\begin{algorithmic}
\Require $n=2$, $M\in\mathbb{N}$ sufficiently large
\Require $X_{1i}$ are iid and continuous with pdf $f$ and cdf $F$ for $i=0,1,...,k-1$
\Require $X_{2i}=1+r$, $r\geq0$ for $i=0,1,...,k-1$
\Require $G_1=\{.1,.2,...,.9\}$
\State $i\gets k$ \Comment{initialize i}
\While{$i>0$}
\State $i\gets i-1$
\For{$a\in\{1,2,...,a_0\}$}
\State $D_{a,i}\gets\Big\{\frac{jm_{a,i}}{M}:j=1,2,...,M-1\Big\}$
\For{$x\in D_{a,i}$}
\State $q_i^*(x,a)\gets 0$ \Comment{initial proposal for $q_i^*(x,a)$}
\State $v_i(x,a)\gets \eqref{eq:gi0}$\Comment{proposal for $v_i(x,a)$, see \ref{s:valg}}
\State $q_1\gets\underset{q\in G_1}{\arg\max}\ \eqref{eq:maxqh}$\Comment{see \ref{s:valg}}
\State $G_2\gets\{q_1\pm .01j: j=-9,-8,...,10\}$
\State $q_2\gets\underset{q\in G_2}{\arg\max}\ \eqref{eq:maxqh}$
\State $V\gets \eqref{eq:maxqh}\vert_{q=q_2}$
\If{$v_{i}(x,a)<V$}
\State $q_i^*(x,a)\gets q_2$ 
\State $v_i(x,a)\gets V$
\EndIf
\EndFor
\EndFor
\EndWhile\\
\Return{$v_i(x,a),\ \boldsymbol\pi_i=(q_i^*(x,a),1-q_i^*(x,a))$ for $a\in\{1,2,...,a_0\}$, $x\in D_{a,i}$ and $i=0,1,...,k-1$}
\end{algorithmic}

\begin{algorithmic}
\Require $n=2$, $M\in\mathbb{N}$ sufficiently large
\Require $X_{1i}$ are iid and continuous with pdf $f$ and cdf $F$ for $i=0,1,...,k-1$
\Require $X_{2i}=1+r$, $r\geq0$ for $i=0,1,...,k-1$
\Require $G_1=\{.1,.2,...,.9\}$
\State $i\gets k$ \Comment{initialize i}
\While{$i>0$}
\State $i\gets i-1$
\For{$a\in\{1,2,...,a_0\}$}
\State $D_{a,i}\gets\Big\{\frac{jm_{a,i}}{M}:j=1,2,...,M-1\Big\}$
\For{$x\in D_{a,i}$}
\State $q_i^*(x,a)\gets 0$ \Comment{initial proposal for $q_i^*(x,a)$}
\State $v_i(x,a)\gets \eqref{eq:gi0}$\Comment{proposal for $v_i(x,a)$, see \ref{s:valg}}
\State $q_1\gets\underset{q\in G_1}{\arg\max}\ \eqref{eq:maxqh}$\Comment{see \ref{s:valg}}
\State $G_2\gets\{q_1\pm .01j: j=-9,-8,...,10\}$
\State $q_2\gets\underset{q\in G_2}{\arg\max}\ \eqref{eq:maxqh}$
\State $V\gets \eqref{eq:maxqh}\vert_{q=q_2}$
\If{$v_{i}(x,a)<V$}
\State $q_i^*(x,a)\gets q_2$
\State $v_i(x,a)\gets V$
\EndIf
\EndFor
\EndFor
\EndWhile\\
\Return{$v_i(x,a),\ \boldsymbol\pi_i=(q_i^*(x,a),1-q_i^*(x,a))$ for $a\in\{1,2,...,a_0\}$, $x\in D_{a,i}$ and $i=0,1,...,k-1$}
\end{algorithmic}
{
    "title": "withdrawal success optimization in a pooled annuity fund",
    "id": "2402.17164",
    "abstract": "consider a closed pooled annuity fund investing in n assets with discrete-time rebalancing. at time 0, each annuitant makes an initial contribution to the fund, committing to a predetermined schedule of withdrawals. require annuitants to be homogeneous in the sense that their initial contributions and predetermined withdrawal schedules are identical, and their mortality distributions are identical and independent. under the forementioned setup, the probability for a particular annuitant to complete the prescribed withdrawals until death is maximized over progressively measurable portfolio weight functions. applications consider fund portfolios that mix two assets: the s&p composite index and an inflation-protected bond. the maximum probability is computed for annually rebalanced schedules consisting of an initial investment and then equal annual withdrawals until death. a considerable increase in the maximum probability is achieved by increasing the number of annuitants initially in the pool. for example, when the per-annuitant initial contribution and annual withdrawal amount are held constant, starting with 20 annuitants instead of just 1 can increase the maximum probability (measured on a scale from 0 to 1) by as much as .15.",
    "categories": "q-fin.mf",
    "doi": "",
    "created": "2024-02-26",
    "updated": "",
    "authors": [
        "hayden brown"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2402.17164"
}
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