In keeping up with my research reading, I came across a white paper from J.P. Morgan titled Defining Absolute Return Investing in Fixed Income.
A graph on one of the last pages caught my eye:
The stunning non-normality of the Barclay’s Aggregate Bond index struck me. “It must be that yield is significantly swamping price return,” I thought.
So I went to recreate the results. I took monthly total returns for the Barclay’s Aggregate Bond index going back to 2/1976.
And I got this:
Well … that looks different. Still not normal – but nothing like what had been shown in the exhibit.
Maybe J.P. Morgan has some other information out there about? So I looked up a classic: J.P. Morgan’s The non-normality of market returns. Within, I found this graph.
Okay – not normal, but nothing like the first exhibit. What in the world was going on?
Sometimes it takes fresh eyes to see the problem. My co-PM Justin Sibears spotted it right away: the bins in the original histogram aren’t equally spaced. Some are 0.25% in width, some are 0.5% in width and some are 1% in width.
Taking this “modified histogram” approach, I recreated the following graph:
Okay … whacky, and still not quite the same – but the number of observations seems to be wildly different, so it’s likely the period we’re looking over.
Fortunately, we can just sum up the number of observations and back out their time frame. The original graph says “As of December 31, 2014” and has a grand total of 176 observations, meaning their study goes from May, 2000 through the end of 2014.
Let’s plot those returns:
Much, much closer. But are the returns as “non-normal” as they claim? Back to an equal-width histogram:
But what in the world is going on with the location of the fit normal distribution? The fits I am getting for the weird bins are obviously wrong, as they aren’t appropriately taking into account the density. Normalizing the weird-bin histogram, we get a better fit:
But there is still something amiss: looking at the original graph, it looks like the mean is around 0.25, whereas the mean in the graph I am plotting is around 0.5.
What has happened? I believe it was a simple math error. Often, index data like this is given in linear returns and we have to convert to log returns. This is done by via the formula log(x+1) where x is the linear return.
Except that most of this index data represents a 1% return as the number 1 instead of the number 0.01. So you really have to take log(x/100 + 1) (and then multiply by 100 if you want to plot 0.0025 as 0.25).
If you took this extra step, your expected value is 0.46 – if you forgot, you get an expected value of 0.27. Which I think is what happened here.
While I am sure the fit normal distribution error is an honest mistake, the varying bin sizes seems intentional. At the very least, if you are going to use varying bin sizes, you should normalize to get a more accurate representation of density.
The lesson is clear: if you’re going to be relying on other people’s research (which, in a field like finance, is somewhat of a must) – make sure you validate their results.