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Two Books on Academic Volatility Research, Reviewed by Alex Castaldo
Half of the 2003 Nobel prize in economics was given to R.F.Engle "for methods of analyzing economic time series with time-varying volatility" (with the other half to Clive Granger "for methods of analyzing economic time series with common trends (co integration)"). Since Engle discovered ARCH models on a trip to England in 1979 there has developed a large (and probably excessive) research literature that tries to refine and extend these models. There has been a proliferation of models with names like GARCH, EGARCH, IGARCH and so on. What is the value of this research to investors?
The basic idea is simple enough: (1) volatility changes over time, (2) volatility is mean reverting, so that if we find that current vol. is higher than normal we can predict a decline in the months ahead, and vice versa when vol. is unusually low the optimal forecast is for a rise, (3) specific equations can be fit to vol. data with a high degree of statistical significance, but these equations are not unique; on the contrary every econometrician with some ingenuity has been able to come up with a slightly different equation, leading to the proliferation of models mentioned above. And this fundamental non-uniqueness not likely to change in the foreseeable future; there is an arbitrary element to the modelling.
P. Rossi, ed: Modelling stock market volatility, Academic Press, 1996
This book is a collection of advanced research papers written during the boom phase of volatility research. Daniel Nelson (of the University of Chicago) is the author of more than half of these papers, he is joined by well known authors such as Engle himself, Hansen, Bollerslev and others. The papers' mathematics are very advanced, at the frontiers of econometrics; I have not been able to fully work through any of the 14 chapters! Most of the book is taken up by proofs of various results.
19-Feb-2006
Sharpe Ratio for Holding Periods, by Kim Zussman
Recently, I saw in a Bodie and Merton text that there is no time diversification, in the sense that holding period does not reduce risk. So here is a study (yet another proving what is already known) using SP500 daily returns from 1950 to present.
The idea was to check what kind of reward/risk ratio was associated with various holding periods, and whether this follows a pattern. Constructed a kind of Sharpe ratio (sans risk free term) defined as [return/(stdev of returns)] for an assortment of non-overlapping holding periods. Periods chosen in days were 1,3,5,10,20,50,100, and 200.
Here are the results, with columns being #days, mean return, stdev of returns for the periods, and "Sharpe" [ (return-1)/stdev ]:
DAYS MEAN STDEV SHARPE
1 1.0003 0.0089 0.0388
3 1.0010 0.0159 0.0658
5 1.0018 0.0211 0.0833
10 1.0035 0.0292 0.1198
20 1.0070 0.0417 0.1689
50 1.0174 0.0625 0.2781
100 1.0346 0.0849 0.4080
200 1.0707 0.1297 0.5449
One can immediately see that reward/risk increases with duration of holding period. Plotting this showed the relationship to be non-linear, but there was a linear regression of Sharpe on days holding (for a reason shown later):
Slope
Coefficients 0.002517
Standard Error 0.00027709
t Stat 9.083688784
P-value 9.99166E-05
OK, the regression thinks the relationship is linear, and, in any case, Sharpe is highly correlated with length of holding period. However, I seem to recall that diversification of a stock portfolio scales with SQRT(# stocks), so I ran same regression with independent variable as SQRT(DAYS)
Slope
Coefficients 0.039411123
Standard Error 0.000677065
t Stat 58.20880773
P-value 1.72725E-09
Which is essentially a perfect fit straight line. So reward/risk scales as SQRT(days holding period); if you want to double your Sharpe (for alpha free strategies), quadruple your days, etc.
So what? For those of us who trade longer term, there could be less pain and associated risk of bad trades by looking at portfolios less frequently. Such shock factor risk declines with SQRT days as above.
For those who do trade frequently and don't have any alpha, the transaction costs will eat your Sharpe. Assuming a very modest -0.1% cost round trip (per period), the table looks like this:
AFTER TRANS COSTS:
DAYS MEAN STDEV SHARPE
1 0.9993 0.0089 -0.0730
3 1.0000 0.0159 0.0031
5 1.0008 0.0211 0.0360
10 1.0025 0.0292 0.0856
20 1.0060 0.0417 0.1449
50 1.0164 0.0625 0.2621
100 1.0336 0.0849 0.3962
200 1.0697 0.1297 0.5372
Comparing this with no transaction costs, Sharpe drops pretty badly for periods less than 20 days.
Professor Charles Pennington comments:
The ratio of return over standard deviation is definitely something you'd expect to go with the square root of time, because the numerator will go with time and the denominator with the square root of time.
If you have a trading strategy that has expected one-day returns of 0.2% and standard deviation of 1%, then first of all please call me, and second of all, its annualized Sharpe ratio (neglecting the risk free rate) would be sqrt(250)*(0.2)/(1.0)=3.2, where the "250" is the number of trading days in a year. I think the usual convention is to annualize the Sharpe ratio, so you can compare apples to apples.
(Of course there's false precision in all this.) |
