Behavioural Finance

Regression

In order to assess the performance of a fund compared to the market, one needs to use a regression analysis. Indeed, on top of providing information regarding the correlation between two variables with the slope coefficient, in the case of fund performance assessment, the intercept represents constant out-, or under-performance. In our analysis the regression that is used is a simple, linear regression, which looks like follows:

R t HedgeFund = α + βRt Index + εt

with α = intercept

       β = the slope coefficient

       ε = error term

As explained above, α represents the performance of the fund compared to the market. If positive, it graphically means that the hedge fund’s returns are constantly outperforming the market by α. The β value, on the other hand, expresses the correlation existing between the two variables. Finally, the error term ε shows the actual fit of the regression. Of the error term is eventually derived the R-squared, and Adjusted-R-squared, expressed as percentages. Those values can be understood as “the model is able to grasp this percentage of the y-variable’s movement”. Also, an F-Test for linear relationship is provided.

More practically, the first step one has to follow to perform a regression analysis is to standardize both variables, making them comparable. In our case, our funds were still expressed as returns, while our index was expressed on a 100-points basis. We therefore transformed the index value into returns as follows:

[pic 1]

1. Convertible Arbitrage

The first fund run against the S&P500 returns was the Convertible Arbitrage one. As its name suggests, this fund should derive arbitrage for its investors; Or, in other words, this fund shout outperform the market significantly. For MatLab to execute the regression the function fitlm was used in the following way:

[pic 2]

The regression results for Convertible Arbitrage looked as follows:

[pic 3]

One can see that, although small, both coefficients (intercept and x1, representing α and β, respectively) are positive. Moreover, when their difference from zero is statistically tested, one can see that the t-stat is enormous and, thus, the p-value, very small. Overall, this means that we can be more than 99% sure that the results we came up with can be replicated in another sample. Additionally, since the x1 (β) coefficient shows the co-movement, or the correlation between the two variables, a value of zero would mean that there are no co-movements, while a value of one would mean perfect correlation. As a result, it is interesting to statistically test whether x1 is different from zero with the following equation:

tStat = (x1 – 1) / SE(x1)

tStat = (0.0969 – 1) / (0.0195) = -46.3128

With such a tStat value, one can estimate that the statistical difference between the observation and the mean is substantial and significant.

In terms of the actual statistical significance of the model itself, one could look to both the adjusted R-Squared and the F-Statistics.

[pic 4]

As previously mentioned, a 0.0822 adjusted R-Squared, would mean that the model explains 8.22% of the movements of the y-variable. Moreover, when the linear relationship - in other words, the model’s worth, is tested, one can see that the p-value shows the model is significant.

In conclusion, Convertible Arbitrage does outperform the market in most the time as its intercept value, although small, is significantly different from zero. Subsequently, it is not correlated much to the market.

1. Dedicated Short Bias and Emerging Markets

Using the same methodology we analyzed two other hedge funds. The first was Dedicated Short Bias, and as its name posits, one would expect that it is negatively correlated with the market. The regression outputs looked as follows:

[pic 5]

First of all, it is interesting to note that the intercept is slightly negative, but when tested against zero, reveal to be insignificant. As a result, one cannot draw conclusions and state that this fund constantly underperforms the market. On the other hand, as our prediction expected, the x1 coefficient is negative and statistically significant. Because testing against +1 would not bring additional information, we choose to test it against -1: