CONSEQUTIVE LOSSES

In two lucid and interesting articles in Traders of January two subjects were discussed, subjects I studied extensively. Luca Barberis tried to give an answer to the question how trading of various markets can enhance the equity curve without optimising individual trading systems. Optimising hides the danger of curve fitting. Though his results are very important a few aspects remains to be answered before applying his method: how many contracts do you trade the various markets, which is not a matter of classical position sizing matters and the question of a possible trade system correlation and what effect this has on risk.

Risk is the main subject of Philip Kahler in Traders. He treats various possible types of risk in trading, especially risk due to consecutive losses and emphasizes its importance to understand how consecutive losses can influence trading results. When studying consecutive losses in series of trades a huge field opens up: many different theoretical frameworks can be set up to master this phenomenon of trading.

There is a clear relation between the win ratio of a trade system and the length of consecutive losses. In an earlier post I stressed the importance of the win ratio in a different context. But it also plays an important role in the existence, the length, and the amount of consecutive losses.

In general can be said that the higher the win ratio the higher the chance becomes for a series of losses of a certain length. In his article Kahler gives some examples and a graphic which can be used to determine 5 or 10 consecutive losses for a given win ratio. It has to be said that these probabilities occur for a given amount of trades performed and will change negatively when the total trades increases.

Though interesting to know the chance for a series of losses it would be even more interesting to know a distribution of series of losses, e.g. the probability for all possible series of losses and moreover the maximum length of consecutive losses (maximum streak) you might expect for a given win ratio.

Below you can see a figure which gives the relation for the maximum streak and the probability of a loss (loss ratio). Total trades is 500, confidence is 0.998%.




From this figure again can be seen the importance of the loss ratio: the higher this becomes (the more losses)then the maximum strak increases exponentially. E.g when q, the probability of a loss is 35% then the maximum expected streak willl be 6 but when increasing to 70% the maximum streak increases to 17 consequtive losses in a row!

LUCK AND CHANGE ON THE MARKET

My partner and I spent the week around Christmas in Mumbai, India. Mumbai is the financial and economic centre of India. The Bombay Stock Exchange (BSE) had a splendid year behind with a year output of more than 33%.

From local newspapers a moderate hosanna mood can be felt. Also internationally India got much attention in the last year. It is particularly the growing middle class that forms the driving force behind the economy and which gets attention of foreign investors.

With an increase of 33% the BSE is one of the best performing markets in the world. Of course it must immediately be said that almost all markets performed very well. The phenomenon that markets are moving more or less in the same direction is called correlation.

Correlation between markets is high: data which I have collected on 15 minutes basis of the futures on the FDAX and the DOW JONES appear to have a correlation of more than 90% over the past 2 years. This is almost complete correlation.

Now the question arises if a result such as that of the Bombay Stock Exchange can be attributed to well performing of the Indian economy or that also chance may play a role in it. It cannot be denied that chance and luck play an important role on the markets.

Let’s have a look at this without going into this matter to deeply. Suppose that an invisible hand would assign the different markets worldwide a return around a certain average. Then you may expect these returns to be divided normally.

A quick lookup at the financial site of Yahoo reveals the following. Of the 17 most important markets there is not one with a negative output over 2006. The output diverges from 10% (London) up to 55% (Moscow) with an average of rounded 24%. The spread, standard deviation amounts to 23%. This means that if the 17 markets are representative for all markets in the world, about 66% of all markets will show an output of 1 standard deviation above or below the average (therefore 24% ± 23%). Of all 17 markets there are only 2 outside this range.

We can look at this also in a different way: we can consider the return of an individual market as an estimate of the world average and look at how well such an estimate would be.

It is known that such an estimate will have a standard error that equals to σ/√T where σ is the annual volatility and T the number of years:

STD = σ/√T (1)


From formula (1) it follows that the standard error of an estimate of the world average decreases with the square root of the number of years.

Let’s take Mumbai as an example. The excess return or α, of this market above the world average is 33-24 = 9%. Though in general the term α refers to the excess return of a fund with respect to the benchmark, it may also be used for the excess return of a market.

To be convinced that an excess return is the consequence of a better performing of the economy and not to pure chance or luck then the return must be bigger than the world average with a significant size. In a one-sided test at the 95% confidence level the difference between the two must be greater than 1.65 standard deviations.

Suppose the volatility to be around 20% on an annual basis for the Bombay market then the standard error according to (1) will be 20%/√1 = 20%. The 9% above the average of the market is clearly smaller than one standard deviation above the average, clearly below 95% confidence level. It lies approximately at 50% what means that 50% of the return of Bombay falls to pure chance. In fact you can say that the 33% return of Mumbai is a good estimate of the world average.

We can carry out this calculation also differently and perhaps it will even become more clearly then. We saw that the standard error of an estimate of the average decreases with the square root of the number of years taken into account.

We know that: α =9% = 1,65* STD. It follows that STD, the standard error, equals 9/1,65 = 5,45. This value substituted for the STD in formula (1) gives √T = 23% /5,45 = 4.22 ==> T = 17,8 year.

This means therefore that the Bombay Stock Exchange for at least 17 years in succession must show a return that 9% or more lies above the world average to be able to say that for 95% this is due to a better performing of the economy and only for 5% to chance.

Around these days various funds present their performances. Some will show better results than others. From the previous it must be clear that luck may play a large role besides skill of the fund managers and in any case is much larger than they will admit. Think to that.