# A Return by Any Another Name…

… would not necessarily smell the same. It is as well known as it is easily forgotten, that financial returns can be computed in several different and all legitimate ways. First of all, returns can be discrete (or arithmetic) i.e.

P(t)/P(t-1) – 1

where P(t) is the price at time t and P(t-1) is the price at time t-1, or they can be continuous (or logarithmic), i.e.

ln[P(t)/ln(P(t-1)]

where ln is the natural logarithm. Logarithms are usually favored by academics because of their algebric properties. Simply stated, the log return from t-2 to t is exactly the sum of the log returns from t-2 to t-1 and from t-1 to t-2. In case you wonder, this does not happen for discrete returns that always beg the question: “how is that if the market has gone down 10% and then it has risen 10%, I am still losing money?”. Well, this does not happen with log returns.

Practioners often use discrete returns and no, it is not because they do not know what a logarithm is. It actually seems likely that thanks also to the growth in the number of business schools, practitioners not only know logarithms, but they have also figured out that discrete returns  are always higher than log returns. Also most widely used performance indices such as the Sharpe ratio, turn out to be higher (i.e. better looking) if computed with discrete returns. Since practioners usually sell investment funds, it goes without saying that this comes in handy.

It is not over yet. Returns can be computed over different time periods: there can be daily, weekly, monthly …, and if you are a high frequency trader you may even want  to compute millisecond returns. Since returns can be computed over different time periods, they need to be chained to obtain  lower frequency returns. This is to say that if you have daily returns and you need annual returns, you have to find a satisfactory way to aggregate the daily returns.  You already guessed it: there are different ways to do it and they do not yield the same result.

Let’assume you have n daily discrete returns called r that you want to aggregate into a compounded annual return. The usual way to do it is this:

CAGR = { ∏ (1+r) } ^ (252 /n) -1          (1)

where CAGR stands for compounded annual growth rate (another way of saying compounded annual return). The exponent in (1) contains 252, because it is usually assumed that financial markets are open for business 252 days in a calendar year and that the series of daily returns  does not include entries for the weekends (if it does, 365 should be used instead of 252). Note that if you have exactly one years (252) daily returns, the exponent collapses to 1. Obviously, if you have monthly returns and you need to compute the CAGR, the exponent in (1) becomes (12 / m) where m is the number of monthly returns used for the computation.

The chaining rule in (1) is computed starting from discrete returns, in fact if what you have are log returns, the computation is much simpler, as they are additive. The annual log growth rate will just be the arithmetic average of the daily log returns times a scaling factor (252 for daily returns, 12 for monthly returns, etc.). However, preactioners tend to stick to the CAGR that always shows a higher value than the corresponding annual log return.

Another way that can be used to convert daily discrete returns into an annual average return is achieved simply taking the arithmetic mean of these returns and rescaling it over 252 days. The result can be very different from the corresponding CAGR, but it does represent an unbiased estimate of the returns for the following period. Indeed some scholars have rightly pointed out that discrete returns should be used as return forecasts when solving for the portfolio efficient frontier. No doubt about it, although   the many heroic assumptions underlying modern portfolio theory on which the efficient frontier concept is built ensure that the result is always only a very general approximation. Return computation becomes understandably even more complex introducing the possibility of inflows and/or outflows during the period over which the performance is computed. As the gist of this post is simply to highlight  that the measurement of financial returns is not the well defined and written in stone procedure that may appear at first sight, I will gladly avoid discussing such a sticky matter as the treatment of inflows and outflows and I refer the interested (if somewhat masochistic) reader to this nice paper.

In all cases, better keep in mind that it may be dangerous (especially for the portfolio) to compare the returns or the performance indices of different investment vehicles without making sure that they are computed consistently with each other. And they almost never are.

References

For a much, much better, but more intellectually challenging ;=) , explanation about log returns, see here.

If you know some R, the package PerformanceAnalytics easily performs most possible return calculations. I’ll come back on the many uses of R in finance in some of the future posts. Maybe.