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Jorion : Model Evaluation of Bankers Trust

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Hi David,
Ref : Jorion ValueAtRisk - Ch6 : Figure : Model Evalution - Bankers Trust

I am not able to follow this example : Kindly Explain the graph if possible (my questions below).

1) The graph is plotted between Daily P&L against 99% VaR. Isn't VaR a single number representing limit. I dont know if the" Diagonal Line" is representing that limit? Very confusing to visualize how that limit is represented by a line for "changing Vol in the X Axis".

2) Assuming that line represents the 99%VaR then the points above the line --> should be the exceptions would be my interpretation. There are 5 points above the line (whereas the author says 4 Exceptions are observed).
2a) Also can you confirm what the author mentions as 2% should lie above the line. Is it 1% loss exceptions and 1% Gain Exceptions on a year (2 days per 250)?

3) Losses and Gains are expressed in Absolute limits to hide the directions of the positions. --> how can profit be in the exception category. VaR is calculated only on the Loss side (One sided confidence limit).

Updated by Nicole to include graph (when referring to a specific graph or image, please provide the image in your question so David and other members know what you are referring to)

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David Harper CFA FRM

David Harper CFA FRM
Staff member
Hi @rajeshtr Great questions! This graph is confusing (I don't think I'd understand it except I've been reading this book/Jorion for ten years ...)

To explain the graph, I'll illustrate. Say we go back -250 trading days to the beginning of the backtest period. On that (initial) day, maybe the updated 99% VaR was 2.33% = 1.0% volatility (ie, assumption) * 2.33 deviate; on a $1,000 position, that's a daily VaR of $23.30. So that would represent the value for the X-axis. In comparison, the day produced an actual P&L gain/loss. If the loss was -30.00, that plots at [x = 23.30, y = +30] which is above the line because the line is y = x. If the gain was +40.00, that also plots above the line at [x = 23.30, y = + 40]. This indifference to the actual/gain loss is per the footnote: "Note that the graph does not differentiate losses from gains. This is typically the case because companies usually are reluctant to divulge the extent of their trading losses. This illustrates one of the benefits of VAR relative to other methods, namely, that by taking the absolute value, it hides the direction of the positions. "

So either the actual -30.00 loss or +40.00 gain plots above the line as an exceedence (exception). This is for presentation (sharing) purposes and presumes the symmetry (lack of skew) inherent to the normal distribution; and we are implicitly assuming a normal by using the 2.33. By assuming a symmetric normal distribution, 99.0% one-tailed confidence equals 98.0% two-tailed confidence. A perfectly calibrated 99.0% VaR (which is always one tailed), if symmetrical, should experience 2%*250 = 5 exceptions above the line. (I really should say if the P&L outcomes are normally distributed) It's not as good as testing only the downside: all five of those exceptions could be gains!

Okay, then now go forward one day from -250 to -249 days. If we didn't update the VaR, then there would be no reason for a scatterplot. The scatterplot implies the VaR is updated daily. Just as we learn to update volatilty by EWMA or GARCH, say the updated GARCH volatility increases from 1.0% to 1.1%. Now the T-249 VaR = $1,000 * 1.1% * 2.33 = $25.63. So the next point plots at [X = 25.63, y = absolute(actual P&L)]. The clusters imply a regime change in volatility.
  1. The line is an abstraction that connect 250 points already on the line, where each point is y = x and x is an updated VaR = VaR[-250 days], VaR[-249 days], ... VaR[yesterday]
  2. I have the actual book and I also see 5 exceptions; I don't know why he says 4. As above, if we include gains, and presume symmetry, then yes we need to expect 2*1%*250 exceptions.
  3. You are absolutely correct that VaR is only calculated on the loss side. The only rationale for this is mentioned above "because companies usually are reluctant to divulge the extent of their trading losses." Internally, it would make no sense to include gains in the backtest of VaR (well, unless you somehow wanted to count abnormal gains as "failures" in a conception of risk as genuine volatility).
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