Equitable Lump-Sum Allocation Across Different Retirement Dates

Summary

The note derives a way to divide a total fund among people who retire at different times so that each allocation grows to the same value at that person’s retirement. It assumes a constant return and annual compounding. Each person’s share is set in proportion to the present value of one unit payable at their retirement date, using the corresponding discount factor. Normalizing these weights so they sum to one ensures the allocations add up to the available total.

The construction directly satisfies both requirements: the shares sum to the fund, and compounding each share to its recipient’s retirement date gives an equal amount. It works for any finite set of retirement dates under the stated assumptions, provided the growth factors are defined and the discount-factor sum is nonzero. The note does not address uncertain returns, differing risk preferences, taxes, or alternative meanings of equitable treatment.

Key ideas

• Weight each recipient’s allocation by the discount factor for their retirement date.
• Normalize the discount-factor weights so their sum equals one.
• The resulting shares sum to the available fund and grow to equal retirement values.
• The solution assumes a constant return with annual compounding.

Tags

# Equitable Allocation


# Equitable Allocation












This questions borders on the actuarial side of things but the general solution should have relevance in several situations. Suppose we have a set of $k$ people who will retire in $\{n_1,...,n_k\}$ years respectively. We say an allocation of $M$ is equitable if each person receives an equivalent lump sum value at retirement. Let's assume the rate of return is constant and equal to $r$ and we will denote each persons allocation as $m_i$ respectively. We then have the following system,

$M=m_1+...+m_k$ $m_i(1+r)^{n_i}=m_j(1+r)^{n_j}$

For all values of $i$ and $j$. It is not clear to me that we can guarantee a solution for general values of k and n_i. Especially as the number of people grow. Any help is greatly appreciated!

## Answer by hjs (score 2)

https://quant.stackexchange.com/a/22995

You're writing it in terms of the growth factors and annual compounding. You want to split up $M$ so that as each piece grows over time, the $i$th person at time $n_i$ gets paid the same amount as the $j$th person gets at time $n_j$. So simply scale by the corresponding discount factors. Let $$ \alpha_i = \frac{(1+r)^{-n_i}}{\sum_j (1+r)^{-n_j}} $$

Then $$ \sum \alpha_i = 1 $$ and $$ \alpha_i (1+r)^{n_i} = \alpha_j (1+r)^{n_j} $$ so you can define $$ m_i = M \alpha_i $$ and then $$ \sum_i m_i = M $$ and $$ m_i (1+r)^{n_i} = m_j (1+r)^{n_j}, $$ as desired.

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