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Modeling and forecasting. Volatility, слайд №1Modeling and forecasting. Volatility, слайд №2Modeling and forecasting. Volatility, слайд №3Modeling and forecasting. Volatility, слайд №4Modeling and forecasting. Volatility, слайд №5Modeling and forecasting. Volatility, слайд №6Modeling and forecasting. Volatility, слайд №7Modeling and forecasting. Volatility, слайд №8Modeling and forecasting. Volatility, слайд №9Modeling and forecasting. Volatility, слайд №10Modeling and forecasting. Volatility, слайд №11Modeling and forecasting. Volatility, слайд №12Modeling and forecasting. Volatility, слайд №13Modeling and forecasting. Volatility, слайд №14Modeling and forecasting. Volatility, слайд №15Modeling and forecasting. Volatility, слайд №16Modeling and forecasting. Volatility, слайд №17Modeling and forecasting. Volatility, слайд №18Modeling and forecasting. Volatility, слайд №19Modeling and forecasting. Volatility, слайд №20Modeling and forecasting. Volatility, слайд №21Modeling and forecasting. Volatility, слайд №22Modeling and forecasting. Volatility, слайд №23Modeling and forecasting. Volatility, слайд №24Modeling and forecasting. Volatility, слайд №25Modeling and forecasting. Volatility, слайд №26Modeling and forecasting. Volatility, слайд №27Modeling and forecasting. Volatility, слайд №28Modeling and forecasting. Volatility, слайд №29Modeling and forecasting. Volatility, слайд №30Modeling and forecasting. Volatility, слайд №31Modeling and forecasting. Volatility, слайд №32Modeling and forecasting. Volatility, слайд №33Modeling and forecasting. Volatility, слайд №34Modeling and forecasting. Volatility, слайд №35Modeling and forecasting. Volatility, слайд №36Modeling and forecasting. Volatility, слайд №37Modeling and forecasting. Volatility, слайд №38Modeling and forecasting. Volatility, слайд №39Modeling and forecasting. Volatility, слайд №40Modeling and forecasting. Volatility, слайд №41Modeling and forecasting. Volatility, слайд №42Modeling and forecasting. Volatility, слайд №43Modeling and forecasting. Volatility, слайд №44Modeling and forecasting. Volatility, слайд №45Modeling and forecasting. Volatility, слайд №46Modeling and forecasting. Volatility, слайд №47Modeling and forecasting. Volatility, слайд №48Modeling and forecasting. Volatility, слайд №49Modeling and forecasting. Volatility, слайд №50Modeling and forecasting. Volatility, слайд №51Modeling and forecasting. Volatility, слайд №52Modeling and forecasting. Volatility, слайд №53Modeling and forecasting. Volatility, слайд №54Modeling and forecasting. Volatility, слайд №55Modeling and forecasting. Volatility, слайд №56Modeling and forecasting. Volatility, слайд №57Modeling and forecasting. Volatility, слайд №58Modeling and forecasting. Volatility, слайд №59Modeling and forecasting. Volatility, слайд №60Modeling and forecasting. Volatility, слайд №61Modeling and forecasting. Volatility, слайд №62Modeling and forecasting. Volatility, слайд №63Modeling and forecasting. Volatility, слайд №64Modeling and forecasting. Volatility, слайд №65Modeling and forecasting. Volatility, слайд №66Modeling and forecasting. Volatility, слайд №67Modeling and forecasting. Volatility, слайд №68Modeling and forecasting. Volatility, слайд №69Modeling and forecasting. Volatility, слайд №70Modeling and forecasting. Volatility, слайд №71Modeling and forecasting. Volatility, слайд №72Modeling and forecasting. Volatility, слайд №73Modeling and forecasting. Volatility, слайд №74Modeling and forecasting. Volatility, слайд №75Modeling and forecasting. Volatility, слайд №76Modeling and forecasting. Volatility, слайд №77Modeling and forecasting. Volatility, слайд №78Modeling and forecasting. Volatility, слайд №79Modeling and forecasting. Volatility, слайд №80
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Modeling and forecasting. Volatility, слайд №1
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Outline Introduction: Why ARCH? ARCH Models Extensions: GARCH, T-GARCH, Q-GARCH, GARCH-M, Box-Cox GARCH Estimation Multivariate GARCH Models: Diagonal Vech, BEKK and CCC Application: Value-at-Risk (VaR) Appendix
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Outline Introduction: Why ARCH? ARCH Models Extensions: GARCH, T-GARCH, Q-GARCH, GARCH-M, Box-Cox GARCH Estimation Multivariate GARCH Models: Diagonal Vech, BEKK and CCC Application: Value-at-Risk (VaR) Appendix

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1. Introduction: Why ARCH?
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1. Introduction: Why ARCH?

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Modeling and forecasting. Volatility, слайд №4
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Some example series: UST10Y
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Some example series: UST10Y

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Dow Jones
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Dow Jones

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U.S. Unemployment rate vs. stock market volatility, 1929-2010
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U.S. Unemployment rate vs. stock market volatility, 1929-2010

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U.S. Realized Volatility (kernel based) 1997-2009
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U.S. Realized Volatility (kernel based) 1997-2009

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Modeling and forecasting. Volatility, слайд №9
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Modeling and forecasting. Volatility, слайд №13
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Skewness
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Skewness

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Kurtosis
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Kurtosis

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Modeling and forecasting. Volatility, слайд №17
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EViews Example – Daily S&P 500 Returns
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EViews Example – Daily S&P 500 Returns

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When we learn about GARCH(1,1)…
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When we learn about GARCH(1,1)…

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We’ll be able to make squared residuals white noise
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We’ll be able to make squared residuals white noise

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Quality of TGARCH predictions: 1% quantiles, VaR(0.01), from August 1, 2007
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Quality of TGARCH predictions: 1% quantiles, VaR(0.01), from August 1, 2007

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2. ARCH Models
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2. ARCH Models

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Modeling and forecasting. Volatility, слайд №23
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3. Extensions
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3. Extensions

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Modeling and forecasting. Volatility, слайд №28
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Modeling and forecasting. Volatility, слайд №30
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I-GARCH If the coefficients of the GARCH model sum to 1, then the model has “integrated” volatility. This is similar to having a random walk, but in volatility instead of the variable itself. Model itself remains stationary (if constant variance model is stationary) Likelihood-based inference remains valid (Lumsdaine, 1996 Econometrica)
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I-GARCH If the coefficients of the GARCH model sum to 1, then the model has “integrated” volatility. This is similar to having a random walk, but in volatility instead of the variable itself. Model itself remains stationary (if constant variance model is stationary) Likelihood-based inference remains valid (Lumsdaine, 1996 Econometrica)

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Summing up (see Appendix for an expanded list)
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Summing up (see Appendix for an expanded list)

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3. Estimation
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3. Estimation

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Maximum Likelihood
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Maximum Likelihood

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Maximum Likelihood (continued)
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Maximum Likelihood (continued)

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Optimization
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Optimization

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Multiple Solutions
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Multiple Solutions

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4. Multivariate models
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4. Multivariate models

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Modeling and forecasting. Volatility, слайд №49
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An example of volatility “contagion’’
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An example of volatility “contagion’’

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5. Application: Value-at-Risk (VaR)
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5. Application: Value-at-Risk (VaR)

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VaR What is the most I can lose on an investment? VaR tries to provide an answer. It is used most often by commercial and investment banks to capture the potential loss in value of their traded portfolios from adverse market movements over a specified period. This potential loss can then be compared to their available capital and cash reserves to ensure that the losses can be covered without putting the firms at risk. VaR is applied widely in capital regulation (Basel)
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VaR What is the most I can lose on an investment? VaR tries to provide an answer. It is used most often by commercial and investment banks to capture the potential loss in value of their traded portfolios from adverse market movements over a specified period. This potential loss can then be compared to their available capital and cash reserves to ensure that the losses can be covered without putting the firms at risk. VaR is applied widely in capital regulation (Basel)

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Value-at-Risk (VaR) VaR summarizes the expected maximum loss over a time horizon within a given confidence interval The VaR approach tries to estimate the level of losses that will be exceeded over a given time period only with a certain (small) probability For example, the 95% VaR loss is the amount of loss that will be exceeded only 5% of the time
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Value-at-Risk (VaR) VaR summarizes the expected maximum loss over a time horizon within a given confidence interval The VaR approach tries to estimate the level of losses that will be exceeded over a given time period only with a certain (small) probability For example, the 95% VaR loss is the amount of loss that will be exceeded only 5% of the time

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Value-at-Risk (VaR) - Continued The simplest assumption: daily gains/losses are normally distributed and independent. Calculate VaR from the standard deviation of the portfolio change, σ, assuming the mean change in the portfolio value is 0: 1-day VaR= N-1(X)σ, with X the confidence level. The N-day VaR equals sqrt(N) times the 1-day VaR.
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Value-at-Risk (VaR) - Continued The simplest assumption: daily gains/losses are normally distributed and independent. Calculate VaR from the standard deviation of the portfolio change, σ, assuming the mean change in the portfolio value is 0: 1-day VaR= N-1(X)σ, with X the confidence level. The N-day VaR equals sqrt(N) times the 1-day VaR.

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Measuring VaR with historical data
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Measuring VaR with historical data

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Assuming a Normal distribution
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Assuming a Normal distribution

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VaR with Normally Distributed Returns
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VaR with Normally Distributed Returns

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Portfolio VaR When we have more than one asset in our portfolio we can exploit the gains from diversification. There are gains from diversification whenever the VaR for the portfolio does not exceed the sum of the stand-alone VaRs (i.e., the VaRs on the single assets). The VaR for the portfolio equals the sum of the stand-alone VaRs if and only if the securities’ returns are uncorrelated.
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Portfolio VaR When we have more than one asset in our portfolio we can exploit the gains from diversification. There are gains from diversification whenever the VaR for the portfolio does not exceed the sum of the stand-alone VaRs (i.e., the VaRs on the single assets). The VaR for the portfolio equals the sum of the stand-alone VaRs if and only if the securities’ returns are uncorrelated.

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An Example Let us consider the following investment US$200 million invested in 5-year zero coupon US Treasury Examine VaR using a daily horizon Assume that the mean daily return is 0.01% Based on past several years of actual returns, the standard deviation is s = 0.295%.
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An Example Let us consider the following investment US$200 million invested in 5-year zero coupon US Treasury Examine VaR using a daily horizon Assume that the mean daily return is 0.01% Based on past several years of actual returns, the standard deviation is s = 0.295%.

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An Example (cont.)
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An Example (cont.)

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An Example of Portfolio VaR Two securities 30-year zero-coupon U.S. Treasury bond 5-year zero-coupon U.S. Treasury bond For simplicity assume that the expected return is zero Invest US$100 million in the 30-year bond Daily return volatility (std dev) s1 = 1.409% Invest US$200 million in the 5-year bond Daily return volatility (std dev) s2 = 0.295%
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An Example of Portfolio VaR Two securities 30-year zero-coupon U.S. Treasury bond 5-year zero-coupon U.S. Treasury bond For simplicity assume that the expected return is zero Invest US$100 million in the 30-year bond Daily return volatility (std dev) s1 = 1.409% Invest US$200 million in the 5-year bond Daily return volatility (std dev) s2 = 0.295%

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An Example of Portfolio VaR 95% confidence level 30 year zero VaR 1.65 * 0.01409 * 100m = $2,325,000 5 year zero VaR 1.65 * 0.00295 * 200m = $974,000 Sum of individual VaRs = US$ 3.299m But US$3.299 million is not the VaR for the portfolio...why?
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An Example of Portfolio VaR 95% confidence level 30 year zero VaR 1.65 * 0.01409 * 100m = $2,325,000 5 year zero VaR 1.65 * 0.00295 * 200m = $974,000 Sum of individual VaRs = US$ 3.299m But US$3.299 million is not the VaR for the portfolio...why?

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VaR of the Portfolio Suppose the correlation between the two bonds is r12=0.88 Remember that Portfolio variance: (100*0.01409)2 + (200*0.00295)2 +2(100*0.01409)(200*0.00295) * 0.88 = 3.797 Portfolio standard deviation: sp = $1.948m Portfolio VaR = 1.65 * 1.948m = $3.214m This is different from the sum of VaRs
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VaR of the Portfolio Suppose the correlation between the two bonds is r12=0.88 Remember that Portfolio variance: (100*0.01409)2 + (200*0.00295)2 +2(100*0.01409)(200*0.00295) * 0.88 = 3.797 Portfolio standard deviation: sp = $1.948m Portfolio VaR = 1.65 * 1.948m = $3.214m This is different from the sum of VaRs

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The problem with Normality: Kurtosis Extreme asset price changes occur more often than the normal distribution predicts. Excess kurtosis (fat tails)
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The problem with Normality: Kurtosis Extreme asset price changes occur more often than the normal distribution predicts. Excess kurtosis (fat tails)

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Fat Tails and underestimation of VaR If we assume that returns are normally distributed when they are not, we underestimate the VaR
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Fat Tails and underestimation of VaR If we assume that returns are normally distributed when they are not, we underestimate the VaR

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Backtesting Model backtesting involves systematic comparisons of the calculated VaRs with the subsequent realized profits and losses. With a 95% VaR bound, expect 5% of losses greater than the bound Example: Approximately 12 days out of 250 trading days If the actual number of exceptions is “significantly” higher than the desired confidence level, the model may be inaccurate. Therefore, in additional to the risk predicted by the VaR, there is also “model risk”
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Backtesting Model backtesting involves systematic comparisons of the calculated VaRs with the subsequent realized profits and losses. With a 95% VaR bound, expect 5% of losses greater than the bound Example: Approximately 12 days out of 250 trading days If the actual number of exceptions is “significantly” higher than the desired confidence level, the model may be inaccurate. Therefore, in additional to the risk predicted by the VaR, there is also “model risk”

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Relevance: Basel VaR Guidelines VaR computed daily, holding period is 10 days. The confidence interval is 99 percent Banks are required to hold capital in proportion to the losses that can be expected to occur more often than once every 100 periods At least 1 year of data to calculate parameters Parameter estimates updated at least quarterly Capital provision is the greater of Previous day’s VAR 3 times the average of the daily VAR for the preceding 60 business days plus a factor based on backtesting results
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Relevance: Basel VaR Guidelines VaR computed daily, holding period is 10 days. The confidence interval is 99 percent Banks are required to hold capital in proportion to the losses that can be expected to occur more often than once every 100 periods At least 1 year of data to calculate parameters Parameter estimates updated at least quarterly Capital provision is the greater of Previous day’s VAR 3 times the average of the daily VAR for the preceding 60 business days plus a factor based on backtesting results

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Summing up A host of research has examined a. how best to compute VaR with assumptions other than the standardized normal b. How to obtain more reliable variance and covariance values to use in the VaR calculations. Here Multivariate GARCH models play an important role in assessing both portfolio risk and diversification benefits. We will see this in the forthcoming workshop
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Summing up A host of research has examined a. how best to compute VaR with assumptions other than the standardized normal b. How to obtain more reliable variance and covariance values to use in the VaR calculations. Here Multivariate GARCH models play an important role in assessing both portfolio risk and diversification benefits. We will see this in the forthcoming workshop

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Thank you!
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Thank you!

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Appendix – GARCH univariate families
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Appendix – GARCH univariate families

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Source: Bollerslev 2010, Engle Festschrift
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Source: Bollerslev 2010, Engle Festschrift

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Modeling and forecasting. Volatility, слайд №78
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APPENDIX II – Software
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APPENDIX II – Software

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Modeling and forecasting. Volatility, слайд №80
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