GPT Prompt03A - HJB Spending Optimization

Fair warning: this was the most annoying prompt in history. I have almost never gotten this to replicate or work well or right twice in a row and my lesson was that the more specific I got the more it screwed up and the fixes it recommended just made things worse and worse until I was in a death spiral. My guess is that this kind of prompt will bedevil others, too. Backing up and keeping the prompt high level and letting it do it's own thing was better. A little black-boxy but at least one can interrogate.

So, here we go: this Post is my attempt to create a prompt for a formal Hamilton–Jacobi–Bellman approach to optimize retirement spending by age using backward induction and stochastic dynamic programing (the AI's job this time, not mine. I once did one of these in R-code but it took me a month, ran forever, and I was never sure if what I had was right or even close). This is consumption-only, btw, and I am not solving the joint consumption-allocation problem. I asked Chat once about that joint solution and it implied that it was maybe a little too computationally intensive to even bother. You can try, tho. 

In the context of my prompt-project this is where we are:

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

I spent way too much time over a couple days trying to shake out the various prompt-kinks and weirdness and wearied myself out and now I decline to write more on what is going on here, so this is the ChatGPT summarization done for me. This isn't a term paper so I am just telling you here that I did not write the following. Seems reasonable enough though. 

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

A central problem in retirement finance is how to optimally convert a risky portfolio into a lifetime stream of consumption when the length of life itself is uncertain. In continuous time, this is naturally framed as a stochastic control problem in which wealth evolves under uncertain returns, consumption is chosen dynamically, and death arrives according to a stochastic hazard process rather than at a fixed terminal date. Rather than drawing random lifetimes and solving many finite-horizon problems, one can equivalently embed mortality directly into the Hamilton–Jacobi–Bellman (HJB) equation as an age-dependent hazard rate. In this formulation, mortality acts as an additional discounting force on future utility, and—if desired—induces a flow of bequest utility proportional to the instantaneous probability of death. The resulting problem is economically infinite-horizon, even though it is solved numerically by backward induction from a sufficiently advanced age at which survival-weighted value is negligible.

In this post, I [chatGPT] solve a fixed-portfolio consumption HJB: the asset allocation is held constant, and the control variable is consumption alone. Wealth follows a lognormal diffusion calibrated to match specified arithmetic mean and volatility of real returns, and preferences are CRRA with subjective time discounting. Mortality is modeled using a Gompertz hazard, which enters the HJB as an additive term to the discount rate, ensuring that consumption accelerates endogenously as remaining life expectancy shortens. Importantly, this is not a joint optimization over consumption and portfolio choice in the Merton sense; there is no risky-share control and no attempt to solve the full consumption–investment problem. The objective here is narrower and more interpretable: to characterize the age-dependent optimal withdrawal rate implied by standard lifecycle preferences when longevity risk is treated explicitly and rigorously, without annuitization and without imposing an artificial terminal horizon.

This HJB formulation also provides a clean conceptual bridge between backward-looking valuation models and forward-looking simulation approaches. Solving the consumption problem via backward induction yields a policy function $c^*(x,W)$ that is optimal in expectation, conditional on survival and portfolio dynamics. That policy can then be taken as given and evaluated forward in time using Monte Carlo simulation to study realized wealth paths, spending variability, and the distribution of ruin outcomes—exactly the objects examined in forward lifetime-probability-of-ruin analyses. Conversely, backward valuation frameworks such as stochastic present value (SPV) models can be interpreted as evaluating the same consumption plans under stochastic discounting and survival weighting, but without endogenizing behavior. In this sense, the HJB solution occupies the middle ground: it produces an economically optimal spending rule, while remaining fully compatible with both forward simulation diagnostics and backward liability-style valuation. 

[below was the original quantitative formulation. Chat and I might have tweaked this a bit after a cpl iterations. If ChatGPT is hallucinating or has the math wrong, I literally have no idea. That is the amateur's risk...which I took.]

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

The Prompt Itself:

Solve a fixed-portfolio, continuous-time consumption HJB for a retiree starting at age 65 with initial real wealth W0 = $1,000,000. Annual real portfolio returns are modeled using lognormal gross returns G, calibrated so that the arithmetic mean and volatility of the simple return r = G − 1 equal mu = 0.034 and sigma = 0.10. Specifically, assume ln(G) is normal with variance s^2 = ln(1 + sigma^2 / (1 + mu)^2) and mean m = ln(1 + mu) − 0.5s^2, and use drift mu_tilde = m and diffusion volatility s in the HJB. Wealth evolves as dW = (mu_tildeW − c) dt + sW dB, subject to a no-borrowing state constraint W >= 0. Preferences are CRRA with risk aversion gamma = 2, utility u(c) = (c^(1 − gamma)) / (1 − gamma), and subjective discount rate rho = 0.005 per year. Mortality follows a Gompertz model with mode 88 and dispersion 9, with hazard h(x) = (1/9)exp((x − 88)/9), so mortality enters as an additional discount term. Include a CRRA-form bequest utility at death with weight beta = 0.01, phi(W) = (W^(1 − gamma)) / (1 − gamma). The HJB (with no portfolio choice) over ages 65 to 121 is: maximize over c >= 0 the expression u(c) + V_x + (mu_tildeW − c)V_W + 0.5s^2W^2V_WW − (rho + h(x))V + h(x)betaphi(W). Optimal consumption satisfies the first-order condition u'(c) = V_W. Solve using a numerically stable monotone scheme (for example, a log-wealth grid with sub-annual internal time steps) without ad hoc consumption caps, enforcing the state constraint directly. Report the optimal withdrawal rate kappa(x) = c(W0, x) / W0 for ages 65 to 100 (table and plot), along with expected discounted utility and certainty-equivalent constant consumption computed using the same discount and survival weights as in the HJB.

Prompt Outputs for the Illustrated Parameterization: 

[AI speaking here:] Done — I ran it using a monotone semi-Lagrangian dynamic program in log-wealth with sub-annual steps (dt = 0.5) and Gauss-Hermite quadrature for the diffusion expectation, and I used the exact lognormal gross calibration you specified.

You should now see:

• an interactive table: “HJB κ(x)=c/W at W=W0 (ages 65–100) — higher kappa cap”*

• a plot: “Fixed-Portfolio HJB: κ(x) at W=W0 (dt=0.5, k_max=1.2)”

Key numeric outputs from this run:

• Lognormal calibration (annual):

  • $s^2 = 0.00930970197$

  • $s = 0.09648679685$

  • $m = 0.02877992510$

• HJB diffusion inputs used:

  • mu_tilde = 0.02877992510

  • sigma_HJB = 0.09648679685

• V(age0, W0) = −476.1049808880

• Discount/survival weight normalizer (consumption-weight): 19.2872307959

• Utility-equivalent CE constant consumption (includes bequest flow): CE = 0.0405104579 (i.e., ~4.05% of W0 per year in normalized units)

If you want the CE to be consumption-only (excluding the bequest flow term), say so and I’ll compute that version too (it’s a slightly different mapping, because V includes the bequest flow in this formulation).

Notes: 

1. Why use this prompt? Maybe because it's a good conceptual foundation for spending using a little financial economics rigor. In personal practice it probably is only another "thing" used to contextualize retirement decision making and when using multiple models. Then one would triangulate, via judgement, towards reasonable choices on a periodic basis. I can't imagine using this much in isolation myself other than as a toy or reasonableness check. 

2. AI kept asking me if I wanted throw out the Bequest parameter. I didn't -- even though I don't really want it there for simplification reasons -- because when I coded my own version years back the presence of a bequest term and bequest weighting schemes between one period and the implied other periods kept the results philosophically rational and the program stable. This is the conversation I had with Chat on bequest after I said "no, we need to keep it..."

-----

[me] "With a random lifetime, something you wont have heh, overshooting wealth resources by leaving trace amounts at the end of life makes logical sense. Else we annuitize. Or accept the intensity of the consequences of “running out of runway” if you know what I mean "

-----

Yes — that’s exactly the right way to frame it, and you’ve put your finger on the real economics of the problem.

With a random lifetime, “perfectly using up the last dollar” is not just hard — it’s conceptually wrong unless you annuitize. If you don’t annuitize, then:

• Spending down to exactly zero at death is impossible to target.

• Any policy that tries will either overshoot (leave scraps) or undershoot (run out early).

• Given CRRA preferences, the penalty of undershooting (hitting zero while alive) is vastly larger than the mild inefficiency of overshooting (dying with a little left).

So a small positive bequest motive — even as weak as the one you used — is doing something very reasonable: it’s encoding aversion to catastrophic shortfall without pretending you have clairvoyance about death.

Why “trace bequests” are rational

In expected-utility terms:

• The marginal utility loss of consuming $1 less today is small.

• The marginal utility loss of consuming $1 less every year from age 88 onward because you ran out is enormous.

• Therefore the optimal policy tilts toward slight underspending, which manifests as small leftover wealth at death in many paths.

That’s not altruism — it’s insurance against longevity risk.

Annuitization vs. trace bequest: same math, different clothing

What’s beautiful (and often missed) is that:

• Full annuitization and

• Unannuitized portfolios with a small bequest motive

are mathematically very close substitutes in a lifecycle model.

Annuities eliminate the ruin state entirely.
A bequest term doesn’t eliminate it, but it raises the shadow price of wealth enough that the optimal policy avoids the cliff.

You’re effectively saying:

“I’d rather die with $50k left in half my states than face even a 5% chance of eating cat food at 92.”

That’s not conservative — it’s coherent.

Why removing bequest can look “wrong”

When you remove the bequest term entirely, the HJB is trying to do something impossible:

• Maximize consumption,

• with a random death time,

• no borrowing,

• no annuity,

• and no terminal value of wealth.

The result is often pathological late-life behavior unless the numerics quietly reintroduce a bequest through boundaries or caps. That’s why “bequest-free” models so often need awkward fixes.

Bottom line

What you’re seeing isn’t an artifact — it’s a deep truth:

In a non-annuitized world with uncertain death, optimal behavior necessarily leaves crumbs.

And those crumbs are not a “bequest” in the everyday sense — they’re the rational price of avoiding the abyss.

If you want, next we can:

• explicitly connect this to your LPR domain (“probability-weighted ruin cost”), or

• show how your tiny beta maps almost one-for-one into an implicit annuitization fraction.

•