Let us assume you want to play the Dow Jones Industrial Average index (DJI30) on a monthly basis and you are concerned mostly with the downside risk, i.e. what is the possibility that you will get a negative return in a month compared to getting a positive return.

Concerning the current market situation and speaking in terms of particular months the empirical probability of September, 2011 to present a negative return stands at 1.33%. So there is about 98% probability the month would end with a positive return. Some possible trading and hedging strategies to deal with such a situation could be seen near the end of this article.

The magnitude of that possible September negative return even if it would have a relatively small probability could vary in a large range and is hardly to be measured. An optional explanation about a possible magnitude of that negative return could be found in the section describing the probability of extreme consecutive negative returns.

If we look further than only the next month however, examining the general statistical characteristics of DJI30 monthly returns presented in Table 1 would be helpful. The sample data of observations [1] covers a period of 994 months - from October, 1928 to July, 2011. As a risk-free rate is used the rate of 3 months T-bill as of 30 July 2011 which is virtually close to zero.

^{Table 1; DJI L30Y - DJI30 for the last 30 years (1981-2011); DJI L30Y2 - DJI30 for the previous 30 years (1951-1981)}

The DJI30 monthly returns seem to follow almost a normal distribution although a bit negatively skewed and with a high excess kurtosis [2]. For the whole examined period the average mean monthly return is 0.51% and the compounded monthly growth rate is 0.37%. The Standard Deviation is 5.31%, Coefficient of Variation is 10.34 and Sharpe ratio is 0.1.

We have to notice that for the last 30 years however the DJI30 index shows higher mean average monthly return (0.80%) and a significantly smaller kurtosis (2.77) than for the whole period. All things equal this would mean the risk of getting monthly returns far away from the mean was getting smaller for the last 30 years compared to the whole sample period.

Digging deeper into the data we would see that for the previous 30 years period (from August, 1951 to July, 1981 and noted in the table with DJI L30Y2) the kurtosis was even smaller (0.83). That period seems the one with the less volatility and smaller possibility of monthly returns far away from the mean.

That leaves the period before August, 1951 (which includes World war II and the Great Depression Era) to be the more volatile one and the one most responsible for the high kurtosis of the data.

Focusing back on the last 30 years of the Dow Jones Industrial Average index monthly performance we see that it was relatively more profitable than both the previous 30 years and the whole sample period even if that difference happens to be statistically insignificant. The DJI30 L30Y managed to achieve both higher monthly return over an unit of risk (higher Sharpe ratio) and an unit of monthly return was achieved with less risk (lower Coefficient of Variation).

The bigger negative skewness for the last 30 years shows a change from the previous period in a way that the smaller positive monthly returns are becoming more likely at the expense of more unlikely but more extreme negative returns. That combined with a higher kurtosis, in a way means there is an increase in the possibility of picking on an extreme negative month (and therefore the risk of negative monthly return) but such a bad timing could be compensated by the further development of returns.

The smaller sample of the last 30 years monthly returns however is a sub-sample of the bigger one. As such it could not be representative of the population of monthly returns of DJI30. In a way the skewness and kurtosis of the smaller sample even fall outside the possible ranges (with certain confidence levels like 95% or 99%) of the population descriptors calculated on the basis of the whole available period sample. That could mean a return to the ranges of the population values is about to be witnessed, i.e. an increase of the kurtosis, a decrease of the negative skew of the distribution and a decrease of the average mean. In other words there could be expected a combination of more small both positive and negative (with the prevailing of negative ones) results concentrated around the mean and a decrease of the extreme values of monthly returns, both negative and positive ones (with the prevailing decrease of negative ones). That could mean a decrease in risk together with a decrease in the magnitude of returns.

Having examined the distribution of Dow Jones Industrial Average monthly returns during the examined years we can focus on the possibility of hitting a downside month ahead.

The historical possibility of having a single negative month stands at about 42%. Which means there is a chance of 58% to get a positive month. That is already better than the normal chance of 50-50 but is not something extraordinary.

Let us examine the consecutive negative months in history presented in Table 2.

Table 2

Historically there were 10 instances of 5 consecutive months with negative returns, 6 instances of 6 consecutive months and by one instance for 7, 8 and 9 consecutive negative months. Calculated from the sample data the historical probability of 5 consecutive months with negative returns stands at 1.33% or about 13 in the history till now. The same probability for 6 months is 0.56% or about 6 times. For 7 consecutive negative months the probability is at 0.24% or about 2 times. For 10 negative months in a row the probability goes to about 0.02%.

So in general the more negative months we have in a row the smaller is the probability that the next one would be negative again.

Now we turn to the current market data.

The current condition of DJI30 shows the index has made 3 consecutive negative months and is about to score its fourth one. Based on the above mentioned historical probability the possibility of scoring a fifth negative month during September stands at 1.33%. That leaves a room of 98.67% for a positive monthly return during September which seems a far better bet than the general probability of 58%.

Let's assume the event occurs and DJI30 scores a negative month in September. Of all the negative returns in the historical sample 42.72% of those are between -2% and 0. So basically the chance of hitting a negative return at the end of September in the range of [-2% - 0%) given the month turns out to be a negative one is about 43%. This unfortunately leaves plenty of room for bigger negative returns.

Fortunately the more extreme becomes a negative return the more infrequent it is. For instance the probability of a fall of DJI30 of more than 10% in September if the month ends negative stands at about 3.34%. That is only given the month turns out to be a negative one again.

Historical data also shows that a highly negative month (with a return of more than -10%) was followed by another month of negative return of the same or bigger magnitude in only 6 cases or in about 0.6% of the observations. The majority (5) of those instances were before year 1938 and one was in 1974.

In the whole period there was only one case where there were 3 months in a row with negative returns more than 10%. That was in year 1932.

An interesting point should be made clear here.

Assume August ends with a negative return of 10% or more. If we account for the probability of 5 negative months in a row and it comes true at the end of September, the probability of September being as bad or worst than August (in means that it also scores -10% or more) increases to about 45%!

At the same time the empirical probability of 6 negative months in a row with the last 3 of them with returns worse than -10% stands at around 18%. Which in general would mean that the greatest danger of a negative return of more than 10% comes in the 5th consecutive negative month given the 4th one scores more than 10% of negative return, too.

There is a behavioral explanation for that phenomenon also, in means of fear and growing panic but that is out of the scope of current article.

Combining all said we could conclude that entering the market around the beginning of September by getting long on DJI30 could prove to be profitable with a relatively small probability of a downside risk. This however will be known for sure only at the end of the September. During the month there could be movements in both directions.

Moreover the magnitude of the possible negative monthly return remains unknown no matter how small is its probability of occurrence. Different trading or hedging strategies could be used to limit the potential loss in such an investment. Apart from hedging the investment itself, one should take care of protecting his/her whole capital by a proper money management. Risking only such a part of the money which one could allow to lose is a sound principle.

If one decides to buy the underlying or some ETF on DJI30 the general method of limiting loss is to put a stop-loss. This however could end up in the stops being hit after which the market could turn positive. This will only generate losses for the investor. A protective put strategy could serve better as a hedge against a bigger decline in the underlying while at the same time would preserve the opportunity for taking part in a possible positive return of the month.

A better strategy might be to enter long in an October CALL option of the underlying as this would limit the downside risk to the premium paid and at the same time one could take advantage of the gearing the option provides should the event of a positive return in September occurs. The reason for preferring October option than a more distant one is simple - the longer ones (for the same strike price) are more expensive now as they would possibly be at the end of September given September happens to be a negative month.

In a conclusion I might say that the risk is almost always present. In most of the cases however, the point is to limit it to a bearable level. The possibility that September will be negative again should be kept in mind. But in general that would only mean the risk of October being negative gets even smaller. Situations of such small downside probabilities could turn easily into a self-fulfilling prophecies.

There is also the possibility that September turns positive and then again October turns negative. The negative October could even erase the positive result of the previous month. Because of this when playing on a monthly basis one should always keep an eye on the current situation and analyze it with respect to its environment. Another tools like Macro economical analysis or Technical one could be also included in the process of developing an insight of the market and its possible movements.

Notes:

[1] Source: Historical price from Yahoo! Finance

[2] High excess kurtosis in general means a distribution with fatter tails which extends the possibility of outcomes far away from the mean in both positive and negative directions.

Disclosure: I have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours.