October 22, 2011

## Growth Optimal Portfolio

Academic financial literature is fond of the notion of No Free Lunch (NFL); which is the same as saying you can not do better than us. If there was a free lunch, we would have eaten it already; there might be some crumbs left but then again...

Long term return expectancies, from what ever trading method you intend to use is simply what the market has to offer which is the average market return. And if you ever do better than average, it most certainly will be due to sheer luck. Skill has very little to do with it, we have found very little evidence of the existence of alpha. And even if we temporarily found some, long term, it would tend to zero. You would be like the exception that confirms the rule (someone has to be at the distribution's extreme, but we don't know who); and we have to confirm that the odds are stacked, really stacked against your over-performance. If we can't do it, be assured you can't either. We have tried hard enough.

Well, not necessarily.

After reading Schachermayer's 2000 lecture notes, a few years back, I had to convert my mathematical notation to more adhere with his concise formulation. In a single expression, Schachermayer had determined the outcome of portfolio management: Σ(H.*ΔS), and voilà: the cumulative sum of all generated profits by what ever trading strategy used over the whole investment period. A single matrix multiplication representing the pay-off of any holding strategy subject to price variations. You could analyze any portfolio size with only two variables: H (the inventory), and ΔS (the securities price variations).

Σ(H.*ΔS)          Schachermayer's pay-off matrix

With a single expression you could view the trading process as a monolithic block. In some software programs, the expression can be written as easily as: sum(H.*S) where S is the matrix of closing price differences. Where before, I would have looked at price movements as independent functions, I was now forced to view the trading process as a matrix of functions. It was not trying to improve on a strategy, but on a method that would span the whole time matrix. What ever trading procedure envisioned, it had to apply to all, it had to fit the matrix. The holding matrix H became a pulsating, vibrant entity instead of remaining a flat surface.

Representing the generated profits for the Buy & Hold investment strategy was very simple:

Σ(H.*ΔS)         Schachermayer's pay-off matrix

Done, no change. You fill H with the initial stock purchases, and that is it. Looking more closely in order to view the initial positions we could use:

H = hojI.

where hoj is the initial quantity purchased in each stock j  and I is a matrix (the same size as H) with all its elements equal to 1. The .* is for element-wise matrix multiplication.

Σ(hojI .*ΔS)       Buy & Hold pay-off matrix

The above equation is just a slightly different representation of the Schachermayer pay-off matrix, but it says exactly the same thing, and produce the same end results.

## The Market Portfolio Matrix

Profits generated by holding all stocks in the marketable universe could be expressed as:

Σ(HM.*ΔSM)         Market pay-off matrix

where  HM  represents holding all shares available of all available stocks. In the same way, you could represent profits generated by holding all stocks of an index:

Σ(HD.*ΔSD)          Dow Jones Industrial Index pay-off matrix

Your portfolio, based on your stock selection would similarity be expressed as:

Σ(HP.*ΔSP)          Portfolio pay-off matrix

The closer your portfolio resembled the market, the more you should expect achieving the same end results, percentage wise.

Σ(HP.*ΔSP)           Σ(HD.*ΔSD)            Σ(HM.*ΔSM)           Profits
───────   →    ───────   →    ───────    =   ───────
HPo*SPo                HDo*SDo                HMo*SMo           Investment

And on that statement alone, the introductory remarks are perfectly valid. It is sufficient to be diversified to have the portfolio's pay-off matrix tend to produce the same performance return as the Dow Jones or the market in general. You can claim alpha points only if you out-perform:

Σ(HP.*ΔSP)           Σ(HD.*ΔSD)           Σ(HM.*ΔSM)
───────
>    ───────   →    ───────
HPo*SPo               HDo*SDo               HMo*SMo

Your portfolio's pay-off matrix must out-perform the indexes or the market averages to show its added value. And this task is formidable; most do not succeed. And therefore, it is understandable that somehow we develop the No Free Lunch theory because it appears as if there is, in fact, no free lunch available.

It appears excessively difficult to escape average portfolio performance. But what if, using trading procedures, we could jump over the efficient market frontier, over the capital market line and easily generate alpha points. Would the no free lunch still hold?

## Mr. Buffett: Portfolio Super Star

Mr. Buffett has outperformed the markets over his illustrious career: some 12 alpha points over the secular trend. Mr. Buffett has shown superior portfolio management skills in the application of his investment strategies. He has also been helped by his fascination for compounded returns. And it is not by luck or fate that he finds himself at the extreme of the performance distribution.

Part of Mr. Buffett's performance can be explained by the reinvestment of his accumulating profits in buying new businesses or increasing the number of shares of companies he already owned. It did not matter that he was buying ice cream or under ware, his main concern was in the cash generation. He was not just “doing” Buy & Hold. He was not content with just reinvesting dividends; he was reinvesting profits as well. This way, he was compounding the interest on his interests instead of leaving unused equity sitting idle in his portfolio doing nothing. The following represent his investment approach:

Σ(HP(1+g)(t-1).*ΔSP)             Σ(HD.*ΔSD)           Σ(HM.*ΔSM)
────────────    >     ───────   →   ─────────
HPo*SPo                         HDo*SDo                HMo*SMo

With the generated profits, he simply acquired more shares (companies) as his profits grew. With a positive reinvestment rate g, his portfolio had to outperform the averages. It was not a matter of luck. It was his understanding of what to do, how to manage his portfolio that made a difference. He was generating alpha points as a byproduct of his methodology.

Also, Mr. Buffett has over time made better stock selections than most which helped in generating another part of his alpha points. This could be expressed as follows:

Σ(HP(1+g)(t-1).*ΔS+P)             Σ(HD.*ΔSD)           Σ(HM.*ΔSM)
────────────    >     ───────   →   ─────────
HPo*SPo                        HDo*SDo               HMo*SMo

where ΔS+P  represents his better stock selection. These two points are sufficient to explain most of Mr. Buffett's over-performance. He did not out-perform by luck, he out-perform by skill, talent, vision of what he could do to better manage his portfolio. He did not win all his bets, but he kept reinvesting his profits, buying more and bigger companies, putting his generated profits to work.

Therefore, based on this argumentation, one might accept that maybe there is a free lunch after all and that alpha point generation is more than possible; it could turn out to be a simple matter of investment procedures, or should I say, re-investment procedures.

## The Jensen Alpha Revisited

Jensen in his 1968 seminal study showed that alpha points were not that big, in fact, they were negative (-1.1). His attempt at separating positive skill from performance results in a sense failed. However, his ratio remained as a measure of over-performance and a comparative performance measure still in use today.

In light of the Buffett example, the Jensen ratio needs to be modified a little.

Rp(t) – rf(t) + α(1+g)(t-1)
J(t) = ────────────────
σ(t)

where Rp(t) is the portfolio's rate of return over time, rf(t) is the risk-free rate, and σ(t) the standard deviation. The Jensen alpha is scaled by the re-investment rate as shown in the Buffett example. Therefore, Buffett is not only generating alpha by his better stock selection, he is increasing his advantage at an exponential rate by reinvesting his profits.

If that was the case, it could easily be seen on a comparative chart showing Mr. Buffett's portfolio performance compared to a major index. What we should see is an increasing separation between the two curves. They should move in parallel, in the same direction at the same time but with an increasing separation.

The increasing separation would say that alpha is not a single number measuring an added skill level. It would be a number scaled by an exponential time function. The chart below comparing Mr. Buffett's portfolio to the S&P index is illustrating that very point.

## Comparative Charts: Buffett vs. S&P

Source: Yahoo

This brings some evidence that not only there are alpha points to be gained, but that by simply re-investing generated profits is sufficient to achieve exponential alpha. The adoption of a simple re-investment procedure is giving access to an exponential Jensen ratio.

This has far reaching implications.