VAR - Converting time horizon

[papillonring]

Hi David

If I were to come across a question that provide the following and asked to provide the daily VAR,

Annual expected return
Anuual std dev

Is it right to compute the annual VaR first [i.e. -Annual ER + Annual SD x Z] and then convert the annual VaR to a daily VAR?

Or should I convert the annual SD and annual ER to a daily value first [i.e. Daily SD = Annual SD / sqrt(250) and Daily ER = Annual ER / 250] before computing the VAR (which will be daily)?

I tried both ways and I got different results. Which is the right way?

 

[papillonring]

Another question:

When I am asked to calculation diversified VAR of a portfolio with 2 assets and Expected Return (ER) is provided, can I do the following:

step1:
VaR(a) = (-ER + SD x z ) x value(a)
VaR(b) = same

Step 2:
Diversified VAR = sqrt (VAR^2(a) + VAR^2(b) + 2 x VAR(a) x VAR(b) x Corr(a,b))

The focus of my question is, can I incorporate ER when calculating VAR(a) and VAR(b)? Or should this only be done only after portfolio VAR is calculated?

 

[David Harper CFA FRM]

Hi Papillionring,

On the first, this ("Is it right to compute the annual VaR first [i.e. -Annual ER + Annual SD x Z] and then convert the annual VaR to a daily VAR? ") will NOT work because the drift scales with time and the volatility/VaR with the square root of time. Therefore, the scaling must be done on each element; the scaling must must be done, effectively, separately on each term. If, for example, we are re-scaling annual return and annual volatility (drift = annual expected return and sigma = annual volatility), then T-day VaR is given by:
Absolute T-day VaR = -drift*T/250 + sigma*deviate*SQRT(T/250)
…. where T = T-day VaR
…. So, put another way, if and when you happen to see the VaR itself scaled with only one VaR term, this must implicitly be a RELATIVE VaR (i.e, without drift or assuming drift = 0) and not an ABSOLUTE VaR (with non zero drift).

On the second, I was not 100% sure so I tested in our (8.b.1. XLS). See here:
http://db.tt/6evef6V
… see test at bottom
… not only won't it work, I can't get it to work. I'd love to come back after the exam and see if there is an analytical reduction
… but in the meantime, we can EXCLUDE step 1. The XLS proves it fails (which makes sense: the drift is infecting a covariance matrix … that doesn't imply, to me, that there is no matrix solution but we can rule out the blind application)

David

 

[johnccarter]

The formula you quoted for diversified VaR is nothing but the formula for adding the standard deviations of two normal distributions. It works for ABSOLUTE VaR because this is simply a constant (Z) times the standard deviation.

So RELATIVE VaR must be added in the same way that the means and standard deviations of two distributions are combined - i.e. with separate formulae:

Mean(p) = Mean(a) + Mean(b)
SD(p) = SQRT(SD^2(a) + SD^2(b) + 2 x SD(a) x SD(b) x CORR(a,b))

Your ER is essentially the "mean".

Mean is treat separately because it is explicitly removed from both the Standard Deviation and the Correlation.

 

[David Harper CFA FRM]

Hi ugli-stix,

I do agree, but please note:

Market risk relative VaR = sigma*deviate;
Market risk absolute VaR = - mean + sigma*deviate
i.e., if drift (mean) is positive, then absolute VaR must be less than (<) relative VaR

(just in case there is confusion over absolute vs relative)
(credit and ops are trickier b/c the drift is negative; EL is a loss, so Credit risk absolute VaR > relative b/c CR abs VaR = -(-mean) + f[UL].)

Thanks, David

 

[cqbzxk]

Hi Papillionring,

On the first, this ("Is it right to compute the annual VaR first [i.e. -Annual ER + Annual SD x Z] and then convert the annual VaR to a daily VAR? ") will NOT work because the drift scales with time and the volatility/VaR with the square root of time. Therefore, the scaling must be done on each element; the scaling must must be done, effectively, separately on each term:
Absolute VaR = -drift*T/250 + sigma*deviate*SQRT(T)
…. where T = T-day VaR
…. So, put another way, if and when you happen to see the VaR itself scaled with only one VaR term, this must implicitly be a RELATIVE VaR (i.e, without drift or assuming drift = 0) and not an ABSOLUTE VaR (with non zero drift).

On the second, I was not 100% sure so I tested in our (8.b.1. XLS). See here:
http://db.tt/6evef6V
… see test at bottom
… not only won't it work, I can't get it to work. I'd love to come back after the exam and see if there is an analytical reduction
… but in the meantime, we can EXCLUDE step 1. The XLS proves it fails (which makes sense: the drift is infecting a covariance matrix … that doesn't imply, to me, that there is no matrix solution but we can rule out the blind application)

David

hello david, I have a little confusion at this point, is this a wrong or right equation Absolute VaR = -drift*T/250 + sigma*deviate*SQRT(T) ?
anther point is, I remembered VaR(N day)=VaR(1day)*sqrt(Nday) right? so is this means relative VaR or Abs VaR?
thank you!

 

[cqbzxk]

Hi Papillionring,

On the first, this ("Is it right to compute the annual VaR first [i.e. -Annual ER + Annual SD x Z] and then convert the annual VaR to a daily VAR? ") will NOT work because the drift scales with time and the volatility/VaR with the square root of time. Therefore, the scaling must be done on each element; the scaling must must be done, effectively, separately on each term:
Absolute VaR = -drift*T/250 + sigma*deviate*SQRT(T)
…. where T = T-day VaR
…. So, put another way, if and when you happen to see the VaR itself scaled with only one VaR term, this must implicitly be a RELATIVE VaR (i.e, without drift or assuming drift = 0) and not an ABSOLUTE VaR (with non zero drift).

On the second, I was not 100% sure so I tested in our (8.b.1. XLS). See here:
http://db.tt/6evef6V
… see test at bottom
… not only won't it work, I can't get it to work. I'd love to come back after the exam and see if there is an analytical reduction
… but in the meantime, we can EXCLUDE step 1. The XLS proves it fails (which makes sense: the drift is infecting a covariance matrix … that doesn't imply, to me, that there is no matrix solution but we can rule out the blind application)

David

Hi david, i remember (1+ Retrun1day)^250=1+Return250day right? but in your equation, you use ER/250=1day return? is this right or not? thanks

 

[ShaktiRathore]

VaR(N day)=-drift+VaR(1day)*sqrt(Nday) is relative var as pointed by david: if and when you happen to see the VaR itself scaled with only one VaR term, this must implicitly be a RELATIVE VaR (i.e, without drift or assuming drift = 0) so drift =0 implies VaR(N day)=-0+VaR(1day)*sqrt(Nday)=VaR(1day)*sqrt(Nday) which is relative var.
Absolute VaR = -drift*T/250 + sigma*deviate*SQRT(T) is an absulote var because here drift is not equal to 0. sigma is daily volatility which is scaled to sqrt of time to calculate T day volatility. similarly drift is scaled to period T but drift is over 250 days so daily drift is drift/250 and over T period its scaled to T as T*drift/250.

thanks

 

[David Harper CFA FRM]

My original equation above contains a scaling inconsistency (sorry, it was in haste, fixed above), now that we have Tex/LaTex, I can show it better :

\( VaR(T,\alpha )=-\mu \Delta t+\sigma \sqrt{\Delta t}{{z}_{\alpha }}\text{ where }\Delta t=\frac{T}{t} \)

The first thing, i think, is to be mindful of the scale of the inputs; it is common for the input return (drift) and volatility to be expressed per annum, or maybe daily, or maybe monthly.

Then just take care to scale the drift directly with time with and scale the volatility by the square root of time, where time is the ratio between your desired horizon and your input horizon.

For example, if we are given E[return] = 10% per annum and annual volatility = 25%, then our input time horizon is annual and, in the above, t = 250 days.

• if we want the i.i.d. implied 1-day VaR, above ΔT = 1/250
• If we want the i.i.d. implied 10-day VaR, ΔT = 10/250
• If we want the i.i.d. implied 3-year VaR, ΔT = 750/250 = 3.0
  (not to suggest it holds up over that long, not realistic!)