My Finance

Assume a stock cost 10$ on March 31, 2015, and 12$ on June 30, 2015. In addition, it paid dividend 0.5$ on June 12, 2015. Price return is computed as
R = 12/10 – 1 = 20%, and total return (including dividends) as Q = (12+0.5)/10 – 1 = 25%. These are arithmetic returns.

These are good for computing portfolio returns: For three stocks A, B, C, with total returns 10%, -5%, and 20%, consider the portfolio investing in them in proportions 40%, 30%, 30%. Then the total return of this portfolio is 0.4*10% + 0.3*(-5%) + 0.3*20% = 8.5%. Same is true if we are speaking of price return.

❌ However, compound interest presents a problem: If in January total return is 10% and in February it is 20%, then the overall total return is not 10% + 20% = 30%. Instead, it is 32%. Why? Because we get February return not simply on the original wealth but on the wealth gained in January. Indeed, if we start with wealth 100 on the New Year’s day then on January 31 we get 110 and on February 28, we get: 110*(1 + 20%) = 132, thus 100 turns into 132 which gives us 32%. Same problem if we got price returns instead of total returns.

Thus analysts sometimes use geometric (logarithmic) returns: Revisit the example at the beginning of this post. The price return is ln(12/10) = ln(1.2) = 18.2%, and the total return is ln(12.5/10) = ln(1.25) = 22.3%.

It is easy to convert arithmetic return into geometric return: If R is the arithmetic return, then G = ln(1+R) is the geometric return. Conversely, R = exp(G) – 1. But do not forget that percentages must always be converted to fractions, for example 0.05 instead of 5%.

This solves the compound interest problem: The total geometric return in January is ln(1 + 10%) = ln(1.1) = 9.5% and in February . Thus the overall total geometric return is 9.5% + 18.2% = 27.7% = ln(1+32%) = ln(1.32).

On the other hand, for geometric returns one cannot compute the portfolio return out of stock returns like we did for arithmetic returns. You need to convert it back to arithmetic returns.

Conclusion: Use geometric returns for analysis of financial time series, and arithmetic returns for portfolio analysis.