Closed-Form Capital Needs for Annual Withdrawals and Fixed Growth
Summary
This document addresses how to choose starting capital so that a portfolio funds a specified stream of annual expenses and reaches zero at a chosen horizon. It describes a forward recurrence: grow the remaining investment by a fixed rate, then subtract each year’s expense. Rather than repeatedly searching for an initial value that produces a zero ending balance, the answer models each withdrawal as an amount set aside at the start of the year and discounts future withdrawals to the starting date.
For level annual withdrawals and a constant growth rate, the required capital is the present value of a geometric series. The answer gives a closed-form expression and checks its interpretation with short-horizon examples. This avoids iterative search under those assumptions. The document does not provide a formula for the original case of expenses that vary by year; those cash flows can instead be valued by discounting each known withdrawal individually. The calculation also depends on deterministic growth and timing conventions, so it does not capture uncertain returns, inflation, or mortality variation.
Key ideas
- With fixed growth, a level withdrawal stream can be valued as a geometric series of discounted expenses.
- A closed-form present value can replace a binary search when the withdrawal and growth assumptions are constant.
- Withdrawal timing matters because funds are set aside at the beginning of each year in the answer’s setup.
- For expenses that vary by year, each known cash flow must be discounted according to its timing.
- The calculation assumes deterministic growth and does not model uncertainty in returns or lifespan.
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# Calculating the ideal initial capital value to optimize a growth model
# Calculating the ideal initial capital value to optimize a growth model
I'm trying to work out a method for finding the initial capital value that allows someone to run out of money at the exact time they reach mortality. Currently, I'm graphing the annual total capital values through an iterative method. I have a pre-calculated list of the annual expenses (discrete values unique to each year), and I know the growth.
The iterative method I use when I know the initial capital value is simple, and works as follows Given an initial value $x_0$ , the next year's total capital value is the growth of the previous year's value $x_0(1 + growth)$ minus the expenses for that year. This value, $x_1$ , is then used to determine the next year's total $x_1(1+growth) - expenses$, and so on until we reach the mortality date.
I now need to find the initial value $x_0$ given that the final value must be 0. I've tried implementing a type of binary-search simulation where I try find the starting value that gives the end value closest to 0, but that is both fairly computationally expensive for a live graphing application as well as quite an ugly solution.
Is there a better method for solving this problem? Or is this simply a boundary condition that cannot be inferred from the end result?
Thanks in advance for your help
## Answer by nbbo2 (score 1)
https://quant.stackexchange.com/a/35550
At the end of each year you have wealth $X_t$ in an investment account which grows at the rate $r$.
At the beginning of each year you withdraw the amount $W$ and keep it in cash to pay for your expenses through the coming year.
For your investment to last $n$ years you need to start with $X=W+W\frac{1}{1+r}+\cdots+W\frac{1}{(1+r)^n}$. In this way you will be able to live for $n+1$ years
Using geomteric series summation formula this can be rewritten in closed form as $X=\frac{W}{r}[1+r-\frac{1}{(1+r)^n}]$
For example to live for 1 year ($n=0$) you need to start with $W$ (which you will immediately withdraw for spending)
To live for 2 years ($n=1$) you need $W\frac{2+r}{1+r}$
And so on.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.