Comparing Pension Contributions with Return Changes Using Fund Sensitivities
Summary
The document models a pension fund as the accumulated value of periodic contributions, where income grows continuously and each contribution earns a continuously compounded return until the valuation date. It derives sensitivities of the fund value to the contribution rate and expected return. The question is how to translate a small change in contribution rate into an equivalent change in return by matching their effects on the fund.
The author asks whether the time factor inside the return sensitivity can be extracted from the summation to yield a simpler expression for the required return change. The proposed shortcut would separate that factor from the contribution sensitivity, but the document provides no answer or validation of that simplification. In general, the terms vary across contribution periods, so their relationship depends on the full sum and its time weights. The practical motivation is to avoid repeatedly evaluating many candidate return changes and selecting the closest fund impact.
Key ideas
- The fund value is represented as a sum of growing contributions compounded to the valuation date.
- The document differentiates fund value with respect to the contribution rate and expected return.
- It seeks an equivalent return change that matches the fund impact of a contribution-rate change.
- The proposed simplification is posed as a question and is not established by evidence in the document.
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# Numerical method to extracting a piece of a summation function?
# Numerical method to extracting a piece of a summation function?
So this is a pension framework. I am trying to code a system and I don't want to have to brute force this answer, but I can't figure out a clean solution.
$$Fund = \sum_{i=1}^t [\cfrac{I\cdot e^{\frac{\pi i}{12K}}}{12K} \cdot C \cdot e^{\frac{Ri}{12K}}]$$
$I = $, annual income, $K = $ pay periods per month, $C =$ Contribution Rate (%), $R =$ expected annualized return (continuous), $\pi =$ expected annual income growth (continuous)
Solving for the derivatives:
$$ \cfrac{dFund}{dC} = \sum_{i=1}^t [\cfrac{I\cdot e^{\frac{\pi i}{12K}}}{12K} \ \cdot e^{\frac{Ri}{12K}}]$$
$$\cfrac{dFund}{dR} = \sum_{i=1}^t [\cfrac{I\cdot e^{\frac{\pi i}{12K}}}{12K} \cdot C \cdot e^{\frac{Ri}{12K}} \cdot \frac{i}{12K}]$$
If $\Delta C = 0.01$, $\Delta Fund_{C} = \Delta C \cdot \cfrac{dFund}{dC}$
How do I solve for $\Delta R$ if I want $\Delta R \cdot \cfrac{dFund}{dR} = \Delta Fund_{C} = \Delta C \cdot \cfrac{dFund}{dC}$?
Basically, is there a way to extract the value of the $\cfrac{i}{12K}$ term within the summation so that it can be expressed outside the summation?
## Edit:
The goal is that by doing so, the problem would easily simplify to $\Delta R \cdot C \cdot \Sigma \frac{i}{12K} \cdot \frac{dFund}{dC} = \Delta Fund_C$, such that I could just solve for $\Delta R = (C \cdot \Sigma \frac{i}{12K})^{-1}$. Currently I using my code to calculate $\Delta Fund_R$ for a large sequence of $\Delta R$ values and then matching the closest $\Delta Fund_R$ to $\Delta Fund_C$. Incredibly inefficient from a resource standpoint.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.