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This the second in my Hedge Fund Hacks series in which I dig just below the surface of some of the common hedge fund performance statistics. Today I take a look at CAGR, or Compound Annual Growth Rate, sometimes referred to as CARR, Compound Annual Rate of Return.

## 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. You can also find an excel formula and what looks to be a javascript calculator.

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 like the wide angle image above, it can give a distorted view. Here are the main short-comings as I see them:

• CAGR is the quintessential summary statistic. It ignores every single bit of return data from the VAMI chart except the very last data point.
• Its value is surprisingly sensitive to the most recent month’s returns. Beware of this when you are screening managers.
• There are timing issues that make it difficult to compare managers’ returns.

## A VAMI Curve

The chart below contains a simulated VAMI curve built using a 12% annual return plus some noise equivalent to a Sharpe Ratio of about 0.8. The noise includes an autocorrelation component of about 0.5 at the first lag, and 0.25 at the second lag. The chart covers a period of 60 months.

The two large dots are the only numbers that go into the calculation of CAGR. The first dot is the datum, in this case, \$1,000 as is typical. As far as information content goes, the only data point that matters is the last. The way in which the VAMI curve got from the first point to the last simply doesn’t matter. Compound annual return is not path-dependent. We are modeling returns as though they followed the dashed line.

## CAGR Is Path Independent

Every single one of the VAMI curves in the following chart has exactly the same compound annual return:

The green lines get across the idea that the timing and current state of draw-down have no effect on the calculated cumulative annual return. The grey line demonstrates that an equity curve with much higher volatility and totally unrelated returns will be reduced to the same CAGR so long as the final values of the VAMI curves are the same.

## CAGR Is Noisy

The next plot serves to demonstrate that calculated compound returns are quite sensitive to noise.

In this example we use the same VAMI curve as before and take a look at what happens if we vary the final point. The purple points are the same (The final purple point has no noise, it is the expected VAMI for a 12% annual return). The green point is the purple point plus 2 monthly standard deviations. The red is the purple point less 2 standard deviations.

If we assume a normal distribution (which we shouldn’t) the last month’s return could have been at the green point or higher with a ~5% probability, and at the red point or lower with ~5% probability. If returns have even slightly heavy tails, the chances are higher. The returns implied by the green point are 13.8% compounded annually. The returns implied by the red point are 10.1%. The purple point gives a CAGR of 12%. I have added the dashed lines to show the “implied” equity curve associated with each of these returns.

The differences between these annual returns are pretty big – the green equity curve implies a rate of return over 37% higher than the red equity curve. That difference is simply noise! If you are looking at the returns of 20 managers, you would expect at least one of them to benefit from the “green curve effect”, and another to suffer from the “red curve effect”. One manager makes it through the first round of screening, the other doesn’t, and it was all just luck!

Just to be clear how likely this is: the Sharpe Ratio of the simulated returns is 0.8, so it’s not super-noisy. This effect could be much bigger with lower Sharpe’s.

## The Effect Of Timing On CAGR

Note that the longer the VAMI curve, the less impact the noise will have (because you are taking the Nth root of the final point in the VAMI curve, where N is the number of periods). You must bear this in mind when comparing a manager with, say a 20 year track record to one with a 3 year track record.

Furthermore, it is possible for a manager to game this effect by cherry-picking the start point for his presentation – effectively lowering the datum to which the final VAMI is compared. This is why your due diligence must be thorough!

## The Hack

Since this post is already longish, I will go into the details of the hack in another post (here it is). In that post I will show how the issues discussed above are mitigated. In the mean time, here is a basic description:

• Fit a linear regression of the log of the VAMI curve, forcing the intercept through the datum (\$1,000, in this case). The slope is the continuously compounded monthly return.
• If you want to get really esoteric, you can figure out the median CAGR between any two points on the VAMI curve. In this way you go a step further than a linear regression and minimize the effect of outliers.

## Conclusion

Hopefully I have highlighted the distortions that come with the extreme wide-angle view of a manager’s performance provided by CAGR:

• It ignores all the detail of the VAMI curve.
• It can be strongly affected by luck.
• Comparing two cumulative growth rates can be tricky.

My feeling is that so long as the number is good enough move on to more informative data.

In the next post I will present a solution (here it is).

Picture Credit: Fish Eye City by juanma1358

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