Wilmott page 462
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Hi Frank,
they are user-selected "areas under the curve." In the parametric approaches, we pick a distribution. Here, it happens to be normal: losses are described by random variable that is normally distributed. If losses are positive ("to the right"), how far over to the right before 95% is under the curve? +1.65 standard deviations if the curve is standard normal. The total area is 1.0 by definition, so if the variable is random, 95% of the time is should fall < 1.65 s.d. If our losses are indeed standard normal, then 95% of the time, losses should not exceed 1.65 s.d.
Having selected the distribution, this is an inverse CDF: how far over to the right before we carve out an area of that is 90%, 95% of the total area under the curve. This is literally the key to all parameteric approaches.
we can pick another distribution and ask same question. For recovery (LGD) say distribution is beta instead of normal, we can still ask an INVERSE CDF question.
assuming my distribution is beta, how far to the right before i've carved out 95% of the area (and therefore, can say with 95% confidence, a random variable will be less than): =BETAINV(95%, alpha, beta).
hope that helps, i think this is worth stewing on, the rest of parametric depends on this inverse CDF...David
they are user-selected "areas under the curve." In the parametric approaches, we pick a distribution. Here, it happens to be normal: losses are described by random variable that is normally distributed. If losses are positive ("to the right"), how far over to the right before 95% is under the curve? +1.65 standard deviations if the curve is standard normal. The total area is 1.0 by definition, so if the variable is random, 95% of the time is should fall < 1.65 s.d. If our losses are indeed standard normal, then 95% of the time, losses should not exceed 1.65 s.d.
Having selected the distribution, this is an inverse CDF: how far over to the right before we carve out an area of that is 90%, 95% of the total area under the curve. This is literally the key to all parameteric approaches.
we can pick another distribution and ask same question. For recovery (LGD) say distribution is beta instead of normal, we can still ask an INVERSE CDF question.
assuming my distribution is beta, how far to the right before i've carved out 95% of the area (and therefore, can say with 95% confidence, a random variable will be less than): =BETAINV(95%, alpha, beta).
hope that helps, i think this is worth stewing on, the rest of parametric depends on this inverse CDF...David
Here is a shot from one of the XLS. View the distribution (normal) as characterizing the random var.
The CDF question is: if I go 1.65 SD to right, what % is under the curve?
The more typical "risk" question is INVERSE CDF: if my var is standard normal (or whatever parametric distribution), then how far to the right do i go before the purple area under the curve is 95% of the total.
http://learn.bionicturtle.com/images/forum/inverseCDF.png
The CDF question is: if I go 1.65 SD to right, what % is under the curve?
The more typical "risk" question is INVERSE CDF: if my var is standard normal (or whatever parametric distribution), then how far to the right do i go before the purple area under the curve is 95% of the total.
http://learn.bionicturtle.com/images/forum/inverseCDF.png
he's un-standardizing to get the VaR
Standard normal: mean = 0, s.d./variance = 1.0. We use standard for the lookup.
Here, 95% gives 1.645 standard deviations
But the 1.64 is unitless. It needs to be applied to the situation; e.g., mean return = 40% with s.d. of 170%
at 95% confidence
40% - (170%)(1.645) = -240%
or at 99% confidence
40% - (170%)(2.33) = -355%
that's the (parametric) VaR. I do better visualizing it. The mean is the peak of the un-standard normal. The expected return = 40%, that's the peak. Now go "left" 1.645 s.d. which is left (1.64)(170%). At that point, only 5% is to the left and 95% is to the right (area under the curve). So, 95% of the time we expect losses won't be less than -240%
David
Standard normal: mean = 0, s.d./variance = 1.0. We use standard for the lookup.
Here, 95% gives 1.645 standard deviations
But the 1.64 is unitless. It needs to be applied to the situation; e.g., mean return = 40% with s.d. of 170%
at 95% confidence
40% - (170%)(1.645) = -240%
or at 99% confidence
40% - (170%)(2.33) = -355%
that's the (parametric) VaR. I do better visualizing it. The mean is the peak of the un-standard normal. The expected return = 40%, that's the peak. Now go "left" 1.645 s.d. which is left (1.64)(170%). At that point, only 5% is to the left and 95% is to the right (area under the curve). So, 95% of the time we expect losses won't be less than -240%
David
right, note Willmott shows two VaRs (without or without subtracting mean). This corresponds to Jorion's:
absolute VaR (subtracts mean): loss relative to zero
relative VaR (doesn't subtract): loss relative to ending (expected) value
best may be to see absolute VaR as the VaR (i.e., the above) where relative VaR is the special case we can use for short (daily) trading, where we can assume exp mean = 0
traditional VaR was "born" here (daily trading), more recently (e.g., capital at risk) it has become a longer term concept (e.g., pensions) where the mean b/c relevant - David
absolute VaR (subtracts mean): loss relative to zero
relative VaR (doesn't subtract): loss relative to ending (expected) value
best may be to see absolute VaR as the VaR (i.e., the above) where relative VaR is the special case we can use for short (daily) trading, where we can assume exp mean = 0
traditional VaR was "born" here (daily trading), more recently (e.g., capital at risk) it has become a longer term concept (e.g., pensions) where the mean b/c relevant - David
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