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

Picture of Changing Volatility

• dowvolplt.m

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

Riskmetrics VaR

• rmdowvar.m

• hrmdowvar.m

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

GARCH Variance Forecast

Days Ahead

Variance

Shock

UnconditionalVariance

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

Correlations

• Moving averages

• Riskmetrics (examples)

• GARCH

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?

Final Suggestions on Volatility

• Options data and implied volatility

• High frequency data

• High/low range data