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Transcript of Value at Risk and Market Risk Eric Falkenstein. Eric Falkenstein 4/14/992 VaR: The Big Picture VaR...
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Value at Risk and Market Risk
Eric Falkenstein
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Eric Falkenstein 4/14/992
VaR: The Big Picture
• VaR is the new standard for measuring market risk across a Variety of security types
• VaR should always be complimented with specific scenario tests
• VaR is integral in any attempt to allocate capital (for both economic and proposed regulatory charges)
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Eric Falkenstein 4/14/993
Subtle Benefits of a VAR System
• Necessitates the ability to mark a book to market, which by itself is a major preventative against unpleasant surprises
• Establishes a clear independent risk monitoring arm
• Helps with existing auditing and risk reviews
• Securities that are not understood well and therefore have a significant risk are flagged as exceptions more often
• If used for capital charge, gives an incentive to economize on risk even when risk limits are not breached
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Eric Falkenstein 4/14/994
Worst-Case Derivative Scenarios
First Reported Company Estimated loss (millions)
11/92 Dell $82/93 Showa Shell $1,7007/93 Nippon Steel $1281/94 Metallgesellschaft $1,3001/94 Codelco $2003/94 Askin Capital Management $6003/94 Minnetonka Fund/Cargill $1004/94 Mead Corp $74/94 Procter & Gamble $1574/94 Gibson Greetings $235/94 Air Products $695/94 ARCO $226/94 Dell $356/94 Virginia Retirement System $666/94 Florida Treasurer's Office $1756/94 Pain Webber Bond Mutual Fund$337/94 Glaxo Holdings $1007/94 CS First Boston Inv't Mgt. $408/94 Piper Jaffray $7008/94 Charles County $28/94 Colonia (Germany) $769/94 Shoshone Indians $59/94 Investors Equity Life $909/94 Odessa College TX $11
First Reported Company Estimated loss (millions)
10/94 Community A Mgmt. $4410/94 Portage County OH $810/94 Sears $23711/94 Todyo Securities Co. $35611/94 3 Farm Credit System Banks $2312/94 Orange County CA $1,70012/94 Chemica Bank $702/95 Barings $1,2003/95 Winsconsin Inv. Board $953/95 MCN Corp $105/95 Five Morgan Stanley clients $287/95 First Capital Strategists $1378/95 Postipankki (Finland) $11010/95 Daiwa $1,100
What Can We Learn From These?
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Eric Falkenstein 4/14/995
• Do not compare VaR to a Platonic ideal, but to feasible alternatives
• Most importantly, by measuring VaR more precisely, the right questions get asked which minimizes “operating risk”
• While VaR won’t catch the next Baring’s, one can also bet that the next Baring’s won’t have a fully operational VaR risk measurement system in place
VaR is the Best Market Risk Metric
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Eric Falkenstein 4/14/996
What Did We Have Before VaR?
Duration
• Duration has a practical application in the measurement of a bond’s risk
% Change in Price = Duration x Change in Yield
• Given a parallel shift in yields, we can know how different securities will react
example:
• A bond with a duration of 5 years will experience a 0.35% change in price if rates rise 7 basis points (5*.07)
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Eric Falkenstein 4/14/997
Problems with Duration
• Duration measures the risk from parallel shifts in rates; rates do not always move in parallel
• Not applicable to FX
• Not informative for securities with significant option characteristics (e.g., mortgages, caps/floors, callable bonds)
• Does not capture risk from changes in credit spreads
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Eric Falkenstein 4/14/998
Problem 2: How do you Measure theRisk of This Portfolio?
Security Notional
Bonds $ 50
Interest Rate Swaps $100
Interest Rate Futures Contracts $150
Caps/Floors $ 45
FX Forwards $130
Total $475
Aggregate Notional is not meaningful in this context
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Eric Falkenstein 4/14/999
Problem 3: Judgement
Expert systems: we should have someone knowledgeable at the helm who can “know” what is happening.
Problem: how do you aggregate a bunch of expert opinions? How do you validate them?
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Eric Falkenstein 4/14/9910
VaR Can Give a MeaningfulAggregate Number
Security Notional VaR
Bonds $100 $ 3
Interest Rate Swaps $100 $ 5
Interest Rate Futures Contracts $100 $ 4
Caps/Floors $100 $ 4
FX Forwards $100 $ 7
Total $11
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Eric Falkenstein 4/14/9911
So What is VaR?
• Assume you had a portfolio that consisted, trivially, of the 30 year Tbond. Your daily P/L would look like this:
Daily Profit and Loss on a 30 yr Tbond
-3
-2
-1
0
1
2
3
1995
Pe
rcent
Ch
an
ge
95% VAR
99% VAR
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Eric Falkenstein 4/14/9912
So What is VaR?
• Converting this information into a histogram, we get:
Daily Profit and Loss on a 30 yr Bond, 1995
0
10
20
30
40
50
60
-2.2 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2
Percent Change Buckets
Fre
qu
en
cy o
f O
ccu
ran
ce
s
95% VAR
99% VAR
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Eric Falkenstein 4/14/9913
In Words
• VaR is the expected least amount lost 5% of the time
or,• VaR is the expected most amount lost 95% of the time • In both of the above examples, you could use a 99% VaR or
any other number, and also use different time horizons (e.g., 95% lost over daily observations, or annual observations)
• VaR takes into account current estimates of the volatility of various fundamental factors (e.g., on-the-run US Treasury rates) and their covariances (especially important when looking at a hedged portfolios)
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Eric Falkenstein 4/14/9914
VaR: One Factor Example
• $1MM 30yr Bond’s DV01=$1400
• Standard Deviation of 30yr bond=4.5 b.p.s
510,10$11400$
5.465.1%95
factorin changeunit $change
deltastdevfactor risk sstdev' of #%95
bpbpVaR
VaR
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Eric Falkenstein 4/14/9915
VaR: 2 Factors
• Portfolio is $1MM 30yr bond
5yr Key Rate DV01=$100
30yr Key Rate DV01=$1300
94.,6,5 5,10305 yyyryryr bpbp
648,13$%95
12870
825
194.
94.112870825%95
1400665.1
100565.1
194.
94.11300$665.1
100$565.1%95
... ofvector factors of correl.factors y tosensitivit port. ofvector %95
VaR
VaR
bpbp
bpbpVaR
VaR
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Eric Falkenstein 4/14/9916
In General
• Calculate all portfolio sensitivities to material risk factors at 95% level
• Aggregate them taking into account correlations
S=an nx1 vector of a portfolio’s change in value to a 95% worst-case scenario change in the n underlying factors
=correlation matrix
example:
• Rate Risk for Bond Option Trading=$86,216
• Volatility Risk for Bond Option Trading=$48,406
• Assume the correlation of those risks is zero
875,98$%95
48406
86216
10
014840686216%95
VaR
VaR
SSVaR
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Eric Falkenstein 4/14/9917
Revaluation Approach forNonlinear Instruments
• Replace 1.65 with the calculated in the value of the derivative for a 1.65 change in the underlying risk factor:
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Eric Falkenstein 4/14/9918
Further Refinements
• Which volatilities to use?
Examples: moving average, exponential moving average, GARCH, and implied volatilities from options
• If a Revaluation approach--Monte Carlo or Historical?
• Do you incorporate spread shocks (e.g., Fed Funds-Libor spread)?
• Do you incorporate volatility shocks (e.g., cap vols rise)?
Always weigh benefits versus costs of implementation
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Eric Falkenstein 4/14/9919
Criticisms of VaR
• If refinements are not addressed, VaR could be a poor representation of risk (e.g., using delta-normal VaR for a book with significant spread risk and optionality)
• Solutions are within the VaR paradigm
• Any good tool needs to be used wisely to add value
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Eric Falkenstein 4/14/9920
Always Compliment with Stress Tests
VaR Up 35 bp Down 35 bp
Portfolio $6.0M -$26.2M $25.8M
• VaR tells you how much you lose with a known probability, but with an unknown scenario
• Stress tests tell you how much you lose in a known scenario, but with an unknown probability
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Eric Falkenstein 4/14/9921
Combine VaR with Stress Tests to get the Complete Picture
VaR
Gives best estimate of worst-case P&L
1) User defined scenarios2) Shows where portfolio loses money 3) Anticipates “non-normal” events
Robust Risk Measurement
Stress Tests
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Eric Falkenstein 4/14/9922
One Should Always be Checking Various Assumptions
Usually VaR measures risk on an end-of-day portfolio. Using historical P&Ls, we can refine VaR estimates to adequately capture:
• Intraday trading
• Peculiar nonlinearities and spreads
• Particular liquidity of the trader portfolio
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Eric Falkenstein 4/14/9923
Validation is Key
• Validation is required for use of VaR as a measure of capital
• Validation also helps one understand the relative importance of various VaR refinements
• Each portfolio or trader will have different relevant risk factors and pricing models, and it is the risk manager’s job to find and quantify these risks
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Eric Falkenstein 4/14/9924
Backtesting
BACKTESTING TYPE 1A test of the validity of the simulation method.
A daily comparison of:
EX-ANTE VAR vs. EX-POST HYPOTHETICAL P & L
BACKTESTING TYPE 2A test of the “goodness” of VAR as a predictor of worst case P/L loss in tradingA daily comparison of:
EX-ANTE VAR 99% Confidence Level vs. EX-POST ACTUAL P&L
Under BIS rules,Multiplicative factor (“K”) depends on number of excesses in one year, where excess means that on a given day:
ACTUAL P/L LOSS > VAR 99% C.L.
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Eric Falkenstein 4/14/9925
R U L E B A S E D"S TA N D A R D M O D E L "
S p ec ificR isk
G en era l M arke tR isk
IN TE R N A L M O D E L SW ith S tan d ard P aram eters
P R E -C O M M ITM E N TA P P R O A C H
A P P R O A C H E S F O RD E TE R M IN IN G
R E G U L A TO R Y C A P ITA L
Regulatory Capital
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Eric Falkenstein 4/14/9926
Regulatory CapitalFOR GENERAL MARKET RISK
- CAPITAL BIS = K * VAR BIS_GENERAL
K DEPENDS ON BACKTESTING
factor is the following:Number of exceptions Factor<=4 35 3.46 3.57 3.658 3.759 3.85>=10 4
- PARAMETERS FOR VAR BIS
– OBSERVATION PERIOD FOR SIMULATION:• AT LEAST ONE YEAR
– HOLDING PERIOD• ONE DAY I.e., assume static portfolio
– CONFIDENCE LEVEL:• 10-DAY “INSTANTANEOUS” SHOCK IN MARKET RATES• 99% CONFIDENCE LEVEL
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Eric Falkenstein 4/14/9927
Regulatory Capital
FOR GENERAL MARKET RISK AND SPECIFIC MARKET RISK:
IF FULLY CAPTURE SPECIFIC RISKCorrelation RiskEvent Risk
THEN: REGULATORY CAPITAL BIS = K * VAR BIS_TOTAL
K DEPENDS ON BACKTESTING IF ONLY CAPTURE CORRELATION RISK
THEN: REGULATORY CAPITAL BIS = K * VAR BIS_TOTAL + VAR BIS_SPECIFIC
K DEPENDS ON BACKTESTING
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Eric Falkenstein 4/14/9928
• The BIS guidelines suggest estimating capital by using 3 or 4 times the VaR plus Specific Risk
• Specific risk is either the Standardized amount or that from internal models (not to be less than 50% of the Standardized amount)
VaR and Regulatory Capital
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Eric Falkenstein 4/14/9929
The Standardized Regulatory Capital Approach for Specific Risk Capital
• 8% against most assets
• 1.6% against OECD bank debt
• 0% against OECD government debt
• 4% against liquid equity portfolio
• 2% against equity indexes
• apply to both long and short positions
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Eric Falkenstein 4/14/9930
Internal Models Calculation of Specific Risk Capital
• If no specific credit risk measurement is operational, 4xMVAR
• If a model demonstrably isolates spread, event, and default risk, 3xSVAR
• If a model can capture some but not all of specific risk, 4xSVAR
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Eric Falkenstein 4/14/9931
Standardized Vs. Internal Models
• For individual security holdings, the standardized regulatory approach allocates less capital for speculative grade bonds and equities
• For portfolios, the internal models approach generates lower capital requirements
• The BIS has therefore created an employment program for quants
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Eric Falkenstein 4/14/9932
• Use 99% VaR over reasonable and flexible time-to-close assumption
• Take into account dynamic strategies in the form of loss limitsThink of an unused loan commitment:
One translates this into a Loan Equivalent Exposure, taking into account the probability of being used in the future.Example: Trader A VaR Loss Limit
$45 $120
Capital=VaR+VaR Equivalent Unused CommitmentCapital=VaR+factor(Loss Limit - VaR)Capital=$45+.5*(120-45)=$82.5
Economic Risk Capital (Not Regulatory)
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Eric Falkenstein 4/14/9933
Economic Risk Capital
A loss limit over the VaR is an unused commitment. Thus, one must translate this into a VaR-equivalent exposure
Example: Trader A 99% VaR Loss Limit
$45 $120
Capital=VaR+factor(Loss Limit - VaR)Capital=$45+.5*(120-45)=$60
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Eric Falkenstein 4/14/9934
• How does one allocate while accounting for diversification?Example: VaR VaR-Equivalent Stand-Alone Unused Commitment Capital
Trader A $42 $26 $68Trader B$23 $13 $36Total $50 ? ?
• We will suppose the total VaR-Equivalent Unused Commitment is reduced similar to the VaR: 50/(42+23)=.77 Total VaR is 77% the sum of the parts
Stand-Alone Capital Allocated CapitalTrader A $68 $52(=68x.77)Trader B $36 $28(=36x.77) Total $80
Loss Limit Risk is Diversifiable Too
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Eric Falkenstein 4/14/9935
Important Notes
• Using VaR and Loss Limits and tying this to a capital charge explicitly incents traders to minimize risk by both lowering their daily VaR and loss limits
• Economic capital different than VaR used in regulatory reporting
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Eric Falkenstein 4/14/9936
Integrating Regulatory and Economic Capital
• Case 1Regulatory Capital = Economic CapitalNo problem
• Case 2Regulatory Capital < Economic CapitalAllocate Economic capital, since economic capital is the binding constraint
• Case 3 Regulatory Capital > Economic Capital Not clear
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Eric Falkenstein 4/14/9937
Return on Equity
• Most successful traders generate exceedingly high ROEs.
• Nonetheless, ROE is a useful performance measure, especially for new traders (e.g., writing calls on the S&P would have been very profitable over many consecutive months, yet on an ROE basis would have been very weak)
• P&L still reigns supreme in performance measurement, and is essential for validation. Risk managers need to know a trader’s P&L and their incentive compensation plan.
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Eric Falkenstein 4/14/9938
2 Ways that Bad Tradersare Exposed
• Blow out. A quiescent market can allow a negative NPV strategy to produce seemingly large returns with low risk for many consecutive months (e.g., Orange County)
• Profit drip. A trader might have very low risk, yet have locked in a negative carry through poor pricing
In both cases, accurate monthly P&L helps highlight these problems
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Eric Falkenstein 4/14/9939
Asset and Liability VaR
• A Balance Sheet Management department has a different risk profile than a trader, as trading horizons are typically 3 months to 1 year in duration
• Accounting measures of risk are the industry norm
• Deposits will not be marked-to-market anytime soon, making VaR more abstract than accounting risk measures
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Eric Falkenstein 4/14/9940
Balance Sheet VaR
• Like trading risk, capital should be allocated against the current risk profile, plus a portion of the maximum feasible risk exposure under corporate policy
• Often duration limits of 1 or 5 years act as the maximum interest rate risk limit
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Eric Falkenstein 4/14/9941
Incorporate Other Risk Factors
• 3 Factors of Yield Curve risk (shift, twist, curvature)
• Spread risk
• Prepayment risk
• Parameter uncertainty
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Eric Falkenstein 4/14/9942
Benchmarks for BSM VaR
• Call Reports are not very informative
• Most banks have duration between 1 and 5 years
• When is a noisy VaR worse than an accounting measure of Risk?
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Eric Falkenstein 4/14/9943
Do Banks Have a ComparativeAdvantage in Riding the Yield Curve?
• Average duration of 3-5 years
• Sharpe ratio of 1.5
• ROE must be greater than cost of equity capital, or, agency problem
• The answer is probably a little of both
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Eric Falkenstein 4/14/9944
An Application of BSM VaR
• The average return of short funding a long maturity bond is positive, as the yield curve is usually upward sloping
• The Sharpe ratio of this risk-return relationship can help one see the relative attractiveness of this strategy
• With derivatives one can see how to optimize a common, simply strategy
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Eric Falkenstein 4/14/9945
1 5 10 20
4
5
6
7
8
Yield Curve
MATURITY
YIE
LD
(%
) RISKPREMIUM
EXPECTEDYIELD
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Eric Falkenstein 4/14/9946
Returns to Riding the Yield Curve
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Eric Falkenstein 4/14/9947
1/83-7/98 Performance
Return from being long the specific maturity,
and short a 6 month security, in basis points
1yr 3yr 5yr 10yr30yr
Avg. Return 36 121 209 300 438
Avg. Volatility 73 161 423 601 876
Sharpe Ratio .49 .75 .49 .50 .50
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Eric Falkenstein 4/14/9948
Replicate the 5, 10 and 30 Year Strategiesby Replicating the 3 Year Return with a
Zero-cost Derivative
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Eric Falkenstein 4/14/9949
Applications are Important
• Wholesale replacement of accounting based measures of risk will not occur soon
• In the meantime, piecemeal applications of VaR to balance sheet management strategies are quite fruitful
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Eric Falkenstein 4/14/9950
Diversification Lowers Risk
• Assume portfolio with Sensitivity to 1 Risk Factor, VaR=100
• Add a portfolio with a sensitivity to a new risk factor
• See new Portfolio VaR by various Correlation and Relative Size combinations
Correlations
0 0.25 0.5 0.75 1.00
10 101 103 105 108 110
size 50 112 122 132 141 150
100 141 158 173 187 200
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Eric Falkenstein 4/14/9951
Conclusion
• VaR is applicable across a variety of instruments, and thus allows a meaningful aggregation. This diminishes the numbers necessary for a complete picture of risk
• For Trading Desks, VaR is a necessary component of a regulatory capital charge
• VaR should always be complimented with stress tests
• While VaR is not as applicable to A/L management vis-à-vis trading desks, piecemeal applications are fruitful