Risk and return

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Ever since Pascal first laid its bases in the 17th century, probability theory has been a matter of avid hate. The question is always the same from Versailles’ gambling rooms to Las Vegas 21: Is the system beatable ?. Unfortunately (and it is a result established by Pascal himself), the answer is no. Random gives the best expected average return on any Gaussian system.

But the theory is more mischievous. In fact, it is possible to follow a strategy which yields a high chance of making more than the average. But this is compensated by a small chance of an abyssal loss. The classical example is gambling even or odd at the Roulette and doubling when you loose. Chances are that you will make a small profit, but if you are very unlucky, you will either hit the gambling limit or loose all your equity.

Since financial markets are the result of the actions of millions of loosely connected actors, they can be seen as Gaussian. And indeed, apart from the base trend (bullish or bearish), they pretty much are. They are thus a wonderful playfield for those who want to use such strategies to beat the monkey. In modern finance, this above average return has a name: it is called the Alpha. But, as the theory says, the risk can be horrendous, even though catastrophic failures are statistically rare (but do happen as shown by LTCM, and others)

Let us crunch a few numbers to show the point. Our hypothesis are the following. The equity market return is Gaussian with a mean of 8.76% and a standard deviation of 18.95% (these values are coming from history between 1871 and 1973). We choose a cost of debt of 5% we and choose various levels of leverage. We do not make any smart investment decision and just follow the market. Our return on equity follows the table below.

One important thing to note is the the numbers grow fast. As a result of that, it is not impressive for a hedge fund leveraged by a factor of 2 to make 15% return. In fact, the fund is underperforming the average.

Where is the catch ? It is in the risk of course. The table below shows the risk that a catastrophe happens (namely the fund looses all itsequity). The risk in minimal when leverage is low but at a factor of 2, it is higher than 1%. Said otherwise, if 80 funds are playing this strategy simultaneously, one of them will probably fail this year.

Does this strategy create value?  Unfortunately not. For each level of leverage, we can use the Modigliani Miller theorem and see what level of return, investors should demand for the level of risk (the computation could be slightly altered by tax but most of these funds do not pay capital gain tax). The results are in the third column. As predicted, this normal level of return is equal to the one achieved by the monkey (i.e. the average).

The example above is crude yet meaningful. Modern finance makes an extensive use of derivatives, short selling and so on wich very often create some level of obligation to the holder and/or increase leverage and/ or risk. The same theroy applies but the estimation of the real risk taken by these investment vehicle is much more difficult. The chase for the monkey will go on.