## On Optimal Portfolio III

November 11, 2011

## Growth Optimal Portfolio III

In my previous notes, I tried to make the case that by adopting administrative procedures, a portfolio manager could achieve an exponential Jensen ratio. All he/she had to do was to re-invest the portfolio's profits as they came in. Not that difficult a task! About the same as re-investing the dividends as they came in. Nothing fancy, I would even say boring, but a task that needs to be done nonetheless.

How come simply re-investing the accumulating profits can increase portfolio performance to such an extent that it would propel the Jensen ratio to an exponential function?  That is the question. Why all the academic financial literature I have read (some 450 thesis), not one, and I mean not one, has ever suggested re-investing the accumulating profits and thereby out-perform the Buy & Hold? Not only out-perform, but achieve an exponential Jensen ratio. Was the concept to hard to grasp? Wasn't it before their eyes all the time?

You could find incremental trading methods, technically betting systems using optimal-f or the Kelly criterion. Using as bet size fixed ratio, percent ratios or equity dependent ratios. There was like a dichotomy between the world of the trader and that of the holder. It was understandable that the trader could vary his trade size at will for what ever reason while the buy and holder had just one mission and that was to hold: no change.

When you try to trade and hold, it gives a new perspective. You start with a Buffett like stock selection processed which is aimed at long term investing and then add a trading component to take advantage of market swings. You use the profits generated by the market swings to accumulate shares for the long term. The effect is an exponential Jensen ratio, simply as a result of administrative investment policies.

Changing the re-investment policy rate will have some effect on the pay-off matrix:

Σ(HP(1+g+)(t-1) .* ΔS+P)    pay-off matrix

In the Buffett example, raising the re-investment rate to 0.85 would increase his CAGR performance results from 20.4% to 20.8%. It might not look like much but on a 200B portfolio, it represents an additional 400M profit per year. The added profits are the result of an investment policy spanning years, not a trading system. It only needed to make the usual selection efforts. The timing of the added trades might be off mark, but in the long run, it would not matter much; the missed opportunity would account for a lot more.

I know, a re-investment policy is no fun; no trading excitement. You install investment procedures where you gradually accumulate more shares. You can try to time some of your entries, it will not matter much long term. You can buy new shares or increase your existing holdings; again it won't matter much in the end. Your ability to forecast 20 years in the future is irrelevant. No one can forecast that far. All you know is that based on all the current information that you could find, the selected stocks seems as a reasonable choice to implement a share accumulation program. And should your analysis in the future prove otherwise then you could always liquidate a position and start a new one in something else.

At the portfolio level, your stock selection should most likely perform a little better than average. Having diversified your portfolio, you know that any one stock can not do that much damage as it represents only a small fraction of your total holdings. However, by accumulating more shares as prices go up, you will be compounding on the generated profits as well as on the rising prices.

Increasing the bet size can also help improve overall performance. In the Buffett example, increasing the bet size to 0.50 from 0.30 will have for effect to increase CAGR from 20.4% to 21.1%; an added 700M profit to the bottom line. Having both an higher re-investment rate and a higher bet size rate would push performance on a CAGR basis from 20.4% to 21.5%: an increase in profits of 1.1B on just two simple administrative procedures. Each implemented gradually in time with no stress on the organization and/or its personnel. You could even automate both procedures and let a computer do all the work.

Mr. Buffett has an average CAGR of about 22%. Maybe he has already applied, and for some time, this kind of re-investment procedures to increase his portfolio performance. Its like saying that the administration of building a portfolio is an ongoing process where you gradually adapt to your growing size. It was your objective in the first place, it enabled you to plan a long time in advance where you wanted to go.

You are increasing portfolio performance not by forecasting prices, not by trying to time the market and not by any sophisticated trading philosophy trying to extract from pass data some anomalies of some sort that might or might not repeat in the future. You are using administrative procedures to execute your trading policies and achieve your goals.

I can understand why market players do not like such methods, they are boring. This is not how they want to do with their money. They need excitement, up and down moves. They need to predict to show how good they can forecast their past with some hindsight.

## The Intricacies of an Accumulative Buy & Hold

You look at the governing equation:

Σ(1+B(t-1))(HP(1+g+)(t-1) .* ΔS+P)   pay-off matrix

where both the higher accumulative rate and the increasing bet size are applied. Their only objective is to increase the inventory of shares as a time function of the generated profits on your better than average stock selection.

There is not much you can do with the Schachermayer pay-off matrix, it has only two variables: quantity and price. You won't be able to change the price differentials so in reality, you are left with only one modifiable variable.

It is by designing a better holding function, an enhanced holding function that you can improve your portfolio performance to exceed the Buy & Hold strategy, as in:

Σ(H+ .* ΔP)  >  Σ(H .* ΔP)

Therefore the real question is: is there a H+ > H? Is there a trading strategy that can do better than the Buy & Hold? That is what needs to be answered. And the answer is: definitely yes.