Cumulative Returns: two approaches to calculation
## [1] "SPY"What’s going on here?
Both of these series represent legit ways of calculating cumulative returns, from the same data, but they look quite different. So what the heck is going on? Which should we use?
Two methods for calculating cumulative returns:
- simple/arithmetic chaining
- geometric chaining
Both are “correct,” but the latter is recommended for what we’re doing, as it takes into consideration the effects of compounding. As we see in the chart above, the the simple method underestimates how long it takes to recover from the effects of the financial crisis. It also underestimates the long run benefits of compounding.
How are the methods different?
The simple chaining method:
To demonstrate, we need some returns. Let’s use SPY prices, to calculate returns, in decimals. We can convert to percent at the end.
We’re familiar with calculating returns from prices using the formula: (new/old) - 1
| Date | SPY Adj Close | Spy Returns |
|---|---|---|
| 2007-01-03 | 106.9321 | NA |
| 2007-01-04 | 107.1590 | 0.0021221035 |
| 2007-01-05 | 106.3043 | -0.0079764069 |
| 2007-01-08 | 106.7960 | 0.0046255427 |
| … | … | … |
| 2020-12-07 | 369.09 | -0.0020549141 |
With simple/arithmetic chaining, we just add up every return from from time 1 to time n, like (r1+r2+…rn). For example:
| Spy Returns | Cumulative Returns (simple) | How to Calculate: |
|---|---|---|
| NA | NA | NA |
| 0.0021221035 | 0.002122104 | 0.0021221035 (it’s just the first return) |
| -0.0079764069 | -0.005854303 | (0.0021221035 + -0.0079764069) |
| 0.0046255427 | -0.001228761 | (0.0021221035 + -0.0079764069 + 0.0046255427) |
| … | … | … |
| -0.0020549141 | 1.53655 | 0.0021221035 + … -0.0020549141 |
We can compare this (1.53, or 153%) with what we get by using the (new/old) - 1 formula to calculate what the buy-hold return is: (369.09/106.9321)-1 = 2.45163 or 245.16%. Clearly these figures are not the same. The reason is that the simple chaining method does not include compounding, leading to a pretty dramatic underestimation of performance.
The geometric chaining method:
Notice that in the chart above, the last value in the series calculated with geometric chaining does agree with our calculation of 245.16%. This is because the geometric method is just a generalization of the (new/old) - 1 formula we’re used to.
Via some math (that i’m omitting), it can be shown that we can extend the (new/old) -1 formula to calculate the cumulative return over more than just 2 time points by rewriting it in the form:
-(Bacon, Carl. Practical Portfolio Performance Measurement and Attribution. Wiley. 2004)
Using the same data as before:
| Spy Returns | Cumulative Returns (geometric) | How to Calculate: |
|---|---|---|
| NA | NA | NA |
| 0.0021221035 | 0.002122104 | 0.0021221035 (it’s just the first return) |
| -0.0079764069 | -0.005871230 | (1+0.0021221035)*(1+ -0.0079764069) = 0.9941288 - 1 = -0.00587123 |
| 0.0046255427 | -0.001272845 | (1+0.0021221035) * (1+ -0.0079764069) * (1+0.0046255427) = 0.9987272 - 1 = -0.001272845 |
| … | … | … |
| -0.0020549141 | 2.451629 | (0.0021221035)* … (-0.0020549141) = 3.451629 - 1 = 2.451629 |
We can see that the result of this method matches what we get with the familiar (new/old) - 1 formula. We’ll be switching over to the geometric method.