Value at Risk and Market Risk Eric Falkenstein. Eric Falkenstein 4/14/992 VaR: The Big Picture VaR is the new standard for measuring market risk across

Value at Risk and Market Risk

Eric Falkenstein

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)


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


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?


• 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


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)


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


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


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?


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


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


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


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)


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


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


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


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:


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


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


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


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


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


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


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.


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


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


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


• 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


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


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


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


• 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)


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


• 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


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


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


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.


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


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


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


Incorporate Other Risk Factors

• 3 Factors of Yield Curve risk (shift, twist, curvature)

• Spread risk

• Prepayment risk

• Parameter uncertainty


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?


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


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


1 5 10 20

4

5

6

7

8

Yield Curve

MATURITY

YIE

LD

(%

) RISKPREMIUM

EXPECTEDYIELD


Returns to Riding the Yield Curve


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


Replicate the 5, 10 and 30 Year Strategiesby Replicating the 3 Year Return with a

Zero-cost Derivative


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


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


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

Documents

Value at Risk and Market Risk Eric Falkenstein. Eric Falkenstein 4/14/992 VaR: The Big Picture VaR is the new standard for measuring market risk across