VaR and Changing Volatility
Jorion, Chapter 8
VaR and the Unreal World
The Pitfalls of VaR estimates
Summary
• Picture of changing volatility
• Moving averages and rolling VaR’s
• Riskmetrics and weighted variances
• GARCH modeling of volatility
• Correlations and portfolios
Summary
• Picture of changing volatility
• Moving averages and rolling VaR’s
• Riskmetrics and weighted variances
• GARCH modeling of volatility
• Correlations and portfolios
Moving Average of Volatility
• Rolling moving average of returns squared
• madowvar.m
2 2
1
1 m
t t jj
rm
Moving Average of Volatility
• Brooks/Persand and Hoppe papers– Tradeoff between small and large samples– Conditional volatility versus large sample size– Small often looks better– Trickier with weightings
• Interesting question– Evaluation? (graphical)
Summary
• Picture of changing volatility
• Moving averages and rolling VaR’s
• Riskmetrics and weighted variances
• GARCH modeling of volatility
• Correlations and portfolios
RiskMetrics VaR
• h(t) = variance at time t
• Smooth weighting of past volatility
21 1
2 2 2 21 2 3
(1 )
(1 )( )
0.94
t t t
t t t t
h h r
h r r r
Summary
• Picture of changing volatility
• Moving averages and rolling VaR’s
• Riskmetrics and weighted variances
• GARCH modeling of volatility
• Correlations and portfolios
GARCH Modeling
• GARCH(1,1): – Complete model for changing variance
1 1
1
20 1 1
(0,1)t t t
t
t t t
r h e
e N
h r h
GARCH Modeling
• Forecasting Variance h(t)2
0 1 1
2 20 1 0 1 1 2
20 0 0
2 2 2 21 1 1 1 2
( )
t t t
t t t t
t
t t t
h r h
h r r h
h
r r r
How Does this Differ from Riskmetrics?
• For 1 horizon, not much
• Multi-horizon is different
• h(t+m) is needed
GARCH Variance T periods in the future
12 11
1 0 11
2 01
1
1 ( )( ) ( )
1 ( )
1
( )1
nn
t T t
t T
E r h
n T t
n E r
RiskMetrics VaRForecasts
• h(t) = variance at time t
21 1
21
1
(1 )
(1 ) ( )
(1 )
t t t
t t t
t t h t
h h r
h h E r
h h h h
Summary
• Picture of changing volatility
• Moving averages and rolling VaR’s
• Riskmetrics and weighted variances
• GARCH modeling of volatility
• Correlations and portfolios
Riskmetrics Correlation Example(rmcorr.m)
12, 12, 1 1, 1 2, 1
12,12,
1, 2,
(1 )t t t t
tt
t t
h h r r
h
h h
Crashes and Correlations
• Large down moves connected to increases in correlations
• Implications for risk management and portfolio construction
• Reliability in the data?
Top Related