Reference 0020

Multiperiod Notation Card

Myopic asks “best book?” A planner asks “best path?” They differ exactly when a path's cost isn't the sum of independent days.

The Notation

state      w_t              the book you wake up with
control    z_t              today's trades
dynamics   w_{t+1} = w_t + z_t          (+ returns drift in full dress)
objective  max Σ_t [ α_tᵀw_t - γ·risk(w_t) - cost(z_t) ]   over z_1..z_H

What Couples Periods (the screening question)

L1 cost κ|z|:   lump κD = split m·κ(D/m)      schedules DON'T matter → myopic OK
impact c·z²:     lump cD² vs split cD²/m          schedules DO matter → plan
alpha decay:     trade slow = trade never          the clock couples periods too

optimal spread of trade Δ over m days:   m* ≈ √(2cΔ/a)
(a = alpha per day per unit;  4× impact ⇒ 2× slower — square-root law)

MPC in Four Lines

loop daily:
    re-estimate  α, Σ, costs
    solve the H-period convex plan  z_t .. z_{t+H}
    execute z_t ONLY;  discard the rest
plan for scheduling; re-solve for honesty about forecasts

Whiteboard Traps

"multiperiod always beats myopic":  false — with linear costs + static alpha they tie
executing the whole plan:           trusts day-30 forecasts made day 1
myopic with L1 costs:               never unwinds a stale position (no future in view)
risk charged on trades not holdings: mixes the impact meter with the risk meter

Use this page with Lesson 0020 (the scheduling lab), Reference 0012 (kink economics), and Reference 0021 (the paper's dialect of this notation).