During the last couple of weeks, it has been raining predictions about the stock markets in 2014. Many of those predictions included 2013, which turned out to be an extraordinary year for equities with a total return on the S&P 500 index (NYSEARCA:SPY) of over 30%. These predictions argued, in some way or another, that because of the fact that 2013 was such an incredible year we should expect another good year for stocks this year. While I do not dismiss this hypothesis, I would like to give a more complete overview of what last year's return could mean for the 'Year After'.

## A favorable starting point

One of the 'predictions' that triggered me to do a little research on the relationship between returns in successive years is that of BCA Research. They found that, since 1870, US equities managed to realize an annual return of over 25% in 30 occasions. Out of these 30 years, 23 were followed by another year of positive returns. This implies a 77% historical probability that a very good year is followed by another year in which equities go up. BCA calculated that the return in those years after averaged 12%.

That sounds very impressive. And, a historical probability of 77% of a positive 2014 with an average expected return of 12% is of course a very favorable starting point. It does not, however, give us any information on how impressive this actually is. It does not tell us if this is the best thinkable starting point for this year, since this is only one possible combination of historical returns.

To make a sound assessment of how favorable this particular starting point is, we should look at other return combinations as well. To calculate different return combinations between Year 1 and the Year After (I will use these terms to refer to the return in the first year and the return in the following year) I use US total return data from Dimson, Staunton & Marsh (2013) starting in 1900. Since they haven't yet updated their database I added last year's total return on the S&P 500 index (32.4%) to include 2013. My analysis is split into two sections; the first focuses on the cases in which Year 1 was positive, the second in which the return in Year 1 was negative. This makes it relatively easy to determine if it matters if you enter the Year After with a positive of negative Year 1.

## Historical probabilities after a positive year

The graph below shows the historical probability of a positive Year After for several 5%-cohorts of positive returns in Year 1. For example, we can determine what has historically been the probability of another positive year after years with a return bigger than 5%, 10%, 15% and so on. By looking at different return cohorts we also get an indication if the size of the return in Year 1 is related to the probability of a positive return in the Year After. I also include the historical probability of a positive year over all years since 1900.

The graph reveals a couple of interesting things. First, despite the fact that my historical period is somewhat shorter than that of BCA, the results are very similar to their research. After a Year 1 with an annual return of 25% or more there has been a 77% probability (the same as BCA Research) of a positive return in the Year After. BCA based their prediction for 2014 on a return of 25% or more since their research was published around the middle of December. Stocks went up a little more afterwards resulting in a total return in 2013 of 32.4%. As can be derived from the graph a total return of 30% or more corresponds with a historical probability of 78% of a positive return the Year After, against the 77% for a return of 25% or more.

But by looking at other cohorts in the graph as well I have to conclude that 78% is not that spectacular from a relative perspective. For example, *if* last year's return would only have been 10% or more, the Year After would have generated a positive return in 76% of the time. That is only 2 percentage points less. More importantly, since 1900 the return on US equities has been positive in 74% of all years. If we relate this to the probabilities we found for the 5%-cohorts, the differences are again small. The historical probability of a positive Year After following a Year 1 that returned 30% or more (78%) is hardly higher than the general chance of a positive return in any given year (74%). The knowledge that we have had an astonishing year in equity markets does not increase the probability of another positive equity year all that much. A positive return of any kind will give you a probability of roughly 75% of a positive return in the Year After. Hence, the positive return probability in the Year After is pretty much uncorrelated to the positive return you made last year.

## Historical probabilities after a negative year

That concludes the analysis for Year 1, *if* it turned out positive. Since 74% of all years since 1900 have resulted in a positive return on US equities, there is still 26% of the years left in which the return in Year 1 was negative. By doing the same analysis, but now for 5%-cohorts of negative returns, we can take a look if the probabilities differ more here. The results are shown in the graph below.

First, the hit ratios of a positive return in the Year After are somewhat lower *if* Year 1 was negative than *if* it was positive. This was to be expected since most positive return-cohorts experienced a slightly higher probability of a positive return in the Year After than the average of all years (74%). Hence, we can now draw the important conclusion that the probability of a positive return in the Year After has been higher when Year 1 was also positive.

## It's the sign that matters not the size

Second, the graph shows that, just like the positive return analysis, the differences in historical probability are small. After a decline in the US stock markets of more than 5%, more than 25% and more than 30% in Year 1, the Year After has been positive in 67% of the time. This compares to a positive return probability of 71% after a Year 1 with over 20% loss (and 74% for any year to yield a positive return). Again, the differences between cohorts are no bigger than a few percentage points. Also, there is no clear relationship between these probabilities and the cohorts. Thus, it is the sign of the return in Year 1 that matters, not the size.

## What return to expect?

Historical hit ratios are of course only one way to look at historical data. They give you information about historical probabilities but not what kind of return you may expect. Therefore, to conclude my analysis, I will calculate the historical average annual returns for the return cohorts as shown in the two previous graphs.

Let's concentrate on cases where Year 1 yielded a positive return, first. The graph below shows the average return on US equities for the 5%-cohorts shown in the first graph.

What immediately stands out is that the differences in average returns are small as well. Equities returned 17.22% in the Year After when Year 1 yielded 10% or more, while the return in the Year After has been 18.37% when Year 1 realized a return of 20% or more. All other return cohorts are in between. A 1.15% bandwidth is narrow the least. The small differences in return are partly explained by the fact that there is quite a bit of overlap when moving from cohort to cohort. For example, by moving from the cohort with a 5% or more return in Year 1 to the cohort of 10% or more return in Year 1 you leave only the years with a return between 5-10% behind. That said; the overlap in the first and last cohort is negligible, but yet the average returns remains very similar. Hence, there is little discrimination in both probabilities and returns between cohorts when Year 1 is positive.

The graph shows another interesting result. Since 1900, US equities have realized an average annual return of 20.4% during all years in which the return was positive. In all cohorts the average return in the Year After has been lower than this average. This implies that *if* Year 1 is positive, you will probably experience a below average positive return the Year After. This sounds a bit complicated and paradoxical. It's important to realize that for the different cohorts the return in the Year After, when positive, is still way above the overall average of all years (positive and negative, which is 11.3%). This is, however, not the relevant benchmark to use here. Since we're focusing solely on the probabilities and returns of years in which the return in the Year After is positive we have to compare the average return in each cohort with the average of all positive years.

## Bad years are followed by very good years

The knowledge that *if* the Year After turns out to be positive after a positive Year 1, its return will probably be below the average return of all positive years leads to the following; *If* the Year After is positive in cases in which Year 1 was negative, you will probably beat the average of all positive years. This is exactly what the next graph reveals.

Every 5%-cohort of negative returns in Year 1 is followed by a very strong year *if* the Year After turns out to be positive. Take for instance a Year 1 in which you lost 25% or more*. If* the next year turned out positive you would have realized an average annual return of over 35%. Again this is only true *if* the Year After is positive. Hence, if you want to experience a real bumper year for equities, your chances rise *if* last year was poor.

Also, notice that for negative year cohorts the differences in return are much more explicit. Moreover, the results seem to suggest that the bigger the loss in Year 1, the bigger the gain in the Year After. The cohort of 30% negative return or more distorts this view, but there are only two years in this cohort.

## Conclusion

I hope I have been able to shed some light on what 2014 could bring for equities markets purely based on historical data. As can be derived from the probability analysis it only really matters *if* last year was positive or negative. Since last year turned out to be positive for equities the probability of a positive 2014 is a little higher than in case of a negative 2013. That the return in 2013 was very impressive is of no real importance. This also goes for negative years. How bad it gets, is not relevant for the probability of a positive Year After. Referring back to the prediction of BCA, just a mention of 2013 being a positive year for equities would have done it.

When looking at returns instead of probabilities the results are similar in cases where Year 1 was positive. The returns in the Year After are very similar for all return cohorts, and show that the size of the return in Year 1 is not related to the size of the return in the Year After. Interestingly, *if* the Year After is positive the return tends to be below the average of all positive years (but still well above the average of all years). When returns are negative in Year 1, the Year After tends to be very strong, well above the average of all positive years. In this specific case there are substantial differences in return between cohorts. The data seems to point out that the size of the loss in Year 1 is of importance. The bigger the loss in Year 1 the better the return the Year After.

Dimson, PR Marsh and M Staunton, Triumph of the Optimists: 101 Years of Global Investment Returns, Princeton University Press, 2002, as updated in E Dimson, PR Marsh and M Staunton, Global Investment Returns Sourcebook 2013, Credit Suisse/London Business School.

**Disclosure: **I have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.