Hybrid Approach Weights Formula

[jairamjana]

Assuming Lambda = 0.96
Window Period= 10 to 200 days

From what I have seen there are 2 formulas for the hybrid approach.

Formula 1 (1-lambda)*(lambda)^(n-1)
&
Formula 2 (1-lambda)*(lambda)^(n-1)/(1-(lambda)^window period)
​

This is a excel working using both the formulas. The window period is irrelevant for Formula 1.

As we increase the window period the Formula 2 approximates Formula 1 because of the denominator. I understood that much. But my question is for shorter period say 10 days it looks like summation is 1 as per formula 2 so ideally that should be used. When is formula 1 used which doesnt depend on window period?

 

[David Harper CFA FRM]

Hi @jairamjana You are exactly correct that two formulas can apply. Under your example, where λ= 0.96, if N = 10 (ie, very few data points), then the sum of the weights is only 33.52%, such that we can divide these truncated weights by (1 − λ^N)/(1 − λ) in order to "lever up" the weights so that they sum to 100%. My view is that for small (N) we should use this divisor. The reason is simply that, with a total weight much smaller than 100%, we will underestimate the variance/volatility!

Consider the simple case (boring) case where daily returns have been 1.0% each day over the last 10 days. If we use the adjusted weights, then the volatility estimate will be appropriately sqrt(0.01^2) = 1.0%, but if we use the un-adjusted weights, the volatility estimate will be 0.58% because we only weighted ~ 33%.

Here is what Linda Allen says; i.e., it doesn't matter for large (N) is how I read her

"We have two choices with respect to this residual weight:

1. We can increase N so that the sum of residual weight is smal;
   
2. or divide by the truncated sum of weights (1 − λ^N)/(1 − λ) rather than the infinite sum 1/(1 − λ). In our previous example this would mean dividing by 16.63 instead of 16.66 after 100 observations.

This is a purely technical issue. Either is technically fine, and of little real consequence to the estimated volatility." -- Linda Allen page 42

 

[jairamjana]

Hi @David Harper CFA FRM ..Thank you for your clarification.. I made a excel file with a 100 day window(rather small for WHS) for MSFT Stock and computed the 96% VAR. If you have time kindly go through the final result. . I used interpolation to get the exact number. Don't know if its done in actual practise. Linda Allen says most industries uses Historical Simulation. I believe Weighted Historical Simulation is an improvement over the normal HS counterpart since it tends to react to volatilities much rapidly whereas in HS they have all have 1/n weights and hence big or small it's just a number. Thank you again for the quick reply..

 

[David Harper CFA FRM]

Hi @jairamjana I copied your tab and edited briefly, see WHS(2). You are using LOG10(.) but you want LN(.) for the continuously compounded return. I'm not sure about your cumulative weights; e.g., just on lookup, your approach implies the -5.84 (worst return) has a weight of zero (100%), I think. However, your individual weights on column H look good to me! Using those, your interpolation approach is acceptable. I showed you the more technical approach (described by L Allen and Hull) in my quick columns (L) to (O). Here the key idea is, for example, cumulative weight of 3.537% is associated with -2.54% = average(-2.86%, -2.21%) because it "straddles" (has a midpoint) at the -2.86% return. We don't necessarily need to do this. I hope that's helpful. Thanks!

 

[jairamjana]

@David Harper CFA FRM
Thank you for showing me a alternate approach.. Of course I am embarrased I took a log 10 by mistake.. My intention was LN and that made the answer vary a lot.. And also I understand Linda Allen approach we average the worst returns rank by rank and probably that enhanced the interpolation but what I didn't understand is why we want to find the return for 3.6% and not 4% (Lamba is 96%) .. Also the Cumulative weights idea I took from a Exercise Spreadsheet associated with Elements of Financial Risk Management by Christofferson.. So I guess there are multiple approaches...This is the spreadsheet from the companion site for your reference.. Of course he ignored interpolation or averaging and took the possibly lower return..

http://booksite.elsevier.com/9780123744487/chapter_data_results/Chapter2_Results.xls

 

[David Harper CFA FRM]

Hi @jairamjana Thanks for the book recommendation. Re: "what I didn't understand is why we want to find the return for 3.6% and not 4% (Lamba is 96%)" sorry, I was just using 3.6% as an example! Of course it's true given 96.0% confidence that we seek the 4.0% quantile. However, the technical approach in Linda Allen used to retrieve the 4.0% is a little tricky. Here is what I mean, how it works. Your XLS shows (my tab) that:

• return -2.86% corresponds to cumulative weight 3.537%, and
• return -2.21% corresponds to cumulative weight 6.030%

Your very natural approach (your tab) to interpolation between these points at 4.0% VaR produces -2.739%. However, the "more technical" approach recalibrates our endpoints as follows:

• return -2.54% (which is the average of 2.86 and 2.21) corresponds to cumulative weight of 3.537%;
  
• return -2.21% corresponds to cumulative weight of 4.783% (which is 3.537% + one-half of 2.49%)

Oh, I see now that I am looking: I mistakenly used 3.6% in my interpolation, sorry! You are correct: in cell P17 the formula should read "=(4%-O16)/(O17-O16)*(L17-L16)+L16" for a result of -2.416%. Good catch, thanks!

 

[jairamjana]

Thank you so much for your time @David Harper CFA FRM ... I learnt a lot atleast about WHS method.. Just a thought.. Have you heard of Oracles Crystal Ball ERM ??