For 5% VaR, it's the 5th lowest return. For 10%, is it the 10th lowest?

[Steve Jobs]

The question is related to topic 1(out of the 9 topics). I could find only one practice question in kaplan books for this particular type of question.

The question provides the below data and asks for the daily VaR at 5% level of significance using the historical simulation method:
Portfolio value = 5m
Lowest returns during last year are: rate1, rate2, rate x

The provided answer is:
1. arrange the rates from lowest to highest; then from the lowest side, select the fifth.
2. fifth lowest rate * 5m

I understand here that because the level of significance is 5%, the 5th lowest return was selected. Is this correct?

 

[ShaktiRathore]

In finding VaR using historical simulation this is the general method ,
Arrange the returns from lowest to highest over a period. if there are 100 returns over a period than the 5th worst return will be the benchmark to find the VaR at a given confidence level of 95%. That is if its -50% than the worst loss that can be exceeded in 5 out of 100 days (95% CL) is -50% of the portfolio value. based on the historical information of returns we try to predict the future returns that is if there are 100 returns than we assume that this returns will replicate in the future also.
Thus for 100 returns over 100 days we assume that this returns will replicate in next 100 days also. SO chance of exceeding 5th worst loss is 5 days out of 100 days that is we are 100% sure that worst loss cannot exceed -50% over 95 days in next 100 days. or in other word we are 95% confident that loss will not exceed 50% on any day.because in the past the worst loss exceeded 50% only in 5 days out of 100 days thus we are sure of the future also based on this information that loss will not exceed 50% over 95 days out of next 100 days. Thus there is 95%(95/100) chance that worst loss will not exceed 50% over next 100 days. Assuming equal probability for 100 days we can say that there is 95% chance that worst loss will not exceed 50% on any given day or that the VaR is -50% only(since here we are giving equal weights to the days for losses recent returns are weighted same as most recent returns).
Var gives the limit of loss that cannot be exceeded over a given period at a given confidence level. Thus if my limit at 95% CL is 100-95 = 5th worst loss. If its a continuous distribution than we find VaR where area to the right of VaR is 5%. But since historical simulation is discrete we normally take the point(5th worst return) as our VaR. Thus its correct to find the vaR this way when we are given some past returns.

thanks

 

[Steve Jobs]

I don't understand the 50% in the third line, where is it coming from?

However, what I understand is that usually the 5% is used in historical simulation and there will be no questions in FRM asking for 10%. I hope this is correct.

 

[ShaktiRathore]

50% is the assumed 5th worst loss that is its an example i have taken

thanks

 

[David Harper CFA FRM]

If n = 100, the 95% HS VaR is also the 6th worst;
in fact candidates, who are using the readings and following the assignment, will correctly determine the 6th worst (per the assigned Dowd) as the discrete HS VaR is given by N*(1-c%)+1 = 6, in this case

GARP is totally aware of the difference (which really stems from Jorion vs Dowd), we settled this three years ago, see (eg)
http://forum.bionicturtle.com/threads/p1-focus-review-6th-of-8-valuation.6430/
... how did GARP resolve? They will not force you to choose, either will be correct; e.g. the value of the 5th and 6th will be the same

The confusion is, I think, because if n = 1,000 (just to change the n) then:

• The 95% ES has only correct one answer, it is unequivocally the conditional mean, regardless of continuous or discrete distribution There is only one correct ES.
• But the 95% VaR has at least three acceptable answers, in a discrete HS: 50th worst, 51st worst, or interpolation between

I personally prefer Dowd's approach b/c it encourages the semantic:

• with a frequency of 5%, we expect the loss will be greater than, or at least, the [51st worst] .... instead of the otherwise identical but semantically different:
• with a frequency of 95%, we expect the loss will not exceed the [50th worst]. Of course, both are valid, but the former emphasizes the existence of the tail, about which nothing is guaranteed.

 

[Steve Jobs]

For 90%, n=1000, is it the lowest 100(101)th return?

 

[ShaktiRathore]

As per david explaination,
you can take the lowest 100th(=alpha*n=10%*1000=100) return or the 101st loss level. But i would prefer to take it as 100th lowest return(When i read the garp readings i saw the formula alpha*n as the standard level to be taken as the limit of VaR. ). TO be more conservative you can take the 101st loss as the acceptable VaR or you can take the interpolation between them to be on the safer side. but follow the text of garp or what david has said.

thanks

 

[Steve Jobs]

Thanks