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Consider the quarterly sales figures and returns in the table below:

Qtr Sales ($m) Return (%)
Q4 19 8
Q1 20 10 25
Q2 20 13 30
Q3 20 15.6 20
Q4 20 23.4 50

The year on year return for Q4 20 is: $$ Ret = \frac{R_4}{R_1}-1 = \frac{23.4}{8} - 1 = 1.925 = 192.5\% $$ Which can also be found geometrically as: $$ Ret = \left(\prod(1+R_n)\right)-1 $$ $$ Ret = (1.25 * 1.3 * 1.2 * 1.5) - 1 = 1.925 = 192.5\% $$ Suppose you want to find year on year performance where each of the quarters had a different weighting. E.g. the Q1-3 at 20% and Q4 at 40%. This should give returns from Q4 more importance.

What is the mathematical method for applying weights to geometric returns?

Pedro Godwand
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1 Answers1

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In this case, you would just have a year-on-year return of $1.2^3 \cdot 1.4 - 1 \approx +142 \%$ return, where I assume you mean that quarters 1, 2 and 3 all have a $20 \%$ quarterly return. Although $1.2 < 1.4$, $1.2^3 > 1.4$ so the $1.4$ does not contribute as much to the total product.

What you probably want is a weighted average where for example, the rate of return for Q4 is twice as important as the other quarters. We can assign quarters $1, 2, 3$ a weight of $1$, and quarter $4$ a weight of $2$. However, since the sum of the weights is now $5$ where it was previously $4$, we need to take the $4/5$th root so that the total sum of weights is the same, which gives:

$$\left(1.2^1 \cdot 1.2^1 \cdot 1.2^1 \cdot 1.4^2 \right)^{4/5} - 1 \approx 165 \%$$

In general, with weightings $w_1, w_2, \cdots w_n$ and growth rates $r_1, r_2, \cdots r_n$, the weighted percentage growth rate will be:

$$({r_1}^{w_1} \cdot {r_2}^{w_2} \cdots {r_n}^{w_n})^{n/(w_1+w_2+\cdots+w_n)} - 1$$

Toby Mak
  • 16,827
  • Thanks for this. I had read about weighted geometric mean but wasn't sure it applied to compounding returns. In your generalised example at the bottom, what is there reason for it being $w_n^{r_n}$ as opposed to $r_n^{w_n}$ – Pedro Godwand Mar 08 '21 at 08:44
  • Thanks for pointing out the correction: it should be ${r_n}^{w_n}$ as you have said. – Toby Mak Mar 08 '21 at 10:19