Price of budget
λ* = objective improvement per extra point of turnover budget, valid locally.
Reference 0013
Turnover Budget ↔ Penalty Equivalence
A hard budget and a per-unit penalty are the same problem: the budget's optimal dual is the equivalent fee. Works for any convex cost or risk term.
The Two Forms
budget: minimize f0(w) s.t. |w - w_prev| <= τ penalty: minimize f0(w) + κ |w - w_prev|
The Bridge (group the Lagrangian)
L(w, λ) = f0(w) + λ(|w - w_prev| - τ)
= [ f0(w) + λ|w - w_prev| ] - λτ
= penalty objective with κ = λ
budget τ, dual λ* <=> penalty κ = λ*
penalty κ, trade τ* <=> budget τ = τ*, dual κ
requires convexity (strong duality); breaks otherwise.
Course Example (f0' = 0.30w - 0.30, target 1.00)
binding upward trade: w* = w_prev + τ dual: λ* = 0.30 (1 - w_prev - τ) τ = 0 λ* = 0.30(1 - w_prev) (max pressure) τ grows λ* decays linearly (Lesson 0010) τ >= 1 - w_prev λ* = 0 (budget loose)
Reading the Dual
Implied fee
λ* is the per-unit fee that would reproduce the same trade with no constraint at all.
Mandate check
λ* far above the real trading cost = the turnover mandate is starving the strategy.
Everywhere pattern
Risk budget σ² <= s with dual γ* = risk penalty γ* · w'Σw. The efficient frontier is this sweep.
Choosing a Form
use budget when: limits come from mandate/compliance,
hard guarantees required
use penalty when: real per-unit cost is estimable,
trade size should respond to signal
translate with: the dual, in either direction
Use this page with Lesson 0013, Reference 0012 (penalty form), and Reference 0009 (dual as local price). Formal source: Boyd & Vandenberghe §5.6, §4.7.4.