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# Hybrid approach to compute VaR

#### sl

##### Active Member
The below example is taken from Linda Allen

Order Return Periods Hybrid weight Hybrid cumul. weight HS weight HS cumul.ago weight
1 −3.30% 3 0.0221 0.0221 0.01 0.01
2 −2.90% 2 0.0226 0.0447 0.01 0.02
3 −2.70% 65 0.0063 0.0511 0.01 0.03
4 −2.50% 45 0.0095 0.0605 0.01 0.04
5 −2.40% 5 0.0213 0.0818 0.01 0.05
6 −2.30% 30 0.0128 0.0947 0.01 0.06

25 days later:
1 −3.30% 28 0.0134 0.0134 0.01 0.01
2 −2.90% 27 0.0136 0.0270 0.01 0.02
3 −2.70% 90 0.0038 0.0308 0.01 0.03
4 −2.50% 70 0.0057 0.0365 0.01 0.04
5 −2.40% 30 0.0128 0.0494 0.01 0.05
6 −2.30% 55 0.0077 0.0571 0.01 0.06

In the first case the VaR is computed as follows

2.80% − (2.80% − 2.70%)*[(0.05 − 0.0479)/(0.0511 − 0.0479)]
=2.73%

In the second case the VaR is computed as
2.35% − (2.35% − 2.30%)*[(0.05 − 0.0494)/(0.0533− 0.0494)]
=2.34%

Why is that in the first case, the author uses the average Hybrid Cumm. weight (0.0479) for returns -2.7% and -2.9% in the numerator, whereas in the second case, he uses the average Hybrid Cumm. weight (0.0533) for returns -2.4% and -2.3% in the denominator

Can you please clarify how we are supposed to compute the VaR using this approach ?

Are we just eyeballing the Hybrid Cumm. weights and picking those that are close to the 5% VaR and performing interpolation?

Thanks
Sundeep

Hi David

Thanks
Sundeep

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi Sundeep,

It hasn't come up before and i admit I don't get why Linda Allen does this. It seems to me, to take the first example, that the linear interpolation is given by:

=(5.0% - 4.47%)/(5.11% - 4.47%) * (-2.70% - - 2.90%) + - 2.90% = -2.73438%
i.e., interpolate the 5% based on the nearest returns @ -2.90% and -2.70%

Linda Allen's example is at least consistent: she is conducting two steps:
1. She interpolates to insert a new point at [return = -2.80%, cumulative hybrid weight = 4.79%]
... this is just a midpoint: as -2.80 is mid between -2.70 and -2.90 so is 4.79% the midpoint between 4.47% and 5.11%
2. Now, once she's done that, then the interpolation operates on the new (-2.80% and 4.47% weight)

I am not sure why she does this?? In this case, the answers are exactly the same. I'm not certain, but i think maybe they would be the same (proof eludes me now, maybe i'm wrong). Maybe there is a good technical reason...?

but in terms of the FRM exam, you would never do this. You make a great observation. At most, we would do a "regular" interpolation between two existing data points

Hope that helps, David

#### hellohi

##### Active Member
dear @David Harper CFA FRM

1) according to Histoeical Simulation as I know when we say that we have 100 observation and want to determine 5% Var . so we just order the returns in ascending order and chose the fifth least worse return and here is -2.40% but Linda Allen solved it as 2.35% (between the 5 and 6 order) why did she do this?

2) according to your solve above : =(5.0% - 4.47%)/(5.11% - 4.47%) * (-2.70% - - 2.90%) + - 2.90% = -2.73438%
you chose the returns that between 4.47% and 5.11%. is this mean you chose this interval because we are looking for the 5% Var, and the 5% in between?

best regards,
Nabil

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @hellohi

1. In simple HS, the 95% VaR is typically the 6th worst (Dowd) but can be the 5th worst (as you say, Jorion) and can also be technically adjusted as Allen does. See my set of calculations labelled "Tech Cumulative". This is because she assume the weights straddle the observation
2. Linda Allen is performing two technical steps. As mentioned, the first is to assume the weights straddle the observations. Second, then she perform a linear interpolation using the Tech Cumulative numbers. The above I wrote years ago. Today, I believe she has an error in her first interpolation such that her first 95% VaR is -2.63% but her second (25 days later) is correct at -2.35%
These are both subtle concepts. It is further confusing because the first step she calls an interpolation, and if we do that, she interpolates, then again interpolates, which is very confusing. But if look at each as a separate step, it eventually makes sense. Recommend you analyze my XLS before further queries, or it won't really make sense, it will just be a word salad . Thanks!

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#### modigliani

##### New Member
1 −3.30% 3 0.0221 0.0221 0.01 0.01
2 −2.90% 2 0.0226 0.0447 0.01 0.02
3 −2.70% 65 0.0063 0.0511 0.01 0.03
4 −2.50% 45 0.0095 0.0605 0.01 0.04
5 −2.40% 5 0.0213 0.0818 0.01 0.05
6 −2.30% 30 0.0128 0.0947 0.01 0.06

is there anything wrong to interpolate this way:

2.80% − (2.80% − 2.70%)*[(0.05 − 0.0479)/(0.0511 − 0.0479)]?

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#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @modigliani That's Linda Allen's textbook example. I think she made a mistake. The calculation you show is not wrong, it's just the result of interpolating twice (i.e., the first time to retrieve 2.80 at 4.79%). If you take two points (i.e., 2.90% and 2.70% weights), and interpolate them to retrieve an between point (in this case 2.80% weight), then additional linear interpolations all produce the same result as if you just did it once. In this case, see below in purple. The VaR answer of -2.73% return is the same as the result of your formula, but it's just one interpolation between 4.47% and 5.11%: =(0.05-0.0447)/(0.0511 - 0.0447)*[-2.70% - (-2.90%)] + (-2.90%).

Why did she do this? She didn't mean to, it's a mistake. She meant to retrieve -2.63% per the two-step that she discussed, which is illustrated on the right. I will explain that further here on a more recent thread, if you don't mind (to be continued): https://www.bionicturtle.com/forum/threads/calculating-revised-var-hybrid-approach.9857/