Hacking Compound Annual Growth Rate

This the third in my Hedge Fund Hacks series in which I dig just below the surface of some of the common hedge fund performance statistics. In the previous post I highlighted some of the ways in which Compound Annual Growth Rate can be distorted by chance. In this post I provide a simple hack to help bring into sharper focus the estimated returns derived from a value added monthly index.

Compound Annual Growth Rate

Nearly every hedge fund tearsheet will prominently feature CAGR. Wikipedia has a good reference. Investopedia has a thorough definition and a narrative about its use.

CAGR is probably the first number anyone looks at when considering a hedge fund investment. True, it provides the big picture of a fund’s performance, but it can give a distorted view. Here are the main short-comings as I see them:

• Path Dependency: CAGR is the quintessential summary statistic. It ignores every single bit of return data from the VAMI chart except the very last data point.
• Noisiness: CAGR is surprisingly sensitive to the most recent month’s returns. Beware of this when you are screening managers.

I hope to show below that applying the simple hack of using linear regression to the time series can mitigate these issues. I will look at each problem separately and throw in a couple extra credit sections to boot!

Path Dependency

Definition of Path Dependency

I want to be clear what I mean by path dependency in the context of this post. The notion is that two completely different return streams, drawn from different distributions, can deliver the same compound rate of return. The VAMI will travel from the same starting point to the same ending point via a path characterized by a different set of returns.

In this specific post, I am not referring to the effect of simply re-ordering a given set of returns. I know this is important, especially to meaningful measures of risk. I will get to that in later posts in the series. We will actually see that the approach I am proposing here may not help differentiate between the same returns in a different (possibly better, possibly worse) order. They will, however, help differentiate between returns drawn from different distributions.

Trend Line: Least Squares Linear Regression

Let’s start out by considering the following two VAMI curves plotted on a Log scale (see my post on Value Added Monthly Index) for refresher on why you might want to do this. In this and all the following charts, the purple dashed line is the equity curve implied by the compound annual growth rate.

The VAMI curves above are not hugely different, one might be perfectly happy to accept the CAGR of 12% as being a fair representation of what one could expect in the future. Looking closely one might assert that the blue line seems stronger in the beginning than towards the end, the green seems strong for the most part but is in a draw down towards the end.

Looking at the green VAMI curve alone and fitting a trend line using a least squares fit we get the following (refer to Wikipedia for a good primer on linear regression):

While you might not want to bank on getting a 15.75% return in the future, the regression line gives you a better idea of the underlying trend, or the central tendency of returns, than the point-to-point 12% CAGR. Now let’s add back in the blue VAMI, this time with a least squares fitted trend line:

What I am struck by here is how dramatically different the fitted trend lines are. The VAMI curves at first glance do not look hugely different. If you saw them on separate data sheets, or even on the same chart but with linear axes, you might barely notice a difference. And the CAGR reported on each manager’s performance capsule would be identical. However, the implied return of the blue fitted trend line is 9% versus the green line at 16%!

A qualitative description of what is going on here is that the trend line is picking up what the VAMI curve is spending the majority of its time doing. In the case of the blue curve it shows a brief spell of strong performance and a long period of performance below the CAGR. In the case of the green curve, it has a long period of out-performing the CAGR, with a short, sharp draw-down.

Be aware that it is not the ordering of events that is driving the difference in the trend lines. As we shall see later, the ordering of the monthly returns can be completely reversed and the linear trend line would be the same. It is the distribution of returns that affects the trend line. In that sense, the least squares fit trend line is capturing more of the information in the VAMI curve than the CAGR.

A conservative approach when considering these two VAMI curves is to use the lower of the CAGR or the fitted trend line.

NB: If you are very familiar with linear regression, you will probably be feeling uncomfortable about the autocorrelation issue to do with time-series. I briefly discuss this in the Extra Credit section at the end of the post.

Trend Line: Fixed Intercept

You might be tempted when fitting a trend line to a VAMI curve to fix the intercept at time zero to the datum, $1,000. After all, the time series must pass through this point, it is the only data point we know for sure! This would be a mistake. The following will show why.

Here is the green line with a new trend line fitted. This time the intercept is forced to $1,000:

Looks ok. The slope indicates an even higher return of 17%, but it looks like it fits. Let’s now add another VAMI curve. This time it is in red. It is the green VAMI curve re-ordered so that the draw-down is near the beginning and the run up comes second. The noise on both curves is identical. I have fitted a trend line with an intercept forced to $1,000:

Uh-oh, we seem to have a problem. By forcing the trend line through $1,000, the draw-down has been unable to exert its influence on the least-squares regression. It has been unable to drag the left hand end of the trend line down. Now the exact same distribution of returns gave us an estimate rate of return of 17% in one case and 9% in the other. Let’s see what happens when we allow the intercept to be driven by the linear regression rather than our mistaken logic:

Now we get identical slopes (but different intercepts) for both VAMI curves.

Don’t misunderstand me: the ordering of returns (the “path” of the VAMI) matters. But there are other statistics that specifically address this issue – we will get to them later in this series. The objective of CAGR is to provide a measure of the central tendency of the distribution of returns (the geometric average to be precise). I would argue that a least squares linear regression does a better job than CAGR.

Noisiness: The Last Data Point Effect

The second gripe I have with CAGR is its sensitivity to the most recent data point. Particularly with short track records, such as emerging managers or new programs, the CAGR can vary dramatically from month to month purely as a result of noise.

Here is an example:

The VAMI curve above was generated to give a 10% CAGR over 60 months with a Sharpe Ratio of 0.6. The final point in purple is exactly on 10% CAGR. The green point is the purple point plus 2 standard deviations. The red is minus 2 standard deviations. Bear in mind that 1 in 10 VAMI curves you see will have a final point that either exceeds the green dot, or is below the red.

The results are shocking, the CAGR indicated by the dashed green line is 12.3%. The red dashed line has a CAGR of 7.5%. That’s a big difference and it could entirely be due to chance. One manager makes the cut, the other doesn’t!

So what happens when we fit a linear trend line to each of the VAMI curves (which are identical except for their last points)?

First note how consistent the trend lines are: they range from 6.5% to 6.9%. So, regardless of the final point on the VAMI curve, the trend lines all came in more or less the same. CAGR ranged from 7.5% to 12.3%. Least squares fit does a great job of de-noising the data.

Be aware that the shorter the time series, the less damping the trend line will do.

Secondly, note how much lower the trend line estimate of rate of return is compared to CAGR. Picture the VAMI curve extending backward in time, oscillating around the trend line. You might be forgiven for thinking the starting point on the VAMI curve was cherry picked to show CAGR in the best possible light!

Extra Credit

Time Series and Trend Lines

I feel a little queasy using linear regression to fit a trend line to a time series. One of the requirements underlying the procedure is that the observations (i.e. the data) be independent and identically distributed. Clearly they are not: the VAMI at time t depends greatly upon the VAMI at time (t – 1). Fortunately, the result is still an unbiased estimator, but the error in the estimator is larger than indicated in the standard error statistics generated by whatever software you use to implement the linear regression.

Median Rate of Return

I came across a method to derive a median rate of return, which I believe was described by Rousseeuw. I have implemented it below from memory. It involves figuring the median of the returns at each time interval from 1 month to 60 months (in this case). Then you take the median of the 60 estimates of returns. The chart looks like this:

All three VAMI curves yield the exact same median estimate of return of 5.8%. The fact that this is even lower than the linear regression value suggests that the outliers tend to be on the high side i.e. the surprises tend to be to the upside. In this example, the median approach is completely immune to the noise!

Conclusion

It is well worth considering using the slope of a least squares linear regression trend line to supplement your understanding of the central tendency of a manager’s historical returns. The approach has the following benefits:

• It can differentiate between return streams that have the same overall CAGR, but are drawn from different distributions.
• Unlike CAGR, it is less affected by the choice of the first data point and noise in the last data point.

Image: Skitterphoto on Pixabay Creative Commons License.