K-Ratio(s)   ... a continuation of Part I

So far we have these magic formulas:
[A]
VAMI = a chart of $1K invested in some stock versus the number of months invested
with points (xk, yk) where xk = k, the number of the month

logVAMI = a chart of log[VAMI]

logVAMI regression line:   y = α + β x.

the pointwise errors are: ek = yk - ( α + β xk)

β = COVAR[x,y] / SD2[x] = Slope of logVAMI regression line

Error2 = Σek2 / n = SD2[y] (1 - r2)
measuring the error between the yk and the regression line

(Standard Regression Error)2 = Σek2 / (n-2)

Standard Error of the Slope = Error / SD[x] sqrt(n-2)

and a few k-Ratios:
[B]
    K-Ratio (Kestner) = (Slope of logVAMI regression line) / n (Standard Error of the Slope)
    where there are n return periods in the monthly return data.
and
    k-ratio#1 = (Slope of logVAMI regression line) / (Standard Regression Error)
and
    k-ratio#2 = (Slope of logVAMI regression line) / (Standard Error of the Slope)
and
    k-ratio#3 = (Slope of logVAMI regression line) / (Standard Deviation of the Errors)
For 10-year's worth of monthly XOM data, we'd get this:
>Hey! k-ratio#3 is a new one!
Yeah, I thought I'd toss that it for good measure. Speaking of measure:
The denominator (for #3) measures the variability of the monthly returns, namely the volatility (or standard deviation) of the errors.

>By "errors", you mean ...
I mean the ek = yk - ( α + β xk).

>Is that good?
I have no idea. However, once we get a few candidates we'll backtest 'em all ... if'n we can figure out how to do that.

>I think you're still confused.
Could be, but my problem now is to see which (if any) k-ratio is "best".
Of course, I gotta figure out what's it's used for, so I look here and find ... among other comments:

I've developed a new measure, the K-ratio, that gauges performance by examining the consistency of returns with respect to time in Kestner's words.
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.
Detects inconsistency in returns. Should be 1.0 or more.
The higher the K ratio, the more consistent return you may expect from the system.
>So you'll test the various ratios, right?
Yes. I was thinking that we should look at some 5-year period (as Kestner suggests), evaluate the ratios, then look at the subsequent 5-year ratio.
>But you just did 10-year monthly data.
Yeah, well now I'm gonna do 5-year.
First we'll do the most recent 5 years, calculating a bunch of k-ratios (as defined above) for a gaggle of stocks.
Then we'll do the previous 5 years and see how each ratio has changed.


>How you gonna do that?
I was thinking of calculating the percentage change, from one 5-year period to the next.
This is what I get for a bunch of stocks, most of which are DOW stocks:


>Uh ... is that consistency?
From one 5-year period to the next? I wouldn't say so.

>By the way, which k-ratio did you use to calculate the percentage change?
Surprisingly, the % changes were the same for all ratios.
That is, k-ratio[Dec/03-Dec/08] / k-ratio[Dec/98-Dec/03] was the same for k-ratio#1, #2, #3 or Kestners' K-Ratio.


>Did you expect that?
Well ... to tell the truth ...

>You had no idea that would happen!
Alas, that's true.
Here's another pretty chart:

Let's look at HD over each of the periods Dec98-Dec/03 to Dec/03-Dec/08:   (That's the one with the smallest, 4% change.)

>Hey! Maybe "consistent" means consistent over a single 5-year period.
Yes, but then that's looking back and may provide little indication of what's ahead.

>But what about HD? There was little change from one 5-year period to the next.
But (to use your words): You had no idea that would happen!

>Alas, that's true. But isn't the K-Ratio supposed to be an alternative to the Sharpe Ratio?
Yes, so here are the comparisons: % changes in K-Ratios and Sharpe Ratios:

>So, what's the conclusion?
How many guesses do I get?
Remember when I said we'd talk about that Standard Error of the Slope? Well, now's the time:

the t-statistic

Here's the idea:

  • We take a bunch of observations, like y1, y2 ... yn.
  • We assume these sample values are taken from some HUGE universe of y-values.
  • We calculate something (we'll call it K), based solely upon the sample.
  • How much confidence should we place on this K-value as a measure of the K-value for the entire population?
>I give up.
Pay attention.

Example 1:

  1. We observe n values of a stock return: yk ... like the last 100 monthly returns.
  2. We calculate the Mean value: M[y] = (1/n) Σyk.
  3. We wonder how well M[y] measures the actual mean of the population: μ.
  4. To do this, we calculate the t-statistic:
    [C]
    t = (M[y] - μ) sqrt(n) / sd[y]
      where sd[y] is the "sample" standard deviation of the observed y-values, namely:
      sd2[y] = Σ(yk - M[y])2 / (n-1)
>You divide by (n-1) instead of n?
Yes, we mentioned that already. Statisticians do this when they regard the observed y-values as a "sample" taken from some larger population of y-values.
Financial types (often) divide by n. Of course, it makes little difference when n is large, so ...

>And all this stuff is approximations, estimations, guesses and wishful thinking, right?
Right. Anyway, we're really interested in this:

Example 2:

  1. We observe n values of monthly stock returns over a 5-year period..
  2. We calculate the logVAMI regression line:   y = α + β x.
  3. We're interested in the slope associated with the entire population of monthly returns (not just our sample).
  4. We wonder how well β measures the actual slope for the entire population: β0.
  5. To do this, we calculate the t-statistic:
    [D]
    t = (β - β0) sqrt(n-2) / (Standard Error of the Slope)
      where Standard Error of the Slope = Error / SD[x] sqrt(n-2)
      ... as we noted above.
>Huh? That looks like Kestner's K-Ratio ... with β0 = 0.
It do, don't it?
Indeed, we're considering the value of (β - β0) and would like to know if we should expect it to be close to 0.
That is, we're asking how confident we should be that: (β - β0) = 0.

Let's consider two sets of data:
  • a bunch of y-values: y1, y2 ... yn and a bunch of z-values: z1, z2 ... zm
  • Imagine the distribution of the two sets of values ... and their Mean values.
  • We ask: "Are these two related?"
    That is: "Can we use M[y] to provide an estimate for M[z]?
How about this if the distributions looked like this?

Figure 2

Figure 1

>I'd pick Figure 1. In Figure 2 they don't look similar at all.
You like Fig. 1 because the distributions overlap quite a bit in Fig. 1, eh?
So our confidence in using the M[y] to estimate M[z] will depend upon how far apart they are ... and the "spreads" in their distributions.

>Big spread means big overlap, right?
Yes ... and we measure "spread" via the Standard Deviation.
So let's try to generate some Measure of Confidence which incorporates these notions:

  1. Our "confidence" will be greater if (M[y] - M[z]) is small.
  2. It will also be greater if the Standard Deviations are large.

>I'd take the first guy and divide by the second guy.
Uh ... that'd measure lack of confidence. If #1 is small and #2 is large, it means (1)/(2) is small.

In practice, one calculates something called the t-statistic and checks to see if it's small.
[E]
t = ( M[y] - M[z] ) / sqrt[ sd2[y] / n + sd2[z] / m ]

Notice that, if m were some infinite universe gargantuan (like m = ∞) then the second term in the denominator would vanish.
If, then, we let M[z] = μ, we'd be left with:
[F]
t = ( M[y] - μ ) sqrt(n) / sd[y]
>Huh? That looks like [C].
It do, don't it?