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Estimating Outperformance

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By TMFShadowDragon
September 2, 2009

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Hi Friendly Foolish Folks,

A little while ago Rich started an "experiment" on mechanical investing based on Ben Graham's "magic formula". The original post and discussion can be found on this thread.

Rich asked if I knew of any way to assess performance that took "risk" into account. At first I cringed since the easiest, most reliable method I knew was "regression analysis" of the performance of the portfolio versus a proxy for the market such as a market index. I cringed, because as soon as you suggest something like this on the boards my experience has been that a bunch of people come out of the word work and to inform me "volatility", "beta", "alpha", etc. are not measures of "risk" and "outperformance". Rest assured I do know the limitations of the method and the arguments on all sides. We have to remember that we're not trying to build a rocket ship to go to Mars or even to predict the future performance of a portfolio; we're just trying to understand did the portfolio do well or poorly because of luck or because of the method. I will try to walk through the basis of the "linear regression" approach including its limitations. It's not the whole answer, but it is a useful partial answer.

First off to assess if Rich's mechanical method or any method, you need to a reference point - a benchmark, such as an index. However, keep in mind that you should use the right index that reflects the markets in which the companies in the portfolio operate. I have a portfolio of Canadian stocks. Against the S&P 500 index, they've done marvelously, but against the Toronto Stock Exchange (TSE) index they've only managed equal the index (except for this year when my portfolio has blown the TSE away by 10%...YES!!!!!!). So with respect to the S&P 500 it's not a case of how of picked my Canadian stocks, but merely that they were Canadian stocks - period! Probably one should create a "combined index" that's reflective of all the markets in which the companies that are part of the portfolio operate, but if you're like me your lazy and that's too much work and you'll want to use just one index and you'll probably use the S&P 500 because everyone does, right? Can we all say, "Baaaaa....Baaaaa" together?

Satisfied with our one index we now want to compare our portfolio's performance to the index. Well that's simple just subtract the percentage return of the index from the portfolio's percentage return....and there we have out- or under-performance. Personally, I've found that to be a very misleading number. It really only uses two points - the value when the portfolio was initiated and the value of the moment of the assessment (i.e., the last market valuation). It ignores how the portfolio performs on average; and "on-average" may be more important unless you sell the entire portfolio at one go and put the money into a "safe" investment (like a mattress). However, in most cases you'll be buying and selling as you go along as well as adding or taking out cash. These will cause confusion with calculating "percentage return".

Suppose you sell a position for $10,000 that cost you $8,000 and take the cash out of the portfolio. You'd see that as part of the return, right? But for how long will you keep that $10,000 in the numerator (return) and the cost ($8,000) in the denominator (amount invested)...for one year, ten years? I find that "total return" is useful for looking at an individual stock position or a portfolio that's not traded, but if the portfolio is actively traded the more distorted that number can become with time; and it can be easily manipulated to by those who might have slightly suspect motives. I'm not suggesting that one doesn't calculate "total percentage return". Just be aware of its limitations.

Then there are other questions that the "total percentage return" does not answer, which are, "was I just lucky?" or "how do I take ‘risk' into account?" For this we need a little more, which is where the regression analysis comes in. However, this requires some kind of view of what were the expected returns of the portfolio - and by "expected" I mean by the market as a whole. You as the owner of the portfolio would, of course, "expect" it to outperform. That's going to require a "model" of how the market "expects" returns; and, unfortunately there is no "proven" model for that. So, the best I can suggest is a simple one. So, here it is for a single company....

(stock return less the fixed return rate) = k * (market return less the fixed return rate) + company specific factors + noise factor

Notice that the returns are in relation to the "fixed return rate" (which I'll denote as Rf from here on). That's because out zero point isn't 0% but Rf since that's what one could have earned in a safe fixed return investment with very little risk.

The equation says that the return on the company's stock is proportional the market's overall return (which I denote by Rm from here on) by a factor of k (I'm not using b so that people won't say, "arrrrgh, beta") plus factors specific to the company and the industry in which it operates plus some other factors I call "noise".

Notice that "k" and Rm do not alone provide enough information to predict the return of a stock. There's also the "company specific factors" that for an individual company will likely dominate the expected return. However, it's still reasonable that to some extent stock returns are proportional to the overall movements of the market. Most stocks fall during a bear market and most rise in a bull market.

"Noise" takes into account the "random" movements of stocks when there's no real news or rationale on which to base values. If you have a large enough portfolio the noise components will cancel out - if the "noise" is not correlated across your portfolio (i.e., overall market irrationality is a "correlated" factor and is therefore not taken into account by this factor).

It's the second component, "company specific factors", about which people argue. The "efficient market" people say that market has taken all "company specific factors" into account so that if you try to predict stock performance on the basis of "company specific factors" you'll be wrong as often as your right. So, for a large enough portfolio these should cancel out....i.e., for a portfolio they view "company specific factors" the same as "noise". Stock pickers, which will be the vast majority of TMF's client base, differ on that regard. They do believe that they can select stocks well enough so that these factors do not cancel out in a portfolio. However, if you glance through TMF's various newsletters, you'll see that those "efficient market" folks do have a point. There are a lot of stocks within each that underperform in addition to those that don't. Take the Inside Value newsletter, which in total is outperforming the S&P by 5.2% as of writing this. In that it has 7 stocks outperforming the S&P by 50% or more while 6 stocks are underperforming by 50% or less. So, even if, as stock pickers, we don't think the market is all that efficient we have to admit that it's mostly efficient or at least that it's not that easy to pick a single stock with a lot of confidence on in terms of how unpredictable "company specific factors" will affect its return. Still, the Inside Value service's position is that despite "mostly efficient" markets, it can pick stocks well enough that as a "portfolio" it will outperform.

As stock pickers, what this means is that, within a reasonably well-diversified portfolio, we can pick a set of stocks so that some "company specific factors" will add together coherently to give us an outperformance (it could also be underperformance - it you can profit from an ability to consistently pick an underperforming portfolio too) which I will call B for the "bias" (i.e., that our stock picking is biased). The remaining "company specific factors will tend to sum to a net zero effect for our "well-diversified" portfolio. I'll lump these factors in with the "noise" factors and term them "N".

So we can write our portfolio's expected return (Rp) as:

(Rp - Rf) = k * (Rm - Rf) + B + N

Stock picking states that B will be a non-zero value while "efficient market" folks and indexers will believe that it will tend to be a zero value.

So, the efficient market version is:

(Rp - Rf) = k * (Rm - Rf) + N

...and, since N also tends to be zero for well-diversified portfolios, the efficient marketers can claim:

(Rp - Rf) = k * (Rm - Rf).

Now you know why "efficient marketers" emphasize "k" (which most of you will recognize as "beta" in financial circles). It's all they have left to play with when constructing portfolios. It's also why they equate estimates of "k" with "risk". The conventional view is that portfolios with "k" < 1 are less risky than the market and those with "k" > 1 are "riskier" than the market. The reasoning being that portfolios with k < 1 will increase or decline less than market increases or declines and those will k > 1 do so more than the market.

However, as "stock pickers" we actually more interested in B (the bias). What we want to know is if our method picking stocks is reliable enough that we can count on B being in excessive of some value that makes our stock picking worthwhile. If we can do that we can reduce our "risk" even though our "k" value may be > 1.

Of course, N only tends to zero if you invest in all stocks. For real portfolios over any given real period of time it will also be non-zero. So we have to collect data from enough periods of time; hoping that this will improve our ability to estimate values for both "k" and "B", which are called "beta" and "alpha" in financial circles.

Note that our estimates are "looking backward" at the performance of our portfolio and the overall market to determine if our ability to pick outperforming stocks was more than luck or that we had picked "risky" stocks in the convention financial use of the term "risk".

This brings us to some important assumptions:

1) The value of "k" for our portfolio is constant. In reality it may very well change in time. This could cause problems in estimating "k" if it does vary over the duration of our "experiment". Unreliable estimates of "k" will affect our ability to estimate "B".
2) The noise (N) won't be zero for any given period of time. Of course, the longer the period the more it will tend to zero, which is why day to day price movements are relatively greater than yearly price movements. The greater the value of N for individual periods the less confidence we can have in both our estimates of k and B.
3) Not all risk can be represented by k or by the uncorrelated, random values of N. There may be factors (unknown unknowns, if you will) that can introduce "correlated noise" to our model of expected returns. These are sometimes referred to as Black Swan events, "fat tail" events, etc. It is possible that they might so dominate our portfolio that estimates of k and B are pretty much a fruitless activity. That's a very interesting topic for discussion but beyond the scope of what I wanted to do here for Rick and Kitkat.

I'll conclude this post on the background to the regression analysis by stating the objective of the analysis for Rich's experiment. The purpose is to compare the "stock picking" view that:

(Rp - Rf) = k * (Rm - Rf) + B + N, where B > 0 for Rich's Ben Graham formula,

In contrast to the efficient market view that:

(Rp - Rf) = k * (Rm - Rf) + N

It is sufficient here to demonstrate that B > 0 with enough confidence to refute the latter version of the equation since that is the conventional financial view.

I'll go through the mechanics of the regression analysis in the next post.


The Dragon
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