Risk Management for Whales Rama Cont and Lakshithe Wagalath - - PowerPoint PPT Presentation

Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion

Risk Management for Whales

Rama Cont and Lakshithe Wagalath

Imperial College, London and IESEG School of Management, Paris

Rama Cont and Lakshithe Wagalath Risk Management for Whales

Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion

Plan

1 Introduction and motivation 2 Modeling liquidation losses and liquidation-adjusted VaR 3 Taming the Whale 4 Conclusion

Rama Cont and Lakshithe Wagalath Risk Management for Whales

Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion

Statistical risk measures for portfolios

Current risk management frameworks focus on modeling/hedging the variation in Mark-to-Market P&L of portfolios. A common approach is to use a statistical model to estimate a risk measure (typically VaR or Expected Shortfall) for P&L distribution

• ver a given risk horizon.

Regardless of their complexity, such risk measures have a positive homogeneity property and hence scale linearly with portfolio size: if notionals are multiplied by 100, “risk” is multiplied by 100.

Rama Cont and Lakshithe Wagalath Risk Management for Whales

Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion

Statistical risk measures for portfolios

The computation of risk-measures enables to compute capital requirements, margin requirements, ..., to face losses in extreme risk scenarios. Examples:

capital requirements for banks margin requirements for derivatives portfolios counterparty risk evaluation

Rama Cont and Lakshithe Wagalath Risk Management for Whales

Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion

Liquidation risk vs market risk

Such losses materialize in risk scenarios where one is forced to liquidate a sizable portion of the portfolio. This points to another, quite different, concept from market value: the liquidation value in a stress scenario. The purpose of margin and collateral requirements is to cover losses

• f the exchange/ clearinghouse in case a clearing member defaults.

To fulfill this purpose, their level should cover liquidation costs in such scenarios.

Rama Cont and Lakshithe Wagalath Risk Management for Whales

Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion

Liquidity risk: under the radar?

Currently margin requirements, counterparty exposures, collateral requirements ... are computed based on a measure of market risk (VaR, ES,...) of the position. No consideration is given to the liquidity of instruments or the size

• f positions relative to market depth.

How can the result reflect potential liquidation costs if it does not use any liquidity measure or market depth as input? The result is a serious, and massive, underestimation of portfolio exposures to liquidity risk.

Rama Cont and Lakshithe Wagalath Risk Management for Whales

Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion

Taking liquidation risk seriously

Several recent risk management fiascos 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. These considerations call for a comprehensive approach for integrating liquidation value into risk measures. Such an approach should:

differentiate between positions across assets of varying liquidity account for limited market depth be scalable to complex, multi-asset portfolios with derivatives

Rama Cont and Lakshithe Wagalath Risk Management for Whales

Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion

Example of endogenous liquidation

Consider a fund with initial size V , equity E and subject to a leverage constraint Lmax:

V E ≤ Lmax.

If the fund’s portfolio experiences a percentage loss l, then its leverage ratio increases to V (1−l)

E−lV .

If l∗ = 1

V ELmax−V Lmax−1 < l < E V then V (1−l) E−lV > Lmax

The fund violates its leverage constraint. Two possibilities:

increase equity → costly after a shock and not ideal due to the debt

• verhang problem

decrease the size of assets → liquidation of a portion 1 − Lmax (E−lV )

V (1−l)

• f its holdings

Rama Cont and Lakshithe Wagalath Risk Management for Whales

Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion

Example of endogenous liquidation

0,2 0,4 0,6 0,8 1 1,2 0,00% 0,30% 0,60% 0,90% 1,20% 1,50% 1,80% 2,10% 2,40% 2,70% 3,00% 3,30% 3,60% 3,90% 4,20% 4,50% 4,80% 5,10% 5,40% 5,70% 6,00% Fund liquidation portion of the fund liquidated portion of the fund liquidated (smoothed) Fund percentage loss

Figure: Portion of the fund liquidated as a function of the percentage loss for a fund with initial leverage ratio V

E = 25 and leverage constraint Lmax = 33

Rama Cont and Lakshithe Wagalath Risk Management for Whales

Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion

Plan

1 Introduction and motivation 2 Modeling liquidation losses and liquidation-adjusted VaR 3 Taming the Whale 4 Conclusion

Rama Cont and Lakshithe Wagalath Risk Management for Whales

Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion

Discrete-time setting: fundamental asset returns

Time step ∆t (typically one day) n assets: value of asset i at date k∆t denoted Si

k

Fundamental return of asset i at period k: Si

k+1 − Si k

Si

k

= ǫi

k+1 =

√ ∆tξi

k+1 + mi∆t

with mi the drift of asset i and (ξk+1)k≥0 a sequence of iid centered random variables such that: Cov(ξi

k, ξj k) = Σi,j

Σ fundamental covariance matrix: captures a structural relationship between asset returns.

Rama Cont and Lakshithe Wagalath Risk Management for Whales

Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion

Discrete-time setting: the fund

Fund with positions α1, ..., αn Benchmark fund value: Vk =

n

• j=1

αjSj

k

Buy-and-hold strategy and no price impact: Vk+1 − Vk =

n

• j=1

αjSj

kǫj k+1

Rama Cont and Lakshithe Wagalath Risk Management for Whales

Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion

Discrete-time setting: fund liquidations

Funds are often subject to a constraint: capital requirement, liquidity ratio, leverage constraint, performance constraint... Loss in asset values → breach of constraint → portfolio deleveraging Portion of the fund liquidated in response to price moves: f Vk V0

• − f
• Vk

V0 +

n

• i=1

αiSi

k

V0 ǫi

k+1

• Assumption of proportional liquidation and linear price impact (Di

market depth of asset i)

Rama Cont and Lakshithe Wagalath Risk Management for Whales

Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion

Discrete-time setting: price dynamics

New price dynamics due to price impact from portfolio liquidation:

Si

k+1 − Si k

Si

k

= ǫi

k+1

• Fundamental return

− αi Di

• f

Vk V0

• − f
• Vk

V0 +

n

• j=1

αjSj

k

V0 ǫj

k+1

• Feedback from liquidation

When the fund value does not drop below a threshold → no fund liquidation and no price impact. In stress scenarios, the deleveraging process may generate significant deviations of price dynamics from fundamentals.

Rama Cont and Lakshithe Wagalath Risk Management for Whales

Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion

Continuous-time price dynamics

Theorem 1 When ∆t goes to 0, the multi-period model converges weakly to a continuous-time limit described by a multi-asset diffusion model: dPi

t

Pi

t

= µi

tdt + (σtdWt)i

1 ≤ i ≤ n where the drift µi

t and the instantaneous covariance ct = σtσ′ t are given by

µi

t = mi + Λi

2 f ′′ Vt V0 < πt, Σπt > V 2 ct = σtσ′

t = Σ + 1

V0 f ′ Vs V0

• [Λπ′

sΣ + ΣπsΛ′] + 1

V 2 (f ′)2 Vs V0 π′

sΣπs

• ΛΛ′

where πt =

• α1P1

t , ..., αnPn t

′ is the dollar allocation of the portfolio and Λ =

• α1

D1 , ..., αn Dn

′ represents the positions expressed as a fraction of market depth.

Rama Cont and Lakshithe Wagalath Risk Management for Whales

Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion

Realized correlations

In a liquidation scenario, realized correlations/covariances are temporarily modified. Realized covariance matrix = fundamental covariance + liquidity and path-dependent excess covariance. When the fund’s positions are large compared to asset market depth, the deviation from “fundamentals” may be significant. → generates larger-than-expected losses and realized volatilities for the fund, precisely in stress scenarios.

Rama Cont and Lakshithe Wagalath Risk Management for Whales

Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion

Numerical results: loss distribution

−0.10 −0.05 0.00 0.05 0.10 0.15 1000 2000 3000 4000 5000 6000 7000 8000 9000

Figure: Distribution of portfolio losses with (blue) and without (red) feedback from liquidations.

Rama Cont and Lakshithe Wagalath Risk Management for Whales

Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion

Numerical results: liquidation-adjusted VaR

4 8 12 16 20 10 20 30 40 50 60 70 80 90 100 110 120 Value-at-risk ($ billion) Liquidity-Adjusted VaR Benchmark VaR

Fund size ($ billion)

Figure: Liquidation-adjusted 99% 1 day Value-at-Risk for a sample portfolio with 3 asset classes.

Rama Cont and Lakshithe Wagalath Risk Management for Whales

Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion

Realized correlations

Feedback from fund liquidations in stress scenarios generate a fat tail in the loss distribution. Value-at-Risk is no more linear but convex in portfolio size. The difference between liquidation-adjusted VaR and “benchmark” Var reflects the liquidity risk of the portfolio. As long as the portfolio is not too large, both VaR are close. For a portfolio with large positions (compared to asset market depths), liquidation-adjusted VaR can be significantly larger then benchmark VaR (up to 10 times, using reasonable liquidity parameters)

Rama Cont and Lakshithe Wagalath Risk Management for Whales

Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion

Impact of the exit strategy on liquidation-adjusted VaR

10 20 30 40 50 60 10 20 30 40 50 60 70 80 90 100 110 120 Liquidation strategy 3 Liquidation strategy 2 Liquidation strategy 1 Value-at-risk ($ billion) Fund size ($ billion)

Figure: Liquidation-adjusted Value-at-Risk for different liquidation strategies. 1: liquidating most liquid asset first. 2: proportional liquidation. 3: liquidating least liquid asset first.

Rama Cont and Lakshithe Wagalath Risk Management for Whales

Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion

Plan

1 Introduction and motivation 2 Modeling liquidation losses and liquidation-adjusted VaR 3 Taming the Whale 4 Conclusion

Rama Cont and Lakshithe Wagalath Risk Management for Whales

Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion

CIO losses

In 2012, JP Morgan’s Chief Investment Office (CIO) experiences

The positions built up by the CIO were complex combinations of long short of various CDS indices of various maturities, over gross notionals of hundreds of billions of dollars. Due to the size of those positions, the trader who built up this portfolio was called the London Whale. Most of the losses materialized while liquidating positions in CDX IG9 index.

Rama Cont and Lakshithe Wagalath Risk Management for Whales

Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion

Underestimating liquidity risk

According to the senate report, positions in CDX IG9 amounted to

Risk of positions was monitored using traditional risk indicators (VaR, CS01, DV01) which all scale linearly with size. Liquidation of the London Whale’s portfolio was expected to generate losses of

losses of

Reports on the CIO losses have focused on mismanagement, lack of transparency inside the organization, mismarking of positions and spreadsheet errors. But the way risk was computed and provisioned for was not the focus of recommendations in any of the reports.

Rama Cont and Lakshithe Wagalath Risk Management for Whales

Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion

Liquidation and correlation breakdown

0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1

Figure: 1-year realized correlation between CDX IG9 and CDX IG10 returns

Rama Cont and Lakshithe Wagalath Risk Management for Whales

Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion

Liquidation-Adjusted VaR for positions in CDX IG9

2 4 6 8 10 12 14 16 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 VaR for positions in CDX IG 9 Liquidity-Adjusted VaR ($ billion) Benchmark VaR ($ billion) Gross notional ($ billion)

Figure: 95% 5-month Value-at-risk for positions in CDX IG9

Rama Cont and Lakshithe Wagalath Risk Management for Whales

Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion

Accounting for liquidity risk

Using our framework, we find a liquidation-adjusted VaR for a

Bn gross notional position in CDX IG9 equal to

than the benchmark VaR (

Difference between liquidation-adjusted VaR and benchmark VaR due to liquidation costs. London Whale losses could have been anticipated if liquidity had been taken into account in risk calculation.

Rama Cont and Lakshithe Wagalath Risk Management for Whales

Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion

Plan

1 Introduction and motivation 2 Modeling liquidation losses and liquidation-adjusted VaR 3 Taming the Whale 4 Conclusion

Rama Cont and Lakshithe Wagalath Risk Management for Whales

Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion

Conclusion

Tractable modeling framework for including liquidation costs in the risk analysis of financial portfolios. This enables to distinguish between liquid and illiquid positions and to account for liquidation risk when calculating the “risk” of a portfolio. Liquidation-adjusted VaR does not scale linearly with portfolio size. Framework (Black & Scholes base model, constant covariance matrix, linear price impact) may be refined and associated to existing benchmark risk-management framework to account for liquidity risk.

Rama Cont and Lakshithe Wagalath Risk Management for Whales

Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion

Research papers

“Risk management for whales”, with Rama Cont, 2016, working paper “Fire sales forensics: measuring endogenous risk”, with Rama Cont, 2016, to appear in Mathematical finance “Running for the exit: distressed selling and endogenous correlation in financial markets”, with Rama Cont, 2013, Mathematical finance, 23 (4): 718:741

Rama Cont and Lakshithe Wagalath Risk Management for Whales