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Tomasz, this both helps clear things up and makes things more
confusing at the same time.
First, it helps clear things up because now I understand why our two
formulas produce the same results (for values of n > 50). The
function Sqrt(SumOfSquaredDeviationsOfBarNumber) quickly reaches a
limit of Sqrt(n)*(n) / Sqrt(12). They are equal to within 2
significant digits when n is 10 or greater, 3 significant digits once
n is about 20, and more than 5 significant digits for n > ~210.
Therefore, my formula does match the AmiBroker K-Ratio results when n
is large enough:
K-Ratio(AmiApprx) = LinRegSlope / StdErr * sqrt(n) / sqrt(12)
This produces same K-Ratio as AmiBroker for values of n > 50. Maybe
some math wizard out there can explain to us why the sqrt(12) is a
magic number/factor when calculating Sqrt
(SumOfSquaredDeviationsOfLinearFunction).
Now for the more confusing part. Why is Sqrt
(SumOfSquaredDeviationsOfBarNumber) part of your K-Ratio numerator?
I have read the Lars Kestner S&C V 14:3 article many times (as well
as any other info I could find on this subject). No where could I
find any mention of the Sqrt(SumOfSquaredDeviationsOfBarNumber)
function, with the exception of the Excel example stated below. I do
not have a copy of his book, but the Kestner article clearly states
that:
K-Ratio(1996) = LinRegSlope / (StdErr * Sqrt(n))
which was later revised to
K-Ratio(2003) = LinRegSlope / (StdErr * n)
In the article's example calculations, the numerator is shown as 0.58
for System 1 and 0.55 for System 2. This is the LinRegSlope for
those two systems. It is not the LinRegSlope * Sqrt
(SumOfSquaredDeviationsOfBarNumber).
To complicate things even more, the Excel example in the Kestner
article introduces the Sqrt(SumOfSquaredDeviationsOfBarNumber)
function in the denominator but provides no reason why. Perhaps this
is explained in his book or perhaps it has something to do with the
way that Excel calculates StdErr of LinRegLine versus the way that
TradeStation does. There also appears to be an error in the
description of the Excel formula as the sqrt(n) is shown as a
multiplier instead of a divisor.
If someone could tell me why the Sqrt
(SumOfSquaredDeviationsOfBarNumber) function is needed, I would
greatly appreciate it.
--- In amibroker@xxxxxxxxxxxxxxx, "Tomasz Janeczko" <groups@xxx>
wrote:
>
> > K-Ratio(Ami) = LinRegSlope / StdErr * sqrt(n) / sqrt(12)
> No it is NOT !
>
> K-Ratio (AMI) is
>
> LinRegSlope * Sqrt( SumOfSquaredDeviationsOfBarNumber ) / (StdErr *
n);
>
> not the one you wrote.
>
> From the HELP FILE (again: read, read, read the help file).
>
> K-Ratio - Detects inconsistency in returns. Should be 1.0 or more.
The higher K ratio is the more consistent return you may expect
> from the system. Linear regression slope of equity line multiplied
by square root of sum of squared deviations of bar number divided
> by standard error of equity line multiplied by square root of
number of bars. More information: Stocks & Commodities V14:3
> (115-118): Measuring System Performance by Lars N. Kestner
>
> Best regards,
> Tomasz Janeczko
> amibroker.com
> ----- Original Message -----
> From: "Ron Rowland" <rowland@xxx>
> To: <amibroker@xxxxxxxxxxxxxxx>
> Sent: Friday, July 27, 2007 7:06 PM
> Subject: [amibroker] Re: K-Ratio Implementation (revisited)
>
>
> > Thanks Thomas, but no, I don't believe that post does not answer
my
> > question. The post below explains the the difference between K-
Ratio
> > (1996) and K-Ratio(2003). However, it does not explain the
> > Backtester results.
> >
> > K-Ratio(1996) = LinRegSlope / (StdErr * sqrt(n))
> > K-Ratio(2003) = LinRegSlope / (StdErr * n)
> > K-Ratio(Ami) = LinRegSlope / StdErr * sqrt(n) / sqrt(12)
> >
> > The sqrt(12) function is typically used to convert a monthly
StdDev
> > (or StdErr) to an annualized one. I do not understand its purpose
> > here.
> >
> > Alternatively, there could be additional errors in mys
undertanding
> > of these functions:
> >
> > LinRegSlope returns the slope for 1 period. Since the slope is
> > linear, it can be annualized by multiplying a daily slope by 252
> > market days in a year. Multiplying by n will provide total
increase
> > over the entire backtest.
> >
> > StdErr returns the standard error function (the 1-period average
of
> > the entire range). Since it is essentially a 1-period standard
> > deviation, to annualize this value you should multiply by sqrt
(252).
> >
> > It seems to me that LRS / StdErr = K-Ratio for one day.
> >
> > To annualize this, it should be multiplied by 252/ sqrt(252).
> > Of course, any number multiplied by its sqrt = its sqrt, to the
> > annualized version can be simplified to
> >
> > LRS/ StdErr * sqrt(252).
> >
> > If for some reason, you do not want an annualized version that
can be
> > compared for various time frames, you can create the cumulative
> > version by replacing 252 with n (the number of observations).
This
> > suggests that sqrt(252) should be in numerator instead of n being
in
> > the denominator.
> >
> > My thinking and/or assumptions must be off somewhere, but I cannot
> > determine where.
> >
> >
> >
> > --- In amibroker@xxxxxxxxxxxxxxx, Thomas Ludwig <Thomas.Ludwig@>
> > wrote:
> >>
> >> TJ once explained that in a posting in this list:
> >>
> >> > Hello,
> >>
> >> > By the way it is NOT surprising that you are getting lower
values
> >> >than before.
> >>
> >> > In his book Mr. Lars Kestner writes:
> >>
> >> > ' The K-ratio is a unitless measure of performance that can be
> >> >compared across markets and time periods. [ - - - ] Traders
should
> >> > search for strategies yielding K-ratios greater than +0.50.
> >> >Together, the Sharpe ratio and K-ratio are the most important
> >> > measures when evaluating trading strategy performance. Note:
When
> >> >I created the K-ratio in 1996, I thought I had created a
> >> > robust measure to evaluate performance. In mid-2000, trader Bob
> >> >Fuchs brought a small error to my attention regarding the
> >> > scaling of the K-ratio. He was correct in his critique and I
have
> >> >corrected the error in this text. Publications prior to 2002
will
> >> > show a different formula for the K-ratio. The updated formula
in
> >> >this book is correct.'
> >>
> >> > Previous AB versions contained old K-ratio formulation [of
1996]
> >> >and newest one contains
> >> > new formulation [from Kestners book of 2003].
> >>
> >> > The difference between those two formulations [i.e. 'trader Bob
> >> >Fuchs brought a small error to my attention regarding the
> >> > scaling of the K-ratio. ' ]
> >> > is just the factor denominator that is now
[NumberOfObservations]
> >> >instead of SQRT[ NumberOfObservation]
> >>
> >> > Since [NumberOfObservations]/SQRT[NumberOfObservations] = SQRT
> >> >[NumberOfObservations]
> >> > it makes it obvious that new K-ratio figures will be SQRT
> >> >[NumberOfObservations] times smaller than previous.
> >>
> >> > The relationship between new and old version can be written as:
> >> >
> >> > KRatio[ NEW2003 ] = KRatio[ OLD1996 ]/SQRT
[NumberOfObservations]
> >>
> >> > You can correspond with Mr. Kestner why 'new' is better
than 'old'
> >> >but do not discuss this with me, because I did not invent it.
> >>
> >> I guess that answers your question.
> >>
> >> Greetings, Thomas
> >>
> >> > I made two posts on this subject a couple of weeks ago, but it
> > seems
> >> > those posts have disappeared.
> >> >
> >> > Anyway. I am now able to duplicate the AmiBroker 4.96 beta CBT
> > results
> >> > for K-Ratio. My implmentation follows and appears to match the
> >> > AmiBroker results for backtest periods lasting from 2 months to
> > 17+
> >> > years.
> >> >
> >> > Eq = Foreign("~~~EQUITY", "C"); // Assign Close of Backtest
> > Equity = Eq
> >> > n = Barcount -1; // Number of periods
> >> >
> >> > // K-Ratio - only valid for non-compounding systems
> >> > EqLRS = LinRegSlope(Eq, n+1); // Linear Reg Slope of entire
(n+1)
> > range
> >> > EqSE = StdErr(Eq, n+1);// Std Err of entire (n+1) range
> >> > EqKR = EqLRS/EqSE; // K-Ratio (unitless measure)
> >> > EqKRAmi = EqKR*sqrt(n+1)/sqrt(12);// AmiBroker 4.96
> > implementation
> >> >
> >> > My question is: Why does it require a divide by sqrt(12) to
work?
> >> >
> >> >
> >> >
> >> >
> >> > Please note that this group is for discussion between users
only.
> >> >
> >> > To get support from AmiBroker please send an e-mail directly to
> >> > SUPPORT {at} amibroker.com
> >> >
> >> > For NEW RELEASE ANNOUNCEMENTS and other news always check
DEVLOG:
> >> > http://www.amibroker.com/devlog/
> >> >
> >> > For other support material please check also:
> >> > http://www.amibroker.com/support.html
> >> >
> >> > Yahoo! Groups Links
> >> >
> >> >
> >> >
> >>
> >
> >
> >
> >
> > Please note that this group is for discussion between users only.
> >
> > To get support from AmiBroker please send an e-mail directly to
> > SUPPORT {at} amibroker.com
> >
> > For NEW RELEASE ANNOUNCEMENTS and other news always check DEVLOG:
> > http://www.amibroker.com/devlog/
> >
> > For other support material please check also:
> > http://www.amibroker.com/support.html
> >
> > Yahoo! Groups Links
> >
> >
> >
> >
> >
>
Please note that this group is for discussion between users only.
To get support from AmiBroker please send an e-mail directly to
SUPPORT {at} amibroker.com
For NEW RELEASE ANNOUNCEMENTS and other news always check DEVLOG:
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For other support material please check also:
http://www.amibroker.com/support.html
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