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# Interpolating VaR by the hybrid method. Why create a false entry? (Allen)

#### Sixcarbs

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I am trying to interpolate the VaR using the hybrid method. Below is Table 1-3 from Allen.

For the initial, (top half), for 5% I interpolate between .0447 and .0511 and -2.90% and -2.70%. I get 2.73%.

.0500 is roughly 17% less than .0511 so 17% of the difference is roughly .03% which gives me the answer 2.73%.

For the bottom half I interpolate between .0494 and .0571 and -2.4% and -2.3%. I get 2.39%.

But the text inserts a third entry for each and gets a different answer for the latter.

The text inserts a return and cumulative weight half way between the two returns, -2.8%/.0479 for the first and -2.35%/.0533 for the latter. It then uses the same interpolation methods I used but comes up with a different answer for the latter.

Why must we create a false entry to interpolate the returns to a cumulative weight of .0500? Shouldn't my method work?

On a related note, I am having trouble generating the weights in excel using the formula in the text.

Trying to set up Excel table with K column and hybrid weight column.

Text says [(1-La)/(1-La^K)]*[La^(K-1)]

I thought K was the entry. Maybe it is just the total number of entries??? I think the second term (Bold) may not be Lamda to the (K-1) power but instead Lamda of the previous K.

Thanks #### Detective

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I didn’t quite follow your interpolation logic. I don’t think you can assume what you’re doing will hold true generally.

Also, sorry I don’t have Allen text in front of me on phone, but that formula doesn’t look right. In Dowd at least chapter 4 (note: this is part 2 reading, in part 1 they only make you read chapter 2), I thought it was explained well.

Formula from Dowd:

w(i) = (lambda^(i-1) * (1-lambda)) / (1-lambda^n)

This gives you the weight “i” periods ago, here n (or your K) should be the length under consideration, in this example it seems n = 1/0.01 = 100.

The idea is that:
w(1) = (1-lambda) / (1-lambda^n)
w(2) = (lambda) (1-lambda) / (1-lambda^n)
w(3) = (lambda^2) (1-lambda) / (1-lambda^n)
...
Notice the ratio between each successive term, e.g. w(2)/w(1) and w(3)/w(2) is lambda, the decay factor.

Also, if you take limit of w(i) as lambda -> 1 (for i<=n) you will get 1/n, which essentially means you weigh all the observations the same. Also called “HS” (historical simulation) in your table above.

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#### Sixcarbs

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I didn’t quite follow your interpolation logic. I don’t think you can assume what you’re doing will hold true generally.
Thanks for the forumlas. I will see if I can make it work now.

With resepct to the interpolation. As far as I know I am looking for the value of the return where the Hybrid Cumulative Weight=.05.

For the initial case I interpolate between row 2 and 3 and come up with 2.73%. For the 25 days later case I interpolate between rows 5 and 6 and come up with 2.39%.

But in the text they insert a Return and Hybrid Cumulative Weight half way between each of those rows, and then each interpolate to .0500 Hybrid Cumulative Weight.

From the text for the first one:

".......To obtain the 5 percent VaR for the initial period, we must interpolate as shown in Figure 1-10. We obtain a cumulative weight of 4.79 percent for the -2.80 percent return. Thus, the 5th percentile VaR under the hybrid approach for the initial period lies somewhere between 2.70 percent and 2.80 percent......"

From the text for the second one:

".......Similarly, the hybrid approach estimate of the 5 percent VaR 25 days later can be found by interpolating between the -2.40 percent return (with a cumulative weight of 4.94 percent) and -2.35 percent (with a cumulative weight of 5.33 percent, interpolated from the values on Table 1-3). ...."

#### Detective

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Thanks for the forumlas. I will see if I can make it work now.

With resepct to the interpolation. As far as I know I am looking for the value of the return where the Hybrid Cumulative Weight=.05.

For the initial case I interpolate between row 2 and 3 and come up with 2.73%. For the 25 days later case I interpolate between rows 5 and 6 and come up with 2.39%.

But in the text they insert a Return and Hybrid Cumulative Weight half way between each of those rows, and then each interpolate to .0500 Hybrid Cumulative Weight.

From the text for the first one:

".......To obtain the 5 percent VaR for the initial period, we must interpolate as shown in Figure 1-10. We obtain a cumulative weight of 4.79 percent for the -2.80 percent return. Thus, the 5th percentile VaR under the hybrid approach for the initial period lies somewhere between 2.70 percent and 2.80 percent......"

From the text for the second one:

".......Similarly, the hybrid approach estimate of the 5 percent VaR 25 days later can be found by interpolating between the -2.40 percent return (with a cumulative weight of 4.94 percent) and -2.35 percent (with a cumulative weight of 5.33 percent, interpolated from the values on Table 1-3). ...."
Thanks for the clarification. It seems goal is to find 95% VaR. I'd say it's ambiguous since there is no hybrid cum. weight that is 5%. I wouldn't be able to fault anyone that just took the closest return in a conservative fashion. Interpolation makes sense to me, but maybe linear is too simplistic.

Anyways assuming we are doing linear interpolation, I agree with you. I get your answer of ~-2.39221% in the second case as well. If you have two points (a,b) and (e,f) and you find the line of best fit (which is just line that determines those two points), it's the case that the midpoint ((a+e)/2,(b+f)/2) will be a point on that line, so the answer you get between linearly interpolating midpoint and endpoint, should be no different. I'm not sure why they add the midpoint; it's not necessary, but maybe it is for extra exposition to make the linear interpolation more visualizable?

What answer is the book saying?

#### Sixcarbs

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Thanks for the clarification. It seems goal is to find 95% VaR. I'd say it's ambiguous since there is no hybrid cum. weight that is 5%. I wouldn't be able to fault anyone that just took the closest return in a conservative fashion. Interpolation makes sense to me, but maybe linear is too simplistic.

Anyways assuming we are doing linear interpolation, I agree with you. I get your answer of ~-2.39221% in the second case as well. If you have two points (a,b) and (e,f) and you find the line of best fit (which is just line that determines those two points), it's the case that the midpoint ((a+e)/2,(b+f)/2) will be a point on that line, so the answer you get between linearly interpolating midpoint and endpoint, should be no different. I'm not sure why they add the midpoint; it's not necessary, but maybe it is for extra exposition to make the linear interpolation more visualizable?

What answer is the book saying?
For the initial, they insert a false entry between 2 and 3:

-2.80% Return and Hybrid Cumulative Weight 0.0479

It then interpolates between that entry and row 3 and comes up with 2.73%, same thing I got.

But the second they insert the false entry:

-2.35% Return and Hybrid Cumulative Weight 0.0533

It then interpolates between that entry and row 5.

-2.40% .0494
-2.35% .0533

For that I get 2.39%.

Now, I just noticed an flagrant error in the book's calculation. Instead of using 2.40% and 2.35%, it used 2.35% and 2.30%. So the book is completely messed up. (It says 2.4% in the paragraph preceding it, but uses 2.30% in the calculation.)

But it still raises the question, why would we have to insert a Return and Cum. Weight exactly halfway between each and then interpolate between that inserted value and the next closes one on the other side of .05 instead of just interpolating to .05 Cum. Weight from the table we have?

I can only think it has something to do with the ealier section when it talked about selecting a value between .05 and .06 in the pure HS method. But here we interpolate to .05 exactly anyway.

#### Detective

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For the initial, they insert a false entry between 2 and 3:

-2.80% Return and Hybrid Cumulative Weight 0.0479

It then interpolates between that entry and row 3 and comes up with 2.73%, same thing I got.

But the second they insert the false entry:

-2.35% Return and Hybrid Cumulative Weight 0.0533

It then interpolates between that entry and row 5.

-2.40% .0494
-2.35% .0533

For that I get 2.39%.

Now, I just noticed an flagrant error in the book's calculation. Instead of using 2.40% and 2.35%, it used 2.35% and 2.30%. So the book is completely messed up. (It says 2.4% in the paragraph preceding it, but uses 2.30% in the calculation.)

But it still raises the question, why would we have to insert a Return and Cum. Weight exactly halfway between each and then interpolate between that inserted value and the next closes one on the other side of .05 instead of just interpolating to .05 Cum. Weight from the table we have?

I can only think it has something to do with the ealier section when it talked about selecting a value between .05 and .06 in the pure HS method. But here we interpolate to .05 exactly anyway.
I think they are just adding midpoint for illustrative purposes rather than any deeper underlying reason. I found source material and you're right they butchered the linear interpolation in the second case.

I think confusion in your formula (initial post) was they were telling you a step by step algorithm and you simply copied over penultimate term: They tell you lambda = 0.98 and K = 100, so you should be able to tie using my formula ( I or Dowd rather calls "K", n).

Second part... #### Sixcarbs

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Sorry, we're going to need another dead horse here since this one is about done.

But, using their inserted midpoint, the .0533 and .0494 are correct. Remember we are looking for .05 between the two. It is the percentages that need to be changed. We want a number between 2.35% and 2.40%.

I still need to insert that Lambda formula into Excel and see if it matches the book. Thank you.

#### Detective

##### Active Member
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Sorry, we're going to need another dead horse here since this one is about done.

But, using their inserted midpoint, the .0533 and .0494 are correct. Remember we are looking for .05 between the two. It is the percentages that need to be changed. We want a number between 2.35% and 2.40%.

I still need to insert that Lambda formula into Excel and see if it matches the book. Thank you.
Lambda formula should work, I tried it out myself.

Also, I see what you're saying but ultimately it doesn't matter here. There are two given points: (-2.4%,0.0494) and (-2.3%,0.0571), which is sufficient on their own to determine what return corresponds to 0.05 cumulative weight using linear interpolation, it should lean towards -2.4% based on these numbers so ~-2.39% makes sense.

They've gone and added a "midpoint" as (-2.35%, 0.05325), now if I use either of the existing points with the midpoint, I will get the same line so it doesn't matter if I use the -2.35% in conjunction with the -2.4% or -2.3%.

In terms of my screenshot, I took a stab at what modifications give the right answer. Ultimately, I can't read their mind, so I don't know where they errored in terms of backing out the second interpolation They were also careless with keeping the negative sign as well.

#### David Harper CFA FRM

##### David Harper CFA FRM
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Hi @Sixcarbs and @Detective We've previously grappled with this at some length, see https://www.bionicturtle.com/forum/threads/calculating-revised-var-hybrid-approach.9857/post-51656 i.e.,
I just wanted to close the loop (because I was answering a similar question here at https://www.bionicturtle.com/forum/threads/hybrid-approach-to-compute-var.4384) in case there is a want for future reference. Shown below, on the left side panels, are Linda Allens assumptions given (in 2.2.7 The hybrid approach) under both the initial date and 25 days later. It is my opinion that she has a mistake in the text. When she computes 2.73% for the 95% hybrid VaR (initial state) she is mistakenly interpolating twice (Schweser's example is above is presumably based on this). These are shown in purple. But it's not consistent with her description of "mass centering" before interpolating, which is shown on the right panel and (according to my calculations) would produce 2.63% (instead of 2.73%). Details after the exhibit, but my evidenced includes the fact that her 25 days later 95% VaR calculates to 2.34% by correctly applying the "mass centering" methodology (but if we were to repeat the naive interpolation/2x interpolation, then we would get 2.39% instead). Specifically, in my opinion,on page 58 instead of:
In contrast, the hybrid approach departs from the equally weighted HS approach. Examining first the initial period, table 2.3 shows that the cumulative weight of the −2.90 percent return is 4.47 percent and 5.11 percent for the −2.70 percent return. To obtain the 5 percent VaR for the initial period, we must interpolate as shown in figure 2.10. We obtain a cumulative weight of 4.79 percent for the −2.80 percent return. Thus, the 5th percentile VaR under the hybrid approach for the initial period lies somewhere between 2.70 percent and 2.80 percent. We define the required VaR level as a linearly interpolated return, where the distance to the two adjacent cumulative weights determines the return. In this case, for the initial period the 5 percent VaR under the hybrid approach is:

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

Similarly, the hybrid approach estimate of the 5 percent VaR 25 days later can be found by interpolating between the −2.40 percent return (with a cumulative weight of 4.94 percent) and −2.35 percent (with a cumulative weight of 5.33 percent, interpolated from the values on table 2.3). Solving for the 5 percent VaR: 2.35% − (2.35% − 2.30%)*[(0.05 − 0.0494)/(0.0533− 0.0494)]
= 2.34%.
the text should read (my changes emphasized to match above exhibit):
In contrast, the hybrid approach departs from the equally weighted HS approach. Examining first the initial period, table 2.3 shows that the cumulative weight of the −2.90 percent return is 4.47 percent and 5.11 percent for the −2.70 percent return. To obtain the 5 percent VaR for the initial period, we must interpolate as shown in figure 2.10. We obtain a cumulative weight of 4.79 percent for the −2.80 percent return and 5.11% for -2.60% (which is the midpoint between -2.70% and -2.50%). Thus, the 5th percentile VaR under the hybrid approach for the initial period lies somewhere between 2.60 percent and 2.70 percent. We define the required VaR level as a linearly interpolated return, where the distance to the two adjacent cumulative weights determines the return. In this case, for the initial period the 5 percent VaR under the hybrid approach is:

2.70% − (2.70% − 2.60%)*[(0.05 − 0.0479)/(0.0511 − 0.0479)]
= 2.63%.

Similarly, the hybrid approach estimate of the 5 percent VaR 25 days later can be found by interpolating between the −2.35 percent return (with a cumulative weight of 4.94 percent) and −2.30 percent (with a cumulative weight of 5.3246 percent, interpolated from the values on table 2.3). Solving for the 5 percent VaR: 2.35% − (2.35% − 2.30%)*[(0.05 − 0.0494)/(0.0533− 0.0494)] [<--notice how this formula is correct; it's just the text that needs editing to match!]
= 2.34%.
My view is is captured in the XLS and includes:
• The "mass centering" is a bit difficult to grok but logical and returns for us the following pre-interpolation datapoints
• Initial: -2.70% return @ 4.79% cuml weight, and -2.60% return @ 5.11% cuml weight
• +25 days: -2.350% return @ 4.937% c.w., and -2.300% @ 5.325% c.w. See right-hand side of above XLS
• Then the interpolated returns are:
• Initial: -2.63%, so this is where the error occurs (text has -2.73% ). Because rather than =2.8% - (2.8% - 2.7%)*((5%-4.79%)/(5.11%-4.79%)) it should be 2.7% - (2.7% - 2.6%)*((5%-4.79%)/(5.11%-4.79%)) = -2.63%, which is consistent with the correctly calculated -2.34% @ +25 days
• +25 days: -2.34%

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#### Sixcarbs

##### Active Member
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Good morning @David Harper CFA FRM

So, the real issue is "Mass centering." Would we need to apply this for simple HS too?

I'm fine with interpolating, as I am sure anyone who got this far is too. It's just a matter of setting up the table now.

We need to add midpoints between all of the observed returns. Also add midpoints of the weights.

Then shift the weights down one row. Each return will be assigned the weight of the return above it.

Then interpolate to a weight of .05 using the new "Mass centering" Return/Weight table.

Correct?

Thank you,

SIxcarbs

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @Sixcarbs Good morning! Yes, agreed, the tricky part is the "mass centering" (as the interpolation is straightforward linear interpolation; you may notice on the left side of the exhibit above, I show how the incorrect 2.73%, which at the time was raised by a reference to Kaplan's study note, can be calculated by interpolating twice, which is an unnecessary thing to do when interpolating).

Re: Would we need to apply this for simple HS too? We do not need to, but we can (L. Allen's text actually illustrates mass centering under simple HS). In the example above, the mass-centered simple HS 95.0% VaR is -2.35% (for both Initial and +25 days later) simply because cumulative 5.0% (which row-wise is horizontally adjacent to -2.40% return) aligns between the mid-point of -2.40% and -2.30%. So, yes, that sounds similar to what you are saying, I hope that's helpful