Building a Daily Rebalanced Index from Weighted Returns
Summary
The document describes how to combine three asset return series into a portfolio index with fixed weights of 50%, 25%, and 25%. It recommends calculating each asset's simple percentage return, taking the weighted sum at each time step, adding one to obtain each period's growth factor, and compounding those factors to form the index. It cautions against directly combining log returns for this construction.
The resulting series represents a portfolio rebalanced back to the target weights every day. The explanation illustrates that rebalancing resets each holding to its weighted share of the updated portfolio value. This is a standard backtest convention, but the document notes that few portfolios actually rebalance daily. It does not compare alternative rebalancing schedules, incorporate transaction costs, or show empirical performance, so the index is a theoretical benchmark for the stated daily-rebalanced rule.
Key ideas
- Combine simple percentage returns using the portfolio's target weights at each time step.
- Compound one plus each weighted portfolio return to construct the index level.
- The method represents a portfolio rebalanced to constant weights every day.
- Daily rebalancing may not reflect how many real portfolios are managed.
- The described construction does not account for transaction costs or compare other rebalancing frequencies.
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Full text
# Index creation from multiple time-series and variable weights
# Index creation from multiple time-series and variable weights
I am trying to compose one index out of several (three) indices with variable weights, 50%, 25% and 25%.
After normalizing and calculating the log returns, what would be the best way to create the final benchmark index. We would also require to re-balance daily?
Many thanks in advance!
## Answer by Air (score 2, accepted)
https://quant.stackexchange.com/a/15308
You can't really combine the assets' log returns. You should calculate percentage returns for the three assets. Then at each time step, the portfolio's total return is:
$r(i) = 0.5 \times \text{asset1_return}(i) + 0.25 \times \text{asset2_return}(i) + 0.25 \times \text{asset3_return}(i)$
Once you've calculated the time series of the portfolio's returns, you create another time series by adding 1 to each return, $r(i)$ -- let's call this series "$s$". It is the daily return of a portfolio with 50% in asset 1, 25% in asset 2 and 25% in asset 3.
Then the portfolio index is created by multiplying each successive element of "$s$" with all the previous elements. This can be written recursively as:
$\text{Index}(i) = \text{Index}(i-1) \times s(i)$
This index then represents what a daily rebalanced portfolio would perform as.
Importantly, the index weights are constant, so if you had \$100 invested the first day, the portfolio would consist of \$50 in the first asset and \$25 each in assets 2 and 3. If on the second day, the portfolio value was \$103, then the procedure above assumes that the portfolio is rebalanced to consist of \$51.50 in the first asset and \$25.75 in the second and third assets.
Note that I've just asked some questions about this procedure elsewhere on this site -- this is the classic way to backtest a strategy but in reality few portfolios rebalance daily like this.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.