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# Relative VaR & tracking error

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Lee just asked me about relative VaR as discussed in this fine article: http://www.financewise.com/public/edit/riskm/rmforinvestors/rmforinvestors-models.htm .

His question here:

"Dear David- I am currently studying for 2007 FRM exam and trying to read articles as to
VAR and other FRM-related articles. In last weekend, I found one interesting article (please refer the
following link) but I couldn'tâ€™t figure out how those figures came up with.

http://www.financewise.com/public/edit/riskm/rmforinvestors/rmforinvestors-models.htm

Can you take a look at Table F (Performance year to date 2000, Jan 1-Oct
25) and Table G (Relative VAR) and teach me how to calculate those figures?
I tried numerous different approach but couldn'tâ€™t figure out the methodology methodology by now."

Lee - While this is a good article, I recommend you avoid it in favor of the assigned readings: Linda Allen Chapter 1 for VaR (but also Jorion, of course) and Noel Amenc Chapter 4 for Tracking Error. Just to be pragmatic. The problem is that, for both of these terms (relative VaR and tracking error), people can have different definitions. As your article does. So, I didn't check the calculations, but here is some high-level help (I hope):

Relative versus Absolute VaR
For our purposes (the exam), relative VaR refers to VaR relative to the expected value of the portfolio. Absolute VaR refers to VaR relative to zero. So, if you start today with portfolio value of $100, expected annual return of 10% and (annualized) standard deviation (of returns) of 10%, the one-year 95% RELATIVE VAR = ($100)(10%)(-1.645) = -$16.45. The one-year 95% ABSOLUTE VAR = ($100)[(-1.645)(10%)+10%] = -$6.45. See the difference? At the end of the year, we expect the portfolio to grow to$110. But $16.45 is "at risk." Relative VaR is the full$16.45 and Absolute VaR, giving credit to the gains that theoretically relate to the risk, speaks to loss versus initial value. Another way to view it: either way, our 1-year 95% VaR (note how i am careful to qualify the VaR with both a time horizon and a confidence level, because there is a different VaR for different combinations of time & confidence) says the risk is a final value of $93.55. The difference is whether we count the loss relative to where we started ($100) or where we expect to end (\$110)

Tracking error
Tracking error has two definitions in practice. Some mean the difference between portfolio return and the benchmark (e.g., S&P;500). Others mean the standard deviation of the series of the difference [portfolio return - benchmark]. Jorion means the first (i.e., tracking error = the difference) and he calls the latter "tracking error volatility (TEV)." (btw, TEV is also called tracking error risk).

But no matter, the assigned reading (and learning outcome) is Amenc. He defines tracking error as the standard deviation of [portfolio returns - benchmark returns]. So, that's our definition. And it's easier this way, because the information ratio has this tracking error as the denominator.