How to estimate the daily return

[Ludwma]

Hi David,

I found this question in your practice on Estimating Volatilities and Correlations (Hull):
"Assume an exponentially weighted moving average (EWMA) model with a lambda parameter of 0.94 (as per RiskMetrics). Yesterday’s DAILY volatility was 1.90%. The price of the stock closed at $10.00 yesterday and closed up today at $11.275. What is the updated EWMA volatility estimate?"

You estimate the daily return by using ln(today price/yesterday price), however in Hull it seems that the suggestion is to use (today price - yesterday price)/yesterday price. Is there any reason why you have used the ln approach?

Thanks,

FS

 

[Aleksander Hansen]

For short horizons the difference is negligible. However, for projection you should always use compound returns: their distribution is approximately symmetrical; you can project to any horizon and then translate back into the distribution of the market prices at the specified horizon.

For allocation decisions you should use linear returns (however, the distribution of linear returns is not symmetrical).
After all, the unit of account in finance is money.

 

[David Harper CFA FRM]

Hi FS,

Agreed, good observation. There is a history here, but the main points are:

• If my question does not specify explicitly, then it's guilty of (wrong for) the same imprecision that I lobby against. The question needs to specify either: continuous (log) returns or simple (aka, percentage) for the daily return. Because there isn't a correct approach, there is a trade-off: log returns add over time; percentage returns add positions across the portfolio
• The reason I have defaulted in the past is that Jorion, Linda Allen (long assigned specifically to this topic) and, to a degree, Dowd default to log (continuous) returns due to its advantages. However, you are absolutely correct: Hull explains why he defaults to daily simple (percentage) returns ... and justifies on the assumption that it's a rounding error for short periods ... but there is a meaningful difference of 0.75% between LN(11.275/10) and 11.275/10 - 1.
• I view this as a special case of the want of a compound frequency standard for the FRM. In November, GARP added to the exam instructions, according to Hend, "a page in the beginning saying so and so abbreviations mean this and that...oh yes, and all rates are annually discounted unless question says otherwise (Thank you David)"
• But, it occurs to me that this hasn't settled the daily return question, so I just added to our ongoing collection for feedback to GARP (that i will summarize and accumulate for their convenience): http://forum.bionicturtle.com/threa...-percentage-or-continuous-in-garch-emwa.5439/

In summary: I agree Hull defaults to daily percentage (simple) return; I have been defaulting to log returns because three of our authors, including Jorion, prefer log returns; but my question is WRONG to omit the specification; GARP is alert to this (compound frequency), the exam question will do better by specifying, although I can't say there is a default method. Thanks,

 

[David Harper CFA FRM]

Hi ahansen,

I didn't see your post. Thank you! Can I ask about your terminology: is your choice of "linear" versus "compound" by preference? I am just curious b/c I don't find "linear" intuitive as it already goes by simple (Allen), percentage (Hull) or arithmetic (Dowd; versus geometric) ...

actually for FRM (new learners), I sort of prefer continuous (log) versus simple, because this makes it part of the compound frequency topic. Just a pedagogical perspective/musing

thanks,

 

[Aleksander Hansen]

Just to clarify, which post did you not see (unable to see?)?
I use total, linear and compound by preference.
Anecdotally, I think this is fairly common for people with more quantitative backgrounds (math, physics) as well as common in the hedge fund industry.
Don't quote me on that, however, at an event in I attended recently both Derman, Litterman, Carr and Dupiére used those terms.
I figured if it's good enough for them...

 

[David Harper CFA FRM]

ahansen, I just meant that i posted before reading your (cross posted), I can see your posts fine. Okay, thank you for actual practice perspective, it's good to know ... re: the "appeal to authority:" well-played sir, i am a buyer. (although, yikes, that is four synonyms ... although frankly, each has a worthwhile perspective ... it's rich to unpack just the semiotics)!