# Back to Basics — Cumulative Returns

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.

Thanks.

• 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. – Tyler Olsen Nov 16 '15 at 20:32
• 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. – noob2 Jan 15 '16 at 21:58