Cumulative Sums and Compounded Returns in Factor Performance
Summary
The document compares two ways to display the historical performance of a long–short factor such as Fama–French HML. A compounded wealth series applies each period’s return to the prior wealth, representing how an investment grows through reinvestment. A cumulative sum instead adds periodic factor returns without compounding, as in the cited HML chart. The question is why analysts might choose that presentation when compounded growth seems more representative of investor wealth.
The response distinguishes cumulative return over a full horizon from an annualized compound rate. It illustrates the distinction with a two-year investment: total growth is reported across the whole period, while the annualized rate expresses an equivalent per-year pace. Cumulative figures can be easier to interpret as total horizon returns; annualized compounded rates are more useful for comparing investments held for different lengths of time. The explanation is simplified and does not fully explore factor-return conventions, rebalancing, or the implications of summing long–short returns rather than tracking investable wealth.
Key ideas
- Compounded returns track wealth by applying each period’s return to the accumulated value.
- A cumulative sum of periodic factor returns adds observations without compounding.
- A total return describes growth over the full holding horizon, while a compound rate annualizes growth.
- Use annualized compound rates to compare investments with different horizons, while cumulative figures can convey total-period performance.
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# Back to Basics -- Cumulative Returns # 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. ## Answer by dnl (score 1) https://quant.stackexchange.com/a/21772 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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