how VaR can be used as a performance benchmark
- Thread starter ajsa
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asja,
They have in common that both are "merely quantiles" (i.e., VaR) but tracking error VaR implies (needs) a benchmark.
Our "standard issue" VaR is what Jorion calls absolute VaR (using Dowd's notion b/c I favor its robustness to adjustments):
absolute VaR = -expected return + normal deviate (as function of confidence) * volatility
e.g., = -5% return + 2.33 deviate @ 95% one-tail * 20% volatility
absolute VaR is the loss from zero or, from where we start (initial value)
relative VaR = normal deviate (as function of confidence) * volatility
relative VaR is the loss from the expected future value and, therefore, must be larger b/c it gives no "credit" for the expected gain
to compare them, please look at bottom (green cells) of:
http://www.bionicturtle.com/premium/spreadsheet/4.a.1_two_asset_var_relative_vs_absolute/
Tracking error VaR, as Jorion has defined it, is relative VaR but replace portfolio volatility with tracking error;
and tracking error = standard deviation of (portfolio returns - benchmark)
so:
tracking error VaR = normal deviate (as function of confidence) * tracking error
e.g., I have simulated example here @ http://www.bionicturtle.com/premium/spreadsheet/1.a.7._simulation_tracking_error_sortino/
the TE VaR above is a parametric sort, right? we can also perform an "historical TE VaR" by sorting the difference between portfolio returns and benchmarket return; e.g.,
portfolio return - benchmark(t-x) = + 6%
portf - benchmark(t-y) = + 5%
portf - benchmark(t-z) = + 3%
and so on ...
if we had a list of 100 of these "deviations," we can look down this list to the 95th to get a tracking error VaR.
based on the historical sample, the worst we expect to underperform the benchmark with 95% confidence is X/.
(But Jorion defines the parametric sort, just trying to show that VaR is just a distributional quantile ... the whole enchilada is generating the distribution)
David
They have in common that both are "merely quantiles" (i.e., VaR) but tracking error VaR implies (needs) a benchmark.
Our "standard issue" VaR is what Jorion calls absolute VaR (using Dowd's notion b/c I favor its robustness to adjustments):
absolute VaR = -expected return + normal deviate (as function of confidence) * volatility
e.g., = -5% return + 2.33 deviate @ 95% one-tail * 20% volatility
absolute VaR is the loss from zero or, from where we start (initial value)
relative VaR = normal deviate (as function of confidence) * volatility
relative VaR is the loss from the expected future value and, therefore, must be larger b/c it gives no "credit" for the expected gain
to compare them, please look at bottom (green cells) of:
http://www.bionicturtle.com/premium/spreadsheet/4.a.1_two_asset_var_relative_vs_absolute/
Tracking error VaR, as Jorion has defined it, is relative VaR but replace portfolio volatility with tracking error;
and tracking error = standard deviation of (portfolio returns - benchmark)
so:
tracking error VaR = normal deviate (as function of confidence) * tracking error
e.g., I have simulated example here @ http://www.bionicturtle.com/premium/spreadsheet/1.a.7._simulation_tracking_error_sortino/
the TE VaR above is a parametric sort, right? we can also perform an "historical TE VaR" by sorting the difference between portfolio returns and benchmarket return; e.g.,
portfolio return - benchmark(t-x) = + 6%
portf - benchmark(t-y) = + 5%
portf - benchmark(t-z) = + 3%
and so on ...
if we had a list of 100 of these "deviations," we can look down this list to the 95th to get a tracking error VaR.
based on the historical sample, the worst we expect to underperform the benchmark with 95% confidence is X/.
(But Jorion defines the parametric sort, just trying to show that VaR is just a distributional quantile ... the whole enchilada is generating the distribution)
David
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