Relative vs. Absolue VaR

[Johnny Firpo]

With reference to the spreadsheet P1.T4a1_2 Asset VaR:

I am confused about the calculation of relative and absolute VaR.
Why is the initial value of 100 instead the expected future value of 100,73 is used to calculate the relative VaR?
Why is the initial value multilied with -0,73% in the calculation of the absolute Var?

Thanks in advance for clarifying this issue.

 

[David Harper CFA FRM]

Hi Johnny,

Much pre-existing discussion already on relative vs absolute VaR
See http://forum.bionicturtle.com/threads/frm-fun-13-absolute-versus-relative-var-versus-ul.6020
... and a search of "absolute VaR" will give many results http://forum.bionicturtle.com/search/3289395/?q=absolute var&o=date

in the XLS, the "cutoff" would be $95.82, as in:

• relative VaR = exp future value of $100.73 - 95.82 = $4.91; i.e., loss relative to expected future value
• absolute VaR = $100 - 95.82 = 4.18; i.e., loss relative to current value

 

[Johnny Firpo]

Actually, I calculated both VaR the way you did in your answer in this thread. Thus, could there be a mistake in the formula of the spreadsheet?

 

[David Harper CFA FRM]

I don't see a mistake, absolute VaR = -drift + volatility * deviate and relative VaR is a special case with drift = 0 (since it is loss relative to future expected value). In the XLS, we have:

• absolute VaR(%) = -18.5%*10/250 + 15%*SQRT(10/250)*1.64 deviate ~= 4.18%; and
• relative VaR = 15%*SQRT(10/250)*1.64 deviate = 4.91%; ie, when drift is positive, as in market risk, the relative VaR > absolute VaR

Abs and relative VaR differ by the drift, in this case, drift = -18.5%*10/250 = 0.73% or $0.73 = $4.91 - $4.18, thanks,

 

[Johnny Firpo]

Understood. However, I am not sure if the initial values used to calculate the VaR in the spreadsheet are correct.
As pointed out above:
Why is the initial value of 100 instead the expected future value of 100,73 is used to calculate the relative VaR (cell D34)?
Why is the initial value multilied with -0,73% in the calculation of the absolute VaR (cell D35)?

Maybe I am completely confused now but I think the spreadsheet calculation does not match your explanation here... sorry for being bothersome but I think this issue is important for my understanding.

 

[David Harper CFA FRM]

It really might help to read some of the many prior threads on absolute/relative VaR that I referenced above.

• The current value, T(0), is $100
• The expected future value after 10 days is, (T + 10 days) = $100.73 = 100 * (1 + 18.5% * 10/252)
• The relative VaR = $4.91 = volatility * deviate (without the drift). This is the first way of looking at the risk: the loss relative to our future expected value. We expect the portfolio to be at 100.73 (the mean) but our 95% worst case is a loss of $4.91 relative to this future expected value (future mean).
• But we start today at 100, so the second way to look at the risk is: relative to our current/initial value;aka, absolute VaR. If the 95% worst is 100.73 - 4.91 = 95.82, that is only a loss $4.18 from our starting point of 100.
• In both cases, the 95% worst future value is 95.82. The difference is how we view a "loss" down to 95.82: is it a relative loss from the expected future value (100.73 - 95.82 = 4.91) or an absolute loss from current value (100 - 95.82 = 4.91)

The absolute VaR can also be expressed: volatility*sigma - drift. But this encourages errors when adding a liquidity term, so my XLS follows Dowd's better expression: absolute VaR = - drift + sigma*deviate. Otherwise, the XLS also does multiply the initial value, but that could be removed for a %. Sorry, I don't know why you say my XLS differs from these explanations? Thanks,

 

[David Harper CFA FRM]

this may help explain why we use the formulas:

E[future value] = 100*(1+drift) = 100*1.073%
the 95% worst future value; aka, "cutoff" is the same regardless of relative/absolute = $95.82 = 100*1.073% - 100*2.99%*1.645 = V*σ*α

So in terms of values, we have three:

• current = V(0) = 100
• expected future = 100*(1+d) = V(0)*(1+d)
• 95% worst (future) = 100*(1+d) - 100*σ*α = V(0)*(1+d) - V(0)*σ*α = V(0)*[(1+d) - σ*α]; but that's a worst future value, not a VaR!
• Absolute VaR = current - worst future value = V(0) - V(0)*[(1+d) - σ*α] = V(0) [1 - (1+d) + σ*α] =V(0)[-d + σ*α]; i.e., the calc in the XLS
• Relative VaR = expected future value - worst future value = V(0)*(1+d) - V(0)*[(1+d) - σ*α] = V(0)*σ*α; i.e., the calc in the XLS

 

[Johnny Firpo]

Thank you David!