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**Identifying Model Risk under Basel 2.5 and the FRTB**

Risk Week Cambridge 23-26 September 2014 Péter Dobránszky Disclaimer: The contents of this presentation are for discussion purposes only, represent the presenter’s views only and are not intended to represent the opinions of any firm or institution. None of the methods described herein is claimed to be in actual use.

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**Generic credit spread curves by a fully cross-sectional method**

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**Evolution of risk aversion premium**

Negative risk premium may appear nowadays for HY

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**Dynamics of the generic credit spread curves**

ln 𝑠 𝑖,𝑡 𝐿𝐺𝐷 𝑖,𝑡 = 𝛼 𝑡 + 𝛽 𝑡 ∙ 𝑑 𝑅 𝑖,𝑡 + 𝜎 𝑡 ∙ 𝜀 𝑖,𝑡 The 𝛼, which is the key driver of the CDS spreads and can be interpreted as market activity rate, is highly correlated with the main implied volatility levels structural model. The 𝛽 which can be interpreted as risk aversion premium, shows a fundamental market structure change as a result of the subprime crisis. The 𝜎, which is the residual volatility, shows that the confidence level of the generic spread estimates is moderate / low. Volatility within a rating is less than 1 notch level difference.

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**Dynamics of the generic credit spread curves**

Variance(1-day Caa3 ret.) = Variance(10-day Caa3 ret.) = Variance(Caa3 level) = 10-day returns are more than 𝑇 times volatile. Positive autocorrelation of daily returns is not in line with general mean-reversion model assumptions. Positive autocorrelation has been observed and investigated in the literature also for transition matrices. This indicates the need to model business cycles and regimes: in times of stress trend of increase without mean-reversion, while recovery is driven by mean-reversion.

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**Autocorrelation of rating transitions**

It is more likely that a downgrade is followed in a year by a downgrade / default and an upgrade is followed by an upgrade. A downgrade in the past three months implies a conditional PD as high as a non-conditional PD 4-5 notches down. 1Y up stay down 15% 78% 7% 10% 77% 14% 6% 71% 23%

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Arrival to defaults Expected vs. unexpected defaults – What is incremental? Implicit equity-credit link Differences for corporates, financials and sovereigns by fundamentals

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Arrival to defaults Consistency and coherency issues between capital charges Potential exposure within a year does not capture that losses in case of a future default have potentially been realised already by CVA VaR when spreads were climbing up – this CVA variation is capitalised now Similarly for IRC vs. VaR – if being long credit for Greece, daily MtM losses were capitalised by VaR, while there was no further loss at the time of default, thus IRC capital charge was questionable Sudden and expected defaults shall be separated and capitalised accordingly

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**Black swan events on the equity market**

assuming normality this should happen once in years StDev(C US 30/12/05-12/09/08) = 2.35% Log-return on 15/09/08 was % -7σ event

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**Mixing observations from various distributions**

Assume stochastic volatility daily log-returns are not from the same population! Normalise daily log-returns by dividing them by the concurrent short-term implied volatility (easily available in Bloomberg). Less extreme events (most extreme is IV) and closer to the Gaussian shape.

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**Through-the-cycle vs. point-in-time**

Case: equally weighted, daily rebalanced, long-only portfolio of Eurostoxx 50. Pure 1-day 99% Historical VaR and Monte-Carlo VaR showed 9-10 excesses in 2008 suggesting “extraordinary” behaviour of the market. Black Swan events: unexpected and inexplicable by a given model. Key property of market crisis: increased market activity rate.

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**Through-the-cycle vs. point-in-time**

Year Unfilt. VaR EWMA HVaR Filt. HVaR Filt. MC VaR 2007 4 2 2008 9 3 2009 1 2010 5

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**Through-the-cycle vs. point-in-time**

Once in 2.3 billion year event– N ( N-1 ( 99% , 60% ) , 20% ) 99% 1-day VaR 60% - average VIX in Q4 2008 20% - 25-year average of VXO/VIX One may need to consider the designed purpose of the model. Using an appropriate weighting scheme, ceteris paribus, the number of excesses may be fine. For such a small linear portfolio, no dependence on the copula choice. However, weighting scheme is incompatible with regulatory requirements. 3x increase in few weeks – potential for large pro-cyclicality. Solution of regulation for capital pro-cyclicality are the Stressed VaR and Stressed ES measures. Economic interpretation of such stressed measures may exist only for simple linear portfolios. As traders follow other risk limits as well, market pro-cyclicality may persist. The emergence of new risk types in times of stress is not addressed by the stressed measures. Divergence between risk models and capital models Even more the case with the liquidity horizons of the FRTB Back-testing issues in case of changing regimes: 1-year TTC measure vs. traffic light approach

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**Heteroscedastic returns seen as co-jumps**

Often said that co-jumps are more frequent in times of stress and are a key characteristic of crises. Pearson correlation assumes that correlation was the same each day. Gaussian correlation structure underestimates the probability of large co-movements in large portfolios. Frequent co-jumps may be captured by HVaR but not by Gaussian copula based MCVaR. Case: example of Eurostoxx 50 In theory: 2-3 times a year that more than 2/3 of constituents jumped In practice: 22 times in 2008

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**Heteroscedastic returns seen as co-jumps**

Accounting for the stochastic behaviour of the business time: Frequency of co-jumps reduces significantly. A large part of observed correlation increase may be explained. Applying no weighting scheme is seen as smoothing: Gaussian copula is seen as even more smoothing (average correlation). Thanks to Stressed VaR and Stressed ES, HVaR and MC VaR disagree now in benign periods also. How reactive Stressed VaR should be?

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**Risk metrics of quadratic payoffs**

Eurostoxx 50 – such well-behaved portfolio hardly exists in practice How the risk metrics scale as a function of risk factor volatilities? No weighting scheme – Stressed VaR is not a PIT measure at the peak of the crisis, but rather a TTC measure through the stress period Around twice as volatile risk factors – similarity to ten-day scaling

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**Risk metrics of quadratic payoffs**

Delta-hedged long straddle position 𝑆=100, 𝜎 𝑖𝑚𝑝𝑙𝑖𝑒𝑑 = 𝜎 𝑟𝑒𝑎𝑙𝑖𝑠𝑒𝑑 =40%,𝑟=𝑦=0,𝑇=1𝑌 ∆ 𝑐𝑎𝑙𝑙 =0.54, ∆ 𝑝𝑢𝑡 =−0.46, thus we short 0.08 shares to delta hedge We are reaching different domains of the same P&L function. 𝑃&𝐿≈∆𝑑𝑆+ 1 2 Γ𝑑 𝑆 2 +Θ𝑑𝑡 Note that in the above Taylor expansion Θ is risk neutral, while 𝑑𝑆 is statistical and may be stressed.

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**Risk metrics of quadratic payoffs**

Combine payoff function with spot return density Remember: 𝑃&𝐿≈∆𝑑𝑆+ 1 2 Γ𝑑 𝑆 2 +Θ𝑑𝑡. If 𝑑𝑆 is normally distributed, the P&L density is approximately shifted chi-square distributed. Strongly skewed, scaled linearly as function of risk factor volatilities. Strong dislocation, not seen for simple long-short positions.

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**Risk metrics of quadratic payoffs**

𝐸 𝑃&𝐿 ≈𝐸 1 2 Γ𝑑 𝑆 2 +𝜃𝑑𝑡 = 1 2 Γ 𝑆𝑡𝑑 2 𝑑𝑆 Γ 𝐸 2 𝑑𝑆 +𝜃𝑑𝑡 Second term is basically zero. If 𝜎 𝑠𝑖𝑚𝑢𝑙𝑎𝑡𝑒𝑑 = 𝜎 𝑖𝑚𝑝𝑙𝑖𝑒𝑑 , theta effect cancels out the gamma effect, but if the two materially differs, there is a string dislocation of the P&L distribution. Dislocation is not diversifiable, aggregated figures may be driven by dislocations – pure tail measures like VaR or ES may be misleading (−0.03% =−0.03%). Specific trading strategies may influence the charge through dislocation. The effect may explain strange SVaR/VaR ratios Depending on the portfolio structure, the most volatile period is not necessarily the one which results in the highest Stressed VaR and Stressed EEPE figures. This puts in question the objective of the period selection requirements. The situation has became even worse with the FRTB where instantaneous shocks are assumed, i.e. portfolio is not aged and it is assumed that any shock affects the price as of today, hence no theta effect is expected to be taken into account. 1-day VaR 4-day VaR 1-day SVaR Mean 0.00% 0.09% 1% VaR -0.03% -0.13%

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**Accounting for risk premium**

Banks usually take over risks from the companies, diversify them and get compensation for systemic risk. Diffusion processes: risk premium over risk is negligible on short-term. Risk premium is usually discarded when calculating market risk VaR. Nevertheless, risk premium related to jump risk and gap risk is priced differently. BB sector 5Y CDS ranged between 100 and 700 bps from the beginning of 2006 until mid 2011. Implied default rate means then % in case of 40% recovery rate vs. TTC default rate of 1%. Rare events are priced with even larger risk premium. Default risk loss distribution is strongly affected by risk premium: 𝑃&𝐿= 𝑃𝑉 𝑡 1 𝜃 𝑡 1 ∙ 𝐷𝐹 𝑡 0 , 𝑡 1 − 𝑃𝑉 𝑡 0 𝜃 𝑡 𝑡 0 𝑡 1 𝐶𝐹 𝜏 ∙𝐷𝐹 𝑡 0 ,𝜏 𝑑𝜏 Visualise the potential time value effect when risk premium is significant: 𝑃&𝐿= 𝑃𝑉 𝑡 0 𝜃 𝑡 1 − 𝑃𝑉 𝑡 0 𝜃 𝑡 𝜕 𝑃𝑉 𝜏 𝜃 𝑡 0 𝜕𝜏 𝑡 1 − 𝑡 0 The EBA guidelines on IRC has already prescribed that instantaneous shocks can be applied without accounting for the theta affect, i.e. the last term can be dropped from the formula for capital computations.

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**Accounting for risk premium**

Case: short protection portfolio of CDSs written on BB rated issuers as of 30 June 2009 Average 1Y CDS spread of the constituents was 600 bps. In case no default or migration event happens, expected portfolio P&L is around 6%. Not accounting for time value, expected portfolio P&L is around -1% (TTC). Numerous default events may occur before any effective loss is realised. Dislocation is hidden behind the IRC figures.

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Recovery rates What are the local and foreign currency recovery rates and probabilities of defaults? Sovereigns may go default on their hard currency and local currency obligations separately Various approaches to adjust the LC recovery rates to account for FX depreciation risk or to interpret the problem as what is the LC bond value in case the HC bond migrate or default. However, FX risk and, accordingly, the risk of FX depreciation in case of default is excluded from IRC and IDR! What are the recovery rates for covered bonds and government guarantees? The rating of issuing bank is taken, which implies “high” PD, but when the issuer goes to default, there is still a pool of assets or another guarantor to meet the obligation. Recovery rates may be adjusted to compensate that “wrong” PDs are used. However, if for instance the recovery rate for covered bonds is 90%, but the dirty price of a covered bond is 82% at the moment, then what is the P&L in case of default? Recovery remarking process – general problem of enforcing that RR < PV. Stochastic recovery rates: One may distinguish stochastic expected recovery rates and stochastic realised recovery rates. If the expected recovery rates are already modelled by a stochastic process and the risk is already captured in the VaR / ES, which may be required in case the institution trades fixed recovery rate exotic deals and / or distressed bonds, then the realised recovery rate is not stochastic anymore except in case of sudden defaults. If the recovery rates are assumed to be stochastic, the related risks are captured in VaR / ES, then the default risk incremental to VaR / ES includes only the rare cases of sudden defaults.

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**Estimation of transition and default probabilities**

Autocorrelation issue has already been mentioned earlier Number and granularity of applied rating matrices may be challenged as well Less or more rating matrices: Trade-off between capturing better the specific risk profiles and basic risk vs. reducing the estimation noise. Which ones? Sovereign and corporate migration matrix? Corporate divided by region (US / Europe) and industry (financial / non-financial)? Relevance for bank portfolio vs. availability of data, i.e. available data often with US concentration. Finer rating grid may reduce the jump of P&Ls on the tails, but it introduces estimation noise. The FRTB has introduced a 3bp floor similarly to the the banking book rules. Binomial proportion confidence interval may answer how reliable is the transition probability estimate: CLT: 𝑝 ± 𝑧 1−𝛼/2 𝑝 1− 𝑝 𝑛 , Wilson interval: 𝑝 + 1 2𝑛 𝑧 1−𝛼/2 2 ± 𝑧 1−𝛼/2 𝑝 1− 𝑝 𝑛 𝑛 2 𝑧 1−𝛼/ 𝑛 𝑧 1−𝛼/2 2 For a 95% confidence interval:

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**Joint defaults and correlated migrations**

Common practice: default correlation modelling through asset value correlation (AVC) The approach is in line with the earlier presented strong equity-credit link. However, this means also model deficiency because sovereigns can hardly be treated as companies. Often, copula approach is mixed with the structural Merton model. However, in the Merton model the default can be triggered only at maturity. Hence, it does not qualify without specific treatments to tackle multiple step simulations and various LHs. Gaussian copula approach is not a consistent continuous-time model. If originally the default events and migration moves were simulated in one time step, then simulations cannot be split into two or more time steps in a way that will result in the original joint law of defaults at the end. The forward density does not exist. What asset value correlation parameter to use for the various time horizons?

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**Joint defaults and correlated migrations**

Fix the AVC and measure the Pearson default correlation for various horizons (annual PD = 2%, 2-state Markov chain with jump-to-default) Similar term structure of default correlations by ratings: The lower the cumulative probability of defaults the lower is the default correlation. Most copula based approaches imply that the defaults of highly rated names are basically independent .

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**Results and key findings of QIS1 **

Content LH in ES Results and key findings of QIS1 Equity correlations and CDS spread correlations – note on sovereigns Structural change of market, pricing and risk management Higher risk aversion rate, rescaling of ratings Multiple conservatism – what is really “incremental” – spread evolution vs. migration and default P&L - stochastic expected and realised recovery rate In case of intensity correlation the joint PD is limited, Sovereign correlation modelling challenge Scaling of 1-day returns is accepted, but back-tests move even further away from application. Amortisation of sensitivities, vega, implied correlation, etc. Long-term modelling in the CCR framework is already a challenge. Hardly possible to back-test. On top of it, CCR focuses only on traditional risk types, while MR exposure is mainly on spreads, bases, for which proper processes can be hardly designed and calibrated with confidence. Option positions cannot be assumed being unhedged, rebalanced for months. It assumes no trade at all until the first LH. There may be illiquidity for some underlyings, but not for all at the same time.

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