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Market Risk ManagementApril 8, 2015

Understanding Market Risk

Effective Risk Management Banks Perspective and Market Risk Managers Perspective

Market Risk Metrics

VaR Models Parametric, Historical Simulation, Monte Carlo Simulation

Stress Testing

Extreme Values Theory

Outline

What Is Market Risk?

Basel Committee

the risk that the value of an investment will decrease due to moves in market factors

National Bank of Romania NBR Regulation No. 5/2013

the risk to incur losses corresponding to on-balance and off-balance positions due to adversemarket movements in prices and interest rates concerning the trading book business, as well asfrom movements in foreign exchange rate and commodities prices for the whole business of thecredit institution (e.g. share prices, interest rate, foreign exchange rate)

Indeed, better risk management may be the only truly necessary element of success in banking. (Alan Greenspan, 2004)

Risk management does not attempt to eliminate risk, but it ensures that the leveland sources of risk are consistent with the risk appetite, risk profile, systemicmateriality and level of capitalization of the credit institution

Risk management does not represent a hindrance for a banks activity

It remains essential to value creation -> a bank with an effective risk managementsystem is more comfortable positioned to undertake higher risks (in line with its risktolerance) that should translate in time into higher return -> maximize the risk-adjusted return on capital

Risk Management - Facilitator Vs. Hindrance

Endow risk management with autonomy, empower risk management with control ofthe oversight process, assure total management support

Ensure excellent reporting framework so that risk management always has accurateinformation and disseminates timely to decision makers

Build/acquire powerful analytical tools that are easy to use, and that enable acommon risk language

High technical skills in the risk management team are a must, so that risk managersunderstand in detail how the portfolio would perform under stressed marketconditions, recognizing that risk in the future is not necessarily the same as in thepast Next Page.

Effective Risk Management Banks Perspective

What Do You Need to Be a Market Risk Manager? Employee Perspective

Be well versed in regulations

and understand

how they affect the bank

Master : linear algebra &

probability including stochastic calculus statistics &

econometrics

Market Risk Manager

To Remember

Overconfidence in numbers and quantitative techniques and in our ability to represent extreme events should be subject to severe criticism because it lulls us into a false sense of security. (Thomas Coleman)

Essentially, all models are wrong, but some are useful. (Box and Draper)

Market Risk Metrics

- Vol & Correl (only when the returns/risk factor returns have normal or Student/ multivariate elliptical distribution) - Traditional risk metrics measure only the sensitivity to a risk factor

Aggregation of individual exposure data on portfolio level based on risk factors multivariate distribution: Value at Risk, Expected Shortfall

Aggregation of individual exposure data on portfolio level based on scenarios of risk factor movements: Stress Tests

- dur, KR dur, convexity, DV01, - Spread dur, CR01- beta of a stock /portfolio - delta and gamma of an option

portfolio

A market risk metric is a single number that captures the uncertainty in a portfolios P&L, or in its return, summarizing the portfolios potential for deviations from a target or expected return

Why Do I Need Risk Models

Purpose: disaggregate portfolio risk into components corresponding to different types of risk factors

Steps:1. Identify the risk factors Risk factor mapping rather than modeling the returns or P&L

distribution directly at the portfolio level2. Calculate portfolio sensitivities to risk factors 3. Identify the risk factors distribution/conditional distribution (or empirical distribution)4. Analyze the distribution is serial correlation present, long term dependence, time-varying

volatility, fat tails, asymmetry.5. Analyze the dependence structure of risk factors.

Note:All risk metrics, including VaR, should take account of portfolio diversification effects when aggregating risks across different types of risk factors.

Risk Factor Mapping In Practice

Interest rate sensitive portfolios have interest rate risk factors and duration or present value of basis point (PV01) sensitivities.

Equity portfolios usually have indices as risk factors and equity betas as sensitivities.

Futures and forwards have interest rate risk factors with present value of basis point sensitivities, in addition to the risk factors of the corresponding spot position.

Options portfolios have many risk factors, the main ones being the underlying assets and the assets implied volatility surfaces. Sensitivities to risk factors are called the portfolio Greeks.

Value at Risk

Currently the most widely applied risk management measure It can be measured at any level, from an individual trade or portfolio, to a single

enterprise-wide VaR measure covering all risks in the firm as a whole It is an universal metric that applies to all activities and to all types of risk

Under the Basel II Accord, banks using internal VaR models to assess their market risk capital requirement should measure VaR at the 1% significance level,

Value at Risk (VaR) is an estimate of the loss from a fixed set of trading positions over a fixed time horizon that would be equaled or exceeded with a specific probability. Parameters:- time horizon

- confidence level

To calculate VaR: Need to know the MTM of your portfolio / position

What is the 1-day 95% VaR for Triangle Asset Management?

12

The pdf for daily profits at TAM can be described by the following function:

1

10

1

100 , 10 0

1

10

1

100 , 0 10

What is the one-day 95% VaR for TAM?

R: 6.84

Expected Shortfall/Expected Tail Loss/Conditional VaR

ETL is the loss we can expect to make if we get a loss in excess of VaR.

VaR is already pass (Kevin Dowd Measuring Market Risk)

The current frameworks reliance on VaR as a quantitative risk metric raises a number of issues, most notably the inability of the measure to capture the tail risk of the loss distribution. [] ES accounts for the tail risk in a more comprehensive manner, considering both the size and likelihood of losses above a certain threshold. (BCBS -Fundamental review of the trading book: A revised market risk framework, October 2013)

ETL

To define ES, let L be a random variable density f and distribution FL ; a confidence level (close to 1).

Average loss beyond -quantileof FL:

ES 1- =E[L|L>VaR1- ]

ES 1- =

ETL is a coherent risk measure.

What is the ES of TAM?

15

R: 7.89

VaR Models and Risk Factor Returns/Asset Returns Distribution

The distribution of risk factor returns is multivariate normal (each risk factor is i.i.d. and normal)

Not adequate for non-linear instruments with asymmetric payoffs

Parametric Linear VaR (Analytical/

Delta-normal)

Does not make any assumption about the parametric form of the distribution

Historical simulated distribution is identical to the returns distribution over the forward looking risk horizon

Historical Simulation VaR

Different assumptions about the multivariate distribution of risk factor returns can be accommodated use copula to model the dependence and specify any type of marginal risk factors return distribution we want

Monte Carlo Simulation VaR

Parametric Linear VaR (1)

h-day VaR at the confidence level (1-)% is given by:,=

(where is the lower - quantile return of the distribution (!,"2) -> P( R

Parametric Linear VaR (2)

Example 1

For the price of a bond, using only the first derivative from the Taylor series:

% %01 '()*+,-).,-)/-0''-+0/

VaR 1-day, = 1(1) PV01 * Daily Yield (bps) ( Assumptions: mean of daily yield changes approximately zero, daily yield changes i.i.d. and standard normal distributed, quantile of a standard normal distribution )

The parametric linear VaR model is applicable to all portfolios except those containing options, or any other instruments with non-linear price functions.-> all portfolio whose returns or P&L is a linear function of its risk factors returns or its asset returns

Parametric Linear Example (3)

Reminder VaR formula:

,=(#"+! )

You look at a portfolio with daily returns that are normally and identically distributed with expectation 0% and standard deviation of 1.5%. What is 1% 1-day VaR, under the assumption that daily excess returns are independent?

VaR 1, 0.01 = 2.33*0.015=0.0349

We can use the square root of time rule:

VaR 1, 0.99 = 10 3.49%=11.035%

Parametric Linear VaR (4)

Example 2

For a portfolio of fixed income securities, with 1the vector of PV01s (key rate DV01) and the covariance matrix of risk factor returns (bps changes in yields):

Interest Rate VaR 1-day, = 1(1) 12 1

Risk factor returns are normally distributed and their joint distribution is multivariate normal, therefore the covariance matrix of risk factors returns is all that is required to capture the dependency between risk factors returns

Parametric Linear VaR Flavours

Analytical formula for linear VaR can be obtained also for risk factor returns with other distribution than normal:

-Student t distribution (accommodate a leptokurtic distribution)

-Normal mixture when market display regime-specific behavior (variance mixture same mean, normal mixture especially when I identify asymmetry in data)

Principal components can be used as risk factors, especially for Interest Rate VaR as the first 3 components have an intuitive interpretation: level, slope, curvature.

Interest Rate PC VaR 1-day, = 1(1) 1234321

Where W* is the matrix whose columns are the first 3 eigenvectors of the covariance matrix of absolute changes in yields and D = diag (first 3 eigenvalues).

Stress tests are very easy to be applied on PC representations of changes in yields.

EWMA Parametric Linear VaR

Exponentially weighted model - places exponentially declining weights on past observations such as:

Most recent observations are given higher weights Todays variance will be positively correlated with yesterdays variance, which captures

the idea of volatility clustering

Estimate volatility:

Parameters- - persistence parameter - the higher the , the bigger

the impact of t1 on t- (1) reflects the intensity of reaction.

Unfortunately, these two are not independent in EWMA model (they are in GARCH).

Obtaining the EWMA Covariance Matrix

EWMA Parametric Linear VaR

Advantages over the equally weighted model:

Volatility reacts faster to shocks in the market as recent data carry more weight

Following a shock, volatility declines rapidly as the weight of the shock observation falls(hence, ghost effect from moving window disappears)

RiskMetrics: 6=0.94 produces good forecast for one-day volatility, and 6=0.97 results ingood estimates for one-month volatility)

Note:

EWMA produces a constant volatility for any future forecast, which is equal to todaysvolatility calculated by EWMA. Hence, EWMA fails to capture the key characteristic ofvolatility time variability and is not appropriate for estimating the market evolution ofvolatility over longer time horizon.

Scaling VaR to Different Risk Horizon

When is it rational to apply square root of time rule?

Returns are normal and i.i.d Yes or No?

Returns are not normal but are i.i.d. (with finite variance) Yes or No? Thing about Central Limit Theorem

Returns are normal but not independent volatility clustering Yes or No?

Correction for serial correlation when daily log returns follow a AR(1):V(789) 79;< 2 >?@ 79; three types of GEV, depending on the : Gumbel, Weibull, Frchet.

Generalized Extreme Value (GEV) distributions

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If =0 =>Gumbel distribution. The corresponding density function has a mode at 0, positive skew and declines exponentially in the tails.

If Weibull distribution. The density converges to a mass at . If >0 we have the Frchet distribution. The density converges to a mass at ,

but it converges more slowly than the Weibull density since the tail in the Frchet declines by a power law.

Generalized Pareto Distribution (GPD)

applies only to a specific tail, i.e. to excesses over a pre-defined threshold -peaks-over-threshold (POT) model.

- scale parameter and is the tail index

How do I compute VaR?

Correlation is a minefield for the unwary(Embrechts at all 2002)

When returns are not assumed to have elliptical distributions Pearsons linear correlation is an inaccurate and misleading measure of association between two returns series.

Why ?

- Correlation is not invariant under transformation of variables.- Feasible values for correlation depend on the marginal distributions.- Perfect positive dependence does not imply a correlation of one.- Zero correlation does not imply independence.- Variances must be finite: this is often not true with heavy-taileddistribution e.g.Cauchy distribution

Correlation Misleading Indicator of Non-Linear Dependence

Source: WikipediaSeveral sets of (x, y) points, with the Pearson correlation coefficient of x and y for each set. Note that the correlation reflects the noisiness and direction of a linear relationship (top row), but not the slope of that relationship (middle), normany aspects of nonlinear relationships (bottom). N.B.: the figure in the center has a slope of 0 but in that case the correlation coefficient is undefined because the variance of Y is zero.

How To Model Dependence in a Multivariate Setting

Volatility and correlation are portfolio risk metrics but they are only sufficient(in the sense that these metrics alone define the shape of a portfolios return or P&Ldistribution) when asset or risk factor returns have a multivariate normal distribution. Whenthese returns are not multivariate normal (or multivariate Student t) it is inappropriate andmisleading to use volatility and correlation to summarize uncertainty in the future value of aportfolio. (Carol Alexander Market Risk Analysis, Vol II)

Questions- What is the multivariate distribution of r.v. with different marginal distributions?

- How can I capture the dependence structure in such a multivariate distribution?Answer- Copula theory A copula is a function that joins a multivariate distribution function to a collection of univariate marginal distributions function. The construction of a joint distribution entails estimating the parameters of both the marginal distributions and the copula. different copulas produce different joint distributions when applied to the same marginals.

Copula Theory

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Two random variables X1 and X2 with continuous marginal distribution functions F1(x1) and F2 (x2) and set u i= Fi(xi) , i= 1, 2.

The bivariate form of Sklars theorem : given any joint distributionfunction F(x1,x2) there is a unique copula function C : [0,1][0,1][0,1] such that:

F(x1,x2) =C(F1(x1) , F2 (x2) )

Why does this help?

Step 1: Choose a copula to represent the dependence structure

Step 2: Estimate the parameters involved

Step 3: Apply the copula to the marginals

Step 4: Use the joint distribution function to estimate any risk measures.

Examples of Copulas(1)

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Examples of Copulas (2)

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A bivariate normal mixture copula density Bivariate Gumbel copula density for =1.25