array9

You have some money, some time periods when you must invest your money in shares of one company, and you know future price movements are always unknown and totally unpredictable.

How should you invest?

Two simple cases:

A. If you feel the price is on average likely to increase from its present value over the totality of the periods:

Spend all the money at earliest opportunity.  

B. If you feel the price is on average likely to decrease from its present value over the totality of the periods:

Spend all the money at latest opportunity.

Proof:  let us say you did not spend all your money in the first period, then you are basically back to the same situation for the remaining money, except you have a starting point now spread out across the first step price movement.  This has to be slightly biased up in case A and down in case B.  This will inevitably reduce the expected number of shares you buy.  QED

You could just say

'Why wait, the starting point is more likely to get slightly worse' for A, and 'why rush, things are more likely to bet slightly better' for B.

This leaves the more tricky question about if you think the price is as likely to up as go down in the long term. ( I am assuming no change in quality of information about the future).  The answer is that it really does not make any difference what you do.  The two extremes get the same average answer and any solution can be seen just as a combination these extremes, even split into individual units of time.

The difference that splitting purchase makes is to reduce the spread of the possible outcomes, not the average of the outcomes.  You may feel safer being averaged across the period.

We considered the purchasing situation here, but the selling case is pretty much identical in structure.

Another way of putting this argument is that if you don't have any information, particularly about any asymetries, then the average for any strategy is exactly the same.  And this applies to buyer and seller.

Or life is easier if you dont have information.

Of course in reality information is not evenly spread across the participants in timing, quantity or quality, and so the game is heavily biased in reality.  Plenty of people will be more worried about the variance, rather than the average.

The argument in favour of splitting purchases across time in money terms is spurious.  There is a reduction in variance, but not in average.  There may be funny cases where the share price falls close to zero.  But the company will be bust.

I guess the 'you are buying more when the price is low' argument is nice but flawed by the obvious comment the you don't want to be buying any bankrupt stock.  

Also it is not obvious what other spreading you can do.  You cannot buy a fixed number of stock each period, as you dont know if you will actually spend all the money or overspend.  Without splitting the money, rather than the shares you basically cannot do it.  So you really have very few uniform strategies, other than equal money, and anyway on average they are all equal in terms of expected average outcome.

The argument for equal spreading only sounds good against the non-existant equal money strategy.

There may also be a temptation to think that the equal money strategy is somehow learning as it goes long.  Unfortunately this is an illusion as there is by definition nothing to learn.

Overall i find it strange that people argue on basis of making more money, rather than avoiding extreme outcomes or variability.  

If you saw it as a game against someoneelse, then exposing your equal money strategy sounds bad as well, as someone may try and take advantage somehow.  A random or single randomed timed purchase strategy might be preferable.

I continue to be surprised that people are so keen on it.

Martin

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