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!


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.

|improve this answer|||||

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.