Bank Regulation, Risk Management,

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Bank Regulation, Risk Management, and Financial
Stability October 2, 2009 Risk
Management Institute (RMI) National University of
Singapore
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• 1. Motivation
• Bank regulators
• Value-at-Risk (VaR) is used to measure the risk
  of trading books and the corresponding minimum
  capital requirements
• Stress Testing (ST) is used to assess whether
  banks withstand extreme events.
• Practitioners
• Banks use VaR and ST to set risk exposure limits
  (survey of Committee on the Global Financial
  System, 2005, and Jorion, 2005).
• Researchers
• VaR is not sub-additive (Artzner, Delbaen, Eber,
  and Heath, 1999)
• VaR does not consider losses beyond VaR (Basak
  and Shapiro, 2001, Rockafellar and Uryasev,
  2002, and Alexander and Baptista, 2006)
• Advocate Conditional-Value-at-Risk (CVaR) it is
  sub-additive, and considers losses beyond VaR.
• Our paper
• Examines the extent of the conflict between (1)
  the popularity of VaR and ST among regulators
  and practitioners and (2) the advocacy of CVaR
  by researchers.
• More specifically, we examine the effectiveness
  of a risk management system based on both VaR
  and ST constraints in controlling CVaR.
• Put differently is the joint use of VaR and ST
  equivalent to the use of CVaR?

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• 2. Main results
• When short selling is disallowed
• VaR and ST constraints lead to the selection of
  portfolios with small CVaRs
• Hence, the joint use of VaR and ST is effective
  in controlling CVaR.
• When short selling is allowed
• VaR and ST constraints allow the selection of
  portfolios with large CVaRs
• Hence, the joint use of VaR and ST is
  ineffective in controlling CVaR.
• Thus, the severity of the conflict between
  popularity of VaR and ST among regulators and
  practitioners and the advocacy of CVaR by
  researchers depends on the allowance of short
  selling.
• Implication since large banks often have
  significant short positions in their trading
  books, the joint use of VaR and ST is unreliable
  in controlling CVaR for these banks.
• Broadly speaking, our results are consistent
  with
• Losses in trading books are sometimes attributed
  to short positions
• Berkowitz and OBrien, 2002 trading books of
  large banks suffer losses surprisingly larger
  than their VaRs
• Basle Committee on Banking Supervision, July
  2008 banks have recently suffered large trading
  losses not captured by VaR.

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3. VaR, CVaR, and ST
For simplicity, consider a portfolio with a
normally distributed return
confidence level
95
Return
VaR at the 95 confidence level (maximum loss
under normal conditions)
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3. VaR, CVaR, and ST
For simplicity, consider a portfolio with a
normally distributed return
confidence level
95
Return
VaR at the 95 confidence level (maximum loss
under normal conditions)
CVaR E loss loss VaR (expected loss under
abnormal conditions)
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3. VaR, CVaR, and ST
For simplicity, consider a portfolio with a
normally distributed return
confidence level
95
Losses in ST events (e.g., crash of 87 and 9/11)
Return
VaR at the 95 confidence level (maximum loss
under normal conditions)
CVaR E loss loss VaR (expected loss under
abnormal conditions)
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• 4. Methodology
• Allocation problem among nine asset classes
• T-bills (assumed to be risk-free)
• Government bonds
• Corporate bonds and
• Six size/value-growth Fama-French portfolios.
• Monthly investment horizon
• Historical simulation
• 73 of banks that disclose methodology to
  estimate VaR report the use of historical
  simulation (Pérignon and Smith, 2007)
• Monthly data during the period 19822006
• ST events (i) 1987 stock market crash and (ii)
  9-11 (CGFS survey, 2005).
• Consider three different risk management systems
  based on
• A single VaR constraint
• Two ST constraints and
• A single VaR constraint and two ST constraints.
• Examine whether each set of constraints
  precludes the selection of all portfolios with
  relatively large CVaRs

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4. Methodology
1. Choose confidence level ? (e.g., 99) and VaR
bound V (e.g., 4) for the constraint.
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4. Methodology
1. Choose confidence level ? (e.g., 99) and VaR
bound V (e.g., 4) for the constraint. 2. Given
VaR constraint, find maximum and minimum feasible
expected returns .
Expected return
E
CVaR
Risk-free return
9
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4. Methodology
1. Choose confidence level ? (e.g., 99) and VaR
bound V (e.g., 4) for the constraint. 2. Given
VaR constraint, find maximum and minimum feasible
expected returns . 3. Determine
Expected return
E
CVaR
Risk-free return
10
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4. Methodology
1. Choose confidence level ? (e.g., 99) and VaR
bound V (e.g., 4) for the constraint. 2. Given
VaR constraint, find maximum and minimum feasible
expected returns . 3. Determine
4. Construct grid of expected returns E0
E E1 E d ... E100 .
Expected return
E100
E50
E0 E
CVaR
Risk-free return
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4. Methodology
1. Choose confidence level ? (e.g., 99) and VaR
bound V (e.g., 4) for the constraint. 2. Given
VaR constraint, find maximum and minimum feasible
expected returns . 3. Determine
4. Construct grid of expected returns E0
E E1 E d ... E100 . 5. For each
value in this grid Ei, find maximum efficiency
loss Mi.
Expected return
E100
Ei
E0 E
CVaR
Risk-free return
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4. Methodology
1. Choose confidence level ? (e.g., 99) and VaR
bound V (e.g., 4) for the constraint. 2. Given
VaR constraint, find maximum and minimum feasible
expected returns . 3. Determine
4. Construct grid of expected returns E0
E E1 E d ... E100 . 5. For each
value in this grid Ei, find maximum efficiency
loss Mi.
Expected return
Mean-CVaR boundary
E100
Portfolio with minimum CVaR
A
Ei
E0 E
CVaR
Risk-free return
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4. Methodology
1. Choose confidence level ? (e.g., 99) and VaR
bound V (e.g., 4) for the constraint. 2. Given
VaR constraint, find maximum and minimum feasible
expected returns . 3. Determine
4. Construct grid of expected returns E0
E E1 E d ... E100 . 5. For each
value in this grid Ei, find maximum efficiency
loss Mi.
Expected return
Mean-CVaR boundary
E100
Portfolio with minimum CVaR
Portfolio with maximum CVaR
B
A
Ei
E0 E
CVaR
Risk-free return
14
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4. Methodology
1. Choose confidence level ? (e.g., 99) and VaR
bound V (e.g., 4) for the constraint. 2. Given
VaR constraint, find maximum and minimum feasible
expected returns . 3. Determine
4. Construct grid of expected returns E0
E E1 E d ... E100 . 5. For each
value in this grid Ei, find maximum efficiency
loss Mi.
Expected return
Mean-CVaR boundary
E100
Portfolio with minimum CVaR
Portfolio with maximum CVaR
B
A
Ei
Maximum efficiency loss Mi
E0 E
CVaR
Risk-free return
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4. Methodology
1. Choose confidence level ? (e.g., 99) and VaR
bound V (e.g., 4) for the constraint. 2. Given
VaR constraint, find maximum and minimum feasible
expected returns . 3. Determine
4. Construct grid of expected returns E0
E E1 E d ... E100 . 5. For each
value in this grid Ei, find maximum efficiency
loss Mi.
Expected return
Mean-CVaR boundary
E100
Portfolio with minimum CVaR
Portfolio with maximum CVaR
B
A
Ei
Maximum efficiency loss Mi
E0 E
CVaR
Risk-free return
For example, if Mi 3, then the VaR constraint
allows the selection of a portfolio with a CVaR
that exceeds the CVaR of the minimum CVaR
portfolio by 3.
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4. Methodology
1. Choose confidence level ? (e.g., 99) and VaR
bound V (e.g., 4) for the constraint. 2. Given
VaR constraint, find maximum and minimum feasible
expected returns . 3. Determine
4. Construct grid of expected returns E0
E E1 E d ... E100 . 5. For each
value in this grid Ei, find maximum efficiency
loss Mi.
Expected return
Mean-CVaR boundary
E100
Portfolio with minimum CVaR
Portfolio with maximum CVaR
A
Ei
B
Maximum efficiency loss Mi (small)
E0 E
CVaR
Risk-free return
-gt More generally, if maximum efficiency loss Mi
is relatively small, then the VaR constraint is
effective in controlling CVaR when the required
expected return is Ei.
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4. Methodology
1. Choose confidence level ? (e.g., 99) and VaR
bound V (e.g., 4) for the constraint. 2. Given
VaR constraint, find maximum and minimum feasible
expected returns . 3. Determine
4. Construct grid of expected returns E0
E E1 E d ... E100 . 5. For each
value in this grid Ei, find maximum efficiency
loss Mi.
Expected return
Mean-CVaR boundary
E100
Portfolio with minimum CVaR
Portfolio with maximum CVaR
B
A
Ei
Maximum efficiency loss Mi (large)
E0 E
CVaR
Risk-free return
-gt However, if maximum efficiency loss Mi is
relatively large, then the VaR constraint is
ineffective in controlling CVaR when the required
expected return is Ei.
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4. Methodology
1. Choose confidence level ? (e.g., 99) and VaR
bound V (e.g., 4) for the constraint. 2. Given
VaR constraint, find maximum and minimum feasible
expected returns . 3. Determine
4. Construct grid of expected returns E0
E E1 E d ... E100 . 5. For each
value in this grid Ei, find maximum efficiency
loss Mi. 6. Compute average and largest
efficiency losses
Expected return
Mean-CVaR boundary
E100
Portfolio with minimum CVaR
Portfolio with maximum CVaR
B
A
Ei
Maximum efficiency loss Mi
E0 E
CVaR
Risk-free return
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4. Methodology
1. Choose confidence level ? (e.g., 99) and VaR
bound V (e.g., 4) for the constraint. 2. Given
VaR constraint, find maximum and minimum feasible
expected returns . 3. Determine
4. Construct grid of expected returns E0
E E1 E d ... E100 . 5. For each
value in this grid Ei, find maximum efficiency
loss Mi. 6. Compute average and largest
efficiency losses
Expected return
Mean-CVaR boundary
E100
Portfolio with minimum CVaR
Portfolio with maximum CVaR
B
A
Ei
Maximum efficiency loss Mi
E0 E
CVaR
Risk-free return
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4. Methodology
1. Choose confidence level ? (e.g., 99) and VaR
bound V (e.g., 4) for the constraint. 2. Given
VaR constraint, find maximum and minimum feasible
expected returns . 3. Determine
4. Construct grid of expected returns E0
E E1 E d ... E100 . 5. For each
value in this grid Ei, find maximum efficiency
loss Mi. 6. Compute average and largest
efficiency losses
Expected return
Mean-CVaR boundary
E100
Portfolio with minimum CVaR
Portfolio with maximum CVaR
B
A
Ei
Maximum efficiency loss Mi
E0 E
CVaR
Risk-free return
For example, if the relative efficiency loss is
100, then the VaR constraint allows the
selection of a portfolio with a CVaR that is
twice as large as the CVaR of the minimum CVaR
portfolio.
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4. Methodology
1. Choose confidence level ? (e.g., 99) and VaR
bound V (e.g., 4) for the constraint. 2. Given
VaR constraint, find maximum and minimum feasible
expected returns . 3. Determine
4. Construct grid of expected returns E0
E E1 E d ... E100 . 5. For each
value in this grid Ei, find maximum efficiency
loss Mi. 6. Compute average and largest
efficiency losses, and average and largest
relative efficiency losses.
Expected return
Mean-CVaR boundary
E100
Portfolio with minimum CVaR
Portfolio with maximum CVaR
B
A
Ei
Maximum efficiency loss Mi
E0 E
CVaR
Risk-free return

• Relative efficiency loss ? 8 as CVaRA ? 0
• In the computation of average and largest
  relative efficiency losses, we only consider
  levels of expected return for which the CVaR in
  the denominator is larger than 1.

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5. Results with short selling disallowed VaR
constraint

• Smaller bound (tighter constraint)
• Large average loss
• Large average relative loss
• Small maximum expected return

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5. Results with short selling disallowed VaR
constraint

• Larger bound
• (looser constraint)
• Larger average loss
• Larger average relative loss
• Larger maximum expected return

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• 6. Variable bounds
• Consider the case where the bound depends on the
  required expected return variable bound.
• Set the value of the variable bound to equal
• the VaR of the portfolio on the mean-CVaR
  boundary with the required expected return.
• Note that
• Using a smaller value for the bound would
  preclude the portfolio on the mean-CVaR boundary
  with the required expected return (which has a
  zero efficiency loss) and
• Using a larger bound would generally increase the
  maximum efficiency loss.
• Hence, this variable bound has two appealing
  features
• it allows (but does not generally force) the
  selection of a portfolio with a zero efficiency
  loss and
• it leads to the smallest maximum efficiency loss
  (given that it allows the selection of a
  portfolio with a zero efficiency loss).

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7. Results with short selling disallowed VaR
constraint (comparison between fixed and
variable bounds)
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8. Results with short selling disallowed
variable bounds (comparison between various sets
of constraints)
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• 9. Allowing short selling
• Weight of each asset class is restricted to be
  between 50 and 150.
• The possibility of short selling and these
  weight restrictions are realistic in the context
  of the trading books of large banks (see Jorion
  (2006)).
• In order to focus on a realistic set of required
  expected returns
• E is set to be equal to the risk-free return
  (i.e., 0.43)
• is set to be equal to the expected return on
  a leveraged position on the asset with the
  highest expected return (i.e., 2.16).
• Effects of allowing short selling
• - Set of feasible portfolios is larger (tends
  to increase losses)
• - Value of the variable bound is possibly
  smaller (tends to decrease losses) and
• - Range of feasible expected returns is larger
  (unclear effect on losses).

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10. Results with short selling allowed variable
bounds (comparison between various sets of
constraints)
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10. Results with short selling allowed variable
bounds (comparison between various sets of
constraints)
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10. Results with short selling allowed variable
bounds (comparison between various sets of
constraints)
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10. Results with short selling allowed variable
bounds (comparison between various sets of
constraints)
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10. Results with short selling allowed variable
bounds (comparison between various sets of
constraints)
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• 11. Robustness checks
• Results are similar when only the interquartile
  range of expected returns E25,,E75 is
  considered.
• Results are similar when
• T-bills are removed from consideration
• Bonds are removed from consideration or
• Both T-bills and bonds are removed from
  consideration.
• Results are similar when is set to be equal
  to the maximum feasible expected return, i.e.,
  3.24 (instead of the smaller value of 2.16).
• Results are stronger (i.e., losses are larger)
  in the case when short selling is allowed and the
  weight of each asset class is restricted to be
  between 100 and 200 (instead of between 50
  and 150).

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• 11. Robustness checks
• Consider the cases where there is a larger
  number of
• ST events (87 crash, 9-11, 97 Asian crisis, 98
  Russian crisis)
• Assets (T-bills, T-bonds, corporate bonds, ten
  size FF portfolios) and
• Both ST events and asset classes.
• confidence level ? 99, short selling allowed
• VaR and ST constraints are still ineffective in
  controlling CVaR when short selling is allowed.

Maximum efficiency loss ()
9 assets 2 ST events
9 assets 4 ST events
13 assets 2 ST events
13 assets 4 ST events
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• 12. Conclusion
• When short selling is disallowed
• VaR and ST constraints lead to the selection of
  portfolios with small CVaRs
• Hence, the joint use of VaR and ST is effective
  in controlling CVaR.
• When short selling is allowed
• VaR and ST constraints allow the selection of
  portfolios with large CVaRs
• Hence, the joint use of VaR and ST is
  ineffective in controlling CVaR.
• Implication since large banks often have
  significant short positions in their trading
  books, the joint use of VaR and ST is unreliable
  in controlling CVaR for these banks.
• Broadly speaking, our results are consistent
  with
• Losses in trading books are sometimes attributed
  to short positions
• Berkowitz and OBrien, 2002 trading books of
  large banks suffer losses surprisingly larger
  than their VaRs
• Basle Committee on Banking Supervision, July
  2008 banks have recently experienced large
  trading losses not captured by VaR.

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• Further research
• We use historical simulation since Perignon and
  Smith (2007) report that this is the methodology
  used by the majority of banks in estimating VaR.
• gt Examine whether VaR and ST constraints are
  effective in controlling CVaR when methodologies
  other than historical simulation (e.g., Monte
  Carlo simulation) are used.
• We consider a parsimonious setting that uses the
  crash of 1987 and 9/11 as the ST events since
  they are reported by the Committee on the Global
  Financial System (2005) to be the most commonly
  used ones.
• gt Investigate the adequacy of using VaR and ST
  in more complex settings with a larger number of
  assets and/or ST events.
• We assume that estimation risk is absent.
• gt Explore how the extent to which the results in
  our paper are affected by the presence of
  estimation risk.

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• Intuition
• The calculation of a portfolios loss in an ST
  event differs from that of its VaR and CVaR in
  two respects
• First, while ST uses asset returns during a
  fixed (historical) event, VaR and CVaR use asset
  returns in a set of states that depends on the
  confidence level and the portfolio
• Second, while the period of time used to find
  the loss in an ST event depends on the event (one
  day for the crash of 87, and several days for
  9/11), the period of time used to compute VaR and
  CVaR is fixed (e.g., several years of monthly
  data).
• Due to these two differences, portfolios with
  similar losses in an ST event may have notably
  different VaRs and CVaRs.

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• Intuition
• When short selling is disallowed
• Relatively small losses in ST events can only be
  achieved by investing in assets with relatively
  small losses in these events (i.e., bonds)
• Hence, ST is effective in supplementing VaR.
• When short selling is allowed
• Relatively small losses in ST events can also be
  achieved by short selling assets with relatively
  large losses in these events (i.e., stocks)
• Hence, ST is ineffective in supplementing VaR.

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Short selling disallowed VaR and ST constraints
with fixed bounds
Short selling allowed VaR and ST constraints
with fixed bounds
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• Motivation for methodology
• An examination of maximum efficiency losses
  captures the idea of being agnostic regarding the
  portfolio selection model that is used in the
  presence of VaR and/or ST constraints.
• The motivation for this idea is two-fold.
• First, we are interested in exploring the
  effectiveness of these constraints to control
  CVaR without making any assumption on the
  portfolio selection model that is used in the
  presence of VaR and/or ST constraints.
• Second, while the use of VaR and ST constraints
  by certain banks is apparent, we do not know
  exactly the models that they utilize for
  portfolio selection. For example, while bank
  managers may have to meet these constraints, they
  may also have incentives to take substantive
  risks in attempting to generate large profits
• Berkowitz and O'Brien (2002) present evidence
  that these managers take on a substantial amount
  of risk. They find that large banks sometimes
  suffer losses in their trading books that are
  notably larger than their VaRs.

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• Definition of ST events
• U.S. stock market of 1987 (October 19, 1987)
• Terrorist attacks in the U.S. of September 2001
  (September 1121, 2001).
• In determining the time periods for the ST
  events involving the Asian and Russian crises, we
  follow RiskMetrics.
• However, our starting date for the Russian
  crisis event is one day earlier than that used by
  RiskMetrics so that the event includes the day
  when the Russian government decided to default on
  its debt.
• Asian crisis (10/23/97-10/27/97)
• Russian crisis (08/14/98-10/08/98)