Recovering Quarterly Returns from Rolling Annual Returns
Summary
The document considers whether individual quarterly returns can be recovered from a sequence of rolling one-year returns. Each rolling figure combines several quarterly returns, so the unknown quarterly values can be represented as a system of equations. The response explains that the system is generally underdetermined when the starting quarterly returns are unavailable: several possible return sequences may fit the observed rolling totals.
If the initial quarterly returns are known along with the rolling cumulative returns, later quarterly returns can be solved sequentially. When those starting values are missing, a pseudo-inverse can provide an approximate solution to the system, but it does not establish that the inferred series is the unique or actual return history. The example supplies sample quarterly and rolling returns, but the response does not work through the calculations. The method also depends on matching the definition and compounding convention used to construct the rolling returns.
Key ideas
- Each rolling annual return imposes a constraint on the quarterly returns within its window.
- Without enough starting observations, the system can have multiple solutions.
- Known initial quarterly returns allow subsequent returns to be recovered from rolling figures.
- A pseudo-inverse can produce an approximation when the system is underdetermined.
- The inferred values depend on the convention used to calculate cumulative returns.
Tags
Full text
# How can I convert rolling annual returns back to quarterly returns? # How can I convert rolling annual returns back to quarterly returns? I have a series of rolling annual returns and would like to convert these back to quarterly returns, which have not been provided. Is this possible formulaically, or is something like Excel's solver required? Sample data below, quarterly returns are provided (still need to calculate column four): ``` +---------+-----------------------+------------------------+--------------------------------+ | quarter | quarterly returns (%) | rolling 1y returns (%) | back out quarterly returns (%) | +---------+-----------------------+------------------------+--------------------------------+ | 1 | 5 | | | +---------+-----------------------+------------------------+--------------------------------+ | 2 | 4 | | | +---------+-----------------------+------------------------+--------------------------------+ | 3 | 3 | | | +---------+-----------------------+------------------------+--------------------------------+ | 4 | 2 | 14.7 | | +---------+-----------------------+------------------------+--------------------------------+ | 5 | 1 | 10.36 | | +---------+-----------------------+------------------------+--------------------------------+ | 6 | 2 | 8.23 | | +---------+-----------------------+------------------------+--------------------------------+ | 7 | 3 | 8.23 | ? | +---------+-----------------------+------------------------+--------------------------------+ | 8 | 4 | 10.36 | ? | +---------+-----------------------+------------------------+--------------------------------+ | 9 | 5 | 14.73 | ? | +---------+-----------------------+------------------------+--------------------------------+ | 10 | 6 | 19.22 | ? | +---------+-----------------------+------------------------+--------------------------------+ ``` Thanks! ## Answer by Kermittfrog (score 2, accepted) https://quant.stackexchange.com/a/53830 Generally speaking, you pose a system of linear equations that is undetermined. If you provided the initial 3 quarterly returns together with the cumulative returns then yes, all other quarterly returns can be backed out. Else, you could try to approximate the other returns using the pseudo-inverse. This answer here has a very nice example on that topic. https://stats.stackexchange.com/questions/67907/extract-data-points-from-moving-average
Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.