Figure 8.1: VaR Term Structures using NGARCH and Monte Carlo Simulation
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Monte Carlo Simulation
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• In Figure 8.1, the VaR is simulated using Monte Carlo on
an NGARCH model
• We use MCS to construct VaR per day as a function of
horizon K for two different values of σt+1
• In the left panel the initial volatility is one-half the
unconditional level and in the right panel σt+1 is three
times the unconditional level.
• The horizon goes from 1 to 500 trading days
• The VaR coverage level p is set to 1%
Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen
Monte Carlo Simulation
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Figure 8.1 shows that the term structure of VaR is initially
upward sloping both when volatility is low and when it is
high
• The VaR term structure is also driven by the term structure
of skewness and kurtosis and other moments
• Kurtosis is strongly increasing at short horizons and then
decreasing for longer horizons
• This hump-shape in the term structure of kurtosis creates
the hump in the VaR as seen in the right panel of Figure
8.1 when the initial volatility is high
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Figure 8.2: ES Term Structures using NGARCH and
Monte Carlo Simulation
Notes to Figure: The left panel shows the S&P 500 ES per day
across horizons when the current volatility is one half its long run
value. The right panel assumes the current volatility is 3 times its
long run value.
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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Monte Carlo Simulation
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• In Figure 8.2, the ES is simulated using Monte Carlo on an
NGARCH model
• Here we plot the ESPt+1:t+K per day, against horizon K
• The coverage level p is again set to 1% and the horizon
goes from 1 to 500 trading days
• Note that the slope of the ES term structure in the left panel
of Figure 8.2 is steeper than the corresponding VaR term
structure in the left panel of Figure 8.1
• The hump in the ES term structure in the right panel of
Figure 8.2 is more pronounced than the hump in the VaR
term structure in the right panel of Figure 8.1
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen