Prompt Portfolios

• Prompt portfolios are investment strategies that use smooth generating functions applied to market weights to provide precise pathwise performance and risk estimates.
• They leverage rank-based stochastic models and hydrodynamic limits to connect market dynamics with explicit master formulas for performance attribution.
• The framework enables practical risk quantification and robust short- to medium-term excess return predictions through deterministic PDE and stochastic SPDE analysis.

A prompt portfolio is a class of investment strategies derived from the theory of functionally generated portfolios, which use smooth generating functions applied to the vector of market weights. Within rank-based stochastic models that describe the evolution of market capitalizations via interacting diffusion processes, prompt portfolios enable precise pathwise performance attribution over finite and asymptotic horizons. Central to this framework are explicit “master formulas” relating the portfolio’s relative value process to the underlying dynamics of market weights, permitting quantitative short- and medium-term risk estimates through hydrodynamic limits and fluctuation theorems (Monter et al., 2018).

1. Functionally Generated Portfolios: Definitions and Formulation

Let $\mu(t) = (\mu_1(t), \ldots, \mu_n(t))$ denote the vector of market weight processes, constrained by $\mu_i(t) \geq 0$ and $\sum_i \mu_i(t) = 1$. A smooth generating function $G : \Delta^n \rightarrow (0,\infty)$ on the open unit simplex $\Delta^n = \{ x \in (0,1)^n \mid \sum_i x_i = 1 \}$ defines two principal classes:

• Multiplicatively generated portfolios: Given by

$$\pi_i^{G, \times}(t) = \left( \frac{\partial}{\partial x_i} \log G(\mu(t)) + 1 - \sum_j \mu_j(t) \frac{\partial}{\partial x_j}\log G(\mu(t)) \right) \mu_i(t)$$

with relative value

$$\log V^{G, \times}(t) = \log \frac{G(\mu(t))}{G(\mu(0))} - \frac{1}{2} \sum_{i,j=1}^n \int_0^t \frac{\partial^2}{\partial x_i \partial x_j}\log G(\mu(s))\,d[\mu_i, \mu_j](s)$$

• Additively generated portfolios: For a concave extension $\widetilde{G}$,

$$\pi_i^{\widetilde{G}, +}(t) = \left( \frac{\partial}{\partial x_i} \widetilde{G}(\mu(t)) - \sum_j \mu_j(t)\frac{\partial}{\partial x_j}\widetilde{G}(\mu(t)) \right) + \mu_i(t)$$

with normalized relative value

$$V^{\widetilde{G}, +}(t) = 1 + \widetilde{G}(\mu(t)) - \widetilde{G}(\mu(0)) - \frac{1}{2} \sum_{i,j=1}^n \int_0^t \partial^2_{x_ix_j} \widetilde{G}(\mu(s))\,d[\mu_i, \mu_j](s)$$

Both constructions yield a portfolio whose trajectory, relative to the market, depends solely on the market weight path $\mu_i(t) \geq 0$0 and the generating function, excluding additional stochastic integration.

2. Rank-Based Stochastic Models and Macroscopic Observables

Market capitalizations are modeled as log-capitalizations $\mu_i(t) \geq 0$1 governed by SDEs of the form

$\mu_i(t) \geq 0$2

where $\mu_i(t) \geq 0$3 is the empirical measure and $\mu_i(t) \geq 0$4. Under mild exponential tail assumptions on the initial data, as $\mu_i(t) \geq 0$5, two regimes emerge:

• Hydrodynamic limit: $\mu_i(t) \geq 0$6 deterministic, $\mu_i(t) \geq 0$7 solving a nonlinear porous-medium PDE,

$\mu_i(t) \geq 0$8

with $\mu_i(t) \geq 0$9, $\sum_i \mu_i(t) = 1$0.

• Fluctuation regime: $\sum_i \mu_i(t) = 1$1, where $\sum_i \mu_i(t) = 1$2 satisfies a linear SPDE capturing stochastic fluctuations around the deterministic mean.

Macroscopic observables take the form $\sum_i \mu_i(t) = 1$3, covering quantities such as the Rényi diversity $\sum_i \mu_i(t) = 1$4 for $\sum_i \mu_i(t) = 1$5 and the market entropy $\sum_i \mu_i(t) = 1$6.

3. Performance Fluctuations: Short- and Medium-Term Theorems

Fluctuations in the performance of functionally generated portfolios are characterized by central limit theorems (CLTs):

• Finite-dimensional CLT for observables: For any smooth $\sum_i \mu_i(t) = 1$7 and test functions $\sum_i \mu_i(t) = 1$8,

$\sum_i \mu_i(t) = 1$9

jointly in $G : \Delta^n \rightarrow (0,\infty)$0 and $G : \Delta^n \rightarrow (0,\infty)$1.

• CLT for hitting times: If $G : \Delta^n \rightarrow (0,\infty)$2 crosses a level $G : \Delta^n \rightarrow (0,\infty)$3 at time $G : \Delta^n \rightarrow (0,\infty)$4 with $G : \Delta^n \rightarrow (0,\infty)$5,

$G : \Delta^n \rightarrow (0,\infty)$6

where $G : \Delta^n \rightarrow (0,\infty)$7.

The main probabilistic arguments invoke Taylor expansions for non-Markovian observables, propagation of chaos, and empirical-measure fluctuation control via Wasserstein metrics.

4. Explicit Portfolio Examples

Two canonical portfolio classes result from specific generating functions:

Portfolio Type	Generating Function $G : \Delta^n \rightarrow (0,\infty)$8	Portfolio Weights
Diversity-weighted	$G : \Delta^n \rightarrow (0,\infty)$9	$\Delta^n = \{ x \in (0,1)^n \mid \sum_i x_i = 1 \}$0
Entropy-based (universal)	$\Delta^n = \{ x \in (0,1)^n \mid \sum_i x_i = 1 \}$1	$\Delta^n = \{ x \in (0,1)^n \mid \sum_i x_i = 1 \}$2

The diversity-weighted portfolio yields an excess growth rate

$\Delta^n = \{ x \in (0,1)^n \mid \sum_i x_i = 1 \}$3

while the entropy-based portfolio’s drift and variance are linked to the market entropy process and can be explicitly computed using the underlying SPDE/PDE for the macroscopic quantities.

5. Practical Implementation and Performance Quantification

The operational workflow for constructing prompt portfolios and assessing their performance comprises:

• Parameter Estimation: Fit the rank-based drift $\Delta^n = \{ x \in (0,1)^n \mid \sum_i x_i = 1 \}$4 and diffusion $\Delta^n = \{ x \in (0,1)^n \mid \sum_i x_i = 1 \}$5 functions to empirical capitalization data, for example via Atlas model fits.
• PDE/SPDE Solution: Numerically solve the porous-medium PDE for $\Delta^n = \{ x \in (0,1)^n \mid \sum_i x_i = 1 \}$6 and corresponding Gaussian fluctuation SPDE.
• Deterministic Path and Derivatives: Evaluate the deterministic observable path $\Delta^n = \{ x \in (0,1)^n \mid \sum_i x_i = 1 \}$7 and its derivative $\Delta^n = \{ x \in (0,1)^n \mid \sum_i x_i = 1 \}$8.
• Variance Quantification: Compute the limiting variance $\Delta^n = \{ x \in (0,1)^n \mid \sum_i x_i = 1 \}$9 using the covariance structure from the fluctuation SPDE.
• Confidence Intervals: Utilize the CLTs and concentration inequalities to derive explicit confidence intervals for relative outperformance over finite horizons. Non-asymptotic bounds on excess growth processes substantiate probability statements about reaching portfolio targets within a predefined timeframe.

In both construction and application, one seeks concave, symmetric generating functions that emphasize small-weight (lower-rank) stocks, as these tend to harvest volatility more effectively. The large-$$\pi_i^{G, \times}(t) = \left( \frac{\partial}{\partial x_i} \log G(\mu(t)) + 1 - \sum_j \mu_j(t) \frac{\partial}{\partial x_j}\log G(\mu(t)) \right) \mu_i(t)$$0 regime enables precise mapping from the choice of generating function to explicit risk and performance predictions.

6. Interpretation and Theoretical Significance

The functionally generated portfolio framework—especially as realized via rank-based models—enables rigorous, model-driven translation of macro-level market observables into actionable, pathwise portfolio rules. The dynamics of prompt portfolios are tractable through both deterministic (hydrodynamic limit) and stochastic (Gaussian fluctuation) perspectives, permitting closed-form approximations and rigorous confidence bounds for both the rate and the time to achieve excess return over the market. A significant theoretical implication is the existence of a well-defined pathway from model calibration, through PDE evaluation, to explicit short- and medium-term performance metrics, supporting both comparative portfolio analysis and robust risk estimation (Monter et al., 2018).