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.
Generic credit spread curves by a fully cross-sectional method
2
One-simensional 5Y generic credit spread curve with calibrated rating distances
Aaa
Aa1
1
Aa2
Aa3
0
A1
A2
-1
A3
Baa1
-2
Baa2
Baa3
-3
Ba1
Ba2
-4
Ba3
B1
B2
-5
B3
Caa1
-6
Caa2
Caa3
-7
Ca
-8
31/10/03
C
31/10/04
31/10/05
31/10/06
31/10/07
31/10/08
31/10/09
31/10/10
31/10/11
31/10/12
31/10/13
2
Evolution of risk aversion premium
1
0
Recovery adjusted 1Y log-spreads and ln ( -ln ( 1 - PD ) )
-1
-2
mean 31Oct03
-3
fit 31Oct03
mean 21Nov08
-4
fit 21Nov08
mean 31Oct13
-5
fit 31Oct13
transformed historical PD
-6
-7
Negative risk premium may appear nowadays for HY
-8
-9
Aaa Aa1 Aa2 Aa3 A1 A2 A3 Baa1Baa2Baa3 Ba1 Ba2 Ba3 B1
B2
B3 Caa1Caa2Caa3 Ca
C
3
Dynamics of the generic credit spread curves
ln 𝑠𝑖,𝑡 𝐿𝐺𝐷𝑖,𝑡 = 𝛼𝑡 + 𝛽𝑡 ∙ 𝑑𝑅𝑖,𝑡 + 𝜎𝑡 ∙ 𝜀𝑖,𝑡
0
5
4.5
-1



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.
4
3.5
-2
3
-3
2.5
2
-4
1.5
-5
Alpha (left)
1
log-VIX (right)
0.5
-6
0
1.2
3
1
2.5
0.8
2
0.6
1.5
0.4
0.2
1
Sigma (left)
Beta (right)
0.5
0
0
4
Dynamics of the generic credit spread curves
1-year rolling correlation of 1-day Caa3 returns
Variance(1-day Caa3 ret.) = 0.00035
0.4
Variance(10-day Caa3 ret.) = 0.00545
0.3
Variance(Caa3 level) = 0.0562
0.2
0.1

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.
0
-0.1
-0.2
1-quarter rolling correlation of 10-day Caa3 returns
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
5
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
up
15%
78%
7%
stay
10%
77%
14%
down
6%
71%
23%
Conditioning default probabilities based on rating migrations in the past 3 months
0
-2
ln ( -ln ( 1 - PD ) )
-4
no condition
upgrade
-6
no change
-8
downgrade
-10
Linear (upgrade)
-12
Linear (no change)
Linear (downgrade)
-14
-16
-18
Aaa Aa1 Aa2 Aa3
A1
A2
A3 Baa1 Baa2 Baa3 Ba1 Ba2 Ba3
B1
B2
B3 Caa1 Caa2 Caa3 Ca
C
6
Arrival to defaults

Expected vs. unexpected defaults – What is incremental?

Implicit equity-credit link

Differences for corporates, financials and sovereigns by fundamentals
7
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
CVA path evolution until default
Potential additional loss
at the time of default
CVA

Total loss disregarding from CVA
∆CVA
Initial CVA
Default event
Time
Maturity

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
8
Black swan events on the equity market
•
•
StDev(C US 30/12/05-12/09/08) = 2.35%
Log-return on 15/09/08 was -16.42%
assuming normality this
should happen once in
3 000 000 000 years
-7σ event
Probability density function of log-returns
fitted Gaussian
log-PDF
4.00
2.00
0.00
-2.00
-4.00
-6.00
-8.00
-10.00
empirical density
-25%
-25%
-20%
-15%
-10%
-5%
0%
5%
10%
0%
15%
25%
20%
25%
C US daily log-returns between 31/12/05-30/09/10
9
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 -4.46 IV) and closer to the Gaussian shape.
C US short-term ATM IV
PDF of normalized log-returns
350
fitted Gaussian
300
empirical density
250
%
200
150
100
50
0
30/12/2005 25/03/2007 17/06/2008 10/09/2009 04/12/2010
-600.0% -400.0% -200.0%
0.0%
200.0% 400.0% 600.0%
C US daily normalized log-returns between 31/12/05-30/09/10
10
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.
Volatility indices normalized to their level on 1st January 2008
400%
350%
VIX Index (S&P)
RVX Index (equity, Russel)
300%
EVZ Index (EUR/USD)
250%
GVZ Index (gold)
200%
OVX Index (crude oil)
150%
100%
50%
0%
01/01/06
01/01/07
01/01/08
01/01/09
01/01/10
01/01/11
11
Through-the-cycle vs. point-in-time
Year
Unfilt. VaR
EWMA HVaR
Filt. HVaR
Filt. MC VaR
2007
4
2
2
2
2008
9
2
2
3
2009
0
0
1
1
2010
5
3
4
4
Various VaR approaches applied on Eurostoxx 50 constituents
14%
Daily losses
VaR as % of the notional
12%
10%
Unfiltered VaR
Filtered Historical VaR
Filtered MC VaR
8%
6%
4%
2%
0%
01/01/07
01/01/08
01/01/09
01/01/10
01/01/11
12
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
13
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
Testing the normality of the equity spot co-movements
50
200%
45
# of extremes (L)
180%
40
VIX (R)
160%
35
140%
30
120%
25
100%
20
80%
15
60%
10
40%
5
20%
0
03/01/06
0%
03/01/07
03/01/08
03/01/09
03/01/10
03/01/11
14
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?
Testing the normality of the equity spot co-movements
50
200%
45
# of extremes (L)
180%
40
VIX (R)
160%
35
140%
30
120%
25
100%
20
80%
15
60%
10
40%
5
20%
0
03/01/06
0%
03/01/07
03/01/08
03/01/09
03/01/10
03/01/11
15
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
Equity volatility index and its 260-day root-mean-squared average
90
80
VIX Index
70
VIX Index 260-day average
60
50
40
30
20
10
0
01/01/07
01/01/08
01/01/09
01/01/10
01/01/11
16
Risk metrics of quadratic payoffs

Delta-hedged long straddle position

𝑆 = 100, 𝜎𝑖𝑚𝑝𝑙𝑖𝑒𝑑 = 𝜎𝑟𝑒𝑎𝑙𝑖𝑠𝑒𝑑 = 40%, 𝑟 = 𝑦 = 0, 𝑇 = 1𝑌

∆𝑐𝑎𝑙𝑙 = 0.54, ∆𝑝𝑢𝑡 = −0.46, thus we short 0.08 shares to delta hedge
P&L function of a delta-hedged straddle position
0.7
1-day potential P&L
1-day PDF spot price
0.5
Probability density
P&L in % of the notional
0.6
1-day stressed PDF of spot price
0.4
0.3
0.2
0.1
0.0
-0.1


94
96
98
100
102
104
106
Spot price
We are reaching different domains of the same P&L function.
1
2
𝑃&𝐿 ≈ ∆𝑑𝑆 + Γ𝑑𝑆 2 + Θ𝑑𝑡

Note that in the above Taylor expansion Θ is risk neutral, while 𝑑𝑆 is statistical and may be stressed.
17
Risk metrics of quadratic payoffs
Combine payoff function with spot return density
1

Remember: 𝑃&𝐿 ≈ ∆𝑑𝑆 + Γ𝑑𝑆 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.
2
P&L distribution of a delta-hedged straddle position
1-day P&L
Probability density

-0.15
4-day P&L
1-day stressed P&L
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
P&L in % of the notional
18
Risk metrics of quadratic payoffs

𝐸 𝑃&𝐿 ≈ 𝐸


1
2
1
1
2
2
Γ𝑑𝑆 2 + 𝜃𝑑𝑡 = Γ𝑆𝑡𝑑2 𝑑𝑆 + Γ𝐸 2 𝑑𝑆 + 𝜃𝑑𝑡
Second term is basically zero.
If 𝜎𝑠𝑖𝑚𝑢𝑙𝑎𝑡𝑒𝑑 = 𝜎𝑖𝑚𝑝𝑙𝑖𝑒𝑑 , theta effect cancels out the gamma effect, but if the two materially differs,
there is a string dislocation of the P&L distribution.
1-day VaR
4-day VaR
1-day SVaR
Mean
0.00%
0.00%
0.09%
1% VaR
-0.03%
-0.13%
-0.03%

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.
19
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.



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 1.7-11% 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:


Risk premium is usually discarded when calculating market risk VaR.
𝑃&𝐿 = 𝑃𝑉𝑡1 𝜃𝑡1 ∙ 𝐷𝐹𝑡0 ,𝑡1 − 𝑃𝑉𝑡0 𝜃𝑡0 +
𝑡1
𝐶𝐹𝜏
𝑡0
∙ 𝐷𝐹𝑡0 ,𝜏 𝑑𝜏
Visualise the potential time value effect when risk premium is significant:


𝑃&𝐿 = 𝑃𝑉𝑡0 𝜃𝑡1 − 𝑃𝑉𝑡0 𝜃𝑡0 +
𝜕𝑃𝑉𝜏 𝜃𝑡0
𝜕𝜏
𝑡1 − 𝑡0
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.
20
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.
IRC loss distribution of a CDS portfolio with 100 BB constituents
Forward repricing
Probability density

Taylor approximation
Taylor appr wo theta
-15
-10
-5
0
5
10
IRC as of 30/06/09 for a 100M issuer risk portfolio
15
Millions
21
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.
22
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:
1
𝑝 1−𝑝
𝑛

CLT: 𝑝 ± 𝑧1−𝛼/2

For a 95% confidence interval:
, Wilson interval:
2
𝑝+2𝑛𝑧1−𝛼/2
±𝑧1−𝛼/2
𝑝 1−𝑝
𝑛
+
1
1 2
𝑧
4𝑛2 1−𝛼/2
2
1+𝑛𝑧1−𝛼/2
Trial
Outcome
Estimate
Lower CI
Upper CI
50
1
2.0%
0.4%
10.5%
100
1
1.0%
0.2%
5.4%
500
1
0.2%
0.0%
1.1%
1000
1
0.1%
0.0%
0.6%
23
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.
k defaults, j pays

Borrower j

•
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?
Both pay
k pays, j defaults
Both default
Borrower k
24
Joint defaults and correlated migrations

Fix the AVC and measure the Pearson default correlation for various horizons (annual PD = 2%, 2state Markov chain with jump-to-default)
Default correlations as function of time horizon for various AVC
Default correlation
4.0%
3.5%
5%
3.0%
10%
20%
2.5%
2.0%
1.5%
1.0%
0.5%
0.0%
0
1
2
3
4
5
6
7
8
9
10
11
12
Time horizon (months)

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 .
25
26

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.
27