Market risk capital (MRC) is derived from value-at-risk (VaR) and a tail-loss penalty based on backtesting results. The Basel II approach to calculating the tail-loss penalty (K-factor) does not adequately account for the magnitude of losses beyond VaR. An alternative approach by Blanco-Ihle defines tail losses as a percentage breach from VaR conditional on a breach. This allows calculating the K-factor based on either the maximum tail loss (KBI.max) or the average tail losses (KBI.mean). Backtesting results for RAS portfolios in early 2012 show the KBI.mean approach provides a more realistic assessment of tail risk compared to the Basel

1. RAS Risk Management
Market Risk Capital
Amit Sinha, James Wang & Tolga Sezer
August, 2012
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2. Agenda
Market Risk Capital: Definition and Requirements
Backtesting Methodology
RAS VaR Backtesting
From VaR to MRC: The Tail-Loss Penalty
RAS MRC Summary
Conclusion
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3. Part I
Market Risk Capital: Definition and Requirements
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4. MRC: Definition and Requirements
Basel II & VaR-based Risk Capital
Basel II sets the regulatory standard for risk-based capital requirements.
Market Risk Capital is derived from the VaR (internal model approach)
and tail-loss penalty.
Internal Model Approach means that the institutions shall be able to use
their in-house VaR models, but are required to conduct extensive
backtesting to validate their performance.
Tail-loss penalty are based on the losses of a portfolio historically beyond
what the in-house VaR model would have predicted.
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5. Market Risk Capital
Definition:
MRCt = max(VaRt(C), K
1
60
t−1
i=1
VaR(i)(C)) (1)
where VaR numbers are expressed in absolute values and K is the
multiplier, the K-factor.
MRC is equal to the maximum of today’s VaR and average VaR
estimated over previous 60 trading days times the multiplier. The
K-factor is a function of the tail-loss, the part of losses which
exceeds our VaR.
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6. Market Risk Capital
Requirement:
VaR and hypothetical PnL backtest in order to measure what
the tail-losses for any portfolio would have been historically.
A penalty-factor function, which incorporates the tail-losses
identified in the backtesting, which results in a K-factor.
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7. Part II
Backtesting Methodology
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8. Backtesting Methodology
A conceptualization of the backtesting routine for any given day
On a specific date (Portfolio Date), we take the actual portfolio positions.
Using 1000 days prior to Portfolio Date, we calculate the 10-day VaR at
T3.
The VaR is compared to the cum. 10-day PnL after T3.
We report if there is a breach and if so the extend of the breach.
Next, we move one day back to T2 and do the analysis again for the
same portfolio.
Once we reach our backtesting horizon, we do the same routine for the
next portfolio.
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9. Part III
RAS VaR Backtesting
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10. RAS VaR Backtesting
10-day VaR (95%) vs 10-day cumulative PnL: RAS Portfolio as of 2012-01-20
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11. RAS VaR Backtesting
10-day VaR (99%) vs 10-day cumulative PnL: RAS Portfolio as of 2012-01-20
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12. Part IV
From VaR to MRC: The Tail-Loss Penalty
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13. The Tail-Loss Penalty
K-factor: How do we penalize tail losses?
Basel II sets forth following K-factor rule:
Condition to calculate multiplier:
KBasel.II = 2. if N <= 3 (2)
= 2 + 0.2(N − 3), if 3 < N <= 4 (3)
= 3 + 0.2(N − 4), if 5 <= N <= 9 (4)
= 4.0, if 10 < N (5)
N here is the number of breaches identified in the backtesting.
This approach only concerns itself with the frequency of breaches.
The magnitude of breach, i.e. the tail loss, is disregarded.
The factors are set without empirical underpinnings.
The penalty factor is floored at 2 and capped at 4.
Critique: Extreme tail-loss risks will be underpenalized and smaller
breaches will be overpenalized.
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14. The Tail-Loss Penalty
K-factor: Alternative approach
The Blanco-Ihle Approach (BI, henceforth) (Dowd,K 2010: ’Measuring
Market Risk’)
TailLosst = {
(Lt−VaRt )/VaRt
0 if Lt >VaRt
Lt <VaRt (6)
Tail-loss is defined as the % breach compared to the VaR estimate
conditional upon a breach.
If the 10-day cumulative loss is greater than the estimated VaR, then the
value is the % deviation. Otherwise the value is zero.
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15. The Tail-Loss Penalty
K-factor: Two ways of deriving the K-factor from the tail-loss
The first way is to take the maximum of tail-losses ever recorded in the
dataset as the basis for the K-factor:
KBI.max = 1 + max(TailLosst) (7)
The second way is to take the average size of tail-losses of all recorded
breaches in the dataset as the basis for the K-factor:
KBI.mean = 1 +
1
N
ΣN
i=1(TailLosst) (8)
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16. The Tail-Loss Penalty
Conclusion
KBasel.II is not very realistic as a tail-loss penalty.
There is a very high risk of understating or overstating tail-risk when
using KBasel.II .
KBI. is a more accurate and realistic approach if one has extensive data
available.
With KBI.max we would assume a single extreme event as the basis for the
penalty, which is very punitive.
KBI.mean is a more balanced approach as it takes the average size of all
tail-loss occurances as the basis.
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17. Part V
RAS MRC Summary
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18. RAS MRC Summary
Examples of Portfolio Backtestings
Portfolio of 2012-01-20
K-Factor (Basel II) K-Factor (Blanco-Ihle Max) K-Factor (Blanco-Ihle Mean)
VaR (USD) 1,398,275 1,398,275 1,398,275
# of breaches 4 4 4
K-factor 3 1.92 1.15
MRC 4,194,825 2,684,688 1,608,016
Portfolio of 2012-02-11
K-Factor (Basel II) K-Factor (Blanco-Ihle Max) K-Factor (Blanco-Ihle Mean)
VaR (USD) 2,373,537 2,373,537 2,373,537
# of breaches 4 4 4
K-factor 3 6.22 3.56
MRC 7,120,611 14,763,400 8,449,792
In the first example, the KBasel.II clearly overstates the tail-risk.
It overstates because empirically, the realized losses were less than 3
times higher than the estimated VaR.
The second portfolio has more cash exposure. KBasel.II understates the
tail-risks, as cash positions can have significant tail risks.
However, the KBI.max is too punitive as it takes the 2008 liqudity crisis
event as the basis for the factor.
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19. RAS MRC Summary
RAS MRC and Capital Charges for Jan-Apr and May-Jul
K-Factor (Blanco-Ihle Mean) K-Factor (Blanco-Ihle Max) K-Factor (Basel.II)
MRC Factor Cap. Charge MRC Factor Cap. Charge MRC Factor Cap. Charge
Jan – Apr 1,987,167 1.84 90,021 4,888,880 7.86 221,714 5,695,849 3.76 251,326
May – Jul 2,325,435 1.05 94,525 2,683,682 1.7 104,377 7,783,110 4 299,991
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20. Part VI
Conclusion
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21. Conclusion
Tail-losses are significant and need to be take account for on top of VaR.
While VaR is lower in Q1, RAS has bigger tail-risks in Q1 ...
... relative to Q2 and
... relative to what KBasel.II would imply.
The KBasel.II would be understating risk due to the cap on the factor.
The Blanco-Ihle approach - together with extensive data mining - enables
us to capture tail-risks more accurately.
It would be very punitive if the KBI.Max approach was used, especially for
cash positions.
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