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Applied Business StatisticsApplied Business StatisticsCase studiesCase studies

Market risk managementMarket risk management

Mauro BufanoMauro Bufano

Risk Management – Risk Management – BancaBanca Mediolanum Spa Mediolanum Spa

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Market riskMarket risk

Market risk can be defined as the risk hat the value of a Market risk can be defined as the risk hat the value of a portfolio, either an investment portfolio or a trading portfolio, either an investment portfolio or a trading portfolio, will decrease due to the change in assets’ portfolio, will decrease due to the change in assets’ pricesprices

Market risk is present in every investment portfolio:Market risk is present in every investment portfolio:• Treasuries’ portfoliosTreasuries’ portfolios• Asset management (funds)Asset management (funds)• Real Estate investmentsReal Estate investments

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Market riskMarket risk

Market risk concept was born in the late 80’s: the 1987 Market risk concept was born in the late 80’s: the 1987 “Black Monday” market crash (in one day Dow Jones “Black Monday” market crash (in one day Dow Jones Industrial Average dropped by more than 22%) showed Industrial Average dropped by more than 22%) showed the importance of a proper market risk quantificationthe importance of a proper market risk quantification

In the early ’90sIn the early ’90s  J. P. Morgan CEO Dennis J. P. Morgan CEO Dennis Weatherstone famously called for a “4:15 report” that Weatherstone famously called for a “4:15 report” that combined all firm risk on one page, available within 15 combined all firm risk on one page, available within 15 minutes of the market close minutes of the market close

Introduction of the most important market risk measure: Introduction of the most important market risk measure: Value at Risk (VaR)Value at Risk (VaR)

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Market riskMarket risk

Market risk is not unique, but there are different sources of Market risk is not unique, but there are different sources of risk:risk:

a)a) Equity riskEquity risk

b)b) Interest rate riskInterest rate risk

c)c) Exchange rate riskExchange rate risk

d)d) Liquidity riskLiquidity risk

e)e) Spread riskSpread risk

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Equity risk: Equity risk: the most known risk, it’s the risk deriving the most known risk, it’s the risk deriving from changes in stock pricesfrom changes in stock prices

Interest rate risk: Interest rate risk: it’s the risk that changes in interest it’s the risk that changes in interest rates affect the value of an asset. It’s generally more rates affect the value of an asset. It’s generally more complex than equity risk because we refer not to a complex than equity risk because we refer not to a single asset, but to different interest rates for each single asset, but to different interest rates for each maturity: the yield curve. Yield curve movements can maturity: the yield curve. Yield curve movements can be summarized into 3 different sources of risk:be summarized into 3 different sources of risk:

1.1. Vertical translations: Vertical translations: parallel movements of the yield curve parallel movements of the yield curve (generally 80% of interest rate risk)(generally 80% of interest rate risk)

2.2. TwistTwist: changes in the slope of the yield curve: changes in the slope of the yield curve

3.3. Interest rate volatilityInterest rate volatility

Market riskMarket risk

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Yield curve: an exampleYield curve: an exampleSwap curve

0

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1y 2y 3y 4y 5y 6y 7y

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)

04/12/2007

19/11/2009

In the chart above we can see different yield curves at different rates. This could impact dramatically in the price of interest rate related assets (e.g. bonds)

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Exchange rate riskExchange rate risk

Exchange rate risk: Exchange rate risk: it’s the risk that assets in foreign it’s the risk that assets in foreign currencies change their value due to movements in the currencies change their value due to movements in the exchange rateexchange rate

Exchange rates generally depends on the difference Exchange rates generally depends on the difference between respective short term interest ratesbetween respective short term interest rates

Therefore exchange rate risk is very difficult to measure, Therefore exchange rate risk is very difficult to measure, also because they are strongly influenced by Central also because they are strongly influenced by Central Bank decisionsBank decisions

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Liquidity riskLiquidity risk Liquidity risk is defined as the risk that assets becomes illiquid, i.e. Liquidity risk is defined as the risk that assets becomes illiquid, i.e.

there is few (or no demand) for those assets and it’s difficult to sell it there is few (or no demand) for those assets and it’s difficult to sell it on the marketon the market

It’s generally measured as the bid-ask spread listed on the market It’s generally measured as the bid-ask spread listed on the market for that particular securityfor that particular security

It depends onIt depends on Market conditions (during 2007-09 market crisis, the amount of liquidity Market conditions (during 2007-09 market crisis, the amount of liquidity

on the market has shrunk and bid-ask spreads have widened)on the market has shrunk and bid-ask spreads have widened) Issuer quality (a EMU government bond is more liquid than a small firm Issuer quality (a EMU government bond is more liquid than a small firm

bond)bond) The amount of that instrument traded on the marketThe amount of that instrument traded on the market The maturity of the asset (longer maturity lower liquidity)The maturity of the asset (longer maturity lower liquidity)

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Liquidity riskLiquidity riskIlliquid bond: the bid-ask spread is high and there are few counterparties. Prices are very different among them

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Liquidity riskLiquidity riskLiquid bond: the bid-ask spread is low and there are many counterparties.

Prices are very close to each others

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Credit spread riskCredit spread risk

Spread risk can be defined as the risk that a decrease in Spread risk can be defined as the risk that a decrease in the credit worthiness of a issuer determines a higher the credit worthiness of a issuer determines a higher discount rate, and therefore a reduced value of the discount rate, and therefore a reduced value of the assets.assets.

Spread risk has increased its importance with the Spread risk has increased its importance with the introduction of the Credit Default Swaps (CDS), i.e. introduction of the Credit Default Swaps (CDS), i.e. insurances against default, that give a precise insurances against default, that give a precise quantification of the credit spread of a issuerquantification of the credit spread of a issuer

It can be generally divided into two sources of risk:It can be generally divided into two sources of risk:• Systemic risk: the risk that credit spreads in the market increase Systemic risk: the risk that credit spreads in the market increase

e.g., due to a reduced risk appetite or a global recession (2008-09)e.g., due to a reduced risk appetite or a global recession (2008-09)

• Specific risk: the risk that credit spread of a firm widens due to Specific risk: the risk that credit spread of a firm widens due to internal problems that increase its probability of default (PD)internal problems that increase its probability of default (PD)

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Credit spreadsCredit spreads

As we can see, credit spreads tend to move together, also for those companies whose credit worthiness remains highAs we can see, credit spreads tend to move together, also for those companies whose credit worthiness remains high Systemic risk, especially in financial industry, generally explain more than 80% of credit spread movementsSystemic risk, especially in financial industry, generally explain more than 80% of credit spread movements

Spread CDS

050

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INTESA SANPAOLO

UNICREDIT SPA

Credit Agricole

MONTE DEI PASCHI SIENA

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VolatilityVolatility In risk measurement, the key parameter that allows risk In risk measurement, the key parameter that allows risk

managers to quantify risk is managers to quantify risk is volatility.volatility. It can be represented as the second moment of asset It can be represented as the second moment of asset

price distribution (being the first moment the asset’s price distribution (being the first moment the asset’s return) and it’s generally measured with the return) and it’s generally measured with the standard standard deviation (deviation (σσ))

The importance of its measurement is trivial, because it’s The importance of its measurement is trivial, because it’s not a fixed parameter but rather time-varyingnot a fixed parameter but rather time-varying

In financial markets we observe In financial markets we observe volatility clusters: volatility clusters: high high volatilities are generally associated with market crashes volatilities are generally associated with market crashes or economic crises, and increase dramatically the risk of or economic crises, and increase dramatically the risk of huge losseshuge losses

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Different measures of volatilityDifferent measures of volatility

Volatility can be measured in different ways:Volatility can be measured in different ways: Historical volatility: Historical volatility: it’s defined as the standard deviation of an it’s defined as the standard deviation of an

asset along a period of time (e.g. 1 year)asset along a period of time (e.g. 1 year) Advantages: easy to measure and to observeAdvantages: easy to measure and to observe Disadvantages: the past is not always the best proxy for the future!Disadvantages: the past is not always the best proxy for the future!

Implied volatility: Implied volatility: it’s the volatility it’s the volatility ((σσ)) embedded in option prices embedded in option prices (call or put), derived from Black & Scholes (B&S) formulation(call or put), derived from Black & Scholes (B&S) formulation

For a call option:For a call option:

For a put option:For a put option:

where where

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Implied volatilityImplied volatility Implied volatility cannot be derived analytically from B&S, Implied volatility cannot be derived analytically from B&S,

but it’s necessary an iterative method (many softwares, but it’s necessary an iterative method (many softwares, i.e. Matlab, encode it)i.e. Matlab, encode it)

Anyway, B&S formulation implies Anyway, B&S formulation implies log-normal distributionlog-normal distribution of the underlying price, that generally not always of the underlying price, that generally not always observed in reality, and it is derived under observed in reality, and it is derived under risk-neutrality risk-neutrality assumptionsassumptions

The implied volatility derived from this formulation is not The implied volatility derived from this formulation is not a real “observable” variable, but rather a proxy of the real a real “observable” variable, but rather a proxy of the real volatility expected by market operatorsvolatility expected by market operators

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Volatility smileVolatility smile

The limitations of B&S formula result in the empirical The limitations of B&S formula result in the empirical observations: there is not a unique volatility but rather a observations: there is not a unique volatility but rather a “volatility smile”, i.e. a different volatility for any strike “volatility smile”, i.e. a different volatility for any strike (see chart below)(see chart below)

Implied volatility

2121.5

2222.5

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90% 95% 97.50% 100% 102.50% 105% 110%

Strike

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%)

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Volatility surfaceVolatility surface Generally, smile volatility is observed for a short time horizon (less than 2 Generally, smile volatility is observed for a short time horizon (less than 2

years). For longer horizons, the smile becomes a “skew”, i.e. volatility is a years). For longer horizons, the smile becomes a “skew”, i.e. volatility is a decreasing function of the smile decreasing function of the smile

Volatility surface is not directly observed, but it has to be interpolated. In this Volatility surface is not directly observed, but it has to be interpolated. In this case, a Local Polynomial Smoothing (LVS) technique has been usedcase, a Local Polynomial Smoothing (LVS) technique has been used

0.81

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StrikeMaturity

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Implied volatilityImplied volatility

Even if it’s not a direct observable variable and it’s derived Even if it’s not a direct observable variable and it’s derived under strong assumptions, implied volatility is used under strong assumptions, implied volatility is used (especially for pricing purposes) because:(especially for pricing purposes) because:

It’s a forward measure and it’s not directly related to past eventsIt’s a forward measure and it’s not directly related to past events It reflects market expectationsIt reflects market expectations The existence of a volatility surface gives the opportunity to The existence of a volatility surface gives the opportunity to

derive “what-if” scenarios or simulation basing on the underlying derive “what-if” scenarios or simulation basing on the underlying priceprice

It can be used to calibrate time-varying volatility models (see It can be used to calibrate time-varying volatility models (see next slides)next slides)

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Volatility simulationVolatility simulation

In some cases, it’s necessary to simulate volatility paths in order to get In some cases, it’s necessary to simulate volatility paths in order to get a distribution of future assets’ or portfolios’ values.a distribution of future assets’ or portfolios’ values.

Different models are available:Different models are available: GARCH models: it’s a time varying simulations based on an GARCH models: it’s a time varying simulations based on an

autoregressive process of variance:autoregressive process of variance:

e.g. GARCH(1,1)e.g. GARCH(1,1)GARCH models are widely used for many applications (especially for GARCH models are widely used for many applications (especially for

VaR or scenario analysis), but is generally not a market model for VaR or scenario analysis), but is generally not a market model for derivative pricingderivative pricing

Its calibration is obtained via maximum likelihood estimationIts calibration is obtained via maximum likelihood estimationA common GARCH(1,1) model is the Risk Metrics one, where:A common GARCH(1,1) model is the Risk Metrics one, where:ωω = 0 = 0αα = 1- = 1- ββββ = 0.94 for daily data, = 0.94 for daily data, ββ = 0.97 for monthly data = 0.97 for monthly data

21

21

2 ttt

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Volatility simulation – Local volatilityVolatility simulation – Local volatility Other models (mostly used for derivative pricing) areOther models (mostly used for derivative pricing) are

Dupire model or local volatility: it’s derived by re-adapting Dupire model or local volatility: it’s derived by re-adapting volatility surface in terms of underlying price and timevolatility surface in terms of underlying price and time

In this way the volatility depends on the path of the In this way the volatility depends on the path of the underlying and on the time. For its calibration it’s underlying and on the time. For its calibration it’s necessary to know the first derivative of the option price necessary to know the first derivative of the option price on time and the derivatives (first and second order) of the on time and the derivatives (first and second order) of the option price on the strike. It’s therefore necessary to have option price on the strike. It’s therefore necessary to have a volatility surfacea volatility surface

),(),( TSmaturityK t

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• Heston model: it’s a time varying process for volatility and the Heston model: it’s a time varying process for volatility and the underlying in which the two processes are correlated (generally, underlying in which the two processes are correlated (generally, in a negative way)in a negative way)

where where WWttSS and and WWtt

V V are correlated Brownian motionsare correlated Brownian motions

Heston model has to be calibrated on the quotations of the Heston model has to be calibrated on the quotations of the options listed on the market. It’s generally used for complex options listed on the market. It’s generally used for complex derivatives’ pricingderivatives’ pricing

Volatility simulation – Heston modelVolatility simulation – Heston model

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CorrelationCorrelation

Another crucial risk measure is correlation, that can be Another crucial risk measure is correlation, that can be either interpreted as asset correlation (in a portfolio) or either interpreted as asset correlation (in a portfolio) or risk factors’ correlationrisk factors’ correlation

Its importance has growth especially in the last crisis, in Its importance has growth especially in the last crisis, in which the risk factors’ correlation has growth in a which the risk factors’ correlation has growth in a dramatic way and causing huge lossesdramatic way and causing huge losses

In fact, if volatility was (in some cases) already In fact, if volatility was (in some cases) already embedded in many instrument and it was possible to embedded in many instrument and it was possible to hedge it, correlation risk was completely mispriced in hedge it, correlation risk was completely mispriced in some basket securities (e.g. CDO)some basket securities (e.g. CDO)

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Correlation measurementCorrelation measurement Until now, the most used correlation measurement is Until now, the most used correlation measurement is historicalhistorical one, one,

with all the limits of this approachwith all the limits of this approach There are, anyway, some instruments that allows to determine an There are, anyway, some instruments that allows to determine an

“implied correlation”“implied correlation” A mathematical instrument widely used to calculate (and simulate) A mathematical instrument widely used to calculate (and simulate)

implied correlation is the “copula function”, derived by Sklar’s implied correlation is the “copula function”, derived by Sklar’s theorem:theorem:

The (1) is a cumulated distribution, the (2) is a density distribution: The (1) is a cumulated distribution, the (2) is a density distribution: this result allows to separate marginal distribution from the “copula”, this result allows to separate marginal distribution from the “copula”, that models the correlation between the 2 variablesthat models the correlation between the 2 variables

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Market risk managementMarket risk management

Market risk management has developed a lot since late Market risk management has developed a lot since late 80’s. We can now observe two different aspects and 80’s. We can now observe two different aspects and uses of market risk managementuses of market risk management

PricingPricing: for some instruments it’s not available a “fair” market : for some instruments it’s not available a “fair” market price, because of their illiquidity (e.g. small firms’ bonds or price, because of their illiquidity (e.g. small firms’ bonds or complex derivatives)complex derivatives)

Risk measurementRisk measurement: it involves risks’ quantification and the : it involves risks’ quantification and the estimation of potential losses of a portfolio. In this case it’s estimation of potential losses of a portfolio. In this case it’s crucial to identify the main crucial to identify the main risk factorsrisk factors

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Pricing - bondsPricing - bondsExample of bond pricing:Example of bond pricing:

A bond price can be determined by discounting cash flowsA bond price can be determined by discounting cash flows Let’s suppose to have a bond with N coupons CLet’s suppose to have a bond with N coupons Ci i and with a nominal and with a nominal

amount 100, to be paid at maturity (T)amount 100, to be paid at maturity (T)

We have three variablesWe have three variables

• r: r: the risk free rate, obtained by interpolation of the swap curve the risk free rate, obtained by interpolation of the swap curve over the coupons’ maturitiesover the coupons’ maturities

• s: the credit spread, obtained by interpolation of the CDS spread s: the credit spread, obtained by interpolation of the CDS spread (when it’s available) or comparables’ spreads (e.g. its rating (when it’s available) or comparables’ spreads (e.g. its rating average spread) over the coupons’ maturitiesaverage spread) over the coupons’ maturities

• l: l: the liquidity spread, representing the liquidity premium for the the liquidity spread, representing the liquidity premium for the single issuance it’s not directly observable but has to be single issuance it’s not directly observable but has to be estimatedestimated

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Pricing - derivativesPricing - derivatives Simple derivative products have “closed formulas” that we can use Simple derivative products have “closed formulas” that we can use

to determine its price:to determine its price: e.g. for a call option on a listed stock or index we can use B&S formulae.g. for a call option on a listed stock or index we can use B&S formula

For more complex derivatives it’s impossible to have a closed For more complex derivatives it’s impossible to have a closed formula their price has to be estimated via Monte Carlo formula their price has to be estimated via Monte Carlo simulationsimulation

e.g. “barrier option”: the option pays a fixed coupon amount until the e.g. “barrier option”: the option pays a fixed coupon amount until the underlying stays below the barrier (generally 50% of its value at issue underlying stays below the barrier (generally 50% of its value at issue date), otherwise it replies the index performancedate), otherwise it replies the index performance

where where DFDFii is the discount factor of the i-th coupon is the discount factor of the i-th coupon

barrierIndexDFIndex

IndexP

barrierIndexDFDFCP

payoff

TT

i

N

iTiii

if*

if *100*

0

1

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Pricing - derivativesPricing - derivatives Monte Carlo pricing: we simulate the underlying price with a Monte Carlo pricing: we simulate the underlying price with a

Geometric Brownian Motion (GBM), under Geometric Brownian Motion (GBM), under risk-neutrality assumptionrisk-neutrality assumption

For every path, we get the payoff For every path, we get the payoff PPii

We repeat the simulation several times (e.g. 100,000 times)We repeat the simulation several times (e.g. 100,000 times) The The fair valuefair value (i.e. the market price) is represented by the average (i.e. the market price) is represented by the average

value of simulated payoffsvalue of simulated payoffs

)*

2exp*

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1 tdWtdrSS ttt

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Risk measurementRisk measurement

As we have already seen, risk measurement can be As we have already seen, risk measurement can be divided into two different families:divided into two different families:

1)1) Expected lossesExpected losses (or expected returns), in average market (or expected returns), in average market conditionsconditions

2)2) Unexpected lossesUnexpected losses, in stressed market conditions, in stressed market conditions

We will focus on the 2), that in market risk is quantified with We will focus on the 2), that in market risk is quantified with the the Value at Risk (VaR)Value at Risk (VaR)

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Value at RiskValue at Risk As we have already seen, Value at Risk (VaR) can be defined as the As we have already seen, Value at Risk (VaR) can be defined as the

maximum potential loss in a time horizon, given a confidence level. maximum potential loss in a time horizon, given a confidence level. Analytically, given a confidence level Analytically, given a confidence level αα, and being L the random , and being L the random variable representing losses within a time horizon, we havevariable representing losses within a time horizon, we have

The VaR is therefore a The VaR is therefore a quantilequantile in the loss distribution in the loss distribution Time horizon of market VaR in trading portfolios is generally 1 day (or Time horizon of market VaR in trading portfolios is generally 1 day (or

10 days for Basel II capital requirements), with confidence level 99%10 days for Basel II capital requirements), with confidence level 99% We have 3 different measures of VaR:We have 3 different measures of VaR:

ParametricParametric (or variance-covariance approach) (or variance-covariance approach) Historical simulationHistorical simulation Monte CarloMonte Carlo

1)Pr( VaRL

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The parametric approach assumes that changes in risk The parametric approach assumes that changes in risk factors are distributed according to a normal distribution factors are distributed according to a normal distribution with a variance of zero and stable volatility over timewith a variance of zero and stable volatility over time

Changes in assets’ values are derived from those in risk Changes in assets’ values are derived from those in risk factors through linear coefficients. More complex factors through linear coefficients. More complex instruments are decomposed in simpler ones (e.g. a instruments are decomposed in simpler ones (e.g. a fixed coupon bond is decomposed in a series of cash fixed coupon bond is decomposed in a series of cash flows)flows)

Changes in risk factors are determined by their Changes in risk factors are determined by their correlation matrixcorrelation matrix

VaR is obtained by multiplying the standard deviation by VaR is obtained by multiplying the standard deviation by a scaling factor a scaling factor αα which depends on the confidence level which depends on the confidence level

Parametric VaRParametric VaR

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Parametric VaRParametric VaR

For each asset we need to know the sensitivities to risk For each asset we need to know the sensitivities to risk factors:factors:

• Interest rate risk: Interest rate risk: modified durationmodified duration• Stock market risk: Stock market risk: betabeta• Derivatives: Derivatives: greeks greeks • Other risk factorsOther risk factors: delta: delta

Then, we need to know the Then, we need to know the variance-covariance matrix variance-covariance matrix of of the risk factors (to determine the risk factors (to determine volatilitiesvolatilities and and correlationscorrelations))

There are many databases (e.g. Risk Metrics) that provide There are many databases (e.g. Risk Metrics) that provide on a daily basis the variance-covariance matrix of risk on a daily basis the variance-covariance matrix of risk factorsfactors

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Parametric VaRParametric VaRExample:Example:

Let’s consider a USD dollar bond with a market value (MV) of 1,000,000 €. We Let’s consider a USD dollar bond with a market value (MV) of 1,000,000 €. We have (at least) 2 risk factors:have (at least) 2 risk factors:

• Interest rate riskInterest rate risk• Exchange rate riskExchange rate risk

Let’s suppose:Let’s suppose:• Modified duration (MD) of 5 yearsModified duration (MD) of 5 years• Volatility of interest rate (Volatility of interest rate (σσii =5%) and volatility of exchange rate =5%) and volatility of exchange rate

((σσee=10%) =10%) • Correlation (Correlation (ρρ) between interest rates and exchange rates equal to 20%) between interest rates and exchange rates equal to 20%• Confidence level 99% (Confidence level 99% (αα = 2.32) = 2.32)

We have:We have: **(MD)*MV iiVaR

**MV eeVaR

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Parametric VaRParametric VaRThe VaR of the asset The VaR of the asset pp will result: will result:

i.e. approximately 688,000 €i.e. approximately 688,000 €

Volatility measure are generally Volatility measure are generally annualized, annualized, if we want the daily VaR we can use if we want the daily VaR we can use the following transformation:the following transformation:

If we consider 252 working days in a year, the daily VaR will be:If we consider 252 working days in a year, the daily VaR will be:

N.B. The previous transformation, even if it’s widely used, is given under the N.B. The previous transformation, even if it’s widely used, is given under the assumption that changes in market factors are independent through time assumption that changes in market factors are independent through time (which has been generally denied by empirical studies!) (which has been generally denied by empirical studies!)

***222eieiP VaRVaRVaRVaRVaR

NVaRVaR tNt **

€43000252

€688000dailyVaR

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Parametric VaRParametric VaRAdvantages:Advantages:

The quantiles of the distribution are already determinedThe quantiles of the distribution are already determined Once the parameters are known, there’s no need of huge Once the parameters are known, there’s no need of huge

datasets to work out the distributiondatasets to work out the distribution Computationally very fast and simple (an Excel spreadsheet Computationally very fast and simple (an Excel spreadsheet

is generally sufficient to compute parametric VaR) is generally sufficient to compute parametric VaR)

Disadvantages:Disadvantages:The estimation of parameters is sometimes difficult and it The estimation of parameters is sometimes difficult and it

trivially depends on the estimation sample that we choosetrivially depends on the estimation sample that we chooseThe output of the model depends on the underlying The output of the model depends on the underlying

distribution chosen (e.g. it’s empirically proven that stock distribution chosen (e.g. it’s empirically proven that stock market returns are leptokurtic, i.e. the Gaussian distribution market returns are leptokurtic, i.e. the Gaussian distribution is not fit!)is not fit!)

The sensitivity of portfolio positions to changes in market The sensitivity of portfolio positions to changes in market factors is linear, but in general this is not truefactors is linear, but in general this is not true

The distribution of risk factors is stable (doesn’t allow The distribution of risk factors is stable (doesn’t allow changes in time) and assumes serial independencechanges in time) and assumes serial independence

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In historical simulation VaR, risk factors changes are assumed to be In historical simulation VaR, risk factors changes are assumed to be represented by their historical empirical distribution:represented by their historical empirical distribution:

We select a sample of risk factors’ changes, related to a given historical We select a sample of risk factors’ changes, related to a given historical period (e.g. last 500 working days)period (e.g. last 500 working days)

Every asset in the portfolio is revaluated at each data applying relative Every asset in the portfolio is revaluated at each data applying relative changes to risk factors changes to risk factors

In this way we have an empirical distribution of portfolio valuesIn this way we have an empirical distribution of portfolio values The VaR is the difference between the current portfolio value and the The VaR is the difference between the current portfolio value and the

percentile of empirical distribution corresponding to the desired confidence percentile of empirical distribution corresponding to the desired confidence levellevel

VaR – Historical simulationVaR – Historical simulation

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VaR – Historical simulationVaR – Historical simulationAdvantages:Advantages:

No need to choose a statistical distributionNo need to choose a statistical distribution Easy to understand: we refer to an historical event, not to a Easy to understand: we refer to an historical event, not to a

quantile in a density distributionquantile in a density distribution Does not require a variance-covariance matrix of risk factorsDoes not require a variance-covariance matrix of risk factors Allows to capture non-linear sensitivities to risk factorsAllows to capture non-linear sensitivities to risk factors

Disadvantages:Disadvantages: We should have a robust dataset of events with several periods We should have a robust dataset of events with several periods

(e.g. in economic analysis, different business cycles with growth and (e.g. in economic analysis, different business cycles with growth and recessions)recessions)

Computationally heavyComputationally heavy The choice of the historical deepness of the dataset is trivialThe choice of the historical deepness of the dataset is trivial Not always the past is the best predictor for the futureNot always the past is the best predictor for the future Past variations of some risk factors depend also on their level, that Past variations of some risk factors depend also on their level, that

could be very different from the current one (e.g. daily variations of could be very different from the current one (e.g. daily variations of volatility tend to be higher when volatility is high) volatility tend to be higher when volatility is high)

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Monte Carlo VaRMonte Carlo VaR

Monte Carlo VaR consists in the simulation of the risk Monte Carlo VaR consists in the simulation of the risk factors with a given stochastic processfactors with a given stochastic process

For each simulation we have a risk factors’ scenario, that For each simulation we have a risk factors’ scenario, that determines a theoretical value of the assets (and of the determines a theoretical value of the assets (and of the portfolio)portfolio)

We repeat the simulation several times (e.g. 10,000 We repeat the simulation several times (e.g. 10,000 times) and we get a portfolio return (and losses) times) and we get a portfolio return (and losses) distributiondistribution

The VaR is the difference between the current portfolio The VaR is the difference between the current portfolio value and the percentile of simulated distribution value and the percentile of simulated distribution corresponding to the desired confidence levelcorresponding to the desired confidence level

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Monte Carlo VaRMonte Carlo VaR Examples of stochastic processes:Examples of stochastic processes:

Equity: Geometric Brownian Motion (slide 26)Equity: Geometric Brownian Motion (slide 26) Interest rates: Cox Ingersoll and Ross (1984) – CIRInterest rates: Cox Ingersoll and Ross (1984) – CIR

it’s a it’s a mean reverting processmean reverting process: the interest rate : the interest rate r r tends to its long term tends to its long term

average average bb with speed with speed aa Exchange rates: GBM with drift given by the difference between domestic Exchange rates: GBM with drift given by the difference between domestic

interest rate and foreign interest rateinterest rate and foreign interest rate Credit spread: martingale process (without drift) with changes depending Credit spread: martingale process (without drift) with changes depending

on the level of the spread (higher spreads – higher spread volatility)on the level of the spread (higher spreads – higher spread volatility)

Volatility: GARCH models, Heston modelVolatility: GARCH models, Heston model

An accurate Monte Carlo VaR would require An accurate Monte Carlo VaR would require correlated processescorrelated processes: : we have to we have to generate correlated brownian motion, using generate correlated brownian motion, using Cholesky decomposition Cholesky decomposition (but the (but the estimation of correlations between different risk factors is sometimes very estimation of correlations between different risk factors is sometimes very difficult – e.g. interest rate vs credit spread)difficult – e.g. interest rate vs credit spread)

dzrdtrbadr )(

SdWds

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Advantages:Advantages: It’s considered the most accurate VaR measurementIt’s considered the most accurate VaR measurement The output portfolio distribution is not given under assumption of The output portfolio distribution is not given under assumption of

normality, but by the stochastic processes of the risk factorsnormality, but by the stochastic processes of the risk factors Allows to capture non-linear sensitivities to risk factorsAllows to capture non-linear sensitivities to risk factors The distribution of the risk factors is based on market expectations The distribution of the risk factors is based on market expectations

and not on historical dataand not on historical data It’s particularly fit for path dependent instruments (e.g. exotic It’s particularly fit for path dependent instruments (e.g. exotic

options)options)

Disadvantages:Disadvantages: Requires calibration of the models used to simulate the underlying Requires calibration of the models used to simulate the underlying

risk factors (sometimes very complex and time consuming)risk factors (sometimes very complex and time consuming) Requires the estimation of risk factors’ variance-covariance matrixRequires the estimation of risk factors’ variance-covariance matrix If we want a stable VaR measurement, we need to generate a lot of If we want a stable VaR measurement, we need to generate a lot of

scenarios it’s computationally heavy (it can takes many hours scenarios it’s computationally heavy (it can takes many hours or days) simplifications are necessary, therefore reducing the or days) simplifications are necessary, therefore reducing the accuracy of VaRaccuracy of VaR

Monte Carlo VaRMonte Carlo VaR

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Before using a VaR methodology for management Before using a VaR methodology for management purposes, this should be purposes, this should be back-testedback-tested

Back-testing means evaluating VaR in the past, Back-testing means evaluating VaR in the past, comparing realized (or simulated) losses with the comparing realized (or simulated) losses with the potential losses of the VaRpotential losses of the VaR

A VaR would pass the back test if the percentage of A VaR would pass the back test if the percentage of violationsviolations (i.e. when the realized loss exceeds the VaR) (i.e. when the realized loss exceeds the VaR) is lower than the complement of confidence levelis lower than the complement of confidence level

Example: let’s consider a VaR with 99% confidence Example: let’s consider a VaR with 99% confidence level. If we test the VaR over 500 days, the methodology level. If we test the VaR over 500 days, the methodology would pass the backtest if the violations are less than 5 would pass the backtest if the violations are less than 5 (1% of the sample)(1% of the sample)

VaR backtestingVaR backtesting

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Limits of VaRLimits of VaR The size of losses:The size of losses: VaR doesn’t provide any information about the VaR doesn’t provide any information about the

sizesize of extreme events. Let’s consider charts below. of extreme events. Let’s consider charts below.

The two portfolios have the same VaR (approx. 240,000) but the The two portfolios have the same VaR (approx. 240,000) but the second one is much more risky (tail events would cause huge second one is much more risky (tail events would cause huge losses!)losses!)

-400000 -300000 -200000 -100000 0 100000 200000 300000 400000Portfolio return

VaR - Portfolio 1

-1400000 -1200000 -1000000 -800000 -600000 -400000 -200000 0 200000 400000 Portfolio return

VaR - portfolio 2

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Expected shortfallExpected shortfall

Therefore VaR cannot be considered a coherent measure of riskTherefore VaR cannot be considered a coherent measure of risk An alternative measure of risk, without the problems listed above, is An alternative measure of risk, without the problems listed above, is

the expected shortfall (ES) or Conditional VaR (CVaR)the expected shortfall (ES) or Conditional VaR (CVaR) It represents the average losses in excess of VaRIt represents the average losses in excess of VaR

where MV is the market value of the portfoliowhere MV is the market value of the portfolio Even if ES has more desirable characteristics than VaR, the latter is Even if ES has more desirable characteristics than VaR, the latter is

still widely used in banks for management purposes and fixing of still widely used in banks for management purposes and fixing of risk limitsrisk limits

VaRMVMVEES |

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ReferencesReferences

1. Resti, A. and Sironi, A., “Risk Management and Shareholders’ Value in Banking”. 2007

2. Hull, J. C., “Option, Futures and other derivatives”

3. Cox, J. C., Ingersoll, J. E. and Ross, S. A., “ A theory of the Term structure of interest rates”, Econometrica, 53 (1985), 385-407

4. Heston, S. (1993): A closed-form solutions for options with stochastic volatility, Review of Financial Studies, 6, 327–343

5. Dupire, B. (January 1994). Pricing with a Smile. Risk Magazine, Incisive Media.

6. Sklar, A. (1959), "Fonctions de répartition à n dimensions et leurs marges", Publ. Inst. Statist. Univ. Paris 8: 229–231

7. Glasserman (2004), “Monte Carlo methods in financial engineering”