“All of life is the management of risk, not its elimination.”— Walter Wriston, former chairman of Citicorp

16 December, 1999

OverviewOverview

ConceptsConcepts

ComponentsComponents

CalculationsCalculations

Corporate perspective Corporate perspective

CommentsComments

I I VALUE AT RISK - VALUE AT RISK - CONCEPTSCONCEPTS

RiskRisk

Financial Risks - Market Risk, Credit Risk, Financial Risks - Market Risk, Credit Risk, Liquidity Risk, Operational RiskLiquidity Risk, Operational Risk

Risk is the variability of returns.Risk is the variability of returns.

Risk is Defined as “Bad” OutcomesRisk is Defined as “Bad” Outcomes• Volatility Inappropriate MeasureVolatility Inappropriate Measure• What Matters is Downside RiskWhat Matters is Downside Risk

VAR measuresVAR measures

Market riskMarket risk

Credit risk of lateCredit risk of late

• VAR is an estimate of the adverse impact on P&L in a VAR is an estimate of the adverse impact on P&L in a conservative scenario.conservative scenario.

• It is defined as the loss that can be sustained on a It is defined as the loss that can be sustained on a specified position over a specified period with a specified position over a specified period with a specified degree of confidence.specified degree of confidence.

Value at Risk (VAR)Value at Risk (VAR)

• Ingredients - Ingredients - • Exposure to market variableExposure to market variable• SensitivitySensitivity• Probability of adverse market movement Probability of adverse market movement

• Probability distribution of market variable - key Probability distribution of market variable - key assumptionassumption

• Normal, Log-normal distributionNormal, Log-normal distribution

Value at Risk (VAR)Value at Risk (VAR)

VARVAR

Daily P&L

VAR

IIII VALUE AT RISK - VALUE AT RISK - COMPONENTSCOMPONENTS

Key components of VARKey components of VAR•Market Factors (MF)Market Factors (MF)

•Factor Sensitivity (FS)Factor Sensitivity (FS)

•Defeasance Period (DP) Defeasance Period (DP)

•VolatilityVolatility

Market Factors (MF)Market Factors (MF)• A market variable that causes the price of an instrument A market variable that causes the price of an instrument

to changeto change• A market factors group (MFG) is a group of market factors A market factors group (MFG) is a group of market factors

with significant correlation. The major MFGs are:with significant correlation. The major MFGs are:– Interest rates, Interest rates, – Foreign exchange ratesForeign exchange rates– Equity pricesEquity prices– Commodity pricesCommodity prices– Implied volatilities (only in options)Implied volatilities (only in options)

• Complex positions can be sensitive to several MFG (e.g. Complex positions can be sensitive to several MFG (e.g. FX forwards or options)FX forwards or options)

Factor Sensitivity (FS)Factor Sensitivity (FS)

FS is the change in the value of a position due to FS is the change in the value of a position due to aa unitunit change in an change in an independent market factorindependent market factor, all , all other market factors, if applicable, remaining other market factors, if applicable, remaining constant.constant.

Other names - PVBPOther names - PVBP

Factor Sensitivity - Zero Coupon BondFactor Sensitivity - Zero Coupon BondWhat is the 1 BP FS of a $2,100 1-year zero coupon bond? (assume What is the 1 BP FS of a $2,100 1-year zero coupon bond? (assume market rate is 5%)market rate is 5%)

MTM Value = $2,100 / (1.05) = $2,000.00MTM Value = $2,100 / (1.05) = $2,000.00

MTM Value = $2,100 / (1.0501) = $1,999.81MTM Value = $2,100 / (1.0501) = $1,999.81

FS = $1,999.81 - $2,000.00 = -$0.19FS = $1,999.81 - $2,000.00 = -$0.19

Market VolatilityMarket Volatility

Volatility is a measure of the dispersion of a market Volatility is a measure of the dispersion of a market variable against its mean or average. This dispersion is variable against its mean or average. This dispersion is called Standard Deviation.called Standard Deviation.

•Variance := average deviation of the mean for aVariance := average deviation of the mean for a historical sample sizehistorical sample size

•Standard deviation : Square Root of the variance Standard deviation : Square Root of the variance

The market expresses volatility in terms of annualized The market expresses volatility in terms of annualized Standard Deviation (1SD)Standard Deviation (1SD)

Estimating VolatilityEstimating Volatility

1.1. Historical data analysis Historical data analysis

2. Judgmental 2. Judgmental

3. Implied (from options prices)3. Implied (from options prices)

Defeasance periodDefeasance periodThis is defined as the time elapsed (normally This is defined as the time elapsed (normally expressed in days) before a position can be expressed in days) before a position can be neutralized either by hedging or liquidatingneutralized either by hedging or liquidating

Defeasance period incorporates liquidity risk Defeasance period incorporates liquidity risk (for trading) in risk measurement(for trading) in risk measurement

Other names - Holding Period, Time horizonOther names - Holding Period, Time horizon

Defeasance Factor (DF)Defeasance Factor (DF)DF is the total volatility over the defeasance DF is the total volatility over the defeasance periodperiod

On the assumption that daily price changes are On the assumption that daily price changes are independent variables (~ correlation zero), independent variables (~ correlation zero), volatility is scaled by the square root of timevolatility is scaled by the square root of time

DF = Daily 2.326 SD * sqrt (DP), orDF = Daily 2.326 SD * sqrt (DP), orDF = Market Volatility * 2.326 *sqrt (DP / 260)DF = Market Volatility * 2.326 *sqrt (DP / 260)DFDF = Annual 1SD * 2.326 * sqrt (DP/260)= Annual 1SD * 2.326 * sqrt (DP/260)

VAR formulaVAR formula

VAR = VAR = zz ppt t * FS* FSWhere:Where:

zz is the constant giving the appropriate one-tailed Confidence is the constant giving the appropriate one-tailed Confidence Interval.Interval. pp is the annualized standard deviation of the portfolio’s return is the annualized standard deviation of the portfolio’s return t t is the holding period horizonis the holding period horizon

FS FS Factor SensitivityFactor Sensitivity

VARVAR

Daily P&L

VAR

IIIIII VALUE AT RISK - VALUE AT RISK - CALCULATIONSCALCULATIONS

Sample VAR CalculationsSample VAR Calculations

Let us consider the following positions:Long EUR against the USD : $ 1 MMLong JPY against the USD : $ 1 MM

Each of these positions has a factor sensitivity of +10,000

Sample VAR CalculationsSample VAR CalculationsAnnual volatility of DEM is 9%Volatility for N days = annual volatility x SQRT(N/ T)where T is the total number of trading days in a year (260)Therefore, 1 day volatility of DEM= 9 x SQRT (1/260) = 0.56%This is 1, so, 2.326so, 2.326 = 2.326 x 0.56% = 1.30% = 2.326 x 0.56% = 1.30%

Sample VAR CalculationsSample VAR CalculationsNow, a 1% change has an impact of 10,000 (FS)Now, a 1% change has an impact of 10,000 (FS)So, a 1.30% change will have an impact ofSo, a 1.30% change will have an impact of1.30 x 10,000 = 13,0001.30 x 10,000 = 13,000This represents the impact of a 2.326 SD change in the This represents the impact of a 2.326 SD change in the market factor over a 1 day periodmarket factor over a 1 day period

Thus, in 1 out of 100 days we may cross actual loss of Thus, in 1 out of 100 days we may cross actual loss of $ 13,000. Our Value at Risk (VAR) is $13,000 on this position$ 13,000. Our Value at Risk (VAR) is $13,000 on this position

Sample VAR CalculationsSample VAR CalculationsSimilarly, for JPY, the annual volatility is 12%Similarly, for JPY, the annual volatility is 12%The 1 day volatility = 12 x SQRT (1/260) = 0.74%The 1 day volatility = 12 x SQRT (1/260) = 0.74%2.326 SD = 2.326 x 0.74 = 1.73%2.326 SD = 2.326 x 0.74 = 1.73%

Impact of a 1% change = 10,000 (FS)Impact of a 1% change = 10,000 (FS)

So, impact of a 1.73% change = 17,310So, impact of a 1.73% change = 17,310

Our VAR on this position is $ 17,310Our VAR on this position is $ 17,310

IVIV VALUE AT RISK FOR CORPORATIONS VALUE AT RISK FOR CORPORATIONS

VAR FOR CORPORATIONSVAR FOR CORPORATIONS• Trading portfoliosTrading portfolios• Longer time horizons for close outsLonger time horizons for close outs• Business risk as opposed to trading riskBusiness risk as opposed to trading risk• Holding period, business time horizonHolding period, business time horizon• VAR as a percentage of CapitalVAR as a percentage of Capital

VAR FOR CORPORATIONSVAR FOR CORPORATIONS

• Identify market variables impacting business Identify market variables impacting business • Map income sensitivity to market variables - Scenario Map income sensitivity to market variables - Scenario analysisanalysis• Based on volatilities of market factors and their correlations, Based on volatilities of market factors and their correlations, arrive at a worst case scenario given the degree of confidencearrive at a worst case scenario given the degree of confidence• Worst case income projection - acceptable or not? Worst case income projection - acceptable or not? • Hedge to reduce VARHedge to reduce VAR

VAR FOR CORPORATIONSVAR FOR CORPORATIONS Hedging tools Hedging tools • Forward FXForward FX• Currency swapsCurrency swaps• Interest Rate swapsInterest Rate swaps• Options on non-INR market variablesOptions on non-INR market variables• Commodity futuresCommodity futures•Commodity derivatives Commodity derivatives

VV VALUE AT RISK- A FEW COMMENTS VALUE AT RISK- A FEW COMMENTS

Significance of VARSignificance of VAR• Applicable mainly to trading portfoliosApplicable mainly to trading portfolios• Regulatory capital requirementsRegulatory capital requirements• Provides senior executives with a simple and effective Provides senior executives with a simple and effective way to monitor risk.way to monitor risk. • VAR incorporates portfolio effects.VAR incorporates portfolio effects.• Uses history to predict near term future. Uses history to predict near term future.

VAR : A Few CommentsVAR : A Few Comments

VAR does not represent the maximum lossVAR does not represent the maximum loss

VAR does not represent the actual lossVAR does not represent the actual loss

It represents the potential loss associated with a specified It represents the potential loss associated with a specified level of confidence. In this case, 99% over 1 daylevel of confidence. In this case, 99% over 1 day

Increased VAR represents increased risk, decrease in VAR Increased VAR represents increased risk, decrease in VAR represents decrease in riskrepresents decrease in risk

VAR limit is related to revenue potentialVAR limit is related to revenue potential

Where to use VAR?Where to use VAR?

Macro measure. High level monitoring, managing, eg. Macro measure. High level monitoring, managing, eg. Regional levelRegional levelCurrently used mainly for trading limits.Currently used mainly for trading limits.Strategic planning - Allocation of resourcesStrategic planning - Allocation of resourcesHowever.. However.. Not an efficient day to day tool.Not an efficient day to day tool.Components - FS, Market volatility, Defeasance period, Components - FS, Market volatility, Defeasance period, Correlations are all integral parts of trading strategy.Correlations are all integral parts of trading strategy.

How to use VarHow to use Var

Stress Testing : * “worst case” scenarioStress Testing : * “worst case” scenario * Multiple Stress Scenarios* Multiple Stress Scenarios * Should include not only price moves * Should include not only price moves In excess of 2SD, but also otherIn excess of 2SD, but also other market events likely to adverselymarket events likely to adversely affect businessaffect businessBack Testing : Compares actual daily P&L movements Back Testing : Compares actual daily P&L movements

predicted variance of P&L predicted variance of P&L

General Market Risk IssuesGeneral Market Risk Issues

Integrity - Rate ReasonabilityIntegrity - Rate Reasonability

- At Inception- At Inception - Revaluation- RevaluationModel CertificationModel CertificationControl Mechanisms / Checks and BalancesControl Mechanisms / Checks and Balances

Corporate Culture!Corporate Culture!

ReviewReview• Loss occurs only if rates move adversely to the positionLoss occurs only if rates move adversely to the position• The loss is proportional to the sensitivity of the positionThe loss is proportional to the sensitivity of the position• The loss is proportional to size of the adverse movementThe loss is proportional to size of the adverse movement• Loss = FS multiplied by the adverse rate movementLoss = FS multiplied by the adverse rate movement• We cannot limit adverse rate movements in the marketplaceWe cannot limit adverse rate movements in the marketplace• We can limit our sensitivity (P&L impact) with FSLWe can limit our sensitivity (P&L impact) with FSL• FSL should be set against potential adverse movementFSL should be set against potential adverse movement• Potential adverse movements estimated through volatilityPotential adverse movements estimated through volatility

Anthony Saunders Anthony Saunders Credit Risk Measurement: New Approaches to VAR and Other ParadigmsCredit Risk Measurement: New Approaches to VAR and Other Paradigms John John Wiley & Sons 1999 Wiley & Sons 1999

Glyn Holton Glyn Holton Value at Risk: Theory and PracticeValue at Risk: Theory and Practice Academic Press 1999 Academic Press 1999

Philip Best Philip Best Implementing Value at RiskImplementing Value at Risk John Wiley & Sons Hardcover, 1998. John Wiley & Sons Hardcover, 1998.

Satyajit Das Satyajit Das Risk Management & Financial Derivatives: A Guide to the MathematicsRisk Management & Financial Derivatives: A Guide to the Mathematics McGraw-Hill McGraw-Hill 1998.1998.

Cormac Butler Cormac Butler Mastering VAR: A Step by Step GuideMastering VAR: A Step by Step Guide Trans-Atlantic Publications Trans-Atlantic Publications

Kevin Dowd Kevin Dowd Beyond VAR: The New Science of Risk ManagementBeyond VAR: The New Science of Risk Management John Wiley 1998 John Wiley 1998

Philippe Jorion Philippe Jorion Value-at-Risk: The New Benchmark for Controlling Market RiskValue-at-Risk: The New Benchmark for Controlling Market Risk, Irwin, 1997, , Irwin, 1997,

Paul Embrechts, Claudia Kluppelberg, & Thomas Mikosch Paul Embrechts, Claudia Kluppelberg, & Thomas Mikosch Modelling External Events for Modelling External Events for Insurance and FinanceInsurance and Finance Springer-Verlag, 1997. Springer-Verlag, 1997.

David Lawrence David Lawrence Measuring & Managing Derivative Market RiskMeasuring & Managing Derivative Market Risk Thompson Press, 1997 Thompson Press, 1997 ((CitibankCitibank).).

Philippe Jorion Philippe Jorion Value at Risk: The New Benchmark for Controlling Market RiskValue at Risk: The New Benchmark for Controlling Market Risk Irwin Professional, Irwin Professional, 1996.1996.

Further Reading on VARFurther Reading on VAR

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