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:
$$M_s = \prod_{u=s+1}^{t+10} (1 + \text{Port}_u)$$
- Explanation: M_s represents the cumulative effect of the portfolio's returns from day s + 1 to t + 10[Important]
Step 2: Compute the Security's Cumulative Contribution
For each security i, calculate its cumulative contribution over the period:
$$\text{CTRi}_{t,t+10} = \sum_{s=t}^{t+10} \left( \text{CTRi}_s \times M_s \right) $$
- Explanation: Multiply each day's contribution of security i (CTRi_s by the corresponding multiplier M_s and sum over all days.
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:
$$\text{Port}_{t,t+10} = \prod_{s=t}^{t+10} (1 + \text{Port}_s) - 1 $$
