Market Efficient Portfolios in a Systemic Economy
aa r X i v : . [ q -f i n . R M ] M a r Market Efficient Portfolios in a Systemic Economy
Kerstin Awiszus a Agostino Capponi b Stefan Weber a March 19, 2020
Abstract
We study the ex-ante maximization of market efficiency, defined in terms of minimum devia-tion of market prices from fundamental values, from a centralized planner’s perspective. Pricesare pressured from exogenous trading actions of leverage targeting banks, which rebalance theirportfolios in response to asset shocks. We develop an explicit expression for the matrix of assetholdings which minimizes market inefficiency, and characterize it in terms of two key sufficientstatistics: the banks’ systemic significance and the statistical moments of asset shocks. Ouranalysis shows that higher homogeneity in banks’ systemic significance requires banks’ portfolioholdings to be further away from a full diversification strategy to reduce inefficiencies.
Keywords:
Systemic economy; systemic significance; price pressure; leverage targeting; marketefficiency.
Forced asset sales and purchases have been widely observed in financial markets. The most popularform of forced trading is that of fire sales, and has been extensively implemented by hedge funds andbroker dealers during the global 2007-2009 financial crisis; see Brunnermeier and Pedersen (2008)and Khandani and Lo (2011) for empirical evidence. a Institut f¨ur Mathematische Stochastik & House of Insurance, Leibniz Universit¨at Hannover, Hannover. email: [email protected], [email protected] . b Department of Industrial Engineering & Operations Research, Columbia University, New York. email: [email protected] .
1n asset is sold at a depressed price by a seller who faces financial constraints that becomebinding, i.e., when the seller becomes unable to pay his own creditors without liquidating the asset.For example, members of a clearinghouse need to post additional collateral if the value of theirportfolios drops by a significant amount (Pirrong (2011)). Similarly, a mutual fund may need toliquidate assets at discounted prices if it faces heavy redemption requests from its investors, anddoes not have enough cash reserves at disposal (Chen et al. (2010)). Banks manage their leveragebased on internal value at risk models (Adrian and Shin (2014); Greenlaw et al. (2008)), and mayneed to liquidate (or purchase) assets if negative (or positive) shocks hit their balance sheets.Forced purchases, despite less emphasized, are also important in financial markets. For instance,empirical evidence (see Coval and Stafford (2007)) suggests that equity mutual funds substantiallyincrease their existing positions if they experience large inflows, thus creating upward pressure inthe price of stocks held by these funds. Such inflow-driven purchases produce trading opportunitiesfor outsiders, who would be able to sell their assets and earn a significant premium.Asset purchases and sales triggered by financial constraints push asset prices away from fun-damental values (Shleifer and Vishny (1992)), a form of inefficiency that we analyze in a systemiceconomy. Typically, when a firm must sell assets to fulfill a financial constraint, the potentialbuyers with the highest valuation for the asset are other firms belonging to the same industry orinvestors with appropriate expertise. Those firms are likely to be in a similar financial situation,and thus unable to supply liquidity. The buyers of these assets are then outsiders, who value theseassets less. A symmetric argument holds if the firm executes inflow-driven purchases.When a firm impacts asset prices through its trading actions, other market participants whohappen to hold the same assets on their balance sheets are also affected, and may in turn violatetheir financial constraints, making it necessary for them to take trading actions. Through thisprocess, the trading risk becomes systemic , i.e, it imposes cascading effects on asset prices andrecursively impacts the equity of market participants through common asset ownership.We consider an economy consisting of leveraged institutions (henceforth, called banks ) thattrack a fixed leverage ratio. Empirically, this behavior has been well documented for commercialbanks in the United States, see, e.g., Adrian and Shin (2010). After a shock hits an asset class,prices change and so does the bank’s leverage ratio. To fulfill the financial constraint of targeting itsleverage, the bank must then liquidate or purchase assets, depending on whether the experienced2hock was positive or negative. Banks trade assets with other non-banking institutions that wemodel collectively as a representative nonbanking sector, assumed to have a downward slopingdemand function as in Capponi and Larsson (2015). The equilibrium price of the asset is uniquelypinned down by the point at which the demand of the banking and nonbanking sector intersect.We study market efficiency, measured by the mean squared deviation of fundamental capitaliza-tion, where all banks’ portfolios are valued at the fundamental values, from market capitalizationwhere banks’ portfolios are valued at the market prices. The latter prices internalize the pressureimposed by trading activities, as banks leverage or deleverage in response to exogenous shocks toasset values. Clearly, the closer prices are to their fundamental values, the more efficient the marketis. We consider the problem of maximizing efficiency from a centralized planner’s perspective. Thisobjective can be micro-founded in terms of increasing the information content of prices, and as aresult, better guiding the resource allocation in the economy. Brunnermeier et al. (2018) providea simple economic setting, in which maximizing social welfare is consistent with the objectives ofminimizing the deviation of asset prices from fundamentals, and of reducing asset market volatility. We develop an explicit characterization of the distribution of banks’ holdings that ex-antemaximize market efficiency. We refer to those as the f-efficient holdings . The key insight resultingfrom our approach is the identification of a sufficient statistic, the systemic significance vector,which captures the contribution of each bank to increased price pressures.The systemic significance depends on the banks’ target leverage, the banks’ trading strategies,and the illiquidity characteristics of the assets. We adopt an “aggregate first then allocate” pro-cedure to construct a solution to the quadratic minimization problem yielding f-efficient holdings.First, using the statistics of asset shocks we construct a vector of auxiliary weighted holdings. Then,we distribute the holdings to the banks based on their contributions to market inefficiency, whichis directly proportional to their systemic significance.We show that portfolio diversification is f-efficient if asset price shocks are homogeneous andbanks are heterogeneous in terms of their systemic significance. Our analysis suggests that as the The important role of prices in aggregating information that is dispersed in the economy has also been con-sidered in the early work of Hayek (1945). Prices thereby facilitate the efficient allocation of scarce resources.In the context of secondary markets, this issue is, e.g., discussed in Leland (1992), Dow and Gorton (1997),Subrahmanyam and Titman (2001), Dow and Rahi (2003), and Goldstein and Guembel (2008). The terminology f-efficient is used to emphasize that the notion of efficiency we consider is related to fundamentals.
Literature Review
There is, by now, a large number of studies that have analyzed the systemicimplications of leverage and risk taking in an economy of leveraged institutions. Existing literaturehas identified two main channels through which banks are interlinked. The first channel is throughthe liability side of the balance sheet. Banks have claims on their debtors, and once they arehit by shocks, they may become unable to honor their liabilities, potentially triggering negativefeedback loops from reduced payments through the system. Seminal contributions in this directioninclude Eisenberg and Noe (2001), which provides an algorithm to measure contagion triggeredby sequential defaults in the network of obligations, and Acemoglu et al. (2015), who analyze thestability of various network structures and their resilience to shocks of different sizes. The second channel is through the asset side of the balance sheet, as banks are interlinkedthrough common portfolio holdings. Financial contagion arises when banks take hits on theirbalance sheets, typically because the price of their assets is subject to pressure due to forcedpurchases or sales (see also the discussion in the introduction). Our paper contributes to thisstream of literature. Our study is related to the study of Greenwood et al. (2015), who calibratea model of fire-sale spillovers, assuming an economy of leverage targeting banks. Their work hasbeen extended by Capponi and Larsson (2015), who consider the higher order effects of fire-salesexternalities in a similar leverage targeting model. Duarte and Eisenbach (2019) construct andempirically valuate a measure of systemic risk generated by fire-sales externalities. The maincomponents of their measure, namely banks’ sizes, leverages, and illiquidity concentration, alsoconstitute the primary determinants of the systemic significance vector in our model. Other workshave considered models where contagion happens both through the asset and liability side of the Other related works include Elliott et al. (2014) and Gai and Kapadia (2010). Glasserman and Young(2015), Capponi et al. (2016), and Rogers and Veraart (2013) account for the impact of bankruptcy costsat defaults in a counterparty network model of financial contagion. Measures of systemic downside riskare analyzed in the works by Chen, Iyengar, and Moallemi (2013), Feinstein, Rudloff, and Weber (2017), andBiagini, Fouque, Frittelli, and Meyer-Brandis (2019). Finally, our work is related to a branch of literature that has analyzed the stability of port-folio allocations, diversification, and heavy tail risks of portfolios. Using a generalized branchingprocess approach, Caccioli, Shrestha, Moore, and Farmer (2014) identify a critical threshold forleverage which separates stable from unstable portfolio allocations (see also Raffestin (2014)). Ina risk-sharing context, Ibragimov et al. (2011) analyze the tradeoff of diversity and diversificationfor heavy-tailed risk portfolio distributions. They show that the incentives of intermediaries (fordiversification) and society (for diversity) are not necessarily aligned. Their findings are confirmed Portfolio similarity has also been considered in other industry sectors than banking. In the insurance industry,Girardi et al. (2019) find a strong positive relationship between the portfolio similarity of pair of insurance companies,and their quarterly common sales during the following year.
5y results of Beale, Rand, Battey, Croxson, May, and Nowak (2011), who analyze the individuallyand systemically optimal allocations in a simplified loss model consisting of a small number ofbanks and assets.
Outline
The paper is organized as follows. Section 2 summarizes the asset price contagion model.Our main contributions start from Section 3, where we define the quantitative measure of marketefficiency, and identify key drivers of this measure such as the vector of banks’ systemic significance.In Section 4, we characterize f-efficient allocations. Section 5 provides case studies for a calibratedversion of our model, which highlight the trade-off between diversification and diversity. Section 6concludes. All proofs are delegated to Appendix A.
To begin with, we introduce a few notations and definitions used throughout the paper. For two(column) vectors u = ( u , . . . , u n ) ⊤ and v = ( v , . . . , v n ) ⊤ , we let u ◦ v = ( u v , . . . , u n v n ) ⊤ denotethe componentwise product. Similarly, uv = ( u /v , . . . , u n /v n ) ⊤ denotes the componentwise ratio.We use Diag( u ) to denote the diagonal matrix with vector u on the diagonal. The identity matrixis denoted by I , the vector or matrix of ones is denoted by , and the vector or matrix of zeros isdenoted by , where the dimension is either specified explicitly, e.g., K ∈ R K , or clear from thecontext.Our analysis is developed within the one-period version of the price contagion model by Capponi and Larsson(2015), which we briefly review in this section. They consider a financial market that consists of twosectors: a banking sector and a nonbanking sector. Each bank manages its asset portfolio to tracka fixed leverage ratio, consistently with empirical evidence reported in the seminal contribution ofAdrian and Shin (2010). The nonbanking sector consists of institutions that are primarily equityfunded (e.g., mutual funds, money market funds, insurances, and pension funds) and thus do notengage in leverage targeting.There are K types of assets available, whose market prices at time t = 0 , P kt . In practice, banks do not immediately revert to the target leverage. Duarte and Eisenbach (2019) estimate thespeed of leverage adjustment, and find that leverage adjustment speed is roughly constant until 2006, before increasingby over 50% and spiking in 2008 due to the greater delevering through balance sheet contraction. Because we considera one period snapshot of the economy, we may simply view the target leverage as a short-term target leverage.
6e write P t = ( P t , P t , . . . , P Kt ) ⊤ for the column vector of asset prices. The aggregate supply of each asset is fixed and given by Q tot = ( Q , . . . , Q K tot ) ⊤ . Each asset k is hit by an exogenous shock Z k , which is modeled asa random variable. We use Z = ( Z , . . . , Z K ) ⊤ to denote the vector of shocks. The vector of fundamental values of the assets at time 1 is P + Z , while we use P to denote the vector ofequilibrium prices which internalize banks’ responses to shocks. The economy consists of N banks, whose stylized balance sheets consist of assets, debt, and equity.Banks manage their balance sheets by buying or selling assets so to keep their leverage ratios(debt to equity ratios) at specified target levels. Banks hold a portfolio of assets at time 0. Then,price shocks occur, and banks purchase or sell assets to restore leverage. These actions imposepressure on prices, and as a result, the market value of bank portfolios at time 1 deviates from itsfundamental value.The quantity (number of units) of asset k held by bank i at time t is denoted by Q kit . Weuse Q it = ( Q it , Q it , . . . , Q Kit ) ⊤ ∈ R K to denote the vector of bank i ’s holdings at t , and Q :=( Q ki ) k =1 ,...,K,i =1 ,...,N ∈ R K × N to denote the matrix of banks’ holdings at time zero. We write A it = ( A it , A it , . . . , A Kit ) ⊤ , where A kit = P kt Q kit is the market value of the i ’th bank’s holdings inasset k at t . The total market value of the i ’th bank is given by ⊤ A it = P Kk =1 A kit .Banks finance purchases by issuing debt. We use D it to denote the total amount of debt issuedby bank i at time t , and assume that the interest rate on the debt is zero. The main behavioralassumption in the model is that each bank i targets a fixed leverage ratio (debt to equity ratio) κ i ,i.e., D it ⊤ A it − D it = κ i , t = 0 , , i = 1 , . . . , N. (2.1)Each bank executes an exogenously specified strategy α i ∈ R K , which specifies how a changein the amount of debt is offset by purchases or sales of the different assets in the portfolio. Hence, Accounting for an exogenous nonzero interest rate would not qualitatively affect the results. Because our focusis on the price inefficiencies caused by banks’ trading responses to shock, we opt for a simpler model that highlightsthese effects.
7t holds that P Kk =1 α ki = 1. For future purposes, let α := ( α ki ) k =1 ,...,K,i =1 ,...,N ∈ R K × N denote thetrading strategy matrix, and let L i = κ i / (1 + κ i ). Hence, by the leverage equation (2.1), it alwaysholds that D it = L i ⊤ A it for t = 0 , Q ki := Q ki − Q ki denotes the change in quantities fromperiod 0 to period 1, admit an explicit expression. Proposition 2.1 (Capponi and Larsson (2015)) . The incremental demand of bank i for asset k isgiven by ∆ Q ki = α ki κ i Q i ⊤ ∆ PP k . (2.2) Nonbanking institutions trade the same assets as the banking sector. This gives rise to additionaldemand, which we refer to as the nonbanking demand and model it in a reduced form. We assumethat demand curves are decoupled across assets, i.e., the nonbanking demand for asset k onlydepends on the price of asset k , and not on the prices of other assets. At time 0, the nonbanking sector holds a quantity Q k, nb0 of asset k , and we write A k, nb0 = P k Q k, nb0 for the corresponding asset value. Unlike the banking sector whose demand function isupward sloping, the nonbanking sector has a downward sloping demand curve: it sells an asset ifits price is above the fundamental value, and purchases an asset if its price is below its fundamentalvalue. Hence, the nonbanking sector acts as the liquidity provider when there are shocks, andexerts a stabilizing force on the pressure imposed by banks. The demand for asset k is given by∆ Q k, nb = − γ k Q k, nb0 ∆ P k − Z k P k , (2.3)where γ k is a positive constant. This choice of demand function admits the following interpretation.Assume no shock occurs, i.e. Z k = 0. Then∆ Q k, nb Q k, nb0 = − γ k ∆ P k P k . (2.4)The parameter γ k can thus be interpreted as the elasticity of the nonbanking demand for asset k , This modeling choice allows us to focus only on the price impact caused by the banks’ needs of tracking theirleverage requirements. κ i α ki in (2.2) for the banking sector. We refer to γ k as the illiquidity characteristic ofasset k . Unlike equation (2.4), (2.3) includes the correction term Z k , because nonbanking demandis due to deviations from fundamental values. We write Q nb t = ( Q , nb t , . . . , Q K, nb t ) ⊤ , and denote by γ = ( γ , . . . , γ K ) ⊤ the vector of illiquidity characteristics. In this section, we review the mechanism used to determine prices. The market-clearing conditionis given by Q nb t + N X i =1 Q it = Q tot , t = 0 , , (2.5)where the vector Q tot of aggregate supply is constant through time. Capponi and Larsson (2015)identify the systemicness matrix S defined by S = P Ni =1 α i γ ◦ Q nb0 κ i Q i ⊤ ∈ R K × K , or, in component-wise form, S kℓ = N X i =1 α ki κ i Q ℓi γ k Q k, nb0 , k, ℓ = 1 , . . . , K, (2.6)as the primary determinant of market prices. This matrix captures how a shock to asset ℓ propagatesto asset k through the banks’ deleveraging activities. Proposition 2.2 (Proposition 2.1. in Capponi and Larsson (2015)) . The dynamics of asset pricesare given by ∆ P = ( I − S ) − Z, (2.7) assuming that the matrix inverse exists. Assumption 1.
We will assume that the spectral radius of S is smaller than one. This yieldsinvertibility of the matrix I − S , as needed in Proposition 2.2, with ( I − S ) − = P ∞ j =0 S j . Thespectral radius ρ ( S ) for non-negative S is bounded from above by (cf. Capponi and Larsson (2015)and Horn and Johnson (1985), Corollary 8.1.29): ρ ( S ) ≤ max k =1 ,...,K P Ni =1 κ i α ki P Kℓ =1 Q ℓi γ k Q k, nb . The spectral radius is small if the size of the nonbanking sector is large in comparison to the size f the leverage targeting banking sector, leverage targets are not too large and price elasticities arenot too small. In this section, we introduce the measure used to quantify price deviation from fundamentals, andcharacterize the key quantities that determine market efficiency. Section 3.1 derives an explicitexpression for the price deviation from fundamentals, and states the objective function of minimiz-ing market efficiency in a systemic economy. Section 3.2 introduces a key sufficient statistic, thebank’s systemic significance, that quantifies the contribution to price pressures of each bank in theeconomy.
Banks actively manage their balance sheets in response to shocks, and this imposes a pressure onasset prices, pushing them away from fundamental values. The value of market capitalization atthe end of period 1 is given by
M C e := Q ⊤ tot P = Q ⊤ tot ( P + ∆ P )= Q ⊤ tot (cid:16) P + ( I − S ) − Z (cid:17) , (3.1)where we recall that Z is the vector of exogenous price shocks (cf. equation (2.7)). In the absenceof leverage-tracking banks, changes in prices are driven solely by changes in fundamentals, i.e., bythe exogenous shocks. The resulting market capitalization is denoted by M C f , and given by M C f := Q ⊤ tot ( P + Z ) . The contribution to realized asset prices arising from the presence of leverage targeting banks,denoted by D := M C e − M C f , is a measure of market inefficiency.Under the conditions discussed in Assumption 1, we obtain the first order approximation to10arket capitalization given by M C a := Q ⊤ tot ( P + ( I + S ) Z ) , where we consider only the first term in the power series expansion of ( I − S ) − .Recall that the i ’th column of the matrix Q denotes the initial asset holdings of bank i . We thenobtain the following first order approximation for the fire-sales externalities: D ≈ M C a − M C f = Q ⊤ tot ( S Z ) = ( Q v ) ⊤ Z, (3.2)where v := Diag( κ ) α ⊤ Q tot γ ◦ Q nb0 ∈ R N . (3.3)We refer to the vector v as the systemic significance of the N banks, and discuss its properties andeconomic implications in Section 3.2. The vector Q v , i.e., the initial allocation of assets within thebanking sector weighted by their systemic significance, is a network multiplier : it describes how aninitial price shock Z is amplified through the network of balance sheet holdings due to the leveragetargeting actions of the banks. Deviation from Efficiency : We use the mean squared deviation criterion E [ D ] ≈ E [( M C a − M C f ) ] =: M SD ( Q ) , to quantify ex-ante these inefficiencies. We use the average of the squares of price deviation fromfundamentals, i.e., we penalize equally positive and negative deviations of market prices from fun-damental values.We use µ and Σ to denote, respectively, the expected value and covariance matrix of shocks,i.e., E [ Z ] = µ = ( µ , . . . , µ K ) ⊤ , and Cov[ Z ] = Σ ∈ R K × K . Henceforth, we impose the followingassumption on the distribution of asset shocks. Assumption 2.
Price shocks Z k , k = 1 , , . . . , K , are uncorrelated with variances σ = ( σ , . . . , σ K ) ⊤ ,i.e., Σ = Diag( σ ) , σ k > , and the initial asset prices are normalized to P k = 1 , k = 1 , . . . , K . This section discusses the properties of the systemic significance of a bank, and its dependence onthe model primitives. We start observing that, for any bank i , its systemic significance equals v i = κ i K X k =1 α ki γ k · Q k tot Q k, nb0 , as it easily follows from Eq. (3.3). The systemic significance v i depends on the targeted leverage κ i , banks’ strategies α ki , the vector γ of price elasticities, and the initial proportion of assets heldby the nonbanking sector Q k, nb0 Q k tot . If bank i liquidates a large fraction of an illiquid asset (i.e., α ki γ k ishigh), then it creates a larger price pressure, especially if it is targeting a high leverage κ i . If thenonbanking sector holds a significant fraction of the assets in the economy, then it will be able tobetter absorb the aggregate demand of the banking sector, and thus the systemic significance ofany bank in the system will be reduced. Remark 3.1.
Observe that both D and the square-root of the mean squared deviation p E [ D ] = p E [( M C a − M C f ) ] are positively homogeneous, when viewed as functions of the systemic sig-nificance vector v . In particular, price pressures vanish and the market becomes efficient when v approaches zero; conversely, as v gets large, inefficiencies are higher. The mean and variance of market capitalization can be uniquely characterized by the matrix ofasset holdings and the systemic significance of the banks in the system, as stated next.
Lemma 3.1.
It holds that E [ D ] = ( Q v ) ⊤ µ, Var[ D ] = ( Q v ) ⊤ Σ ( Q v ) . Moreover, the mean squared deviation of price pressures equal
M SD ( Q ) := E [ D ] = Var[ D ] + E [ D ]
12 ( Q v ) ⊤ ( µµ ⊤ + Diag( σ )) | {z } =: G ∈ R K × K ( Q v ) . (3.4)The formulas above indicate that the mean squared deviation is a function of Q v , i.e., theinitial allocation of assets within the banking sector weighted by their systemic significance, and ofstatistics about fundamental shocks collected in the matrix G . This is consistent with intuition:larger shocks require banks to trade a higher amount of assets to restore their leverage targets. Asa result, through the network multiplier Q v , these shocks are amplified more and lead to a higherpressure on prices. In this section, we study the impact of banks’ portfolio holdings on market efficiency, and developan explicit construction of holding matrices which minimize the deviation of market capitalizationfrom its fundamental value. We refer to those matrices as f-efficient holdings . Section 4.1 introducesa two-step procedure, “aggregate first then allocate” to construct f-efficient holdings. Section 4.2discusses the relation between f-efficient holdings and the diversification benchmark where eachbank fully diversifies its portfolio holdings.
Taking the initial budget of the banks as fixed, we compute the matrix Q of initial holdings whichis f-efficient, i.e., which minimizes the mean squared deviation M SD ( Q ). Let q ∈ R K be the vector of total initial holdings of the banking sector, and b ∈ R N the vectorof banks’ initial budgets. The set of initially feasible asset allocations within the banking sector isthen given by D = D ( q, b ) := { Q ∈ R K × N | Q N = q and ⊤ K Q = b ⊤ } , where we have assumed that the initial prices of all assets are normalized (see Assumption 2). We In Appendix C, we study the sensitivity of market inefficiencies to the matrix α of banks’ trading strategies,taking the initial holdings as given. Consistent with intuition, we find that it is f-efficient for banks to sell solely themost liquid asset. K X k =1 q k = N X i =1 b i =: T. Example 1.
Full portfolio diversification corresponds to the holding matrix Q diversified := 1 T qb ⊤ = b q T · · · b N q T ... . . . ... b q K T · · · b N q K T ∈ D . (4.1) In the special case that initial holdings are the same for all assets, i.e., q = q = . . . = q K , thenfull diversification is characterized by the holding matrix Q diversified = b K · · · b N K ... . . . ... b K · · · b N K . Definition 1.
A feasible allocation matrix Q ∗ ∈ D such that Q ∗ ∈ argmin Q ∈ D ( Q v ) ⊤ G ( Q v ) | {z } = MSD ( Q ) (4.2) is called f-efficient . f-efficient holdings are those which minimize, ex-ante, the mean squared deviation of asset pricesfrom fundamentals among all feasible allocations. To exclude trivial cases and make the probleminteresting, we make the following assumption, which is typically satisfied in practice. Assumption 3.
There exist some i, j ∈ { , . . . , N } such that v i = v j , and it holds that b ⊤ v = 0 . Remark 4.1. If v = . . . = v N then, for each given asset, it does not matter how it is distributedacross the banks because all have the same systemic significance. As a result, M SD ( Q ) is constantfor all Q ∈ D , taking into account the constraint Q = q . The requirement that systemic signif-icance is not identical across banks is satisfied by any economy, which is not fully homogeneousin terms of targeted leverage and trading strategy. Empirically, Duarte and Eisenbach (2019), seeTable 4 therein, finds substantial variation in banks’ leverage targets, with a size-weighted average f 13.6, an equal-weighted average of 11.5, and a standard deviation of 3.9. If b ⊤ v = 0 , then theremust exist banks in the system which are short some of the assets. Then, the negative price pressureimposed by some banks in the system would be compensated by a positive price pressure created byother banks. In this case, diversification would be f-efficient, and lead to zero deviation of assetprices from fundamental values, i.e., M SD ( Q diversified ) = ( T qb ⊤ v ) ⊤ G ( T qb ⊤ v ) = 0 . In practice,however, b ⊤ v > because the budget and the systemic significance of any bank in the system areboth positive. Banks are long their assets, including consumer loans, agency, non-agency securities,municipal securities, etc., see, again, Table 4 in Duarte and Eisenbach (2019). We next describe the two-step procedure used to construct f-efficient holding matrices: • Step 1: Aggregation.
In this step, we recover the network multiplier y that minimizes themean squared deviation, and which is consistent with the market structure. Concretely, let D y := { y ∈ R K | ⊤ K y = b ⊤ v } . Such a multiplier is obtained from banks’ initial holdings (andbudgets), upon weighting them with the systemic significance vector, i.e., y ∗ ∈ argmin y ∈ D y y ⊤ G y. (4.3)Because of this weighting, this multiplier accounts for the banks’ leverage tracking behav-ior, their liquidation strategies, and illiquidity characteristics of the assets. We refer to theminimizing vector y ∗ as the aggregate v -weighted holdings. • Step 2: Allocation.
In this step, we identify an allocation of asset holdings to banks, whichis consistent with the vector of aggregate v -weighted holdings obtained from the previousstep. Specifically, we denote by Q ∗ ∈ D the matrix, consistent with the budget and supplyconstraints, which distributes the aggregate v -weighted holdings to individual banks accordingto their systemic significance, i.e., Q ∗ v = y ∗ .The decomposition discussed above presents both conceptual and computational advantages. Froma conceptual perspective, observe that the matrix G of shock statistics and the vector v of banks’ Their sample includes the largest 100 banks by assets every quarter, in a sample period from the third quarter of1999 to the third quarter of 2016 at the quarterly frequency. They also find that 5% and 95% of the leverage targetdistribution are, respectively, 6.8 and 16.9, and that there is more cross-sectional than time-series variation. G , while v only enters into the constraint set. The allocationstep instead, takes the aggregate holdings computed from the previous step as given, and deter-mines the holding matrix only on the basis of v . From a computational point of view, observe thatfinding the f-efficient holdings requires solving a K × N dimensional quadratic problem with linearconstraints. Using the proposed decomposition, we first solve a K dimensional quadratic problemwith one linear constraint, and subsequently solve a simple K × N dimensional unconstrained linearsystem.The following proposition states that the two-step procedure described above identifies f-efficientholdings. Proposition 4.1.
Let N ≥ , K ≥ . The following statements are equivalent:(i) Q ∗ ∈ D is f-efficient.(ii) y ∗ = Q ∗ v for some Q ∗ ∈ D , and y ∗ solves the problem (4.3) . In the rest of the section, we discuss in more detail each step of the procedure above, andhighlight the key economic insights.
We start showing that the minimization problem (4.3) admits a unique solution.
Lemma 4.1.
The unique solution to problem (4.3) is given by y ∗ := b ⊤ v ⊤ K z · z, where z := G − K ∈ R K . The above expression indicates that aggregate v -weighted holdings are determined by the vec-tor v , capturing the systemic significance of banks, weighted by the budget each bank is endowedwith, and by the inverse of the matrix G which captures the size of the exogenous price shocks.Specifically, the aggregate systemically weighted budget vector b ⊤ v is split into the K availableassets through the inverse of G . 16 xample 2. Consider the special case of zero mean shocks, i.e., set µ = 0 . Then y ∗ k = ( b ⊤ v ) · /σ k P Kℓ =1 /σ ℓ . Hence, the higher the variance of price shocks σ k , the lower the fraction of asset k inthe aggregate v -weighted holdings portfolio y ∗ . This is intuitive: an asset that creates high pricepressure when banks manage their assets to restore their target leverage should, in aggregate, beinvested less. In the second step, the aggregate v -weighted holdings y ∗ ∈ R K are allocated to the individualbanks. Proposition 4.2.
For every N ≥ , K ≥ , there exists a matrix Q ∗ ∈ D satisfying Q ∗ v = y ∗ . In the proof of the proposition, we construct a particular solution Q ∗ , and describe the structureof the linear subspace of solutions. The following theorem quantifies the mean squared deviationachieved by a matrix of f-efficient holdings, and addresses the uniqueness of the allocation. Theorem 4.2.
Let
N, K ≥ .a) There exists an f-efficient holding matrix Q ∗ with mean squared deviation M SD ( Q ∗ ) = ( b ⊤ v ) ⊤ K G − K . (4.4) b) The f-efficient holding matrix is unique, if and only if there are exactly N = 2 banks. In thiscase, the unique f-efficient holding matrix is given by Q N =2 := 1 v − v (cid:18) v q − y ∗ y ∗ − v q (cid:19) ∈ R K × . (4.5) In this subsection, we provide the conditions under which full diversification is f-efficient.
Theorem 4.3.
Full diversification Q diversified is f-efficient, if and only if q and z = G − K arelinearly dependent.
17 direct consequence of the above theorem is that full diversification is f-efficient if the systemis completely homogeneous, i.e., all asset shocks have the same mean and variance, and the totalinitial holdings of the banks are the same. Full diversification is no longer f-efficient if a littleamount of heterogeneity is introduced in the system.
Corollary 4.1.
If eithera) q = . . . = q K , σ = . . . = σ K , µ = . . . = µ K − and µ K = µ + ε with ε = − Kµ , orb) µ = . . . = µ K , σ = . . . = σ K , q = . . . = q K − and q K = q + ε , orc) q = . . . = q K , µ = . . . = µ K , σ = . . . = σ K − and σ K = σ + ε ,then Q diversified is f-efficient, if and only if ε = 0 . The result in the above corollary is consistent with intuition. If assets are fully homogeneousand the total holdings of the banking sector in each asset are the same, there is no reason to preferone asset over the other. In this case, full diversification minimizes the portfolio liquidation riskand is optimal. However, as soon as assets no longer have identical characteristics, an f-efficientallocation requires to allocate assets to banks in accordance with their systemic significance.
In this section, we construct case studies to analyze the structure of f-efficient holdings. In the firstpart of this section, we consider the stylized case N = K = 2, and study the distance of f-efficientholdings from diversification as a function of key model parameters. For any matrix Q ∈ R K × N ,we measure the distance from full diversification by the Frobenius norm d ( Q ) := k Q − Q diversified k F . In the second part of the section, we study the minimal distance of f-efficient holdings from fulldiversification for the case N = K = 3. In the third part, we compare the impact of f-efficiencyand diversification on the distribution of market capitalization under different economic scenarios.18 .1 The Case N = K = 2 According to Theorem 4.2 b), the f-efficient holding matrix for N = K = 2 is unique and given by Q × := 1 v − v v q − y ∗ y ∗ − v q v q − y ∗ y ∗ − v q . (5.1)We normalize the total supply of assets within the banking sector, and the budgets of banks to q = q = b = b =: x >
0; following Capponi and Larsson (2015), we choose x = 0 .
08, where thetotal supply is normalized to 1 for each asset. Hence, the the total size of the banking sector is 8%of the total size of the system.
We start with an exploratory analysis, where we plot the f-efficient holdings of bank 1 and thedistance of f-efficient holdings from diversification d ( Q × ) as a function of the riskiness of the firstasset. Consistent with Corollary 4.1, full diversification is f-efficient if and only if σ = σ = 0 . -b Q x k , d ( Q x ) Q Q Q diversified1,1 Figure 1: f-efficient holdings of bank 1 Q × k, for assets k = 1 , d ( Q × ) = k Q × − Q diversified k F , as a function of σ . We fix σ = 0 . µ = (0 , ⊤ , v = (0 . , . ⊤ , q = b = (0 . , . ⊤ . 19cf. Corollary 4.1). Figure 1 additionally suggests that a systemically more significant bank wouldhave lower f-efficient holdings in the riskier asset than a systemically less significant bank: as σ increases, bank 2 decreases its holdings in asset 1, while bank 1 increases it holdings in that asset.We rigorously formalize these observations in the following lemma. Lemma 5.1.
Let q = q = b = b = x > and µ = µ .a) For σ > , we have ∂∂σ d ( Q × ) < , if σ < σ , > , if σ > σ .b) If v > v > , then ∂ Q × ∂σ > for σ > . As in the previous subsection, we start with a graphical illustration of the sensitivity of bank 1’sf-efficient holdings and their distance from diversification to changes in the expected shock size µ .Figure 2 indicates that, as the expected (absolute) size of price shocks for asset 2 increases, the -1 -0.5 0 0.5 1 -b Q x k , d ( Q x ) Q Q Q diversified1,1 Figure 2: F-efficient holdings of bank 1 Q × k, for assets k = 1 , d ( Q × ) = k Q × − Q diversified k F as a function of µ for fixed σ = (0 . , . ⊤ , µ = 0 , v =(0 . , . ⊤ , q = b = (0 . , . ⊤ . 20-efficient holdings of the least systemically significant bank (i.e., bank 1) increase. This can be,again, understood in terms of the banks’ systemic significances: a more systemically significantbank, i.e., one that tracks a higher leverage ratio or which trades a larger fraction of illiquid assets,should reduce its holdings of asset 2, because the trading actions in response to the shock imposea higher pressure on the price, and hence large deviation of prices from fundamentals.The distance from a full diversification strategy is minimal when the system achieves the highestpossible degree of homogeneity, i.e., µ = µ . As the system becomes more heterogeneous, thedistance increases. As shown in Corollary 4.1, the minimal distance converges to zero as thesystem becomes fully homogeneous, i.e., σ = σ . We formalize these observations via the followinglemma. Lemma 5.2.
Let q = q = b = b = x > , σ = σ and µ = 0 .a) It holds that ∂∂µ d ( Q × ) < , if µ < , = 0 , if µ = 0 , > , if µ > .b) If v > v > , then ∂ Q × ∂µ > , if µ < , = 0 , if µ = 0 , < , if µ > . We analyze how heterogeneity in systemic significances impacts the degree of diversification ofbanks’ holdings. Figure 3 highlights that, as the two banks become closer in terms of systemicsignificance, the distance of f-efficient holdings from diversification increases. This highlights thefundamental role of systemic significance on banks’ f-efficient holdings: if two banks are similarin terms of systemic significance (for instance because they adopt similar trading strategies), thenit is beneficial to sacrifice diversification benefits to reduce portfolio overlapping, and thus pricepressures. These intuitions can be formalized via the following lemma.21 .02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 v -b Q x k , d ( Q x ) Q Q Q diversified1,1 Figure 3: F-efficient holdings of bank 1 Q × k, for assets k = 1 , d ( Q × ) = k Q × − Q diversified k F as a function of systemic sensitivity v for fixed σ = (0 . , . ⊤ , µ = (0 , ⊤ , v = 0 . q = b = (0 . , . ⊤ . Lemma 5.3.
Let q = q = b = b = x > , v > and | z | 6 = | z | . For < v = v , it holdsthat ∂∂v d ( Q × ) > , if v < v , < , if v > v . N = K = 3 Having established the results for a system consisting of two banks and two assets, we analyzenumerically how the findings would change for a larger economy. If the number of banks is
N > N = K = 3. Noticeably, thequalitative findings remain similar to the setting N = K = 2. The f-efficient holdings get fartheraway from a full diversification strategy if heterogeneity in banks’ systemic significance decreases.The intuition behind the result remains unchanged, i.e., in a system where banks are systemically A sufficient condition for | z | 6 = | z | is that µ = µ and σ = σ . d ( Q min ) from diversification for the f-efficientholding matrix Q min with the smallest distance to diversification. We vary the systemic significanceparameters v , v and v , and keep fixed shock characteristics, i.e., µ = (0 . , . , . ⊤ and σ = (0 . , . , . ⊤ . To ensure comparability with the results of Section 5.1, we choose q k = b i = 0 . i, k = 1 , ,
3, and normalize the total supply of each asset to 1.very close, a full diversification strategy for each bank may lead to larger price pressures becauseall banks rebalance their portfolios in a similar fashion to meet their leverage targets.In Appendix B, we analyze the f-efficient holdings in an economy with more than two banks.We find that any f-efficient allocation prescribes the most systemically significant bank to havehigher holdings of the less risky asset, and the least systemically significant bank to have higherholdings of the more risky asset. We also show that for a system of three banks, given a fixedf-efficient allocation, any other f-efficient allocation is obtained by shifting the holdings within eachbank based on the difference in systemic significances of the other two banks.
In this section, we analyze how the distribution of banks’ holdings depends on banks’ systemicsignificance, heterogeneity in the distributions of initial shocks and illiquidity characteristics of theassets. We also validate the accuracy of the first order approximation of market capitalization usedthroughout the paper. Our results indicate that the mean squared deviation of the actual market23apitalization from its fundamental value is low if the matrix of bank holdings coincides with thef-efficient holdings. This indicates that the solution to the (approximate) optimization problem,i.e., where the first-order approximation of the systemicness matrix is used, yields a low value forthe actual objective function where the exact expression of the systemicness matrix is used.In the analysis below, we consider three asset classes, each consisting of assets with identicalcharacteristics, for a total of ten assets. As a result, we demonstrate that the methodology proposedin this paper scales well to economies larger than those considered (analytically) in earlier sections.
We consider a financial market consisting of N = 2 banks. We choose K = 10 assets, normalizethe total supply of each asset to 1, and set the holdings of banks in each asset to 0 . The twobanks are assumed to have the same budget ( b = b = 0 . κ = (9 , ⊤ .Banks are assumed to follow a proportional liquidation strategy, i.e., α ki = 1 /
10 for all assets k = 1 , . . . ,
10, and both banks i = 1 ,
2. The assets belong to three different groups: Assets 1 and 2belong to group 1, assets 3 through 8 to group 2, and assets 9 and 10 to group 3. Within each ofthe three asset groups, the illiquidity characteristics of the assets, and the expectation and varianceof asset price shocks are equal.We consider three economic scenarios, each characterized by a certain value of the shock varianceand liquidity of the assets. We refer to the three scenarios as liquidity, intermediate crisis, and highrisk high illiquidity scenarios. Across all scenarios, we set µ = (0 . , . , . , . , . , . , . , . , . , . ⊤ .The numerical values of the asset illiquidity characteristics should be thought as normalized tothe corresponding characteristics of a reference asset, and are broadly consistent with the estimatesreported in Table 4 of Duarte and Eisenbach (2019). Consistent with empirical evidence, thehigher the illiquidity of the asset (i.e., the lower γ ), the larger the variance of the exogenous assetprice shocks, capturing the fact that more illiquid securities have a higher volatility than liquidones. All parameters in this section are consistent with Capponi and Larsson (2015). Their estimates are based on the Net Stable Funding Ratio of the Basel III regulatory framework. Their illiquidityparameter is the reciprocal of ours, i.e., in their setting a larger value corresponds to a higher illiquidity of the asset.They take US Treasuries as the reference asset, i.e., the price impact of U.S. Treasuries is normalized to 1.
Liquidity Scenario:
Banks invest in liquid assets, i.e, assets with high price elasticities, or equivalently, low illiq-uidity characteristics. The first asset class has the highest price elasticity and the lowest shockvariance, and the third asset class has the lowest price elasticity and the highest variance.The second asset class has an intermediate value for those two quantities. Specifically, wechoose γ L = (9 , , , , , , , , , ⊤ , σ L = (0 . , . , . , . , . , . , . , . , . , . ⊤ . (I) Intermediate Crisis Scenario:
In contrast to scenario (L), the third asset class is signifi-cantly more illiquid (i.e., lower price elasticity) and has higher shock variance. This situationis typical of the beginning of a crisis period, where one asset may experience a severe shockand then become hard to sell quickly due to the lack of outside investors (the nonbankingsector in our model) willing to purchase the asset. The parameters corresponding to theother asset classes are not altered. This scenario is specified by γ L = (9 , , , , , , , , , ⊤ , σ L = (0 . , . , . , . , . , . , . , . , , ⊤ . (H) High Risk High Illiquidity Scenario:
Banks invest in assets with high illiquidity andvolatility. This captures, for instance, a situation where banks invest in securities such asnon-agency based mortgage, municipal bonds, or commercial and industry loans. All assetclasses are thus characterized by a lower price elasticity and higher shock variance comparedto the previous two scenarios: γ H = (3 , , , , , , , , , ⊤ , σ H = (0 . , . , . , . , . , . , . , . , . , . ⊤ . Next, we compute the f-efficient holdings and banks’ systemic significances for each of the abovedefined scenarios. • In the liquidity scenario (L), the systemic significance of the banks equal v L ≈ . < . ≈ For instance, the volume of agency mortgage backed securities, typically highly liquid assets, declined substantiallyfrom 2008 to 2014, which is an indicator of worsening liquidity. L . The second bank—tracking a higher leverage ratio—is systemically more significant thanthe first bank. f-efficient holdings are given by Q ∗ ,L ≈ − . − .
79 0 .
87 0 .
87 0 .
87 0 .
87 0 .
87 0 .
87 1 .
37 1 . .
87 3 . − . − . − . − . − . − . − . − . ⊤ . The systemically more significant bank 2 is endowed with a higher number of assets of class 1(high elasticity, low variance); the least significant bank 1 holds a larger portion of the otherassets (smaller elasticity, higher variance). • In scenario (I), bank 2 is still systemically more significant than bank 1, i.e., v I ≈ . < . ≈ v I ; in comparison to scenario (B), both banks’ systemic significances increase due tothe increased illiquidity and shock variances of assets from group 3. The f-efficient holdingsare given by: Q ∗ ,I ≈ − . − .
87 1 .
07 1 .
07 1 .
07 1 .
07 1 .
07 1 .
07 0 .
87 0 . .
95 3 . − . − . − . − . − . − . − . − . ⊤ . Again, the higher the systemic significance of the bank, the lower its holdings of the saferasset relative to the riskier asset. • In scenario (H), bank 2 remains systemically more significant than bank 1, with v H ≈ . < . ≈ v H , and the significance parameters are higher than in the two other scenarios. Thef-efficient holdings are: Q ∗ ,H ≈ − . − .
41 0 .
10 0 .
10 0 .
10 0 .
10 0 .
10 0 .
10 0 .
31 0 . .
49 0 . − . − . − . − . − . − . − . − . ⊤ . In this scenario, there is little heterogeneity in the riskiness of the assets, and high hetero-geneity in banks’ systemic significance. As a result, the f-efficient holdings are more evenlydistributed, i.e., closer to full diversification (see also the values of the distances given in Table1 below). 26 .3.2 Diversification and f-Efficiency
In this section, we compute the exact market capitalization, i.e.,
M C e = Q ⊤ tot ( P + ( I − S ) − Z ) , both under f-efficient ( Q ∗ , · ) and fully diversified ( Q diversified ) holding matrices. We suppose thatthe vector of shocks Z follows a multivariate normal distribution with mean vector µ , which isthe same across all scenarios, and covariance matrix Diag( σ ), where σ depends on the consideredscenario. In each scenario, we draw 100 ,
000 independent samples of the shock vector Z .We report the relevant statistics in Table 1. Figure 5 also provides a comparison of the prob-ability density functions of market inefficiency and the corresponding box plots across all threescenarios. Consistent with intuition, the variance of market capitalization is the smallest in theliquidity scenario, and the highest in the high risk high illiquidity scenario. We find that diversifica-tion results in a higher variance than f-efficiency across all market scenarios. While this difference isnot very significant in the liquidity scenario, it becomes considerable in the high illiquidity high riskscenario, especially if (as in the intermediate crisis scenario) the assets are shocked heterogeneously.Observe, from the first row of the table, that the distance between f-efficient and fully diversi-fied holding matrices is the lowest in the scenario (H). Nevertheless, the mean-squared deviationcriterion and variance of the exact market capitalization is the highest (in absolute terms) in sucha scenario. Taken together, these two observations imply that if banks invest in high risk high illiq-uid assets, it suffices to only slightly distort the holdings from f-efficiency to induce large increasesin the variance and mean squared deviation of market capitalization from its fundamental value.This is because holdings that are not f-efficient, such as fully diversified holdings, create a pricepressure which becomes increasingly larger as we move towards balance sheets with highly volatileand illiquid assets. We developed a model to examine the ex-ante asset holdings which minimize market efficiency ina systemic multi-asset economy. Price pressure arises in our model from the exogenous trading27
10 -8 -6 -4 -2 0 2 4 6 8 10
Market Inefficiency P r obab ili t y D en s i t y F un c t i on E s t i m a t e PDF Estimates of Market Inefficiency f-efficient (L)diversified (L)f-efficient (I)diversified (I)f-efficient (H)diversified (H) (a) f-efficient (L) diversified (L) f-efficient (I) diversified (I) f-efficient (H) diversified (H)-10-5051015 M a r k e t I ne ff i c i en cy Box Plots of Market Inefficiency (b)
Figure 5: Probability density function estimates (a) and box plots (b) of market inefficiency
M C e − M C f for f-efficient ( Q ∗ , · ) and fully diversified ( Q diversified ) holdings, generated from 100,000samples of Z ∼ N ( µ, Diag(( σ · ) )), in the three considered scenarios: (L) liquidity, (I) intermediatecrisis, and (H) high risk high illiquidity. 28 I H I/L H/L d ( Q ∗ , · ) 8.61 8.74 1.08 1.02 0.13 E ( M C e ) for Q ∗ , · E ( M C e ) for Q diversified M C e ) for Q ∗ , · M C e ) for Q diversified E [( M C e − M C f ) ] for Q ∗ , · E [( M C e − M C f ) ] for Q diversified d ( Q ∗ , · ) measures the difference between diversified and f-efficientholdings. The last two columns present the numbers relative to the liquidity scenario.actions of banks which manage their portfolios to target specific leverage ratios. In the model,we quantify efficiency in terms of the mean squared deviation of market capitalization from cap-italization measured using fundamental values. We find that inefficiencies are low if banks arenot systemically significant, but become substantial if the overall systemic significance is high andbanks are not sufficiently heterogeneous in systemic significance. We develop a procedure whichconstructs f-efficient holdings, and show that these depend on two key sufficient statistics, namelythe banks’ systemic significance and the first two moments of exogenous asset shocks. Our anal-ysis reveals that increasing heterogeneity in banks’ systemic significance moves f-efficient holdingscloser to full diversification, while heterogeneity in the distribution (expectation and variance) ofexogenous asset value shocks moves f-efficient holdings away from diversification. In balance sheetscenarios characterized by high risk and illiquidity, deviating from f-efficient holdings would resultin high inefficiencies and large variance of market capitalization.29 ppendix A Proofs A.1 Proofs of Section 2
A.1.1 Proof of Proposition 2.1
This proof is obtained by specializing Proposition 1.1 in Capponi and Larsson (2015) to a settingwith only one period, and assuming zero revenue shocks therein, i.e., ∆ R i = 0. The fundamentalcash-flow equation is given by: P k ∆ Q ki = α ki ∆ D i , k = 1 , . . . , K, (A.1)Writing the cash-flow equation (A.1) in vector form yieldsDiag( P )∆ Q i = α i ∆ D i . Substituting for D it in the above equation the expressions for L i , we obtainDiag( P )∆ Q i = α i (cid:16) L i ⊤ A i − L i ⊤ A i (cid:17) = α i (cid:16) L i ⊤ ( A i − A i ) + ( L i − L i ) ⊤ A i (cid:17) (A.2)Rearranging the above expression leads to (cid:16) Diag( P ) − L i α i P ⊤ (cid:17) ∆ Q i = α i L i Q i ⊤ ∆ P. (A.3)The matrix multiplied by ∆ Q i can be inverted using the Sherman-Morrison formula. First, since ⊤ α i = 1, we have 1 − L i P ⊤ Diag( P ) − α i = 1 − L i ⊤ α i = 1 − L i = 0 , so invertibility is guaranteed. The inverse is given byDiag( P ) − + κ i Diag( P ) − α i P ⊤ Diag( P ) − , P ) − ( I + κ i α i ⊤ ). From (A.3) we therefore obtainDiag( P )∆ Q i = ( I + κ i α i ⊤ ) (cid:16) α i L i Q i ⊤ ∆ P (cid:17) = (1 + κ i ) α i (cid:16) L i Q i ⊤ ∆ P (cid:17) , where the second equality uses the identity ( I + κ i α i ⊤ ) α i = (1+ κ i ) α i , which follows from ⊤ α i = 1.Noting that (1 + κ i ) L i = κ i , we obtainDiag( P )∆ Q i = κ i α i Q i ⊤ ∆ P, and the stated expression (2.2) follows. A.1.2 Proof of Proposition 2.2
This proof is obtained by specializing Proposition 2.1 in Capponi and Larsson (2015) to a settingwith only one period, and assuming zero revenue shocks therein, i.e., ∆ R i = 0. First, we use themarket-clearing condition (2.5), and then the expressions (2.2) and (2.3) for the demand functionsto get = P ◦ ∆ Q nb + N X i =1 P ◦ ∆ Q i = Diag( γ ◦ Q nb0 ) ( Z − ∆ P ) + N X i =1 α i κ i Q i ⊤ ∆ P. Multiplying from the left by Diag( γ ◦ Q nb0 ) − and rearranging yields " I − N X i =1 α i γ ◦ Q nb0 κ i Q i ⊤ ∆ P = Z. (A.4)The left-hand side is thus equal to ( I − S )∆ P . We now simply multiply both sides of the equality(A.4) from the left by ( I − S ) − to arrive at the stated price change (2.7).31 .2 Proof of Section 4 A.2.1 Proof of Proposition 4.1 • We first prove the direction (i) ⇒ (ii): Let Q ∗ ∈ argmin Q ∈ D ( Q v ) ⊤ G ( Q v ) be an f-efficient holdingmatrix. Since the objective function does only depend on the vector Q v ∈ R K , the followingstatements are equivalent: Q ∗ ∈ argmin Q ∈ D ( Q v ) ⊤ G ( Q v ) ⇔ e y := Q ∗ v ∈ argmin y ∈ e D y ⊤ G y, (A.5)where e D := { y ∈ R K | ∃ Q ∈ D s.t. y = Q v } . Since every vector of the form y = Q v for one Q ∈ D satisfies ⊤ K y = ⊤ K Q v = b ⊤ v, it obviously holds that e D ⊆ D y = { y ∈ R K | ⊤ K y = b ⊤ v } yielding min y ∈ e D y ⊤ G y ≥ min y ∈ D y y ⊤ G y. However, as shown by Proposition 4.2, if y ∗ ∈ D y solves the aggregate problem, i.e., y ∗ ∈ argmin y ∈ D y y ⊤ G y , then there exists a matrix e Q such that y ∗ = e Q v , i.e., y ∗ ∈ e D . Hence, e y ⊤ Q e y = min y ∈ e D y ⊤ G y = min y ∈ D y y ⊤ G y = ( y ∗ ) ⊤ G y ∗ , i.e. e y = Q ∗ v solves the aggregate problem (4.3). • Second, we prove the direction (ii) ⇒ (i): Let y ∗ = Q ∗ v ∈ argmin y ∈ D y y ⊤ G y solve the aggregateproblem. Since min y ∈ e D y ⊤ G y ≥ min y ∈ D y y ⊤ G y , with e D as defined in the previous step, andsince y ∗ = Q ∗ v for some Q ∗ ∈ D , we have that y ∗ = Q ∗ v ∈ argmin y ∈ e D y ⊤ G y , which, finally, isequivalent to Q ∗ being f-efficient, see (A.5). A.2.2 Proof of Lemma 4.1
The KKT conditions for minimizing y ⊤ G y subject to y ∈ D y read as G K ⊤ K | {z } =: M · yλ = K b ⊤ v . (A.6)32he inverse of the KKT matrix M is by blockwise inversion (see, e.g., Proposition 2.8.7 in Bernstein(2005)) given by M − = G − − G − K ( ⊤ K G − K ) − ⊤ K G − G − K ( ⊤ K G − K ) − ( ⊤ K G − K ) − ⊤ K G − − ( ⊤ K G − K ) − =: A BC D , with A ∈ R K × K , B ∈ R K × , C ∈ R × K and D ∈ R . Note that this inverse matrix exists because,firstly, G is invertible because it is symmetric and positive definite, with inverse given by theSherman-Morrison formula G − = Diag( σ ) − µσ ( µσ ) ⊤ µσ ) ⊤ µ , and, secondly, ⊤ K G − K = ( σ ) ⊤ K − ( ⊤ K µσ ) µσ ) ⊤ µ = 0 , since with c := ( σ ) ⊤ K > σ ) ⊤ K − ( ⊤ K µσ ) µσ ) ⊤ µ = 0 ⇔ c (1 + ( µσ ) ⊤ µ ) − ( ⊤ K µσ ) = 0 ⇔ c + c · ( µσ ) ⊤ ( µσ ) − ( µσ ) ⊤ σ ( σ ) ⊤ µσ = 0 ⇔ ( µσ ) ⊤ ( I − c · σ ( σ ) ⊤ | {z } =: E ) µσ = − , and this cannot be fulfilled for any µσ ∈ R K since E ∈ R K × K is a positive semidefinite matrix dueto its only eigenvalues 0 and 1 (cf. Dattorro (2005), Appendix B.3).Hence, the f-efficient solution to problem (4.3) is derived from multiplying both sides of equation(A.6) by M − yielding y ∗ = A · K + B · ( b ⊤ v ) = b ⊤ v · G − K ( ⊤ K G − K ) − = b ⊤ v ⊤ K z z, where z := G − K . 33 .2.3 Proof of Proposition 4.2 We define the matrix Q p := v − v [ v q − y ∗ − ( N X i =3 ( v − v i ) b i ) e K ] , v − v [ y ∗ − v q − ( N X i =3 ( v i − v ) b i ) e K ] , b e K , . . . , b N e K ! ∈ R K × N , (A.7)where e K = (1 , , . . . , ⊤ ∈ R K . Without loss of generality, we have assumed here that v = v (cf. Assumption 3). This matrix satisfies Q p ∈ D as well as y ∗ = Q p v , for y ∗ = b ⊤ v ⊤ K z z , the solutionof the aggregate problem (4.3) derived in Lemma 4.1: • y ∗ = Q p v : It holds:( Q p v ) = 1 v − v v v q − y ∗ v − v N X i =3 ( v − v i ) b i + y ∗ v − v v q − v N X i =3 ( v i − v ) b i ! + N X i =3 b i v i = 1 v − v ( v − v ) y ∗ − ( v − v ) N X i =3 b i v i ! + N X i =3 b i v i = y ∗ , and for k = 2 , . . . , K :( Q p v ) k = 1 v − v ( v v q k − y ∗ k v + y ∗ k v − v v q k ) = y ∗ k . • Q p N = q : We obtain( Q p K ) = 1 v − v v q − y ∗ − N X i =3 ( v − v i ) b i + y ∗ − v q − N X i =3 ( v i − v ) b i ! + N X i =3 b i = 1 v − v ( v − v ) q + ( v − v ) N X i =3 b i ! + N X i =3 b i = q , and for k = 2 , . . . , K : ( Q p N ) k = 1 v − v ( v q k − y ∗ k + y ∗ k − v q k ) = q k . • ⊤ K Q p = b ⊤ : Obviously, it is ( ⊤ K Q p ) i = b i , for i = 3 , . . . , N . For the first and second entry,34t holds that ( ⊤ K Q p ) = 1 v − v v ⊤ K q − ⊤ K y ∗ − N X i =3 ( v − v i ) b i ! = 1 v − v v N X i =1 b i − N X i =1 b i v i − N X i =3 v b i + N X i =3 b i v i ! = 1 v − v ( v ( b + b ) − ( v b + v b )) = b , where in the first step, we have used that P Kk =1 q k = T = P Ni =1 b i and that ⊤ K y ∗ = b ⊤ v .Using the same arguments, we obtain( ⊤ K Q p ) = 1 v − v ⊤ K y ∗ − v ⊤ K q − N X i =3 ( v i − v ) b i ! = 1 v − v N X i =1 b i − v N X i =1 b i v i − N X i =3 b i v i + N X i =3 v b i ! = 1 v − v ( b v + b v − ( v b + v b )) = b , which completes the proof. Remark A.1.
Due to our two-step solution method proven in Proposition 4.1, finding an f-efficientholding matrix Q ∗ ∈ R K × N is equivalent to solving the linear system ⊤ K · · · ⊤ K . . . ...... . . . . . . · · · ⊤ K I K I K · · · I K v I K v I K · · · v N I K | {z } =: F ∈ R K + N × KN vec( Q ∗ ) = bqy ∗ , (A.8) where vec( Q ∗ ) ∈ R KN denotes the vectorized version of the matrix Q ∗ , obtained by stacking itscolumns on top of one another. The null space corresponding to the matrix F is spanned by the K − N − column vectors of the matrix (w.l.o.g. v = v , cf. Assumption 3) O := v − v v − v (cid:16) − ⊤ K − I K − (cid:17) v − v v − v (cid:16) − ⊤ K − I K − (cid:17) · · · v − v N v − v (cid:16) − ⊤ K − I K − (cid:17) v − v v − v (cid:16) − ⊤ K − I K − (cid:17) v − v v − v (cid:16) − ⊤ K − I K − (cid:17) · · · v N − v v − v (cid:16) − ⊤ K − I K − (cid:17)(cid:16) ⊤ K − − I K − (cid:17) K × K − · · · K × K − K × K − (cid:16) ⊤ K − − I K − (cid:17) . . . ...... . . . . . . K × K − K × K − · · · K × K − (cid:16) ⊤ K − − I K − (cid:17) ∈ R KN × ( K − N − . (A.9) The fact that the linearly independent column vectors of O lie in the null space of F is easily checkedvia direct calculation; the fact that the dimension of the null space is equal to ( K − N − followsfrom a rank-nullity argument given in the proof of Theorem 4.2 b) below. Hence, we are able tofully characterize the set of f-efficient holding matrices as Q ∗ ∈ R K × N | vec( Q ∗ ) = vec( Q p ) + ( K − N − X j =1 λ j C j ( O ) , λ , . . . , λ ( K − N − ∈ R , with particular solution Q p as defined in (A.7) and where C j ( O ) , for j = 1 , . . . , ( K − N − ,denotes the j -th column vector of the null space matrix O . A.2.4 Proof of Theorem 4.2 a) The existence of an f-efficient holding matrix for every
N, K ≥ y ∗ = b ⊤ v ⊤ K z z , the solution of the aggregate problem (4.3) derived inLemma 4.1 with z = G − K . The mean squared deviation under an f-efficient holding matrix Q ∗ is given by: M SD ( Q ∗ ) = ( Q ∗ v ) ⊤ G ( Q ∗ v ) = ( y ∗ ) ⊤ G y ∗ = ( b ⊤ v ) ( ⊤ K z ) z ⊤ G z = ( b ⊤ v ) ( ⊤ K z ) z ⊤ GG − K = ( b ⊤ v ) ( ⊤ K z ) ( z ⊤ K ) = ( b ⊤ v ) ⊤ K z = ( b ⊤ v ) ⊤ K G − K . b) As outlined above in Remark A.1, finding an f-efficient holding matrix Q ∗ ∈ R K × N is equiva-36ent to solving the linear system (A.8). We have the following result: The rank of the matrix F is equal to 2 K + N −
2. This can be proven via standard Gaussian elimination. First,resorting the rows of F and adding the new first K rows multiplied by − v to the second K rows yields I K I K · · · I K v I K v I K · · · v N I K ⊤ K · · · ⊤ K . . . ...... . . . . . . · · · ⊤ K → I K I K · · · I K K × K ( v − v ) I K · · · ( v N − v ) I K ⊤ K · · · ⊤ K . . . ...... . . . . . . · · · ⊤ K . Numbering the rows in this last matrix as r , . . . , r K + N , we observe: r K +1 = K X i =1 r i − K + N X i =2 K +2 r i and r K +2 = 1 v − v K X i = K +1 r i − K + N X i =2 K +3 ( v i − K − v ) r i ! , where, as above, w.l.o.g. v = v (cf. Assumption 3). Hence, these two rows can be eliminatedfrom the matrix yielding the row-echelon form: I K I K I K I K · · · I K K × K ( v − v ) I K ( v − v ) I K ( v − v ) I K · · · ( v N − v ) I K ⊤ K ⊤ K · · · ⊤ K ⊤ K ⊤ K · · · ⊤ K ⊤ K ⊤ K ⊤ K · · · · · · ⊤ K . . . ...... . . . ... . . . . . . · · · · · · ⊤ K , F possesses the rank 2 K + N −
2. The rank-nullity theoremnow gives rise to the dimension of the null space (a basis is given by the column vectors ofthe matrix O in Remark A.1 above): KN = rank( F ) + null( F ) ⇒ null( F ) = KN − (2 K + N −
2) = ( N − K − . and, hence, the dimension of the null space is zero, i.e., the solution Q p stated in the proof ofProposition 4.2 is unique, if and only if either K = 1 or N = 2; i.e., if we additionally assumethat K ≥
2, then this is equivalent to N = 2. Finally, the formula for Q N =2 directly followsfrom the definition of Q p in (A.7). A.2.5 Proof of Theorem 4.3
Recall that Q diversified = T qb ⊤ = ⊤ K q qb ⊤ . Thus, it holds that y ∗ = Q diversified v ⇔ b ⊤ v ⊤ K z · z = 1 ⊤ K q q ( b ⊤ v ) b ⊤ v =0 ⇔ ⊤ K z z = 1 ⊤ K q q, i.e., if and only if q and z = G − K are linearly dependent. A.2.6 Proof of Corollary 4.1 • For the proof of statements a) and c), we first observe that if q = . . . = q K , then1 ⊤ K z z = 1 ⊤ K q q ⇔ ⊤ K z z = 1 q · K q · K ⇔ z = 1 K · K ⊤ K z ⇔ K · z = ( K ⊤ K ) z, i.e., that z is an eigenvector to the eigenvalue K of the all-one matrix K × K = K ⊤ K . Thiseigenvector is given as z = c · K for a constant c ∈ R . Hence, z k = z ℓ for all k, ℓ = 1 , . . . , K is equivalent to Q diversified being f-efficient under the assumption that q = . . . = q K .38o prove part a), we now set σ = . . . = σ K . According to the proof of Lemma 4.1, the vector z is in this situation given by z = G − K = 1 σ · K − ( σ ) · µ ⊤ K σ · µ ⊤ µ · µ. This means that the condition z k = z ℓ for all k, ℓ = 1 , . . . , K is equivalent to z k = 1 σ − ( σ ) · P Kj =1 µ j σ · P Kj =1 µ j · µ k ! = 1 σ − ( σ ) · P Kj =1 µ j σ · P Kj =1 µ j · µ ℓ = z ℓ ⇔ ( σ ) · P Kj =1 µ j σ · P Kj =1 µ j · µ k = ( σ ) · P Kj =1 µ j σ · P Kj =1 µ j · µ ℓ ⇔ ( X Kj =1 µ j ) · µ k = ( X Kj =1 µ j ) · µ ℓ , for all k, ℓ = 1 , . . . , K , i.e., µ k = µ ℓ or P Kj =1 µ j = 0. Hence, if µ = . . . = µ K − , and µ K = µ + ε , then, finally, Q diversified being f-efficient is equivalent to either ε = 0, yielding µ k = µ ℓ for all k, ℓ = 1 , . . . , K , or ε = − Kµ , yielding P Kj =1 µ j = 0.To prove part c), we set µ = . . . = µ K , which (see the proof of Lemma 4.1) leads to thevector z given as: z = G − K = 1 σ − µ · ( σ ) ⊤ K µ · ( σ ) ⊤ K · σ . Thus, the condition z k = z ℓ for all k, ℓ = 1 , . . . , K reads as z k = − µ · P Kj =1 1 σ j µ · P Kj =1 1 σ j σ k ! = − µ · P Kj =1 1 σ j µ · P Kj =1 1 σ j σ ℓ = z ℓ ⇔ σ k = σ ℓ for all k, ℓ = 1 , . . . , K . Hence, if σ = . . . = σ K − and σ K = σ + ε , this condition is fulfilledif and only if ε = 0. 39 For the remaining proof of part b), observe that if µ = . . . = µ K and σ = . . . = σ K , then z = G − K = 1 σ · K − ( µ σ ) · K µ σ K · K , in particular: z = . . . = z K . Hence:1 ⊤ K z z = 1 ⊤ K q q ⇔ z · K z · K = 1 ⊤ K q q ⇔ K ⊤ K q = K · q which is equivalent to q being an eigenvector to the eigenvalue K of the all-one matrix K × K = K ⊤ K , i.e., q = . . . = q K . Hence, if q = . . . = q K − and q K = q + ε , this conditionis equivalent to ε = 0, which completes the proof. A.3 Proofs of Section 5
A.3.1 Proof of Lemma 5.1
First, we derive the explicit formulas for Q × and d ( Q × ) in the special case N = K = 2, q = q = b = b = x . Note that, here, the unique f-efficient holding matrix is given by Q × = 1 v − v v q − y ∗ y ∗ − v q v q − y ∗ y ∗ − v q = x ( v − v )( z + z ) v z − v z v z − v z v z − v z v z − v z , due to y ∗ = x · v + v z + z · z . Since, Q diversified = x · × , it holds that Q × − Q diversified11 = x · v z − v z − ( v − v )( z + z )2( v − v )( z + z ) = x · v z − v z − v z + v z v − v )( z + z )= x · ( z − z )( v + v )2( v − v )( z + z ) = Q × − Q diversified22 , and Q × − Q diversified12 = x · v z − v z − ( v − v )( z + z )2( v − v )( z + z ) = x · v z − v z + v z − v z v − v )( z + z )40 x · ( z − z )( v + v )2( v − v )( z + z ) = Q × − Q diversified21 . Hence, d ( Q × ) = k Q × − Q diversified k F = s · x · ( z − z ) ( v + v ) v − v ) ( z + z ) = x · s ( z − z ) ( v + v ) ( v − v ) ( z + z ) . (A.10)Moreover, a direct calculation of z = G − K in the 2-by-2-case shows that Q × = x · v ( µ + σ − µ µ ) − v ( µ + σ − µ µ )( v − v )( µ + µ − µ µ + σ + σ ) , (A.11)and that d ( Q × ) = x · s ( µ − µ + σ − σ ) ( v + v ) ( v − v ) ( µ + µ − µ µ + σ + σ ) . (A.12)a) Under the conditions q = q = x and µ = µ , it holds that d ( Q × ) = 0 as a function of σ ,if and only if σ = σ (cf. Corollary 4.1), and it is strictly positive everywhere else. Hence,we can equivalently analyze the monotonicity behavior of d ( Q × ) . According to (A.12), itsderivative with respect to σ is given by ∂∂σ d ( Q × ) = 8 σ σ ( σ − σ )( v + v ) x ( σ + σ ) ( v − v ) < , if σ < σ ,= 0 , if σ = σ , > , if σ > σ ,for σ > x > v > v >
0. The derivative of (A.11) with respect to σ is given by ∂ Q × ∂σ = x · σ ( µ + σ − µ µ )( v + v )( v − v )( µ + µ − µ µ + σ + σ ) µ = µ = x · σ σ ( v + v )( σ + σ ) ( v − v ) > , for σ > .3.2 Proof of Lemma 5.2 a) As in the proof of Lemma 5.1, d ( Q × ) as a function of µ is non-negative and, under thegiven assumptions, strictly positive except for the case µ = µ (cf. Corollary 4.1). Hence, wecan again equivalently analyze the monotonicity behavior of the squared distance d ( Q × ) .Its derivative with respect to µ is given by ∂∂µ d ( Q × ) = x · µ − µ + σ − σ )( − µ σ + µ ( µ + µ − µ µ + σ − σ ))( v + v ) ( µ − µ µ + µ + σ + σ ) ( v − v ) σ = σ ,µ =0 = x · µ σ ( v + v ) ( µ + 2 σ ) ( v − v ) < , if µ < , if µ = 0, > , if µ > µ is given by ∂∂µ Q × = x · ( − µ σ + µ ( µ + µ − µ µ + σ − σ ))( v + v )( µ + µ − µ µ + σ + σ ) ( v − v ) σ = σ ,µ =0 = x · − µ σ ( v + v )( µ + 2 σ ) ( v − v ) > , if µ < , if µ = 0, < , if µ > v > v > x > σ = σ and µ = 0. A.3.3 Proof of Lemma 5.3
The derivative of (A.10) with respect to v is given as ∂∂v d ( Q × ) = x · v ( v + v )( v − v ) · s ( v + v ) ( z − z ) ( v − v ) ( z + z ) . Under the given assumptions x, v > | z | 6 = | z | , and v = v , this term is strictly positive for v >
0, if v < v and strictly negative if v > v . This is the statement of the lemma.42 ppendix B Non-Uniqueness of Asset Holdings We provide an example to show how the interplay between systemic significance and asset riski-ness influences the structure of f-efficient holdings. We also discuss the intuition behind the non-uniqueness of holdings when we move from an economy with N = 2 banks to one with N > N = 3 banks and K = 3 assets, where we normalize the totalsupply of assets within the banking sector and budgets of banks to q = q = q = b = b = b = x with x := 0 .
08. The total supply of each asset is normalized to 1. Asset 1 constitutes the leastand asset 3 the most risky asset: µ = (0 , , ⊤ and σ = (0 . , . , . ⊤ . The three banks aredifferent in their systemic sensitivity parameters v = (0 . , . , . ⊤ , i.e., bank 1 is the most andbank 3 the least significant to the system. According to Theorem 4.2 b), f-efficient holdings in thisfinancial system are not unique. Every f-efficient holding matrix is of the form Q ∗ + λ − − −
10 0 0 + λ − − − , λ , λ ∈ R , (B.1)where Q ∗ = x ·
23 13
13 13 13
13 23 is a particular solution: the f-efficient holding matrix with the smallest Frobenius distance to fullydiversified holdings Q diversified = x · · × . Hence, Q ∗ represents a lower bound on how farholdings need to move away from the classical diversification benchmark in order to become f-efficient. Setting λ = x · / λ = − x · / Q ∗ = x · , Q diverse = x · I . Note, first, that this type of diverse holdings are only defined in the case N = K . Second, observe that holdings with the smallest distance to diversity do not maximize thedistance to diversification. The example provides the following insights: • In both Q ∗ and Q ∗ , the most risky asset 3 is held in the largest proportion by the leastsystemically significant bank 3. Conversely, the least risky asset 1 is held in the largestproportion by the most significant bank 1.This can be generalized as follows: For every f-efficient holding matrix (see (B.1)), the holdingsof the most significant bank 1 in the least risky asset 1 ( x · / λ + λ ) are larger than theholdings of the least significant bank 3 in this asset ( λ + λ ). Conversely, for every f-efficientmatrix, the holdings of the least significant bank 3 in the most risky asset 3 ( x · / − λ ) arelarger than bank 1’s holdings in this asset ( − λ ). • Note that the null space in Equation (B.1) is equivalently written as (cf. Remark A.1 inAppendix A): λ − ( v − v ) v − v − ( v − v ) v − v − ( v − v ) v − v + λ − ( v − v ) v − v − ( v − v )0 0 0 v − v − ( v − v ) v − v for λ , λ ∈ R . Hence, any transfer between two assets in each bank’s holdings that is donesomewhat proportionally to the differences between the systemic significances (of the othertwo banks) does not alter the mean-squared deviation. This is the reason for the uniquenesscriterion we found: If there are only two banks in the system, there simply are no “other twobanks”, and thus, there are no transfers which are neutral with respect to the mean-squareddeviation. Since λ , λ in formula (B.1) are unbounded, the distance to diversification within the set of f-efficient holdingsis unbounded. ppendix C f-Efficient Liquidation Strategies C.1 General Derivation
We want to minimize the mean squared deviation (3.4) as a function of α = (cid:18) α | · · · | α N (cid:19) ∈ R K × N , for α i ∈ R K with ⊤ K α i = 1 and α i ≥ K for all i = 1 , . . . , N. Thus, each bank i is allowed to choose its own personal liquidation strategy α i and we do not allowfor short-selling. In the following lemma, we rewrite the minimization problem as a function ofvec( α ) := ( α ⊤ , . . . , α N ⊤ ) ⊤ ∈ R KN , i.e., vec( α ) denotes the vectorization of the matrix α . Lemma C.1.
Minimizing the mean squared deviation as a function of the liquidation strategymatrix α with α ≥ is equivalent to the following problem: min vec( α ) ∈ R KN vec( α ) ⊤ ( C ⊗ Q tot γ ◦ Q nb0 ( Q tot γ ◦ Q nb0 ) ⊤ )vec( α ) (G) s.t. ⊤ K . . . ⊤ K vec( α ) = N , vec( α ) ≥ KN , where C = ( C ij ) i,j =1 ,...,N ∈ R N × N with C ij := 2( Q Diag( κ ) e i ) ⊤ ( µµ ⊤ + Diag( σ ))( Q Diag( κ ) e j ) , e i ∈ R N denotes the i ’th basis vector (i.e., e ij = 1 for j = i and zero, otherwise), and ⊗ denotesthe Kronecker product .Proof. It holds that Q Diag( κ ) α ⊤ Q tot γ ◦ Q nb0 = Q Diag( κ ) · ( N P i =1 ( α i ⊤ Q tot γ ◦ Q nb0 ) e i ) . Inserting this expression For two matrices A ∈ R M × N , B ∈ R P × R , the Kronecker product is defined by multiplying every entry of thematrix A by the entire matrix B , i.e., A ⊗ B := A B · · · A N B ... .. . ...A M B · · · A MN B ∈ R MP × NR . M SD ( α , . . . , α N ) = ( N X i =1 ( α i ⊤ Q tot γ ◦ Q nb0 )) Q Diag( κ ) e i ! ⊤ ( µµ ⊤ + Diag( σ )) ( N X j =1 ( α j ⊤ Q tot γ ◦ Q nb0 )) Q Diag( κ ) e j = N X i =1 N X j =1 (cid:16) ( α i ⊤ Q tot γ ◦ Q nb0 ) Q Diag( κ ) e i (cid:17) ⊤ ( µµ ⊤ + Diag( σ )) (cid:16) ( α j ⊤ Q tot γ ◦ Q nb0 ) Q Diag( κ ) e j (cid:17) = N X i =1 N X j =1 α i ⊤ Q tot γ ◦ Q nb0 ( Q tot γ ◦ Q nb0 ) ⊤ α j ( Q Diag( κ ) e i ) ⊤ ( µµ ⊤ + Diag( σ ))( Q Diag( κ ) e j )= 12 N X i =1 N X j =1 α i ⊤ ( C ij · Q tot γ ◦ Q nb0 ( Q tot γ ◦ Q nb0 ) ⊤ ) α j . This proves the formula for the objective function. The linear constraint follows from ⊤ K α i = 1 forall i = 1 , . . . , N .The following proposition now provides a locally f-efficient liquidation strategy. Its interpreta-tion is given in Remark C.1. Proposition C.1.
Let m := max k ∈{ ,...,K } γ k Q k, nb0 Q k tot denote the maximum entry in the vector γ ◦ Q nb0 Q tot and denote by k m = { k ∈ { , . . . , K } | γ k Q k, nb0 Q k tot = m } the corresponding index set with cardinality k m . For every fixed Q ∈ R K × N , a local minimizer ofthe mean squared deviation as a function of the liquidation strategy matrix is given by the matrix α ∗ which is defined by its columns: α ik ∗ := | k m | , if k ∈ k m , , otherwise, ( k = 1 , . . . , K, i = 1 , . . . , N ) . Proof.
It is easily checked that the triplet (vec( α ∗ ) , λ ∗ , s ∗ ), defined as follows, solves the KKTconditions belonging to the optimization problem (G): vec( α ∗ ) = ( α ∗⊤ , . . . , α N ∗⊤ ) ⊤ as defined in46roposition C.1, λ ∗ = ( λ ∗ , . . . , λ ∗ N ) ∈ R N and s ∗ = ( s ∗ , . . . , s N ∗ ) ∈ R KN , where λ ∗ i = P Nj =1 C ij m , s i ∗ = P Nj =1 C ij m ( Q tot γ ◦ Q nb0 − m K ) , for all i = 1 , . . . , N. Next, we need to check f-efficiency. Let g ik ( α ) := − α ik , and h i ( α ) := P Kk =1 α ik − g ik ( α ) ≤ h i ( α ) = 0 for all i = 1 , . . . , N and k = 1 , . . . , K . Let d = ( d ⊤ , . . . , d ⊤ N ) ∈ R KN ,where d i ∈ R K for all i = 1 , . . . , N and define F ( α ) := { d = | ∇ g ik ( α ) ⊤ d = 0 , k / ∈ k m ≤ , k ∈ k m , and ∇ h i ( α ) ⊤ d = 0 , ∀ i ∈ { , . . . , N } , ∀ k ∈ { , . . . , K }} . Applying the second order sufficiency conditions (cf. Theorem 5.2 in Freund (2016)), the KKTpoint (vec( α ∗ ) , λ ∗ , s ∗ ) constitutes a local minimum if for all d ∈ F ( α ∗ ) it holds that d ⊤ ( C ⊗ ( Q tot γ ◦ Q nb0 )( Q tot γ ◦ Q nb0 ) ⊤ ) d >
0. We have F ( α ) = { d = | d ik = 0 , k / ∈ k m ≥ , k ∈ k m , and K X k =1 d ik = 0 , ∀ i ∈ { , . . . , N }} = ∅ for all α and, hence, the second order sufficiency condition is always fulfilled. Thus, (vec( α ∗ ) , λ ∗ , s ∗ )constitutes a local minimum. Remark C.1. • Note that we may characterize liquidity of asset k by its product of elasticityand supply in the nonbanking sector weighted by total supply, i.e., by γ k · Q k, nb0 /Q k tot . Propo-sition C.1 thus shows that the f-efficient liquidation strategy of banks is given by selling solelythe most liquid asset. We will refer to this f-efficient strategy as the most-liquid-strategy . • Note that the most-liquid-strategy depends on (the row sums of ) the holding matrix Q sinceit holds that Q k, nb0 = 1 − P Ni =1 Q ki for all assets k = 1 , . . . , K . • In the special case that we ex ante assume that all banks follow the same liquidation strategy,the most-liquid-strategy even constitutes a global minimizer of the mean squared deviation.This example is analyzed in Appendix C.2. .2 Liquidation Strategy Example In this case study, we ex ante assume that all banks act homogeneously in that they liquidate theirportfolios in the exact same way. This assumption leads to the following structure of the liquidationstrategy matrix: α = ( e α | · · · | e α ) , for a vector e α ∈ R K , with ⊤ e α = 1, specifying the banks’ liquidation of each asset. For the meansquared deviation, this leads to the equation M SD ( e α, Q ) = ( e α ⊤ Q tot γ ◦ Q nb0 ) · ( Q κ ) ⊤ ( µµ ⊤ + Diag( σ ))( Q κ ) . For a fixed given holding matrix Q , we define an f-efficient bank-independent liquidation strategy e α as a minimizer of M SD ( · , Q ) over all e α ∈ R K ≥ with ⊤ e α = 1. We have the following result. Proposition C.2.
For every fixed Q ∈ R K × N , the most-liquid-strategy constitutes a globally f-efficient bank-independent liquidation strategy.Proof. Minimizing
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