# Optimal Position Size with Transaction Costs given Forecast Mean and StDev

I have rather a challenging question. I'm hoping that someone can share their experience. I will build up the problem in steps.

Let's start our thinking with the idea of a buy and hold strategy of an stock whose returns are expected to be 10% (annualised) in excess of the market and a standard deviation of 20% (annualised). If we have £1000 to invest, Chan suggests an optimal leverage to maximise expected growth of mean/stdev^2 = 0.10/0.04 = 2.5 times. In other words we should borrow £1500 to invest along with our £1000 in order to get the best growth. Yes there is a chance that this may blow up so in reality we may choose a smaller multiple such as half of that. So far so good.

Ok, so now let's consider the next step. We have just hired a guru quant who believes that each day he can provide an estimate of the next day's mean and standard deviation of returns. Assuming there are no transaction costs, it actually simulates well if we use our same formula of mean/stdev^2 to resize each day.

Ok, next we discover that we need to pay a transaction cost of 2bp on any changes we make to position size. This condition actually changes things substantially. Clearly in this case we don't want to be chopping and changing our position size too much otherwise it will get expensive quickly.

We think about the problem for a while and decide that in order to beat this, we need to look further into the future, not just at the next bar's forecast. We give our guru quant a payrise and he's able to now not just give us an estimate for the mean and stdev of the next bar, but also n bars ahead, all the way out to N days. Each day he gives us N mean predictions and N stdev predictions.

The question becomes, how do we optimally size based on this information?

One simple solution is each day to examine the mean and standard deviation predictions at each of N bars in the future. For each we calculate the Sharpe Ratio, ie mean/stdev, for each incorporating transaction costs and chose the highest. We then trade this in quantity mean/stdev^2 for that duration.

Any ideas?

-
what do you mean with mean and standard deviation of return in this particular context? you said next day's expected return and standard deviation can be estimated with high certainty. So you do not deal with a mean but an expected return and expected standard deviation. Are we on the same page or do I misunderstand something? Also, is this all for one stock? If yes then your setup makes no sense, or rather the question. Simply invest if your expected Sharpe is high enough otherwise not. If you can chose from different stocks then this becomes a truly interesting question. Please clarify!!! – Matt Wolf Nov 14 '12 at 3:43