# How to combine various equity measures into a single measure (vector magnitude)

I have several measures:

1. Profit and loss (PNL).
2. Win to loss ratio (W2L).
3. Avg gain to drawdown ratio (AG2AD).
4. Max gain to maximum drawdown ratio (MG2MD).
5. Number of consecutive gains to consecutive losses ratio (NCG2NCL).


If there were only 3 measures (A, B, C), then I could represent the "total" measure as a magnitude of a 3D vector:

R = SQRT(A^2 + B^2 + C^2)

If I want to combine those 5 measures into a single value, would it make sense to represent them as the magnitude of a 5D vector? Is there a better way to combine them? Is there a way to put more "weight" on certain measures, such as the PNL?

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What is the purpose of combining them? –  shabbychef Feb 6 '11 at 20:48
@shabbychef A model with multiple metrics has to juggle the various alphas. The OP could use a method that selective applies alphas based on pre-conditions, though it's much easier to combine them somehow. –  chrisaycock Feb 6 '11 at 21:10
I tend to agree with shabbychef. But if you have to have your own factor, you should look into principal component analysis (PCA). This will allow you to find the best linear combination of these factors. –  Richard Herron Feb 6 '11 at 21:10
@shabbychef, I'm trying to figure out a way to "collapse" the vector so I don't have to compare each metric separately. I have thousands of instances to compare and I want to do it as fast as possible. Comparing 5 dimensions is going to be much slower than just comparing a single value. –  Lirik Feb 6 '11 at 21:36

One approach would be to rescale these metrics so that they are approximately normally distributed with unit variance under the null hypothesis that the stock's price is an unbiased geometric random walk (equivalently that the log returns are zero mean). This rescaling is effectively going to 'downweight' the statistics with a large amount of variance. Once they have been rescaled to approximate normality, one *could *combine them as you have done, in which case the sum of their squares would be a Chi square with 5 degrees of freedom under the null. It would probably be more appropriate, however, to simply take their mean, because sign should not be discarded, I think.

The first two metrics should be easy to rescale. The metrics involving maximum drawdown, however, are a bit tricky. You probably want to estimate the variance of these statistics under the null via a Monte Carlo simulation.

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A multi-alpha trading model ranks each asset according to the individual signals. For example, if I have two metrics and three stocks, I could just create this reverse-sorted table:

Rank| PNL  W2L
----| ---------
3   | AAPL AAPL
2   | MSFT YHOO
1   | YHOO MSFT


Because this ranking/sorting method is non-parametric, I can just average each metric's rank by stock:

stock| score
-----| -----
AAPL | 3.0
MSFT | 1.5
YHOO | 1.5


And now it's easy to make a weighted average of the ranks; if I want PNL to be 2/3 of the value and W2L to be 1/3, I have:

stock| score
-----| -----
AAPL | 3.000
MSFT | 1.667
YHOO | 1.333

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