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port_stage

no-port-cons

Portfolio allocation stage with post-decision correlated shocks.

This stage implements portfolio choice where the agent allocates wealth between risky and risk-free assets before shocks are realized.

Stage Overview

Feature Description
Prestate k (investable assets)
Control ς (risky share ∈ [0,1])
Shocks Joint post-decision (Ψ, θ)
Poststate m (cash-on-hand)

ADC Structure

[ARRIVAL] ───g_ad───▶ [DECISION] ───g_de───▶ [CONTINUATION]
    k                   k_d, ς       Ψ,θ           m
              identity          portfolio+shocks
  • Arrival → Decision: Identity transition k_d = k
  • Decision → Continuation: Portfolio return with shocks realized

Economic Problem

The portfolio stage solves: $$ V(k) = \max_{\varsigma} \mathbb{E}_{\Psi,\theta} \left[ V^{e}(m) \right] $$

where cash-on-hand after portfolio choice and shock realization is: $$ m = k \cdot (\varsigma \Psi + (1-\varsigma) R) + \theta $$

  • ς = risky share (fraction in risky asset)
  • Ψ = risky return shock (log-normal)
  • θ = transitory income shock (log-normal)
  • R = risk-free gross return

Stage YAML

name: port_stage

symbols:
  spaces:
    Xk: "@def R++"
    Xm: "@def R++"
    Π:  "@def [0,1]"
    Rp: "@def R++"
    Θ:  "@def R++"

  prestate:
    k: "@in Xk"

  states:
    k_d: "@in Xk"

  controls:
    ς: "@in Π"

  # Shocks realized after the portfolio decision (post-decision timing)
  exogenous:
    Ψ:
      - "@in Rp"
      - "@dist LogNormal(μ_Ψ, σ_Ψ)"
    θ:
      - "@in Θ"
      - "@dist LogNormal(μ_θ, σ_θ)"

  poststates:
    m: "@in Xm"

  values:
    V[<]: "@in R"
    V: "@in R"
    V[>]: "@in R"

  parameters:
    R: "@in R++"
    μ_Ψ: "@in R"
    σ_Ψ: "@in R+"
    μ_θ: "@in R"
    σ_θ: "@in R+"
    ρ_ζ: "@in [-1,1]"   # correlation parameter

equations:
  # No shocks between arrival and decision
  arvl_to_dcsn_transition: |
    k_d = k

  # Joint post-decision shocks realized in dcsn→cntn
  dcsn_to_cntn_transition: |
    m = k_d*(ς*Ψ + (1-ς)*R) + θ

  # Decision value: max-over-expectation (post-decision shocks)
  cntn_to_dcsn_mover:
    Bellman: |
      V = max_{ς}(E_{Ψ,θ}(V[>]))
      ς = argmax_{ς}(E_{Ψ,θ}(V[>]))

  # No expectation at arrival (deterministic arvl→dcsn)
  dcsn_to_arvl_mover:
    Bellman: |
      V[<] = V

Key Features

Post-Decision Shock Timing

Unlike the noport stage, the portfolio decision ς is made before shocks are realized. The shocks appear in the dcsn_to_cntn_transition, meaning:

  1. Agent observes k at arrival
  2. Agent chooses portfolio allocation ς
  3. Shocks (Ψ, θ) are then realized
  4. Cash-on-hand m is computed

Max-over-Expectation Structure

The Bellman equation has the form: 

V = max_{ς}(E_{Ψ,θ}(V[>]))

This is a max-over-expectation (rather than expectation-over-max), which is characteristic of portfolio problems where the decision must be made before uncertainty resolves.

Joint Correlated Shocks

The shocks Ψ (risky return) and θ (income) can be correlated via the ρ_ζ parameter, allowing for realistic modeling of return-income correlations.

Period Usage

This stage is used in:

  • port_cons_period: port → cons (portfolio choice, then consumption)
  • cons_port_period: cons → port (consumption first, then portfolio)

See Port-with-Shocks for composition details.