Compounding Daily Security Contributions into Portfolio Return
Summary
The document explains how to aggregate daily security-level contributions to return (CTR) across a multi-day period while preserving the portfolio total. It first sums security contributions for each day to obtain the portfolio’s daily return, then compounds daily portfolio returns across the period.
To allocate the compounded result back to securities, it weights each security’s daily CTR by the portfolio growth multiplier from the following days through the end of the period, and sums those adjusted contributions over time. The proposed allocation is intended to sum to the portfolio’s compounded return. The source provides the equations but no worked numerical example or empirical evidence. Its notation has a potential indexing ambiguity: the multiplier excludes the current day, and the portfolio-return equation should be read as a compounded return less one when returns are expressed as rates.
Key ideas
- Daily portfolio return is the sum of security-level contributions for that day.
- Multi-day portfolio performance compounds the daily portfolio returns.
- A security’s period contribution is formed by weighting each daily contribution by subsequent portfolio growth.
- The allocated security contributions are intended to add up to the portfolio’s cumulative return.
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Full text
# Contribution to Return - from security to portfolio
# Contribution to Return - from security to portfolio
I have the Contribution to Return (CTR) of all securities in a portfolio for a number of days. I would like to compute the portfolio and securities total return over this period.
The total return of the portfolio for day $t$ can be computed as $Port_{t} = \sum_{i} CTR_{i}$
The portfolio's total return over a number of days can then be computed as $Port_{t, t+10} = \prod_{i} (1+Port_{i})$
How would you compute security $CTR_{i, t, t+10}$ (the total contribution to return of security i) over the period in such a way that $\sum_{i} CTR_{i,t,t+10} = Port_{t, t+10}$?
## Answer by Marut Dubey (score 1)
https://quant.stackexchange.com/a/80598
To compute the total contribution to return (CTR_i₍ₜ,ₜ₊₁₀₎) for each security over a period (from day t to t + 10) such that the sum of all securities' contributions equals the portfolio's total return over that period (Port₍ₜ,ₜ₊₁₀₎), you can use the following method:
Step 1: Calculate the Multiplier for Each Day (Mₛ)
For each day s from t to t + 10, calculate the cumulative multiplier Mₛ, which accounts for the compounding effect of the portfolio's returns in the remaining days:
$$M_s = \prod_{u=s+1}^{t+10} (1 + \text{Port}_u)$$
Step 2: Compute the Security's Cumulative Contribution
For each security i, calculate its cumulative contribution over the period:
$$\text{CTRi}_{t,t+10} = \sum_{s=t}^{t+10} \left( \text{CTRi}_s \times M_s \right) $$
Step 3: Verify the Sum Equals the Portfolio's Total Return
Ensure that the sum of all securities' cumulative contributions equals the portfolio's cumulative return over the period: $$\sum_i \text{CTRi}_{t,t+10} = \text{Port}_{t,t+10}$$
- Portfolio's Total Return:
$$\text{Port}_{t,t+10} = \prod_{s=t}^{t+10} (1 + \text{Port}_s) - 1 $$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.