I recently came across a chart of Fama-French's (FF) HML factor cumulative performance. I first saw this in an article by AQR's Cliff Asness: http://www.institutionalinvestor.com/Article/3315202/Asset-Management-Equities/The-Great-Divide-over-Market-Efficiency.html#/.VkntwZ0o5Ms

I went to Ken French's data library in an attempt to replicate it. I was simply compounding the growth of $100, by doing a straightforward time series of 100*(1+r). After no success on such a deceptively simple task, I eventually found that Asness was showing the cumulative performance, as the cumulative sum. See figure 9.1 Quantitative Equity Investing by Lasse Heje Pedersen (on google books). Here's the footnote to that chart:

Cumulative performance of the value factor HML, 1926—2012. The figure shows the cumulative sum (i.e., without compounding) of the long–short value factor HML constructed based on stocks' book-to-market ratios.

My specific question is why use this "cumulative sum" instead of the compounded wealth growth? What is the use of this data presentation? To me this does not say much about the true cumulative performance of the factor, one needs to see this in terms of compounding.


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    $\begingroup$ You need to pay careful attention to which definition of "returns" is being used. Computing cumulative performance is different based on whether log returns or simple returns are used. If the factor was computed using log returns, then cumulative log-return performance is equivalent to the cumulative sum of log returns. If simple returns are used, then the multiplicative accumulation of wealth is appropriate. I couldn't see the figure to which you referred due to preview page restrictions, so I can't say for sure if this was your issue. $\endgroup$ – Tyler Olsen Nov 16 '15 at 20:32
  • $\begingroup$ So if you plot cumulative log returns, you are basically plotting the log of wealth. Which is probably a good idea over long periods of time. Think of long term plots of a country GDP, they are usually plotted logarithmically, so they look nicer and more understandable. It is the accepted way to plot a constantly increasing thing. $\endgroup$ – noob2 Jan 15 '16 at 21:58

The cumulative return tells you how much 1€ grew over the investment horizon, whereas the compound return is typically annualized.

Consider you invest 100€ for 2 years. At the end of year two, your investment grew to 110€. the cumulative return is then (110-100)/100=10%. What is the compound return? Obviously, it must be lower than the 10% as we have a two-year investment and the compound return is expressed annualy. The compound return is obtained from 110=100(1+r)^2 which results in r=sqrt(110/100)-1=4.88%.

Now image the other way round: If somebody tells you he made 10% cumulative return over the last two years, you can easily calculate without a computer that his 1€ investment grew to 1.1€ during that period. If he would tell you he made 10% compound return (annually), you would have to calculate 1€x1.1x1.1 to come up with the value of the 1€ investment after two years. This calculation is more "difficult" the longer the period.

To summarize, for humans it may be more easy to think in terms of cumulative returns as a return over the whole investment horizon. To compare two investments with differing investment horizons, you would go for compound rates.


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