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- 1. LECTURE FOUR a. Introduction to market risk b. Modelling volatility c. VaR Models 1
- 2. Part 1INTRODUCTION TOMARKET RISK a. Overview b. Risk measurement c. Classification of risks d. Sources of market risks 2
- 3. 1. Overview Types of financial risk •Market risk: movements in prices or volatility •Liquidity risk: losses when a position is liquidated •Credit risk: counterparty cannot fulfil contractual obligations Can interact •Operational risk: related to an inadequate internal process, or caused by an external eventPart 1. Introduction to market risk Settlement risk Operational risk Example The time difference of two parties In the settlement date there is a delayed the payment one day Currency swap blackout that lasts a couple of hours Liquidity risk Credit risk One of the currencies is Iraqi dinar Market risk One of the counterparties goes to bankruptcy During any day the exchange rate can changeLecture 4 The Value at Risk is key to measure market risk : It includes probability and scenario analysis
- 4. 2. Risk measurement systems • From market data, construct the distribution of risk factors • Collect portfolio positions and collect then onto risk factorsPart 1. Introduction to market risk • Use the risk engine to construct the distribution of portfolio profits and losses over the period. Fixed income: linear VaR This is the reason why it is so important Back Options: non linear VaR Testing and Stress testing (scenario analysis) Additional consideration Position based (Risk of positions) VS Returns based (VaR based on returns): • Offer data for new instruments, market and managers • Capture style driftLecture 4 • More realistic BUT • More expensive (technologically)
- 5. 3. Classification of risks Known Knowns • All factors are identified • All factors are measured correctly • Appropriate description of distribution of risk factorsPart 1. Introduction to market risk Losses explained by: • Bad luck • Too much exposure VaR should be viewed as a measure of dispersion that SPX yearly return should be exceeded with some regularity Conditional VaR here is really important! • Losses once the VaR is brokenLecture 4
- 6. 3. Classification of risks Known UNKnowns (Some problems about known risks factors) • Management ignores important risk factors (i.g. political stress) • Inaccurate distribution for a specific factorPart 1. Introduction to market risk SPX volatility from 2004-2007 is extremely low to forecast 2007’s. Wrong distribution !! Mapping process could be incorrect UNKnown UNKnownsLecture 4 • Events totally outside the scenario: i.g. sudden restriction to short sales
- 7. 4. Sources of market risks Bond Where risk comes from? 1. Exposure to the factor 2. Movement to the factor itself Interest rate 3. Risk of the system Volatility or …Part 1. Introduction to market risk Market loss = Exposure x Adverse movement Currency Risk Fixed Income Risk 1. Volatility 1. Inflation: WHY??? 2. Correlations 2. Correlations among bonds 3. Cross-rate volatility: 3. Short term bonds have little price risk (durat.) Two currencies tied by a base currency 4. Reference rate (driven by expected inflation) 5. Credit spread: Bonds VS risk freeLecture 4 Note: TERM SPREAD. Long term-Short term
- 8. 4. Sources of market risks Equity Risk Commodity Risk 1. Volatility 1. Volatility 2. Future risksPart 1. Introduction to market riskLecture 4
- 9. Part 2MODELLINGVOLATILITY a. Intro. models b. Volatility standard approach c. Garch (1,1) d. EWMA e. Risk Metrics ™ f. Details 9
- 10. 1.Volatility Models Standard Approach to Estimating Volatility ARCH(m) Model EWMA Model GARCH model Maximum Likelihood Methods 2. Standard Approach to Estimating VolatilityPart 2. Volatility models • Define sn as the volatility per day between day nt-1 and day nt, as estimated at end of day nt-1 • Define Si as the value of market variable at end of day i • Define Ri= ln (Si/Si-1) {KNOWN from previous lecture}Lecture 4 1 m s 2 n m 1 i 1 ( Rn i R) 2 1 m R Rn i m i 1 A measure of divergence from average m-1 because there are m-1 returns
- 11. 3. Generalized AutoRegressive Conditional Heteroskedasticity 3. Garch (p,q) approximation • GARCH (p, q) and in particular GARCH (1, 1) • Autoregressive: tomorrow’s variance (or volatility) is a regressed function of today’s variance — it regresses on itself • Conditional: tomorrow’s variance depends—is conditional on — the most recent variance. An unconditional variance would notPart 2. Volatility models depend on today’s variance • Heteroskedastic: variances are not constant, they flux over time • GARCH (1, 1) “lags” or regresses on last period’s squared return (i.e., just 1 return) and last period’s variance (i.e., just 1 variance).Lecture 4
- 12. 3. Garch (1,1) approximation In GARCH (1,1) we assign some weight to the long-run average variance rate s gVL + aR 2 n 2 n 1 + bs 2 n 1Part 2. Volatility models Since weights must sum to 1 g + a + b 1 Setting w gV, the GARCH (1,1) model s n w + aRn1 + bs n1 2 2 2Lecture 4 w And: VL 1 a b
- 13. 3. Garch (1,1) approximation p q s w + a i R 2 n 2 n i + b js 2 n j i 1 j 1 substitutefor s n -1 in GARCH(1,1) 2 s n 1 w + aRn21 + b (w + aRn22 + bs n 2 ) 2 2 w + bw + aRn 1 + abRn 2 + b 2s n 2 2 2 2Part 2. Volatility models substitutefor s n -2 2 s n 2 w + bw + b 2w + aRn21 + abRn22 + ab 2 Rn23 + b 3s n 3 2 2 The weights decline exponentially at rate βLecture 4 Tomorrow’s variance is a weighted average of the long run variance!!
- 14. 3. Garch (1,1) approximation Note the Mean reversion! The GARCH(1,1) model recognizes that over time the variance tends to get pulled back to a long-run average level The GARCH(1,1) is equivalent to a model where the variance V follows the stochastic process VL dV a(VL V )dt + VdzPart 2. Volatility models where time measured in days,a 1 - a - b , a 2Lecture 4 14
- 15. 3. Garch (1,1) approximation s n gVL + aRn1 + bs n 1 Volatility of past periods (lagged variance) 2 2 2 Long Run Variance Returns of past periods Long Term Variance (Av. Variance of all) 0,050163% Gamma 0,2 Alpha 0,3 Beta 0,5 Weights of each factor Garch 0,0275% From here we 1st: (1-lambda) ca n ca lculate 2nd: last*lamb s i mple va riance Yesterday Date Close Daily Return Return to 2 Weights 23/10/2012 41,33 -1,822% 0,0332%Part 2. Volatility models 22/10/2012 42,09 -0,545% 0,0030% 6,00% 0,0002% 19/10/2012 42,32 -1,617% 0,0262% 5,64% 0,0015% 18/10/2012 43,01 -0,718% 0,0052% 5,30% 0,0003% 17/10/2012 43,32 1,138% 0,0129% 4,98% 0,0006% 16/10/2012 42,83 1,056% 0,0112% 4,68% 0,0005% 15/10/2012 42,38 1,810% 0,0327% 4,40% 0,0014% 31/10/2011 34,76 -5,404% 0,2920% 0,00% 0,0000% 28/10/2011 36,69 -0,895% 0,0080% 0,00% 0,0000% 27/10/2011 37,02 7,982% 0,6371% 0,00% 0,0000%Lecture 4 26/10/2011 34,18 2,039% 0,0416% 0,00% 0,0000% 25/10/2011 33,49 -3,174% 0,1007% 0,00% 0,0000% 24/10/2011 34,57 3,383% 0,1145% 0,00% 0,0000% 21/10/2011 33,42 To weighted returns : recent past 0,014942% will affect more!
- 16. 4. Exponentially-Weighted Moving Average 4. EWMA approximation Garch (1,1) s n w + aRn1 + bs n1 2 2 2 = 0 and ( + ) =1: The equation simplifies to s (1 b ) R 2 n 2 n 1 + bs 2 n 1Part 2. Volatility models This is now equivalent to the formula for exponentially weighted moving average (EWMA): EWMA s (1 ) R 2 2 n 1 + s 2 n 1Lecture 4 n In EWMA, the lambda parameter now determines the “decay:” a lambda that is close to one (high lambda) exhibits slow decay.
- 17. 3. EWMA approach EWMA s n s n1 + (1 ) Rn1 2 2 2 replace s n 1 2 s [s 2 n 2 n2 + (1 ) R 2 n2 ] + (1 ) R 2 n 1 (1 )( R 2 n 1 + R 2 n2 )+ s 2 2 n2 substitutes 2Part 2. Volatility models n2 s (1 )( R 2 n 2 n 1 + R 2 n2 + R 2 2 n 3 )+s3 2 n 3 continuing the way givesLecture 4 m s n (1 ) i 1 Rn2i + ms n m 2 2 i 1
- 18. 3. EWMA approach For the large m, ms n m is sufficient ly small 2 the equation is same as the quation of weight scheme where a i (1 )i 1 or a i +1 a i Advantages • Relatively little data needs to be stored • We need only remember the current estimate of the variance rate and the most recent observation on the marketPart 2. Volatility models variable • Tracks volatility changesLecture 4
- 19. 3. EWMA approximation Example EWMA Calculation From here we 1st: (1-lambda) ca n ca lculate 2nd: last*lamb s i mple va riance Today Yesterday Weights of sq. Date Close Daily Return Return to 2 Weights Weights Returns 23/10/2012 41,33 -1,822% 0,0332% 6,00% 0,0020% 22/10/2012 42,09 -0,545% 0,0030% 5,64% 0,0002% 6,00% 0,0002% 19/10/2012 42,32 -1,617% 0,0262% 5,30% 0,0014% 5,64% 0,0015% 18/10/2012 43,01 -0,718% 0,0052% 4,98% 0,0003% 5,30% 0,0003% 17/10/2012 43,32 1,138% 0,0129% 4,68% 0,0006% 4,98% 0,0006% 16/10/2012 42,83 1,056% 0,0112% 4,40% 0,0005% 4,68% 0,0005% 15/10/2012 42,38 1,810% 0,0327% 4,14% 0,0014% 4,40% 0,0014% 31/10/2011 34,76 -5,404% 0,2920% 0,00% 0,0000% 0,00% 0,0000% 28/10/2011 36,69 -0,895% 0,0080% 0,00% 0,0000% 0,00% 0,0000% 27/10/2011 37,02 7,982% 0,6371% 0,00% 0,0000% 0,00% 0,0000%Part 2. Volatility models 26/10/2011 34,18 2,039% 0,0416% 0,00% 0,0000% 0,00% 0,0000% 25/10/2011 33,49 -3,174% 0,1007% 0,00% 0,0000% 0,00% 0,0000% 24/10/2011 34,57 3,383% 0,1145% 0,00% 0,0000% 0,00% 0,0000% 21/10/2011 33,42 To weighted returns : recent past Volatility 0,016037% Volatility 0,014942% will affect more! in a exponentially Is the summatory of weighted declining fashion squared returns --Proportional decay -- We do not have to calculate the complete series EWMA s n (1 ) Rn1 + s n1 2 2 2Lecture 4 0,016037% Lambda 0,94
- 20. Lecture 4Part 2. Volatility models Example 3. EWMA approximation
- 21. 3. Risk Metrics 5. Risk Metrics RiskMetrics is a branded form of the exponentially weighted moving average (EWMA) approach: The optimal (theoretical) lambda varies by asset class, but the overall optimal parameter used by RiskMetrics has been 0.94. In practice, RiskMetrics only uses one decay factor for all series:Part 2. Volatility models • · 0.94 for daily data • · 0.97 for monthly data (month defined as 25 trading days)Lecture 4
- 22. 6. Some important details EWMA is (technically) an infinite series but the infinite series elegantly reduces to a recursive form: R R RPart 2. Volatility models R RLecture 4
- 23. Lecture 4Part 2. Volatility models 6. Some important details
- 24. Part 3VALUE AT RISKMODELS a. Overview b. Initial considerations c. Var Models (intro) d. VaR Historical e. Parametric Approach f. Monte Carlo Approach g. Basel 2 24
- 25. 1. Overview VaR There are many models that measure risk. However the Value at Risk is the most popular and also answers all requirements in a financial institution Definition: .VaR is a measure of the 1. worst expected loss that a firm may suffer 2. over a period of time that has been specified by the user, 3. under normal market conditions andPart 3. VaR Methodologies 4. a specified level of confidence. Specifically, it is the maximum loss which can occur with X% confidence over a holding period of n days. It has several advantages, but the most important ones are:Lecture 4 • It gives a clear number (only one) and • It is easy to implement and interpret.
- 26. 1. Overview VaR Limitations: 1. These methods use past historical data to provide an estimate for the future. What happened in the past does not mean that will happen again in the future 2. VaR number can be calculated by using several methods. These methods try to capture volatilitys behavior. However, there is an argument on which is the method that performs best.Part 3. VaR Methodologies 3. Methods for computing it are based on different assumptions. These assumptions help us with the calculation of VaR but they are not always true (like distributional assumptions). 4. There are many risk variables (political risk, liquidity risk, etc ) thatLecture 4 cannot be captured by the VaR methods.
- 27. 2. Additional Considerations • Does NOT describe the worst loss • Only describes the probability that a value occurs • VaR number indicates that 1% of days of a period of time, the losses could be higher • The previous VaR depends on history!. So it will be very important that data have at least one crisis.Part 3. VaR Methodologies VaR Conditional VaR. Conditional VaR The potential loss when the portfolio is hit beyond VaRLecture 4 In JPM case it is $116.000
- 28. 2. Additional Considerations VaR: additional considerations Portfolio: $1.000.000 = $48.404 SD: $23.300 Prob: Normal distribution. 1.65Part 3. VaR MethodologiesLecture 4
- 29. 2. Additional Considerations Maximum drawbackPart 3. VaR Methodologies 70% 45% Main limitation: not comparable among portfolios There is no one VaR number for a single portfolio, because different methodologies used for calculating VaR produce different results.Lecture 4 The VaR number captures only those risks that can be measured in quantitative terms; it does not capture risk exposures such as operational risk, liquidity risk, regulatory risk or sovereign risk.
- 30. 2. Additional Considerations VaR parametersPart 3. VaR Methodologies Short time Long time To check a specific portfolio To avoid bankruptcyLecture 4
- 31. 3.VaR Models Forecasts n paths and find the VaR Order numbersPart 3. VaR Methodologies and obtain quintiles and using history, losses could be…Lecture 4
- 32. 4. Historical simulation Definition Tries to find an empirical distribution of the rates of return assuming that past history carries out into the future. • Uses the historical distribution of returns of a portfolio to simulate the portfolios VaR.Part 3. VaR Methodologies • Often historical simulation is called non-parametric approach, because parameters like variances and covariances do not have to be estimated, as they are implicit in the data. • The choice of sample period influences the accuracy of VaR estimates.Lecture 4 • Longer periods provide better VaR estimates than short ones.
- 33. 4. Historical simulation The methodology: • Identifying the instruments in a portfolio and collecting a sample of their historical returns. • Calculate the simulated price of every instrument using the weights of the current portfolio (in order to simulate the returns in the next period). • Assumption: returns follow is a good proxy for the returns in the next period.Part 3. VaR Methodologies Illustration: We have 1 M pounds in JPM Stocks, and we want to figure out what could be the value at risk of this position 1. Obtain the data. In this case from 2000 Jan to 2012 Oct 2. Calculate daily return (or weekly), depend on the VaR 3. Estimate daily (weekly) gain/lossLecture 4 4. We can construct a frequency distribution of daily returns 5. We can calculate our value at risk
- 34. 4. Historical simulation VaR Deviation from the average returnPart 3. VaR Methodologies With 99% prob, With 99% prob, the loss won’t the loss won’t be higher than be higher than $75.000 per M $75.000 per MLecture 4 I have N observations, and it is easy to find what observation is 1%
- 35. 4. Historical simulation Advantages • It does not depend on assumptions about the distribution of returns. I would avoid fat tails issues • There is no need for any parameter estimation. • There are not different models for equities, bonds and derivatives DisadvantagesPart 3. VaR Methodologies • Results are dependent on the data set from the past, which may be too volatile or not, to pre-dict the future. • Assumes that returns are independent and identically distributed. • It uses the same weights on all past observations. If anLecture 4 observation from the distant past is excluded the VaR estimates may change significantly.
- 36. 5. Parametric approach Variance – Covariance approach Definition This approach for calculating the value at risk is also known as the delta-normal method. • This is the most straightforward method of calculating Value at Risk.Part 3. VaR Methodologies • It is the method used by the RiskMetrics methodology, the VaR system originally developed by JP Morgan. • Assumes that returns are normally distributed. It ONLY requires that we estimate two factors • expected (or average) return and • a standard deviationLecture 4 Using them it could be possible to plot a normal distribution curve. We use the familiar curve instead of actual data
- 37. 5.Variance – Covariance approach Wealth Volatility (sd) of the portfolioPart 3. VaR Methodologies Confidence level (normal equivalent) Variance-Covariance matrix showing wealthLecture 4 $ Positions x VaR-CoV x $ Positions
- 38. 5.Variance – Covariance approach Variance Covariance VaR Goog Nok Portfolio Value $ 100,00 Weights 1/3 2/3 Stock worths $ 33,33 $ 66,67 Volatility 1,703% 3,879% Correlation 0,26109 Pearson Correlation Var-Cov Matrix Goog Nok X Var-Cov Matrix XPart 3. VaR Methodologies Goog 0,000290 0,000172 33,33 66,67 0,000290 0,000172 33,33 NoK 0,000172 0,001505 0,000172 0,001505 66,67 1X 2 2X 2 2x 1 Variance = Covariance: Volatility * Volatility 0,02116696 7,778 σxy=ρxy σx σy 0,10608465 Variance $ 7,78 Volatility $ 2,79Lecture 4 Confidence 95% Critical value 1,645 VaR $ 4,59
- 39. 5.Variance – Covariance approach Advantages • Easy to capture relations among data DisadvantagesPart 3. VaR Methodologies • assuming normal distribution of returns for assets and portfolios with non-normal skewness or excess kurtosis. Using unrealistic return distributions as inputs can lead to underestimating the real risk with VAR.Lecture 4
- 40. 6. Monte Carlo methods Overview • Monte Carlo simulation try to simulate the conditions, which apply to a specific problem, by generating a large number of random samples • Each simulation will be different but in total the simulations will aggregate to the chosen statistical parametersPart 3. VaR Methodologies • After generating the data, quantities such as the mean and variance of the generated numbers can be used as estimates of the unknown parameters of the population • It is more flexibleLecture 4 • Allows the risk manager to use actual historical distributions for risk factor returns rather than having to assume normal returns.
- 41. 6. Monte-Carlo Simulation approach Theory • Consider a stock S, with a price of $20 • The price can only rise (drop) $1 each day for successive days • Forecast instrument: a coin: this is a RANDOM VARIABLE • The more days, the more simulation paths What can we assure? • The EXPECTED mean of the price will be $20 (no matter how many periods ahead!!!) • It is possible to calculate standard deviation and probabilisticPart 3. VaR Methodologies statements We cannot determine what could be the price at the end of a periodLecture 4
- 42. 6. Monte-Carlo Simulation approach TheoryPart 3. VaR MethodologiesLecture 4
- 43. 6. Monte-Carlo Simulation approach Geometric Brownian Motion Continuity: The paths are continuous in time and value. (stock prices can be observed at all times and they are changing). We assume that traders and systems are working weekends and nights Markov process: GBM follows a Markov process, meaning that only the current stock’s price history is relevant for predicting futurePart 3. VaR Methodologies prices (stock price history is irrelevant). Weak form of the efficient market hypothesis. No momentum when occurs a trend No technical analysis Normality: the proportional return over infinite increments of time for a stock is normally distributedLecture 4 The price of a stock is lognormally distributed
- 44. 6. Monte-Carlo Simulation approach Geometric Brownian Motion S Very short period of time t + s t S Certain component Uncertain component The return is uncertain or random Stochastic component Deterministic component (drift)Part 3. VaR Methodologies • ε is the ranom component of the • μ is the expected rate of return standard normal distribution (mean • If the price of the stock today is S0, 0 and sd of 1). then its price ST at time T in the • σ is the volatility future would be: • The longer the time interval, the ST=S0 e(μT) more variable the returnLecture 4 S We already know how to prove this!! N ( t , s t ) S
- 45. 6. Monte-Carlo Simulation approach Geometric Brownian Motion S t + s t S • S=$10 • μ=12% per year • σ=40% per year • t= 1 day, that is 0.004 of a year What this S number 0.12 * 0.004 + 0.4 * 0.8 * 0.0632 2.07% means ?Part 3. VaR Methodologies BUT: S Draws for e will be sometimes negative, the proportional return can be positive and negativeLecture 4
- 46. 6. Monte-Carlo Simulation approach Prices Log-normal distributed The natural log of S are normally distributed S s2 ln N T , s T S 2 The price path will be S t + t s2 ln t + s t S 2 Part 3. VaR Methodologies t s2 St + t St exp t + s t 2 Lecture 4
- 47. 6. Monte-Carlo Simulation approach Prices Log-normal distributed Apply my formula Expected return (yearly) 20% Daily return 0,08% Volatility (yearly) 40% Price Daily volatility 2,52% Day Random Uniform Normal Count 10 Time t (in days) 1 1 0,28878 -0,557 9,65851 Stock price $ 10 N(0,1). Exp value of 0 and sd of 2 0,53095 0,07766 9,71073 1 3 0,70087 0,52692 10,0446 s 2 ( ) (yearly) 12,000% Standard normal cumulative 4 0,44846 -0,1296 9,96742 2 distribution. Value btw -3 and 3. 5 0,15337 -1,0221 9,34798 6 0,16773 -0,9632 8,79975Part 3. VaR Methodologies To randomize my volatility 7 0,47247 -0,0691 8,7656 8 0,2168 -0,783 8,34607 9 0,39029 -0,2786 8,20426Lecture 4
- 48. 6. Monte-Carlo Simulation approach Other models (interest rates)Part 3. VaR MethodologiesLecture 4
- 49. 6. Monte Carlo simulation approach Advantages • Able to model instruments with non-linear and path- dependent payoff functions (complex derivatives). • Moreover, is not affected as much as Historical Simulations VaR by extreme events • We may use any statistical distribution to simulate the returns as far as we feel comfortable with the underlying assumptions thatPart 3. VaR Methodologies justify the use of a particular distribution. Disadvantages • The main disadvantage of Monte Carlo Simulations VaR is the computer power that is required to perform all the simulations • Cost associated with developing a VaRLecture 4
- 50. 7. Basel 2 (a quick view) Qualitative Criteria VaR is a robust Risk Measurement and Management Practice Banks can use their own VaR models as basis for capital requirement for Market Risk Regular Back-Testing Initial and on-going Validation of Internal Model Bank’s Internal Risk Measurement Model must be integrated intoPart 3. VaR Methodologies Management decisions Risk measurement system should be used in conjunction with Trading and Exposure Limits. Stress Testing Risk measurement systems should be well documentedLecture 4 Independent review of risk measurement systems by internal audit Board and senior management should be actively involved
- 51. 3. Basel 2 (a quick view) Quantitative Parameters : VaR computation be based on following inputs : • Horizon of 10 Trading days • 99% confidence level • Observation period – at least 1 year historical data Correlations : recognise correlation within Categories as well as across categories (FI and Fx, etc)Part 3. VaR Methodologies Market Risk charge : General Market Risk charge shall be – Higher of previous day’s VaR or Avg VaR over last 60 business days X Multiplier factor K (absolute floor of 3) MRCt = Max (Avg VaR over 60 days, VaR t-1) + SRC SRC – Specific Risk ChargeLecture 4 K>3 is a multiplier
- 52. LECTURE FOUREnd Of The Lecture 52

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