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We propose a risk modeling framework for financial portfolios that integrates market risk with liquidation costs which may arise in stress scenarios. Our model provides a systematic method for computing liquidation adjusted risk measures for a portfolio. Calculation of Liquidation-adjusted VaR for real and simulated portfolios reveal a substantial impact of liquidation costs on VaR for portfolios with large concentrated positions.

1 Introduction

Quantitative models commonly used in financial risk management have mainly focused on the statistical modeling of variations in the (mark-to-)market value of financial portfolios, in order to estimate a risk measure { such as Value-at-Risk or Expected shortfall{ over a given time horizon. These risk measures are then used for determining capital requirements, margin requirements, reserves, in order to provision for losses in extreme risk scenarios. Typically, when such losses materialize, the financial institution is led to liquidate a sizable portion of its portfolio and the realized liquidation value may be quite different from the (pre-liquidation) market value used in the model. The difference {the liquidation cost {can be significantly if the portfolio contains large, concentrated positions. Not accounting for this liquidation cost in risk calculations may result in a serious underestimation of portfolio losses in a stress scenario. Several risk management _ascos have been associated with the miscalculation of risk for large positions. When the institutions tried to unwind their positions, the realized losses were much larger than what their risk models had anticipated. A spectacular example was provided by the JP Morgan CIO losses in 2012, when the bank sued a $ 6.2 billion loss while unwinding CDS index positions amounting to several hundred billions of dollars in gross notional [U.S. Senate, 2013, JP Morgan, 2014]. These considerations call for a comprehensive approach for integrating liquidation value into portfolio risk measures; this issue is particularly relevant for financial institutions managing large portfolios.

2 A model for liquidation Losses

Consider a portfolio with positions in n assets classes, whose values at date tk = kΔt are denoted S1k,...,Snk . We assume that, in the absence of systematic effects from large trades, the \fundamental" return of asset I between tk and tk+1 is equal to a random variable pΔt_ik+1 +miΔt, where mi is the 1Electronic copy available at: http://ssrn.com/abstract=2739227 drift of asset i and the sequence of random vectors (_k+1)k_0 is iid, with mean 0 and covariance matrix _: Sik+1􀀀 SikSik= ϵik+1 =pΔt_ik+1 + miΔt where E(_ik) = 0; Cov (_ik; _jk) = _i;j : We call the fundamental covariance matrix: it captures the structural relations between asset returns. We assume that is slowly varying over the period of interest. In the examples below, _ will be taken constant but in practice one may also incorporate some dynamics for _ in our model. In absence of market impact, the value of a buy-and-hold portfolio with _i shares of asset i, i = 1::n is given by: Vk=Σni=1_iSik and changes according to Vk+1 􀀀 Vk=Σni=1_iSik ϵi k+1 Institutional portfolios are often subject to a constraint {capital requirement, liquidity ratio, leverage constraint, performance constraint. If the portfolio is subject to a large loss in asset values, the constraint maybe breached, in which case the portfolio maybe deleveraged, i.e. some assets need to be sold over a shorttime period in order to comply with the constraint. This is the phenomenon of distressed asset sales or_re sales [Cont and Wagalath, 2013, Shleifer and Vishny, 2011]. Consider for example the case of a portfolio with initial equity /capitalE and subject to a leverage constraint Lmax; representing the maximum allowed leverage ratio. Initially VE_ Lmax: If the event of a loss of l (%), the leverage ratio increases from VE toV(1 􀀀 l) E 􀀀 l V It is then straight forward to show that if the loss exceeds a threshold l_=1VELmax 􀀀 VLmax 􀀀 1< l < EV then the leverage constraint Lmax is breached and the fund needs to deleverage, i.e. liquidate a portion 1􀀀 Lmax(E􀀀lV) V (1􀀀l) of its holdings. The volume of assets sold thus depends on the magnitude of the loss: Figure 1 shows the portion of the fund liquidated for a fund with initial leverage VE = 25 and leverage limit Lmax= 33 as a function of the loss l (blue line). In practice, the deleveraging policy may deviate from this simple linear example, but it qualitative features remain valid: the volume of asset sales is zero for losses below a threshold, increases with the loss size and saturates beyond a certain loss level (which represents total liquidation). We represent this through a response function f, which represents the proportion of the portfolio which is deleveraged, as a function of the portfolio loss (red curve in Figure 1).Thus, the fraction of the fund liquidated at period k in response to price moves isf(VkV0)􀀀 f(VkV0+Σni=1_iSikV0ϵik+1). We assume that assets are liquidated proportionally to the initial holdings (this assumption maybe relaxed, see next section). Liquidation of large quantities of assets has an impact on the market price: assuming this impact to be linear [Cont et al., 2014, Kyle, 1985, Obizhaeva, 2012], this leads to the following price dynamics, where the new 2

Figure 1: Example of fund liquidation as a function of fund losses terms correspond to the price impact of deleveraging where D_i represents the market depth for asset class i, estimated using the methodology proposed in [Obizhaeva, 2012]. Equation (1) gives a decomposition of the asset returns into a "fundamental" component and an endogenous -or self induced{ component which is generated by the fund's own deleveraging and depends on asset liquidity. This endogenous component is zero in 'normal' scenarios, but when the portfolio experiences large losses leading it to liquidate par of its holdings this term may become non-zero, generating larger-than-expected portfolio losses and an increase in observed correlations, as described below.

            As shown in [Cont and Wagalath, 2013], as ∆t goes to 0, the multi-period model described above converges (weakly) to a continuous-time limit described by a multi asset diffusion ('local volatility') model where the drift _ⁱ_t and the instantaneous covariance are given by where is the dollar allocation of the represents the positions we have denoted by the transpose of a scenario, the dependence structure of asset returns is (temporarily) modified: the realized covariance matrix is equal to the fundamental covariance matrix  plus a (liquidity-dependent) excess covariance term which depends on the volume of assets liquidated in each asset class relative to market depth. This generates larger-than expected losses and realized volatility for the fund, precisely in bad scenarios where it is compelled to engage in fire sales.

3. Liquidation adjusted VaR

As shown above, if a portfolio has large positions (relative to market depth), one cannot ignore the impact of possible liquidations on market dynamics when assessing the portfolio's risk. This impact is size dependent and, unlike usual risk calculations based on VaR or Expected Shortfall to a nonlinear scaling of portfolio risk with the size of its positions. Our model provides a systematic approach for taking into account liquidation risk when assessing the risk of a portfolio. As the following example shows, the resulting adjustments to portfolio risk can be quite substantial. Consider for instance a portfolio, with leverage constraint L_max = 33, initial leverage 25 and positions in three asset classes with independent returns, assumed to be Gaussian with respective annualized volatilities 10%, 20% and 30%. The market depths for these asset classes are taken to be $1,000, $100 and $10 Bn respectively. a benchmark, the estimated market depth for the SPY, the main ETF tracking the S&P500, is close to $1,000 Bn. We simulate a loss distribution for this portfolio using our model (1) and define the Liquidation-adjusted VaR as the 99% quantile of this loss distribution. Figure 2 shows that the price impact of liquidations leads to a fat tailin the loss distribution. Figure 3 displays the one-day 99% Value- at-Risk for the portfolio as a function portfolio size, when all notional positions are increased proportionally. Here, Value at-Risk is calculated over 10,000 scenarios and can be compared to a benchmark Value-at-Risk, based on (2), as commonly calculated by financial institutions.

            Whereas the traditional benchmark VaR is, as expected, linear in portfolio size, the liquidation-adjusted VaR computed using our model is not: it is convex as a function of portfolio size and is much larger than a linear VaR for large portfolio. The difference between the two numbers reflects the liquidity risk of the portfolio. For a portfolio with small positions relative to market depth, liquidity-adjusted VaR is close to a traditional VaR measure. However, for a leveraged portfolio with large, concentrated positions comparable to or larger than market depth, liquidation-adjusted VaR can be significantly (in our example, up to 10 times) larger than the usual VaR. The previous calculations are based on the assumption of proportional liquidations. In practice, financial institutions may choose other deleveraging strategies.

Figure 6 comparesthe 95% 5-month Value-at-Riskwithandwithoutliquidation adjustments, forapositioninCDX IG9, as afunctionof gross notional. Foranotional valueofaround $278 billion inCDX IG9, whichForwithisthesizeofthe London Whale’s positionsinQ1 2012, wefindaliquidation-adjusted Value-at-Risklargerthan $12 Bn, significantlylargerthanthe benchmarkValue-at-Risk, whichisoftheorderof $2 Bn. thedifferencebeingentirely attributabletoliquidation costs.Thissuggeststhatthe ”London Whale” lossescouldhavebeenanticipatedinamorerealisticmannerifliquidation costshadbeenproperlyaccountedforintheriskcalculations. 5 ConclusionWehaveproposedatractablemodelingframeworkfor  includingliquidation costsintheriskanalysisoffinancialportfolios: ourapproachconsistsinmodelingthe impactofliquidationondemand/supplyviaaresponse functionandaddingapriceimpacttermbasedonthisresponsefunctiontothebasemodeldescribingthedynamicsofas- 0 2 4 6 8 10 12 14 16 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 VaRforpositionsin CDX IG 9 Liquidity-AdjustedVaR ($ billion) Benchmark VaR ($ billion) Grossnotional ($ billion) Figure 6: 95% 5-month Value-at-riskfor positionsin CDX IG9 setreturns. Theseingredientsareblendedintoanoperationalframeworkforcalculatingportfolioriskmeasureswhichcorrectlyaccountforliquidation costs, exhibitnonlinearscalingwithportfoliosizeanddistinguishbetweenliquidandilliquid positions. In the particularexamplesdescribedabove, the’base model’isa (Black-Scholes) modelwithconstantcoefficientsandIID returnsandthe priceimpactislinear, butthisissimplyanexampleandboththeseingredientsmaybemodified.

For example, one may establish a pecking order in which various asset classes are liquidated; this is part of the Basel 3 requirements for financial institutions when establishing their ”living will”. For any such liquidation strategy, one may repeat the same analysis as above, and calculate a liquidation-adjusted loss distribution, which will then depend on the chosen liquidation strategy. Figure 4 shows that the resulting (liquidation-adjusted) portfolio risk depends on the chosen exit strategy. In this example, we see, as expected intuitively that it is more (resp. less) optimal to liquidate more (resp. less) liquid assets first than implementing proportional liquidations. This suggests, not surprisingly, that the risk posed by a stress scenario to financial institution depends on its plans for dealing with this stress scenario. In 2012, J P Morgan’s Chief Investment Officer (CIO) experienced losses over  $6billion while unwinding CDS indexpositions.  A major part of those losses materialized while liquidating positions in the IG9 CDS Index. According to the Senate report [U.S. Senate, 2013, JP Morgan, 2014], the positions of the London Whale inCDX IG9 reached $278billion in gross notional value, which accounted for fourty times the average daily volume for the index. Liquidation of the London Whale’s portfolio occurred between end of March and August 2012 and was expected to generate losses of around $500million, while in fact resulting in reported losses of $6.2billion. Reports on the CIO losses have focused on mismanagement, lack of transparency inside the organization, mismarking of positions and spread sheet errors [U.S. Senate, 2013]. But the way risk was computed and provisioned for was not the main focus of recommendations in any of the reports. Yet, according to the report by JP Morgan’s Management Task Force [JP Morgan, 2014], the risk of these positions was evaluated using a Value-at-Risk metric which scales linearly with portfolio size and liquidation risk was not provisioned. Figure 5 shows the one-year realized correlation between CDX IG9 and CDX IG10, two closely related indices, calculated on a rolling window. The breakdown of this correlation after March 2012, i.e.  when the CIO starts liquidating its massive positions in CDX IG9, is evidence of the market impact of CIO’s trading and suggests that the concepts of endogenous risk and liquidation-adjusted risk measureare relevant for this case. The liquidation of the portfolio lasted around 5months [U.S. Senate, 2013]. We use the length of the liquidation period, together with the size of the position, to calibrate the slope f(.) in (2).