**An alternative way to estimate the Expected Returns**

June 16, 2018

The expected return is a fundamental parameter to estimate the future returns of a given asset class.

Financial statistics predominantly use statistical indicators based on mean and variance.

The average is commonly calculated as an average annual return, but it works well enough when considering annual returns while it works poorly if you use daily or monthly returns (expecially if used on short time series such us three years).

When I joined finance at the end of 1998, they expert taught me that the average annual return on the stock market was 14%; but in 2009, as a result of the crises of 2000-2003 and 2007-2009, the same expert evidenced that the average had fallen to 6% annual average.

This change is not correct in my opinion, the annual Mean must be virtually constant, sometimes it differs due to Variance (the volatility of returns to understand), otherwise if the annual Mean must vary continuously the sense of estimating the Mean is lost along with the Variance.

But the return in itself is very variable, especially for the stock markets and not to mention the emerging crypto, so it is not always easy to correctly estimate its future Expected Return.

So I enjoyed taking the historical series of the S&P 500 and verify that at the end of 1998 the market in the last 20 years had obtained an annual average of 13.5%, while 11 years later or in December 2009 the average annual yield of the last 20 years was actually fell to 5.9%.

The first consideration I made is that to calculate a consistent annual average of 20 years is a very short time window and it is necessary to extend it at least to 40 years.

If the window for calculating the average annual yield is 40 years, then the values are very close, because in the 12/1958 to 12/1998 window the average annual yield was 7.95% per year while from 12/1969 as of 12/2009, the average annual return was 6.63% (also due to the fact that 30 out of 40 were in common, whereas in the example above 10 years out of 20 were in common).

This brief analysis shows that the historical period 1978-1998 was "out of the ordinary" and therefore the annual average of the stock markets is much lower than an average annual rate of 14%.

However, it is also clear that calculating the simple compound annual average is not stable except for very long periods of time, because the volatility of the markets determine the starting point and the point of arrival for its correct calculation.

In 2014 with prof. Ruggero Bertelli we tried to solve the problem by creating a statistical indicator that was not related to the concept of Mean or Variance so dear to statisticians and mathematicians because they simplify (perhaps too much) the complex structure of financial markets.

The statistical indicator of which I speak, of a deterministic and non-random matrix, is called (perhaps with little imagination) DIAMAN Ratio.

The DIAMAN ratio was created with the aim of better estimating the expected long-term and short-term returns (in this case assuming the persistence of the current trend).

Its formula is very simple and elegant mathematically, because it is simply DR = Beta * Rsquare, with the Beta calculated as the linear regression of the historical series of prices with respect to time (which is very different from the Beta of returns compared to other typically used returns to estimate the correlation and the link with another historical series).

Why is it multiplied by Rsquare? Because this parameter is the so-called Determination Coefficient, if this is equal to 1 it means that the Beta and the historical series coincide, otherwise if the historical series deviates far from the regression line. The Rsquare tends to 0 and therefore lowers the estimation capacity of the DR.

Let's take an example to better understand the logic

Let's assume we have to estimate the yield of a fund, if the first year has a yield of 1.2%, at the same rates, the following year with the DIAMAN Ratio it is really easy to estimate the yield, in fact calculating the DR = 1.2% (since the Rsquare is equal to 1 because there are no deviations between the regression line and the monetary fund), consequently the following year will be exactly at 10.24 as a value and the DR has perfectly estimated the yield expected from the monetary fund.

The result is different if the historical series has a fluctuating trend such as an Equity fund, since the Rsquare is less than 1 the DR estimate will be lower because the beta (equal to the regression line is corrected by Rsquare); this makes the future estimate more truthful on average, the more volatile the higher the uncertainty of obtaining the expected return rises.

If we take the example of the S & P 500 first and calculate the Diaman Ratio of forty years in the two periods taken into consideration, the DR of the period 1958-1999 is equal to 5.7% against a real 6.6% of the period subsequent, while in the period 1969-2009 it is equal to 8.2%.

It would seem to be a contradiction, of course, but we actually share the story from 2009 to 2018, and in the period 1978-2018 the average annual yield of the S & P 500 was 8.8%, therefore slightly higher than the one estimated in 2009 from Diaman Ratio.

Coincidence? It may be, but also not.

We believe very much in deterministic indicators, because they are not based on the axiom that the markets are random, but on the fact that the markets write history and even with short-term noise, there is a well-defined logic behind that can be described better with regressions rather than with Gaussians.

But then the question arises spontaneously, if the Diaman Ratio is able to estimate future expected returns, when it has a value equal to zero or even negative, what should we do?

Exit form the long position of course, and we can show that this logic allows you to make a very efficient timing, better than moving averages, especially in the world of Crypto Assets, but I do not want to anticipate too much a study that we will publish soon ...

DB