GPT Prompt05 - Perfect Withdrawal Rates

We are here in my GPT Prompt Series

• GPT Prompt 01 - Spending Strategy Comparisons
• GPT Prompt 02 - Lifetime Probability of Ruin (LPR)
• GPT Prompt 03 - Dynamic HJB Spend Optimization 
• GPT Prompt 04 - Portfolio Longevity Heat Map 
• GPT Prompt 05 - Perfect Withdrawal Rate (PWR) <-- [this post]
• GPT Prompt 06 - Stochastic Present Value (SPV)

In 2015, Suarez, Suarez and Waltz, out of Trinity University, wrote a paper "The Perfect Withdrawal Amount: A Methodology for Creating Retirement Account Distribution Strategies" that describes the Perfect Withdrawal, the amount, with perfect foreknowledge of returns, that can be periodically taken from a portfolio over some interval (a life or horizon) and then end at some terminal boundary (say death) with zero dollars (or alternatively with a remainder bequest).  From the paper, the core math is this: 

where n is in their case a fixed horizon, ri... is a known sequence of returns and K(E) is a bequest and K(s) is the starting endowment. We can set K to zero in the no bequest case and we can also animate r via simulation while also making N a life contingent distribution of possibilities using actuarial math. That latter N thing was my innovation added onto Suarez by the way, it was not in the paper. That version can be represented more like this (Suarez writes it in accumulated-value form; I wrote it in discounted-value form):

This approach allows us to represent w not just as a discrete number but rather as a distribution, a distribution that can now be inspected and where w can be "evaluated" based on policies related to risk preferences. In the evaluation process the right tail is less interesting than the left tail because the return path and it's "sequences" can constrain w to low values which is where all the fun and angst is in retirement. 

The life-contingent terminus will distort the distribution (vs a fixed horizon) because in those simulated lives where the end is in the next year or two, with perfect foreknowledge, the math will spend it all this year or two so the life-contingent distribution will have a higher preponderance of very high spend rates. But then again PWR can be run either way or both. 

Here is the AI generated narrative just for fun:

----- AI narrative start -----

Overview

The Perfect Withdrawal Rate (PWR) is the constant real spending rate that exactly exhausts a retirement portfolio at a specified terminal boundary, given a stochastic return path. Unlike heuristic “safe withdrawal rules,” PWR is defined path-by-path and is therefore a distribution, not a scalar.

Formally, PWR solves the retirement budget constraint exactly: wealth reaches zero (or a specified bequest) at the terminal date and not before.

This domain is concerned with sustainability, not valuation. It answers:

Given a return process and horizon, what constant spending rate is feasible?

[ math definition is redacted because we already covered that]

Interpretation

• The denominator is the present value cost of sustaining a constant real withdrawal of 1 per year, plus the discounted value of the bequest.

• The PWR is simply the reciprocal of that cost.

• Each Monte Carlo path produces one exact feasible withdrawal rate.

• Aggregating paths yields a distribution of feasible spending rates.

There is no notion of “failure” in this domain. Every path has a PWR by construction.

Random Lifetime Extension

Let$N\sim \text{Gompertz}(x)\; be\; a\; stochastic\; remaining\; lifetime,\; conditioned\; on\; survival\; through\; the\; first\; year.$

Then:

$$\boxed{ w^\ast(\omega) = \left( \sum_{t=1}^{N(\omega)} \frac{1}{\prod_{k=1}^{t}(1+R_k(\omega))} \right)^{-1} }$$

Each simulation draw now includes both a return path and a lifetime realization.

This transforms PWR into a life-contingent sustainability distribution, directly comparable to annuity factors and mortality-weighted SPV results.

Relationship to Other Domains

PWR vs. SWR

•    SWRs ask: “What spending rate rarely fails?”
•    PWR asks: “What spending rate exactly fits this world?”

PWR vs. SPV

•    SPV prices a spending plan.
•    PWR solves for the spending plan that prices to one unit of wealth.
•   They are mathematical inverses under stochastic discounting.

Why the Distribution Matters

The mean or median PWR hides critical structure:

• Left tail = fragile sequences (early drawdowns)
• Right tail = unusually favorable return paths
• Width = sensitivity to volatility and sequencing

[ etc etc it goes on but that covers the basics. ]

----- AI narrative end -----

The Prompt

Objective:

Simulate the distribution of perfect withdrawal rates — the constant real withdrawal rate that exactly depletes wealth at death — where lifetime T is random per the Gompertz survival model.

---- Prompt begin: -----

You are an expert in stochastic simulation and retirement finance.

Simulate the distribution of the Perfect Withdrawal Rate (PWR) — the constant real withdrawal rate that exactly depletes wealth at death — with random lifetime.

Definitions:

For each path i:

1 = w_i \sum_{t=1}^{T_i} \prod_{k=1}^{t}(1+r_{k,i})^{-1}

Lifetime T_i is drawn from a Gompertz survival distribution

S(a)=e^{-e^{(a-m)/b}},\quad {}_t p_x = S(x+t)/S(x)

with x=67, m=88, b=9 and survival conditioned on year 1 (p₁ = 1).

Parameters:

   •   Portfolio: μ = 0.04 (real arithmetic), σ = 0.12, annual lognormal returns

   •   Start wealth = 1.0

   •   Spending: constant real rate w_i (solved per path)

   •   Annual rebalancing

   •   Simulations = 100 000 paths

Outputs:

   •   Distribution of w_i with P5, P25, P50, P75, P95

   •   PDF and CDF charts annotated with μ, σ, and Gompertz parameters

---- Prompt end -----

Output Examples

Notes

• The axes can be scaled differently by asking Chat
• The horizon can be fixed if you ask
• The two distributions, fixed and life contingent, can be overlaid and compared
• The 5th percentile of the distribution can be likened to a fail-rate policy metric and is of interest
• The way it's written: withdrawals for a life of 1 year can be greater than the endowment (1)
• The 5th percentile at 3.66% seems reasonable for a 65 year old in the current environment (imo)
• Returns are "real" so 4%/12% might be a 60/40-like portfolio
• The return distribution could be skewed if you wanted 
• The life distribution could be pushed hard to something like an individual annuitant table

Comments

Do I use this is my own life? Not really. It is a tool among many used to give triangulated insight into the retirement spending problem. It is confirmatory or inspires new questions that PWR or other tools might answer. Since all retirement tools use the same inputs related to return, vol, horizon, etc. they are often all just answering the same thing but coming at it from different angles. Read the paper and see what the authors were thinking. 

Some Caveats

- The prompting process seems unstable for each re-prompt and yields varying results and is also conditional on the platform. I used ChatGPT's $20/month platform "Plus." Fwiw, I also save a physical copy so that I can re-force AI to more or less start from scratch each time rather than trusting it to remember stuff. 

- AI's hallucinate and you have to take output carefully and/or with a grain of salt,

- Chat may evolve and obsolete the prompts,

- It helps to know the underlying theory and methods to second guess what it is doing. Otoh, one can have a dialogue now to figure it out, 

- There are certain tasks ChatGPT will NOT do. For example, it said once that it could do HJB on 2-asset allocation optimization and then when I said "ok, now run it," it said "sorry, bro, can't; it's too hard and will time out." It later created an acceptable work around but one needs to know what it can and can't do. It probably just want's me to pay more,

- I did this prompt series more or less by the "seat of the pants" method so there might be assumptions Chat used for me during the many iterations that won't be resident in memory for you if you test drive. No idea...

- These are fairly trimmed down prompts and focus more on simple versions of the theory. No fine tuning for fees or taxes or investor idiosyncrasies etc but those are probably easy enough to retrofit,

- Rigorous testing against a previously coded model has NOT been done. I guarantee I have probably mis-prompted somewhere in here and I don't even know it.