VAR

[notjusttp]

How to Solve this problem?

Consider a portfolio with a one-day VAR of $1 million. Assume that the market is trending with an autocorrelation of 0.1. Under this scenario, what would you expect the two-day VAR to be?
Choose one answer. a. $2 million
b. $1.414 million
c. $1.483 million
d. $1.449 million

 

[David Harper CFA FRM]

Hi notjusttp,

This is a unfair question! (can you share the source, please?)
This would never (IMO) be an exam question because you can't intuit the choice between 1.483 and 1.449; i.e., you should be expected to know that autocorrelation implies a 2-day variance that is greater than SQRT(2) but more precision requires the calculation....

if you try to use Linda Allen's (1+b^2)*1-day variance, you'd get:
(1+1.1)*$1 = $2.1 variance over 2-days, such that
SQRT($2.1) = $1.449 MM, but that's not correct....(d) is tempting...

This first learning XLS has an input for autocorrelation, see http://www.bionicturtle.com/premium/spreadsheet/1.a.1._intro_to_var/
From 1.a.Intro To VaR
http://learn.bionicturtle.com/images/forum/aug28_autocorrelate.png

Note standard deviation with autocorrelation = 1.483% ...
cell C19 has the scaling factor, which equals 2.2 over 2-days with autocorrelation = 0.1
the 2.2 replace the 2.0 in scaling such that
2-day VaR = $1 MM * SQRT(2.2) = $1.483

Key points for exam:
* scaling with square root rule (e.g., 2-day vol = SQRT(2)* 1-day vol) assumes i.i.d.
* both mean reversion (negative autocorrelation) and positive autocorrelation violate independence in i.i.d and render square root rule inaccurate
* positive autocorrelation implies n-day VaR greater than (>) SQRT(n)*1-day VaR
* mean reversion (negative autocorrelation) implies n-day VaR less than (<) SQRT(n)*1-day VaR

David

 

[notjusttp]

I am overwhelmed by the answer you reached (which i had deliberately kept in dark to notice your method of arriving at the correct answer)

Bingo the answer is just as given in the source which happens to be from Jorion Handbook Chapter 14 Example 14.1 Time Scaling.

Knowing that the variance is V(2-day) = V(1-day) [2 + 2*rho], we find VAR(2-day) = VAR(1-day) 2 + 2*rho = $1 sqrt(2) + 0.2 = $1.483, assuming the same distribution for the different horizons.

Based on this background Can you pls clarify

1) How does this formula come V(2-day) = V(1-day) [2 + 2*rho],

2) What is the "b" in Linda Allens formula (1+b^2) and why is it not correct in the given context?

3) You arrived at the correct answer based on your excel sheet. how do we arrive at the same in exam as we will not be having excel and in cell c19 you say "don't worry about this formula, the point is that autocorrelation increases the VaR".

3)You are assuming mean return to be 0. In the same question if given mean rtn=2% how would the answer change. Also i am not clear the figure which you arrive at why is the same deducted from mean rtn ?

Thanks a ton for brilliant assistance
Amit

 

[David Harper CFA FRM]

Hi Amit,

thanks for the pointer. Wow, I still consider that an almost impossible question; e.g., I would not have thought to apply the variance formula shown and therefore, without Excel, I would have been unable to select between (c) and (d).

1) Unlike the scaling factor in my XLS, which can go beyond 2 periods, the handbook solution appears elegant to me for this special case of only 2 periods. You can see once again Gujarati's variance properties are helpful! Jorion takes a 2-variable variance; ie..,
variance (A+B) = variance(A) + variance(B) = 2*Covariance(A,B), where here the variables are consecutive returns, so that
variance(return yesterday + return today) = variance (return yesterday) + variance (return today) + 2*covariance(return yesterday, return today)
....now a big assumption: assume both returns have same (constant) volatility/standard deviation
....and given that covariance = correlation * stdDev() * stdDev()
variance(return yesterday + return today) = variance(return) + variance (return) + 2*correlation*StdDev(return)*StdDev(return)
= 2*variance(return) + 2*correlation * variance(return)
= 2*variance(return) * [1 + correlation]

2) That is a great question, I am not sure. Here is my initial guess: I don't agree with Jorion's use of "rho" in 14.9, I think that deserves to be a "b" like Linda Allen such that rho = autocorrelation and b = slope of a regression (i.e., the slope of an autogressive function where we are regressing a current return on a previous return). So, just like in a regression, the slope of the regression line is not identical to the correlation coefficient, here too, the autocorrelation (rho, the correlation between returns) is not the same as the slope of the regression line. In this case, they are close, both are near 1.1 or 0.1 (1.1 -1) and I *think* the difference can be explained by a standard deviation line that is slighty different than 1.0; ie.., slope = rho * StdDev(return)/StdDev(return-1) and i think maybe the autocorrelation makes the StdDev ratio almost but not quite 1.0 creating a bit of a discrepancy...but i am not sure, i haven't tried to reconcile as i am just sort of satisfied that autocorrelation has one meaning compared to the slope of an autoregressive function...Put another way, I do not see how 14.9 leads to 14.10, rather I perceive the "rho" to have two different definitions...I could be missing something, maybe somebody else can improve on this?

3) As I suggested above, IMO, the question is unfair...I don't see how you could unless you knew (or could get to) the handbook equation 14.11 which is indeed elegant, but it's not in the assigmed readings...
the lesson i get from the question is: if you get stuck, don't forget to wonder whether Gujarati's variance and covariance properties might be used to save the day.

3b) The mean return enters/exits for the VaR, i don't think it plays a role in the variance issues...it would impact an "absolute VaR" but I don't think it would impact the scaled volatility....so maybe i don't follow...

thanks, David

 

[babu]

Hi David

please explain in detail, how we can calulate single VAR number by taking all risks like market, credit, counterparty etc..