case. Extreme failure, particularly the all failoutcome, is very unlikely. On the other hand, when n = 1 the pos-sibility of a panic exists, which may yield a considerable number

of failures, with non trivial odds of as much as 80% of investorsgoing under.

The intermediate and easy panic cases bring new light to ourresults. In the moderate panic case, for realistic values of b thelikelihood of the all-fail outcome is maximized at n = 4 in the inter-mediate case, n = 5 in the easy panic one. In other words, interme-diate levels of diversification which were an attractive option withoutpanic now seem particularly harmful, because such levels are not effi-cient enough in smoothing the wealth shocks faced by portfolio hold-ers, but provide linkages through which shocks may spread acrossassets. ‘‘Medium–low’’ levels also appear quite undesirable, withodds that g ¼ 9 and g ¼ 8 significantly above zero when n = 2 orn = 3.

Fig. 13 on desirability confirms this: in the intermediate paniccase and easy panic one, n = 4 and n = 5 become the least desirableoption for respectively b ¼ 0:42 and b ¼ 0:3, which implies a unitcost of failure in the all fail case over that in the one fail only of4.38 and 1.48.

Another interesting feature of the intermediate and easypanic cases is that the cost is a perfectly concave function ofthe diversification level for nearly all values of b. This leadsthe extreme outcomes ‘‘high diversification’’ and ‘‘no diversifica-tion’’ to become relatively more desirable, particularly the for-mer. Indeed in the intermediate case there are values of b,though very high ones, for which ‘‘no diversification’’ becomesthe preferred option, contrary to the no panic set-up. In bothcases very high levels of diversification are no longer the leastdesirable option even with a high exponentiality of mass failure,but remain the most desirable one with a low b. A high diversifi-cation level n = 9/10 is optimal if b < 0:36 in the easy panic caseand up tob < 0:7 in the intermediate one, implying a ratio of2.55 and 54, respectively.

On these intervals, diversification thus provide the first-best alloca-tion if it ‘‘goes all the way’’, but the worst one if it ‘‘stops halfwaythrough’’. This suggests that there is a critical threshold level of diver-sification, past which diversification is desirable, but below which nodiversification should be preferred. These conclusions complementthose of Caccioli et al. who find that when leverage is above a crit-ical value, diversification has non-monotonic impact on the odds ofa cascade of default, which they define as 5% or more of investorsfailing.

11 The expressions change slightly when N is even, for instance in the

case assets prices change per period: covðDpj;DpkÞt ¼r2

2Pq¼ðN=2�1Þ

q¼1 cos

4. Covariances and extensions

In this section we solve the model to derive an analyticalexpression for covariance at a given period Rt , across time Rtot ,and between investors across time RI , in order to see how covari-ances move with each parameter. This has interesting implicationsat the investor level, and allows us to relate the paper to the liter-ature on endogenous risk, and confront our form it to evidence. Thediscussion on covariances will also pave the way for a systemicimpact of changes in parameter N, and a change in the pattern ofasset holdings (4.2).

2pðk�jÞqN

1�cos 2pnq

Nð Þ1�cos 2pq

Nð Þ

� �2t

þ n4t þ cosðpqÞ 1�cos 2pnqNð Þ

1�cos 2pqNð Þ

� �2t!

12 A technical point: h ¼ msmtEðr2

tþtHÞ, where Eðr2

tþtH Þ is the assessment of the individualprice variance. If this assessment were correct, i.e., Eðr2

tþtH Þ ¼ r2tþtH , then deriving the

covariance r2tþtH would imply solving a fixed-point problem, as the expression for

r2tþtH would depend upon itself. However this fixed point problem is complex, and the

question of whether the equilibrium is rational or not is not the main goal of paper.We thus let Eðr2

tþtH Þ as a parameter whose value may be r2tþtH or not, noting only that

it enter positively in the expression for covðDpj;DpkÞt , that is when MT investorexpect a high volatility asset prices covariance is higher, as expected.

13 Parameter r2F rises all covariances uniformly, while a high h=r increases more the

covariances and correlations with more remote assets/investors, as it implies that agiven shock reaches points that are further away in the network.

4.1. Endogenous covariances

4.1.1. Analytical expressionDrawing the general element of Rt ; Rtot , and RI involves diago-

nalizing matrix A|A. This is made quite straightforward by the factthat this matrix is circulant. The demonstration is of mostly tech-nical value, so that we relegate it to Appendices B–E. We obtainthe following expressions for the covariance between k and j,which represent respectively asset price change per period, in total,and total investor wealth change:

cov Dpj;Dpk

� �t¼r2

Nr=hn2

� �2t

2Xq¼ðN�1Þ=2

cos2pðk� jÞq

� �1�cos 2pnq

� �1�cos 2pq

� � !2t

ð10Þ

cov Dpj;Dpk

� �tot¼r2

1� r=h

� �2

þ2Xq¼ðN�1Þ=2

cos2pðk� jÞq

� �1

1� r=hn2

1�cos 2pkn

Nð Þ1�cos 2pk

Nð Þ

� �0BB@

1CCAð11Þ

covðIk ; IjÞtot ¼r2

� 1n2

1� r=h

� �2

þ2Xq¼ðN�1Þ=2

cos2pðk� jÞq

� �1�cos 2pkn

� �1�cos 2pk

� �� 1� r=h

1�cos 2pkn

Nð Þ1�cos 2pk

Nð Þ

� �� 2

ð12Þ

in the case in which N is odd.11

These covariances are defined simultaneously and depend uponthe same factors: the conditions on the market12 r=h, the funda-mental risk r2

F , the distance k� j in the financial network, the numberof assets N, and of course the level of diversification n. As we will see innext section, the two expressions on which we focus, (11) and (12),will in fact move closely together.

The expressions above are in line with the literature on endog-enous risk: the actions of the market participants lead to correla-tions between assets that would not exist otherwise. Parametersr2

F and r=h in particular can be related to Danielsson et al. and Contet al., who provide expressions of the covariance matrix thatdepend on the fundamental covariance structure and some eco-nomic variable that governs to the transmissibility. We add the‘‘network’’ parameters n, N and ðk� jÞ.

What if we allow for panic? Since the covariance matrix is sim-ply the weighted average between that in the linear h/r = 0.6 caseand the h/r = 0.9 one, we have covðDpj;DpkÞ ¼ acovðDpj;DpkÞ

Pþð1� aÞcovðDpj;DpkÞ

NP for all three forms. The impact of each factorwill be split into how it impacts covariances covðDpj;DpkÞ in agiven regime and how it impacts the likelihood a of each regime.

4.1.2. CommentWe study the effect of n, N and ðk� jÞ. Impact studies of r=h and

r2F are available on request.13 For each factor we first discuss its

impact in the intermediate no-panic model, then that in the easypanic case which in our opinion provides a better description ofthe reality of crises. We keep the parameter set from Section 3.

4.1.2.1. Impact of n. As expected, diversifying amounts to transferringsome variability from the neighboring assets/investors to the more dis-tant ones. As shown by Fig. 14, as n rises all assets/investors

become more and more related and all covariances converge to asimilar value,14 as shown by Fig. 14.

Note that a direct connection between two assets/investorsdoes not necessarily enhances their covariance. For instance mov-ing from n = 2 to n = 3 lowers the covariance between j and j + 2when h=r ¼ 0:9. This reflects the fact that when h=r ¼ 0:9 andn = 2 the losses on troubled assets snowball to the point wherethe gain from spreading wealth shocks through diversificationexceeds the cost of a direct connection.

Fig. 15 describes the panic set-up. As you recall panic occurswhen fire sales get passed a given threshold. As n rises wealthshocks and thus fire sales become weaker, this threshold becomesharder to reach. The likelihood of the no-panic case thus rises, andaverage covariances fall to their h/r = 0.6 levels, in which transmis-sibility is lower. However the patterns of convergence across assetsremain the same. This figure is also consistent with our finding thatintermediate levels of diversification are particularly harmful, assuch levels provide a unique mix of homogeneity of the covariancesand significant likelihood of panic.

4.1.2.2. Impact of N. The impact of N, which we also consider as anindicator of the completeness of the markets, is contingent on thelevel of diversification. To see why let us picture the financial net-work as a circle and a shock stemming from a particular asset as awave running through it. When a wave has ran through all assets itreturns to the asset which originated the shock, which we label as‘‘second round’’ impact. In our home biased network the wavemoves slowly as it goes from neighbor to neighbor, and thus such‘‘second round’’ effects have little impact when n is low, even withh/r = 0.9. However when n rises and the speed at which the shockcompletes the circle increases, second round effects becomeimportant and losses converge to the same value across assets/investors.

In this case adding more assets is very desirable from a covarianceminimizing perspective as it divides the total fall across more asset/investors. When N ¼ þ1 , i.e., markets are effectively complete,the covariances tend to 0. Fig. 16 shows this by plotting the covari-ance between assets/investors j and k = j + 2 for n 2 j½1;10�j, in theh/r = 0.75 linear case.

In the panic case the impact of N becomes a priori ambiguous.On one hand as n rises fire sales are less likely as each investor issafer since the covariances between assets in his portfolio arelower. On the other more assets means that the likelihood of a sin-gle extreme movement increases. From this perspective Fig. 17,which plots the covariance between assets j and k = j + 2, is inter-esting. First and foremost the trend is still to a decrease. Thismeans the negative relationship between N and covariances ineach regime, similar to Fig. 16, dominates the impact of N on theprobability of panic. This relationship remains monotonic, so thatincreasing N stills unambiguously lowers the covariances.

Nevertheless, the impact of N on the probability of panicchanges the patterns of falls. Looking at the highest n, the functionbecomes concave on some segments of N 2j ½10;20� j. On such seg-ments the fall is then lower than its no panic-counterpart, implyingthat the marginal impact of N on the likelihood of panic is positive.It is logical that this occurs for high levels of diversification sincefor those levels the ‘‘anti-panic’’ impact of N is low, as investorsare already quite insured against shocks through diversification.When the likelihood of panic approaches 1 however, both effectson panic die out and the function retakes its ‘‘no-panic’’ evolution.Interestingly, the levels of diversification for which N lowers covari-

14 Note that compared to investors, each asset keeps a larger variance. This comesfrom the fact that at t = 0 its price changes are independent, while portfolios are not.

ances the most are the intermediate ones, which appeared particularlydetrimental from a systemic risk perspective.

4.1.2.3. Impact of ðk� jÞ. The role played by distance also dependson n, as when more and more investors/assets become connectedthe initial position in the network becomes irrelevant. Nonethelessdistance is highly explicative for diversification levels below orequal to N/2, as Fig. 18 shows in the h/r = 0.75. For instance inthe n = 4 case, asset covariance between direct neighbors is twicethat with most distant ones. The relative magnitudes are very sim-ilar for h/r = 0.6 or h/r = 0.9, so that the panic case gives the samepattern except for the fact that the average covariance falls witha similar to Fig. 15.

This impact of distance is in line with empirical evidence. Manypapers, such as Lane and Milesi-Ferretti (2004) have demonstratedempirically that informational distance is strongly negatively corre-lated with bilateral equity holdings. Gravity approaches to estimat-ing correlation usually provide good fits. On the link betweencovariances/correlations and distance during crises, the evidenceis not as definite. This may well be due to the fact that the real impactof a financial crisis is often higher in emerging countries, so thatfundamental factors dominate endogenous ones there. Surprisinglylittle work has been undertaken on correlations during the Subprimecrisis, but we may note that by Naoui et al. (2010) who suggest thatcorrelations with the US increased uniformly by about 80% in devel-oped countries during the crisis, while the rise is much variable foremerging ones. Anecdotal evidence that the crisis originated in theUS also suggests that distance is important in studying contagion.

The negative relationship between covariances and distancemay also be related to the so-called home bias puzzle: since moredistant assets have lower covariances, investors apparently forgodiversification benefits by failing to include them in their portfo-lios. In our model investors not only potentially miss out on lowerfundamental covariances, but effectively take on endogenous riskwhich is at least partly avoidable.

Fig. 19 shows the variance benefits of following a non biasedstrategy in our network with home bias. There are substantial privatebenefits from deviating from home biases in the early stages of diversi-fication. Such benefits are no longer significant around the N/2 level.At the investor level, accounting for endogenous risk thus increases theincentive to acquire costly information or pay higher fees.

4.2. Systemic risk impact of a change in the network

The impact of the completeness of markets and distance oncovariances raises the question of the systemic implications of awider network, i.e., with a larger N, and a non-biased network inwhich distance becomes irrelevant. For conciseness we only dis-cuss the easy panic case.

4.2.1. Optimal networkLet us imagine a network in which each investor picks the portfo-

lio that minimizes his endogenous risk rather than the assets closestto the one he holds. For instance if n = 3, an investor of type I wouldhold the asset i he is endowed with and pick the two assets that arethe least correlated with his, that is the assets that other investors oftype I hold the least. Through arbitrage the average quantity byinvestors of type I of all assets other than their own should thus con-verge. Applying this reasoning to all investors we obtain the assetcovariance matrix, whose elements outside the diagonal will all beequal, and lower than the elements on the diagonal for n < N.

Such a network thus brings more homogenous covariances butless risky portfolios, as we saw in previous section. Therefore mov-ing to an optimal network involves weighing up the same costs andbenefits that those of increasing the level of diversification: highercontagion costs versus individually sounder benefits.

Fig. 20 plots the difference between densities in the optimal net-work and those in the previous one. The higher individual resilienceis reflected in the fact that the likelihood of the ‘‘all-survive’’ out-come is higher in the optimal network for all levels of diversification.Yet the faster contagion cost is also visible: for low levels of diversi-fication we have a higher probability of total failure g ¼ N, thoughthe cost of this increase is largely compensated by a strong fall inthe odds of intermediate number of failures. For low levels of diver-sification, the desirability of moving to an optimal network shouldthus depend on how exponential the cost of financial failures is.For instance in the N = 10 and n = 3 case the ‘‘optimal’’ network ispreferable only when b < 0:32, which means a unit cost of failurethat is 1.78 times larger when g ¼ N than when g ¼ 1.

Yet the contagion cost weakens as n rises as a new effect startskicking in: an optimal network spreads sales across assets whichlowers the likelihood of panic. Therefore past n = 4, moving to anoptimal network appears unambiguously desirable. Interestingly,the levels of diversification for which the shift appear the mostdesirable are the intermediate ones, which were the most dangerouspreviously this easy panic case.

From a technical standpoint, note that these two networks areequivalent in Caccioli et al., but not in our set-up. Changing quali-tatively the network to account for different heuristics thus has asignificant impact on our results, as we shall verify in the followingsection.

4.2.2. Wider networkWe move from a N = 10 baseline network to a N = 20 one, as in

Cont and Wagalath.15 We choose16 to compare only similar absolutelevels of diversification, i.e., n 2 ½1;10� even for N = 20, and normalizefailures as the proportion of investors going under, i.e., g ¼ 10 withN = 10 is equivalent to g ¼ ð19;20Þ when N = 20. Importantly, ourchoice of absolute n and relative g yields the situation in whichthe desirability of N ¼ 20 over N ¼ 10 is minimized. The resultspresented are thus a minima.17

Fig. 21 plots the densities with N = 10 minus those with N = 20in the linear h/r = 0.75 case.

The likelihood of the extreme ‘‘all fail’’ outcome is higher whenN = 10. Again, the difference is most pronounced for the intermedi-ate diversification levels which were previously deemed particu-larly dangerous. The ‘‘all survive’’ outcome is also more likely.This results from the fact that increasing the number of assetinduces more independence across investors, thus reducing theodds of perfectly symmetric situations. In response, the intermedi-ate levels of failure become less likely when N = 10, with variouspatterns depending on n. When n is fairly low, positive but moder-ate proportions of investors going bankrupt bear most of theadjustment, which probably reflect the fact that each investor issafer when N = 20 due to lower covariance across assets. As n risesand this independence naturally falls, the difference becomes morehomogenous across all intermediate levels of failures.

In any case , the fact that the ‘‘all-fail’’outcome is significantlymore likely in the N = 10 case implies that N = 20 ought to be moredesirable, even for a relatively low exponentiality of the cost to societywith the number of failures. This finding is in contrast with that ofCaccioli et al. who find an ambiguous impact of rising the N/n ratio.

15 Greenwood et al. (2012) use a number of asset of 42, but in this paper thecomputational burden rises exponentially with N, and the simulations on N = 20 casealready took 10 days to complete.

16 Putting in perspective both cases requires choices. Should we compare the pointðn ¼ 10;g ¼ 10) when N = 10 to ðn ¼ 10;g ¼ 10) or to ðn ¼ 20;g ¼ 20) when N = 20?

17 One should also note that increasing N acts upon the covariance structure, butalso has a combinatorial side which may not be relevant here. The n = 1 shows thiscombinatorial effect, as with no diversification the problem simply amounts topicking g out of N investors. A reassuring sight is that this value of n appears quiteindependent of the others, which implies the impact of N goes primarily through thecovariance structure for n > 1.

4.2.3. Wider and betterFig. 22 shows the difference between an optimal and a biased

network when N = 20 in the linear h/r = 0.75 case, to be comparedwith Fig. 20 which gives the same information when N = 10.

Comparing this figure to Fig. 20 shows that the systemic riskreducing impact of increasing N is higher without home bias. In par-ticular we observe that the magnitude of the increase in the likeli-hood of the all-fail outcome for low levels of diversificationbecomes relatively smaller when N = 20, making the set of param-eters for which an optimal network may be undesirable evensmaller.Therefore a ‘‘wider’’ and a more ‘‘optimal’’ network are com-plementary. This is consistent with our finding in Section 4.1 thatincreasing N is particularly desirable when the financial shocksspread quickly through the system, which is the case in an optimalnetwork.

Fig. 23 looks at the desirability of the wider and better network inthe panic set-up, to be compared with the easy panic case in Fig. 13.

The contrast is evident: the no diversification case remains themost undesirable option for b 2�0;0:48�, which includes value of bfor which no diversification was the preferred option in the base-line scenario of Fig. 13. The ‘‘worst’’ levels of diversification whenb > 0:48 remain very low at n = 2/3. The figure also suggests thatintermediate levels, which were quick to become the least desirableoption, are now quick to become the best one. This finding is in accor-dance with the significant impact that rising N and changing thepattern of asset holdings have had on the likelihood of failures.

In the ‘‘baseline’’ easy panic case, systemic risk was maximizedaround n = 5, past which more diversification was desirable, but neverthe preferred option with b > 0:36. Here the maximum is reachedmuch earlier, and the first-best level of diversification is unambigu-ously beyond it. In other words when markets are complete and inves-tors are not biased, the window for which diversification may beharmful is much reduced, and the first best allocation is a reachablediversified one. Therefore, the threshold past which diversificationbecomes preferred to no diversification falls with the completenessof markets and the ‘‘efficiency’’ of asset-holdings.

5. Conclusion

This paper uses a new bottom-up approach to study systemicrisk and asset covariances, in line with the emerging endogenousrisk literature. We are able to provide a thorough analysis of theimpact of diversification on systemic risk, for any possible levelsand number of defaults. We find that in equilibrium, diversificationincreases the probability of mass failure but decreases that of allother non-zero failure outcomes. This leads to diversification tobe generally desirable during business as usual periods with lowtransmissibility of shocks, but not when transmissibility risesand the cost of mass failure rises is high.

However in this paper the link between diversification and sys-temic risk edges on a deeper question: is contagion driven primar-ily by human instincts or by hard-wired features of the financialmarkets? Indeed, as soon as we introduce heuristics which in ouropinion provide a better description of the reality of crisis, a newdesirable feature of diversification appears. Lower fluctuations toinvestors wealth brings lower selling movements, minimizing thepossibility of panic.

In this context the relationship between diversification and sys-temic risk is a concave function whose maximum is reached forintermediate levels of diversification, as such levels create connec-tions between investors without going far enough in minimizingindividual risk and the scope for panic. The optimal levels of diver-sification then appear to be either ‘‘high’’ or ‘‘none’’, depending onthe transmissibility of shocks and on how costly mass failure is,implying there exist a threshold level past which diversification

18 We derive P in Appendix C.19 An example of matrix V when N = 4 is provided in Appendix D.

becomes desirable, but below which no diversification should bepreferred.

Interestingly this threshold seem to fall with the efficiency offinancial markets. When markets are more complete and investorsare not biased, the systemic risk maximizing level moves fromintermediate to low, even in our most pessimistic scenario of ‘‘easypanics’’. The point at which diversification becomes ‘‘worth it’’ isreached quickly, so that intermediate levels of diversification actu-ally become the most desirable option.

Let us look at the subprime crisis under this light. Credit backedassets were in fact much more closely fundamentally related thanexpected by the banks which held them. As these correlations werehigh, banks were in essence holding a single credit backed asset,their actual diversification level was not high. This led the wealthshocks stemming from adverse movements on ABS to be large,which in turn triggered panic through increased counterpart riskand rising risk aversion. An aggravating factor may have been thatinternational investors tend to be biased towards US securities, acountry with low informational distance to any other. Accordingto our set-up a lower level of diversification would have been pref-erable, for risk would not have spread, and a higher diversificationwould have the first-best allocation, as banks would have beenable to digest the losses from ABS markets without triggeringpanic, while being safer during BAU periods.

Two policy implications may be drawn from this discussion. Thefirst one is quite general: if one agrees with the Financial StabilityBoard (2009) that the most important factor to prevent a systemiccrisis is maintaining confidence, then the failure of any institutionconstitutes a systemic event, so that favoring an approach based onmicro soundness may be still be the best option from a policy per-spective. Second, in this paper a key to enhancing systemic stabil-ity is diversifying both more and better. Some of the progresses inthis direction are bound to come from investors themselves, whoshould start accounting more for endogenous risk of their portfolio,but policy makers may also encourage this trend. In particular thepromotion of international diversification though the lowering ofinformation costs and/or taxes appears useful, as it permits boththe spreading and the minimization of endogenous risk acrossassets.

Appendix A

Fig. 24 shows the impact of lowering the amount of rising thedaily fundamental standard deviation rF by 25% and capital/threshold for default K by 25% also, which implies moving to astandard deviation of 0.0310 and a capital ratio of 6%. We representthe intermediate linear case r/h = 0.75 and the easy panic one.

As mentioned in the paper both actions have a comparableimpact on systemic risk as they rise the marginal risk attached toeach investor. With or without panic, all significant levels of failureare more likely, but the strongest relative increase is on mass fail-ure, which rises sharply with n. Note that though we do not pro-vide the figures, in both cases this relative rise in odds isinversely related to the transmissibility r/h. For K, this is becauser/h rises the total variance of portfolio, thus making the increasein the rise in the threshold for default relatively small. For rF port-folio are 25% more risk regardless of variance, the evolution in oddsthus reflects the difference in the marginal impact of rising thestandard deviation on densities.

Appendix B

Drawing the general element of Rt ; Rtot , and RI involves

diagonalizing matrices ðA|AÞt ;Pt¼þ1

t¼0r=hn2

h it, and 1

n APt¼þ1

h it. Our method is to work our way up from a simple

circulant matrix Z. This appendix gives the outline of the proof,while Appendices C–E provides details on certain steps.

B.1. From the cyclic permutation matrix Z to A

Z is the ‘‘cyclic permutation matrix’’, whose element zi;j ¼ 1 ifi � j� 1 [N], zi;j ¼ 0 otherwise. Taking Z to the power n shifts theone-diagonal n � 1 spots to the right. For instance if N = 5:

0 1 0 0 00 0 1 0 00 0 0 1 00 0 0 0 11 0 0 0 0

0BBBBBB@

1CCCCCCA

0 0 1 0 00 0 0 1 00 0 0 0 11 0 0 0 00 1 0 0 0

0BBBBBB@

1CCCCCCA

The eigenvalues of Z are readily obtained from the observationthat ZN ¼ 1N , where 1N is the identity matrix. Using the Cayley-Hamilton theorem, the characteristic polynomial of Z is of the formQ ½x� ¼ xn � 1 , and its eigenvalues are the nth-roots of unity. Thusthe matrix similar to Z is Dz ¼ diagðx0;x; . . . ;xN�1Þ where xk ¼e

2ipkN ¼ cos 2pk

� �þ i sinð2pk

N Þ and x�k ¼ cos 2pkN

� �� i sin 2pk

� �according

to Euler’s identity. Note also that since xN ¼ 1, we havexaðN�kÞ ¼ xaN�ak ¼ x�ak.

From the properties of Z, it appears that any circulant matrix Cmay be expressed as a polynomial in Z: C ¼ RðZÞ. This impliesC ¼ RðPDZP�1Þ ¼ PRðDZÞP�1, where P is the change of basis matrix.Therefore:

– wk, the kth eigenvalue of any circulant matrix C is a polynomialin xk, the kth eigenvalue of Z.

– P is the change of basis matrix for all circulant matrices.

In particular for A, the circulant matrix which represents ournetwork:

A ¼Xs¼n�1

Z ¼ PXs¼n�1

!P�1 ð13Þ

where n is the level of diversification. The matrix similar to A is thusgiven by DA ¼

Ps¼n�1s¼0 Ds

Z . Its general element is the inverse Fouriertransform wk ¼ RðxkÞ ¼

Ps¼n�1s¼0 xks. wk is the sum a geometric series

of n terms and common ratio e2ipk

N , so that:

wk ¼1�xkn

1�xk¼

1� cos 2pknN

� �� i sin 2pkn

� �1� cos 2pk

� �� i sin 2pk

� �except when k = 0, in which case w0 ¼ n. Note that this expressionalso implies w�k ¼ wN�k.

B.2. From A to ðA|AÞt

To move from matrix A to A|A, we use some of the numeroususeful properties of the change of basis matrix18 P:

(1) P is unitary: i.e., P�1 ¼ Py , where Py is P’s conjugate trans-pose of general term x�jkffiffiffi

(2) P ¼ P| and P�1 ¼ ðP�1Þ|, so that A| ¼ P�1DAP(3) P2 ¼ P�2 ¼ V where V is a near reversal matrix.19

We may then write A|A ¼ P�1DAPPDAP�1 ¼ P�1DAVDAP�1.Pre-multiplying by PP�1 we get the desired result:

A|A ¼ PðVDAÞ2P�1 ð14Þ

which implies ðA|AÞt ¼ PðVDAÞ2tP�1. The matrix similar to ðA|AÞt is

thus ðVDAÞ2t . The covariance between assets price changes at a given

period is then Rt ¼ Eð½DPt � EðDPtÞ�½DPt � EðDPtÞ�|Þ ¼ r2F

PðVDAÞ4tP�1, where we used the fact that EðDeF1DeF|

1 Þ ¼ diagðr2F Þ

and that ðA|AÞt is symmetric, so that ðPðVDAÞ2tP�1Þ|

¼ PðVDAÞ2tP�1.The expression of matrix V implies that ðVDAÞ2’s element on row

kþ 1 > 1 may be expressed,20 if k > 0, as /k ¼ /N�k ¼ wkw�k ¼1�xkn

1�xk � 1�x�kn

1�x�k , which will be positive for any k. Using (13) andrearranging:

/k ¼ /N�k ¼1� cos 2pkn

� �1� cos 2pk

� �and /0 ¼ n2 if k = 0. Note that this implies /0 ¼ n2 is the maximumeigenvalue of A|A, we thus verify the convergence condition r < hgiven in Section 3.

Summary of the diagonalization of A|A:

– matrix Z’s eigenvalues are the roots of unity– matrix A is a polynomial of Z, so its eigenvalues are a polynomial of

Z’s eigenvalues– all circulant matrices have the change of basis matrix P where

PP ¼ V , V a near reversal matrix– this implies A|A ¼ P�1DAVDAP�1 ) A|A ¼ PðVDAÞ2P�1

B.3. From A|A toPt¼þ1

t¼0r=hn2

Xt¼þ1t¼0

� �A|A

� �t

¼ PXt¼þ1t¼0

� �t

ðVDAÞ2t

!P�1 ð15Þ

Pt¼þ1t¼0

tðVDAÞ2t

� �, which we note Dtot , is then the matrix

similar toPt¼þ1

t¼0r=hn2

h it. The vector of total price movement

after the shock is thusPt¼þ1

t¼0 DPt ¼ PDtotP�1eF

1, and its covariance

matrix is Rtot ¼ r2F PðDtotÞ2P�1, using the fact thatPt¼þ1

t¼0r=hn2

h itis a linear function of symmetric matrices and

is thus symmetric itself.

The general element of Dtot is thus nk ¼ Rt¼þ1t¼0

r=hn2 /k

t, i.e., the

sum of a geometric series of common ratio r=hn2 /k

. As we consider

total price shifts from t = 0 to t ¼ þ1, the formula for the sum ofthe terms of geometric series yields:

nk ¼1

1� r=hn2 /k

1� r=hn2

1�cos 2pknNð Þ

1�cos 2pkNð Þ

� �

with n0 ¼ 11�r=h since /0 ¼ n2 . nk ¼ nN�k since /k ¼ /N�k.

B.4. FromPt¼þ1

t¼0r=hn2

h itto 1

n APt¼þ1

t¼0r=hn2

Having diagonalized A and A|A, we use (13) and (15) to write:

AXt¼þ1t¼0

� �A|A

� �t

PðDAÞP�1PðDtotÞP�1

PðDADtotÞP�1 ð16Þ

20 See Appendix D for a derivation through an N = 4 example.

DADtot is thus the matrix similar to APt¼þ1

t¼0r=hn2

h it. The

covariance matrix of the vector of total wealth change can thus

be re-expressed as RI ¼r2

Fn2 ½PDADtotP

�1�½PDADtotP�1�| ¼ PDADtotVDA

DtotP. Post multiplying by PP�1 we obtain RI ¼r2

Fn2 P ðDADtotVÞ2P�1.

The general element of ðDADtotVÞ2 is .k ¼ wknkw�kn�k ¼ /kn2k ,

that is:

.k ¼1� cos 2pkn

� �1� cos 2pk

� �� 1� r=h

1�cos 2pknNð Þ

1�cos 2pkNð Þ

� �� 2

We now have derived the eigenvectors of and matrices similar

to ðA|AÞt ;Pt¼þ1

t¼0r=hn2

h it, and 1

n APt¼þ1

t¼0r=hn2

h it, so that we

may draw the general element of the covariance between assetsprice changes at a given period Rt , the covariance matrix of totalprice movement Rtot , and the covariance matrix of the vector oftotal wealth change RI . Appendix E goes through the algebra.

Appendix C

The change of basis matrix P is the same for all circulant matri-ces. We thus look for the change of basis matrix for Z.

ZP¼ PDZ()

0 1 . . . 0 00 0 1 . . . 00 0 0 . . . 00 0 0 0 11 0 0 0 0

0BBBB@

1CCCCA

p1;1 . . . p1;N. . . . . . . . .

pN;1 pN;N

¼p1;1 . . . p1;N. . . . . . . . .

pN;1 pN;N

x0 0 . . . 00 x1 . . . 00 0 . . . 0 00 0 0 xN�2 00 0 0 0 xN�1

0BBBB@

1CCCCA()

p2;1 p2;2 . . . p2;Np3;1 p3;2 p3;N. . . . . . . . .

pN;1 pN;Np1;1 p1;N

0BBBB@

1CCCCA

p1;1x0 p1;2x1 . . . p1;N�1xN�2 p1;NxN�1

p2;1x0 p2;2x1 :: p2;NxN�1

. . . . . .p2N;1x0 pN;2x1 pN;NxN�1

this yields p2;1 ¼ p1;1x0; p3;1 ¼ p2;1x0 ) p3;1 ¼ p1;1ðx0Þ2; p4;1 ¼p3;1x0 ) p4;1 ¼ p1;1ðx0Þ3p2;2 ¼ p1;2x; p3;2 ¼ p2;2x) p3;2 ¼ p1;1ðxÞ

p4;2 ¼ p3;2x) p4;2 ¼ p1;1ðxÞ3p2;N ¼ p1;NxN�1; p3;N ¼ p2;NxN�1) p3;N ¼

p1;NðxN�1Þ2; p4;N ¼ p3;NxN�1) p4;N ¼ p1;NðxN�1Þ3:And so on, the general form is thus: piþ1;j ¼ ðxj�1Þip1;j. Normal-

izing by 1ffiffiffiNp we obtain an orthonormal base for the eigenvectors of

the discrete inverse Fourier transform matrix of coefficient 1ffiffiffiNp ,

whose general term is piþ1;jþ1 ¼ ðxj�1ÞiffiffiffiNp , with ði; jÞ 2 0;1; . . . ;N � 1f g2:

P ¼ 1ffiffiffiffiNp

1 1 : : 1

1 x : : xN�1

1 xN�1 xðN�1Þ2

0BBBBBBB@

1CCCCCCCA

Appendix D

Case in point with N = 4

VDA ¼

1 0 0 00 0 0 10 0 1 00 1 0 0

w0 0 0 00 w1 0 00 0 w2 00 0 0 w3

1CCCA¼

w0 0 0 00 0 0 w3

0 0 w2 00 w1 0 0

where wk ¼Ps¼n�1

s¼0 xks ¼ 1�xkn

1�xk

DA|A ¼ ðVDAÞ2 ¼

w0 0 0 00 0 0 w30 0 w2 00 w1 0 0

ð/0Þ 0 0 00 ð/1Þ 0 00 0 ð/2Þ 00 0 0 ð/1Þ

where /0 ¼ ðw0Þ2; /1 ¼ /3 ¼ w3w1; /2 ¼ ðw2Þ

Since w�k ¼ wN�k, w3 ¼ w�1; w2 ¼ w�2.

Thus we verify that for k 2 ½0;3�; /k ¼ wkw�k ¼ 1�xkn

1�xk � 1�x�kn

1�x�k ,

which yields /k ¼1�cos 2pkn

Nð Þ1�cos 2pk

Nð Þ.

Appendix E

Here we focus on the general element of asset per period covari-ance. The method is the same for total asset and investors one, onlyð/kÞ

t is replaced respectively by nk and nkwk. From the expression of

ðVDAÞ2 in Appendix D, and that of P in Appendix C, we may write

2tPðVDAÞ4tP�1 as:

Rt ¼r2F

� �2t

1 1 1 1

1 x x2 x3

1 x2 x4 x6

1 x3 x6 x9

0BBBB@

1CCCCAð/0Þ

2t 0 0 0

0 ð/1Þ2t 0 0

0 0 ð/2Þ2t 0

0 0 0 ð/1Þ2t

0BBBBB@

1CCCCCA

1 1 1 1

1 x�1 x�2 x�3

1 x�2 x�4 x�6

1 x�3 x�6 x�9

0BBBB@

1CCCCA

Rt ¼ r2F

� �2t

ð/0Þ2t ð/1Þ

2t ð/2Þ2t ð/1Þ

ð/0Þ2t xð/1Þ

2t x2ð/2Þ2t x3ð/1Þ

ð/0Þ2t x2ð/1Þ

2t ð/2Þ2t x2ð/1Þ

ð/0Þ2t x3ð/1Þ

2t x2ð/2Þ2t xð/1Þ

0BBBBB@

1CCCCCA

1 1 1 1

1 x3 x2 x

1 x2 1 x2

1 x x2 x3

0BBBB@

1CCCCA ¼ r2

� �2t

ð0Þ ð1Þ ð2Þ ð1Þð1Þ ð0Þ ð1Þ ð2Þð2Þ ð1Þ ð0Þ ð1Þð1Þ ð2Þ ð1Þ ð0Þ

0BBBB@

1CCCCA

where we used the fact that xk ¼ xN�k. Using Euler’s identity

xðk�jÞðN�qÞ/N�k þxðk�jÞq/k ¼ 2cos 2ipðk�jÞqN

ð0Þ ¼ ð/0Þ2t þ 2ð/1Þ

2t þ ð/2Þ2t

¼ ðn2Þ2t þ 21� cos 2pn

� �1� cos 2p

� � !2t

þ1� cos 2�2pn

� �1� cos 2�2p

� � !2t

ð1Þ¼ ð/0Þ2tþðx3þx1Þð/1Þ

2tþx2ð/2Þ2t ¼ðn2Þ2tþ cos

� �þ isin

� ��

þcos2p4

� �� isin

� ��1� cos 2pn

� �1� cos 2p

� � !2t

þ½cosðpÞþ isinðpÞ� 1� cosðpnÞ1�cosðpÞ

� �2t

¼ðn2Þ2tþ2cos2p4

� �1� cosð2pn

4 Þ1� cos 2p

� � !2t

� 1� cosðpnÞ2

� �2t

ð2Þ ¼ ð/0Þ2t þ ðx2 þx2Þð/1Þ

2t þ ð/2Þ2t

¼ ð/0Þ2t þ ðx2 þx�2Þð/1Þ

2t þ ð/2Þ2t

¼ ðn2Þ2t þ 2cos2 � 2p

� �1� cos 2pn

� �1� cos 2p

� � !2t

þ ½cosð2pÞ� 1� cosðpnÞ2

� �2t

The general expression for the covariance between k and j givenin the paper (for N even) is verified:

covðDpj;DpkÞt ¼r2

Nr=hn2

� �2t

ðn4t þ 2Xq¼N=2�1

cos2 � ðk� jÞqp

� �"

1� cos 2pnqN

� �1� cos 2p

� � !2t

þ cosð2ðk� jÞpÞ 1� cosðpnÞ2

� �2t35

References

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diversification and systemic risk

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