P1.T4.27.1 Scaling VaR

[The Great Khan]

Hi All,

27.1. Dowd defines an arithmetic, absolute value at risk (VaR) given by VaR(%) = -drift + volatility*deviate. For a portfolio with current value of $1.0 million, expected return of 15.0% and volatility of 40% per annum, which of the following is nearest to the 99.0% confident 20-day absolute VaR (assume T = 250 days per year)?

a) $88,750 b) $103,500 c) $188,400 d) $251,200

Given the answers offered, I was able to deduce it was D by first scaling the ev and vol to 20 days then calculating var
VaR(%) = -15%*20/250 + 40.0%*SQRT(20/250)*2.33 = 25.1412%

However, my first thought was to calculate yearly VaR and the scale it using the square root rule. In that case
VaR(%) = -.15+.4*2.33 = .782
Scaled - .782 * sqrt(20/250) = 22.1%

My understanding is that a reasonable approximation VaR can be scaled using this rule. Is this an improper use of the rule and how can I determine in the Exam if they want me using the rule or not?

 

[Dr. Jayanthi Sankaran]

Hi @The Great Khan,

As far as I can see, your first method seems to be the right one, because you have adjusted the 15% per annum expected return by a factor of 20/250:

$1,000,000*(-15%*(20/250) + 40%*2.33*SQRT(20/250)) = $251,609

In your second method, the adjustment that you have made to the expected return is not correct, because you have scaled it by SQRT(20/250). Only the volatility should be scaled by SQRT(20/250). Your answer in the second method turns out to be $22,118,300. This is incorrect.

Hope that helps
Thanks!

 

[The Great Khan]

Hi Jayanthi,

I understand that the difference between the two methods is that in effect the second method is using the same scalar for return and volatility.

However, if we ignore the inputs for a moment, imagine the question was:

The 1 year absolute VaR is $782,000. What is the 20 day absolute VaR?

Is it not in general considered valid to use the square root rule to scale VaR in this situation?

 

[Dr. Jayanthi Sankaran]

Hi @The Great Khan,

As far as I know, expected return is not scaled by SQRT(T) where T = no. of days or days expressed as a % of the year, as in the case above. Only the volatility is scaled by SQRT(T) when returns are i.i.d. I think (not 100% sure) that this accounts for the heteroskedastic nature of time varying variance.

Hope that helps!
Thanks

 

[The Great Khan]

But lets forget about scaling of parameters for a second, square root rule for scaling VaR has an entire section in the study notes are there a lot of questions regarding this concept.

If I run up against a question in the exam such as "The 1 year absolute VaR is $782,000. What is the 20 day absolute VaR?" am I unable to use the square root rule unless I know that the expected return is zero?

Is this simply a handy rule that is very inaccurate? i.e. use it on the exam if it's the only possible way of getting to an answer, but if there's another way the square root rule will be incorrect?

 

[Dr. Jayanthi Sankaran]

Hi @The Great Khan,

I will let David answer the question. Thanks!

 

[S666]

Perhaps the square root rule of scaling VaR only works for relative VaR and not absolute?...I'd be interested in the answer too...

 

[David Harper CFA FRM]

Hi @The Great Khan @Jayanthi Sankaran is correct: Absolute VaR nets (combines) drift and volatility (ie, standard deviation). If mu (µ) is 1-day drift and sigma(σ) is 1-day volatility, then h-day drift = µ*h, but h-day volatility = σ*sqrt(h) because h-day variance = h*σ. Just to be silly for illustrative purposes, imagine that 1-day aVaR = -µ + σ^2*α instead of the correct aVaR = -µ + σ*α. If aVaR did equal -µ + σ^2*α, then you would be able to scale the aVaR to h-days by multiplying it by (h) because both terms would have the same multiplier (distributive property). But as they do not, strictly speaking you cannot scale a summarized, single absolute VaR value with a multiplier (because you don't know the relative contribution of drift and volatility).

@Jayanthi Sankaran also makes a key point: you can only scale with the square root rule, in the first place, if you assume i.i.d. returns (my question above, 27.1, looks like one of my early questions and it's an imprecise question because it omits the key i.i.d assumption).

So, strictly speaking, the square root rule (i.e., scaling VaR by multiplying the square root of time) only applies to relative VaR when then returns are i.i.d. (and to get really specific, assuming [parameteric] standard deviation informs the VaR; you couldn't scales a simulated quantile). But in market risk, when the horizon is daily, it's typical to assume zero drift (if zero drift, then you have a relative VaR). So, for exam purposes, this tends to matter only for credit or operational VaR because they have longer horizons where the zero drift assumption is less likely. I hope that helps!