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# VaR calculation: Short doubt

#### monsieuruzairo3

##### Member
Hi @David Harper CFA FRM CIPM

I came across the following question
Tim Jones is evaluating two mutual funds for an investment of $100,000. Mutual fund A has$20,000,000 in
assets, an annual expected return of 14 percent, and an annual standard deviation of 19 percent. Mutual fund B
has $8,000,000 in assets, an annual expected return of 12 percent, and an annual standard deviation of 16.5 percent. What is the daily value at risk (VAR) of Jones’ portfolio at a 5 percent probability if he invests his money in mutual fund A? A)$1,668.
B) $13,344. C)$38,480.
D) $1,924. Now my approach was 1) calculate Annual Var = 100,000 (-0.14 + 1.65*0.19)= 17,350 2) scale to Daily var = 17,350/sqrt(250) = 1,097.31 I am surprised that my answer does not even match closely with any one of the options. In your opinion, am I missing a trick here? KR Uzi #### hamu4ok ##### Active Member I might be wrong but I think for VAR we would need to first scale down parameters from annual to daily, and then deduct from expected return so-many standard deviations as required at given confidence level. Step-1. Daily parameters. daily mu=0.14/250 = 0.00056; daily sigma= 0.19/SQRT(250) = 0.01202 Step-1. Calculate VAR: (mu - alpha*sigma) * P = (0.0056 - 1.65*0.01202)*100,000 =$ 1926.75 which is close to D.

In your formula for VAR you assume that expected positive return is the same as expected loss (negative mu), which is wrong as per my understanding. Losses will be experienced on the left-side of the bell-shaped distribution when deviations would exceed the mean.
Also time scaling for mean is division by number of days, instead of square root rule for sigma, since it is itself a square root of variance.

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#### Alex_1

##### Active Member
I might be wrong but I think for VAR we would need to first scale down parameters from annual to daily, and then deduct from expected return so-many standard deviations as required at given confidence level.
Step-1. Daily parameters. daily mu=0.14/2500 = 0.00056; daily sigma= 0.19/SQRT(250) = 0.01202
Step-1. Calculate VAR: (mu - alpha*sigma) * P = (0.0056 - 1.65*0.01202)*100,000 = \$ 1926.75 which is close to D.

In your formula for VAR you assume that expected positive return is the same as expected loss (negative mu), which is wrong as per my understanding. Losses will be experienced on the left-side of the bell-shaped distribution when deviations would exceed the mean.
Also time scaling for mean is division by number of days, instead of square root of days (for sigma, since it is square root of variance).

Hi, I agree with @hamu4ok , just wanted to point out a minor typo: you wrote: "Daily parameters. daily mu=0.14/2500 = 0.00056", should rather be mu = 0.14/250. But I think this was your intention anyway. Best regards.

#### hamu4ok

##### Active Member
Hi, I agree with @hamu4ok , just wanted to point out a minor typo: you wrote: "Daily parameters. daily mu=0.14/2500 = 0.00056", should rather be mu = 0.14/250. But I think this was your intention anyway. Best regards.
thanks for spotting the typo, of course it should be 250 not 2500 days. It is also possible to use 252 days. In such case VAR = 1,919, also I did not rounded numbers, which might also cause some difference in numbers calculated.

#### monsieuruzairo3

##### Member
Brilliant!! Thanks man

#### Abhishek...

##### New Member
Subscriber
As per my understanding the basic formulas are the same but it's the form that the data is in that changes the formulas,
basically
L/P data are a simple transformation of P/L data:
L/P = −P/L,
L/P observations assign a positive value to losses and a negative value to losses, and we will
call these L/P data ‘losses’ for short. Dealing with losses is sometimes a little more convenient
for risk measurement purposes because the risk measures are themselves denominated in loss
terms.
These are the formulas list i made for understanding this better, PLEASE SOMEBODY VERIFY :
Var = -mu(p/l) + sigma *z -> data are in P/L form
Var = mu(l/p) + sigma *z -> data are in L/P form
FOR Arithmetic Returns Data
Var = P*(mu - sigma*z) -> returns are in P/L form
Var = P*(-mu +sigma*z) -> returns are L/P form
FOR Exponential returns Data
Var = P * (1- exp[mu- sigma *z] -> returns are P/L
Var = P * ( 1 -exp[-mu +sigma *z] -> returns are L/P

So basically after scaling mean and standard deviation in the above question which is in the form of P/L we should be using the formula for P/L arithmetic returns.
PLEASE CORRECT THIS IF I AM WRONG.

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#### hamu4ok

##### Active Member
As per my understanding the basic formulas are the same but it's the form that the data is in that changes the formulas,
basically
L/P data are a simple transformation of P/L data:
L/P = −P/L,
L/P observations assign a positive value to losses and a negative value to losses, and we will
call these L/P data ‘losses’ for short. Dealing with losses is sometimes a little more convenient
for risk measurement purposes because the risk measures are themselves denominated in loss
terms.
.....
So basically after scaling mean and standard deviation in the above question which is in the form of P/L we should be using the formula for P/L arithmetic returns.
PLEASE CORRECT THIS IF I AM WRONG.
I believe that it's better to think graphically, always picture the distribution shape (bell shape), if expected return is expressed in loss term, it is still a mean & median of the bell shape. To the left of it will be more losses, to the right will be less losses. Since VAR is about worst possible loss at given CL, the we need to go to the left, that is ( mu(exp.loss) - alpha*sigma)
The formula where (-mu+alpha*sigma) is going the other direction, to the right, where there is less loss.
Hope that I am correct on this one, but by common sense, I should be...

#### Abhishek...

##### New Member
Subscriber
I believe that it's better to think graphically, always picture the distribution shape (bell shape), if expected return is expressed in loss term, it is still a mean & median of the bell shape. To the left of it will be more losses, to the right will be less losses. Since VAR is about worst possible loss at given CL, the we need to go to the left, that is ( mu(exp.loss) - alpha*sigma)
The formula where (-mu+alpha*sigma) is going the other direction, to the right, where there is less loss.
Hope that I am correct on this one, but by common sense, I should be...

The formula where (-mu+alpha*sigma) is when the profit or loss data is in P/L form, it is for the same reason I made the formula list, even I got confused in the beginning. I got confused because of question 69.1 but here https://www.bionicturtle.com/forum/threads/l2-t5-69-parametric-value-at-risk-var.3643/#post-30579 I understood it.
I totally agree with you on the "thinking in terms of graphs" but still the form the data is in is important else the whole calculations will be wrong, as it were in the link i provided above.

#### Abhishek...

##### New Member
Subscriber
The formula where (-mu+alpha*sigma) is when the profit or loss data is in P/L form, it is for the same reason I made the formula list, even I got confused in the beginning. I got confused because of question 69.1 but here https://www.bionicturtle.com/forum/threads/l2-t5-69-parametric-value-at-risk-var.3643/#post-30579 I understood it.
I totally agree with you on the "thinking in terms of graphs" but still the form the data is in is important else the whole calculations will be wrong, as it were in the link i provided above.

@David Harper CFA FRM CIPM Sir could you please comment on whether the formula list i made is correct or not

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @Abhishek...

I always think this sort of listing is risky (I almost think it makes it more complicated than it needs to be). Dowd utilizes an approach in which the final VaR is expressed as a positive number, regardless of the input as P/L or L/P. Please note two things:
1. As a pure input (mu) can be negative or positive, but in market risk the drift will positive (if it's not rounded down/assumed to be zero).
2. Sigma as volatility is always a positive input
Let's work with:
• In P/L form: Mu = 5% and sigma = 20% (i.e., we expect market risk assets to drift up) corresponds to
• In L/P form: Mu = -5% and sigma = 20% (we still expect market risk assets to drift up, but that's counter-intuitively negative in the L/P scale!)
Yes, these are correct:
Var = -mu(p/l) + sigma *z -> data are in P/L form
Var = mu(l/p) + sigma *z -> data are in L/P form
... because they both return the same VaR = -5%+20%*1.645 = 27.90%

FOR Arithmetic Returns Data, same as above:
Returns are in P/L form --> Var = P*(-mu + sigma*z); i.e., 27.9%*P
Returns are in L/P form --> Var = P*(mu +sigma*z) -> i.e., 27.90%*P

FOR Exponential returns Data
Var = P * (1- exp[mu- sigma *z] -> returns are P/L. Correct. Given above, returns +20.46%
Var = P * ( 1 -exp[-mu - sigma *z] or P * ( 1 -exp[-(mu + sigma *z)]-> returns are L/P; in order to return same 20.46%

I hope that helps. Personally, I think memorizing this list is unnecessarily difficult (I don't mean to suggest that is your intent). Rather, if you just understand visually the axis swap and that upward drift (in market risk) counteracts the volatility as risk, there is really just one basic idea plus the arithmetic versus lognormal difference. But, actually, if the purpose is the exam, and it's too time consuming to master, I probably would just focus on the arithmetic which is 10x more likely to be tested. I hope that helps,

#### Abhishek...

##### New Member
Subscriber
Thanks...it's clear now. I got confused in converting theory to formulas, really appreciate your timely reply. It would really be beneficial if you could put this in study notes as this is the best explanation of the concept that I found.

#### hamu4ok

##### Active Member
It is much easier to remember that VaR is about the worst loss, you should select that sign plus or minus that would produce the highest VaR amount. Any amount less because of the sign'' is a step into wrong direction.
That intuitive way, no need to remember 4-5 forms of the same VaR Formula, IMHO

#### southeuro

##### Member
quick question... if we have 3 VaR for market op and credit risks, can we simply not add them up to get aggregate var?
I just ran into a question where the values are squared then added then square rooted... what am i missing? thanks!

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@southeuro The above thread refers to market risk; the reason that matters is that credit risk and operational risk have negative drifts (expected loss rather than expected positive return). You are referring to the aggregation of VaR, which depends a lot on what we are talking about. For example, Basel adds the risk charge for credit, market and operational risk and therefore does not give any credit for diversification. In the mean-variance framework (normal distributions), the general form to which you refer is:

VaR(P) = SQRT[VaR(1)^2 + VaR(2)^2 + 2*VaR(1)*VaR(2)*correlation]

... which is based on the familiar two-asset portfolio volatility, except it is VaR. So there are two extreme cases:
• if correlation is 1.0, then VaR(P) = SQRT[VaR(1)^2 + VaR(2)^2 + 2*VaR(1)*VaR(2)*correlation] = VaR(1) + VaR(2)
• if correlation is zero, then VaR(P) = SQRT[VaR(1)^2 + VaR(2)^2] <<-- to which you refer, so this is a fairly special case of both independence and elliptical distributions

#### southeuro

##### Member
That helps a lot David! Many thanks! 