Spreadsheet 4.A1

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David/

I have a question about time- scaling volatility. I want to make sure I’m very clear on doing this correctly. On page 9 of your PDF accompanying video 4.a, Valuation and Risk Models, when you scale down to a 10 day volatility from and annual volatility, I believe you do the following: annual vol * (10/252)^.5 => 15% *((10/252)*^.5) = 2.99%. However, in Chapter 3 of Understanding Market, Credit and Operational Risk: The Value at Risk Approach by Linda Allen, et al when she calculates the one week vol from an annual 20% vol she simply takes 20%/ (52^.5) = 2.77%. If I was to use your method, I would have done the following, 20% * (5/252)^.5 = 2.817. A number close to hers but still different than the number generated under your method. Similarly, when Allen computes one day VaR she says it corresponds to approximately 1.25% per day. Again using your method, I would compute 20%*(1/252)^.5 = 1.26%, so very close but again slightly different.

I may be splitting hairs here, but I want to avoid a simple mistake on the exam. Any insight here is greatly appreciated.

 

[David Harper CFA FRM]

Hi Chris,

Under i.i.d. (unrealistic!), variance is linear, so T-day variance of (V) becomes an n-day variance of V*n/T; and volatility (sigma) = SQRT(V*n/T) = sigma*SQRT(n/T).
So, scaling volatility, being mindful that independence is assumed, is given by SQRT(n/T).
e.g., it's okay to scale a 52 week volatility to 1 week by *SQRT(1/52); or a 50-week to 1-week with * SQRT(1/50).
or, for that matter, nothing stops us from scaling a 9-week vol to 2 years with *SQRT(104/9) ... although as the FRM has conditioned me to think in days, I would probably use *SQRT(750/45) or *SQRT(756/45).

So, both the (n) and (T) need to be specified. However, the FRM tends to follow Hull and deal NOT in weeks but in years with T = 250 or T = 252 trading days/year. For example Question 5 from 2012 Sample P2:

A portfolio manager owns a portfolio of options on a non-dividend paying stock RTX. The portfolio is made up of 10,000 deep in-the-money call options on RTX and 50,000 deep out-of-the money call options on RTX. The portfolio also contains 20,000 forward contracts on RTX. RTX is trading at USD 100. If the volatility of RTX is 30% per year, which of the following amounts would be closest to the 1-day VaR of the portfolio at the 95 percent confidence level, assuming 252 trading days in a year?

See how the question really needs to give you both the n (1 day) and the T (252)? Without those, it's imprecise. They could give you T = 250, also. I hope that helps, thanks!

 

[Rui]

On this matter of annalisation of volatilities using the square root(n) rule a couple of days ago I saw a very interesting article written by Andreas Steiner, from Andreas Steiner Consulting GmbH in Switzerland, where it is shown that by using it one tends to overestimate the true volatility when annualising the daily volatility.

Quite an interesting article if anyone is interested:

http://www.andreassteiner.net/consulting/downloads/ResearchNoteAnnualizedVolatility.pdf

I hope you find it useful.

Kind regards,

Rui

 

[LL]

Most of my questions are answered in the forum. I just have to browse through it !