EssDeeAitch wrote:So in short, my portfolio had a lower volatility than the benchmark and returned 6.26% more than the benchmark with a 71% correlation to the benchmark (the selection of the appropriate benchmark is clearly key in assessing performance). Does that sum it up adequately?
It's getting there.
From the helpful links that dspp posted, the performance values are clearly percentage returns but over what length period is still not clear. I'll return to the importance of this in a moment.
Let's deal with your "6.26% more than the benchmark" first. A property of the line of best fit is that it always passes through the centroid of the data. So recalling that the data comprises a number of pairs of values and we can abbreviate a general pair to (B, P), the performance values of the benchmark and portfolio, respectively. The centroid is simply the point (B^, P^), where B^ is the average of all the B values in the pairs of values and P^ is the average of the P's. Because, the centroid is always on the line of best fit, it follows that
P^ = alpha + beta*B^
The P^ value here is the average return from your portfolio per period over the 3 (or whatever) years and, the important thing to note is that it is derived from actual data rather than estimates. So the equation is telling you how your average return per period (P^) relates to B^, the average return per period from the benchmark. The point being that the difference between your average return and the benchmark's (ie P^ - B^) is not simply alpha as your statement implies, rather it is
P^ - B^ = alpha + beta*B^ - B^ = alpha + (beta-1)*B^
So although alpha is part of the difference, there is also that additional bit (beta-1)*B^. Since your beta is less than 1, then beta-1 is negative, -0.32. If the average benchmark performance (B^) has been positive over the 3 year period then (beta-1)*B^ will be negative so the difference between your portfolio's return and that of the benchmark will be less than alpha. On the other hand, if B^ is also negative then the difference will be greater than alpha. Alpha on its own isn't the whole story, despite what some authors may claim. Strictly, alpha is the expected return from your portfolio when the benchmark return is zero.
Ok, onto that 71% correlation. Correlation, in the statistics world, has a precise meaning which is related to (one might almost say correlated with) the everyday usage. In statistics, correlation (or Pearson's Correlation Co-efficient [PCC] to give it the full monty) is a measure of the linear relationship between paired variables. It varies between -1 and +1, and is often abbreviated as the Greek letter rho or the Roman letter, r. Perhaps you can see where this is headed. It turns out that in a simple linear regression model, r-squared is the square of the PCC between the two variables. So, to be strictly accurate, the correlation [PCC] is the square root of the r-squared value, ie 84% rather than 71%.
l will now return to that point about the lengths of the periods over which the returns B and P are measured. Beta is a dimensionless number. However, alpha is not. It has dimensions of % return per period. So understanding the period length is vital to correctly understanding alpha. It is also possible that an alpha value which has been derived from say quarterly data is then restated as an annualised value for reporting purposes. I simply don't know enough about MPT and its accepted norms and conventions to say if this is the case, but it is another gotcha to be aware of. If values are restated for reporting purposes then the analysis above about P^ - B^ would need to take this into account by converting alpha back to a per period basis.
Finally, dspp's references have also lead me to the definitions of tracking error and the information ratio. Going back to the pairs of values (B, P), the tracking error is simply a measure of how much the difference, P-B has varied over the observed pairs of values. The measure of variation used is the standard deviation. If these differences were Normally distributed about the mean difference (estimated as P^-B^) then in a large sample of observations around two-thirds of the differences should lie within one standard deviation of the mean and 95% within two standard deviations. The information ratio is simply the mean difference divided by the tracking error. Statistician's use the term "normalisation" to describe this process of dividing something by its standard deviation. The resulting value has a standard deviation of 1, which is quite handy. Quite what MPT uses it for I haven't investigated but a statistician would use it for testing a hypothesis such as whether the mean difference is significantly different from zero. A value of 0.91 is too low to reject the no difference hypothesis, but the hypothesis test sets a high bar and if the number of data points is low, the chances of false negative are considerable (see
viewtopic.php?f=9&t=14959#p187124 for more about significance, hypothesis testing, etc). The differences P-B, have an estimated mean value of 0.91*4.88 = 4.44 and standard deviation of 4.88, so that should provide some additional insight into the relationship between your portfolio's returns and that of the benchmark.
If alpha hasn't been restated then you can plug the mean difference value of 4.44 into the equation above and with a bit of algebraic jiggery-pokery work out the P^ and B^ values. I make them 5.69 for B^ and 10.13 for P^, which leaves me wondering whether the P, B and difference values might also have been annualised prior to model fitting, etc.