5.8 Constraints & Transaction Costs

Basic Prerequisites: Mean–Variance Optimization & the Efficient Frontier and Multivariable Calculus & Convex Optimization. (these are the folder-level prerequisites for pages 02–06; page 01 states its own, smaller, entry requirements)

1. Intuition & Practical Objective

Mean–variance optimization returns a mathematical point: $w^{\star }=\frac{1}{\delta }{\Sigma }^{-1}\mu$. On six assets with realistic inputs that point is $[-2.26,+3.33,0.28,0.78,0.30,4.96]$ - a $12\times$-levered, short-saturated book that no mandate can hold, and whose construction from an equal-weight portfolio costs $562\%$ of notional in one rebalance. Constraints and transaction costs are the difference between that point and a portfolio you can actually run. This folder is about the second problem: given a forecast and a mandate and a cost model, what do you hold, and what do you trade?

Everything in the folder is one of four objects:

• A feasible set. Long-only, position caps, group/sector bounds, turnover budgets - a convex polytope $\mathcal{C}=\{w:1^{\top }w=1,w_{\min }\le w\le w_{\max },Aw\le b\}$. Its shadow prices are the price of your mandate.
• A cost function. Linear ($c|\Delta w|$, the spread: Roll’s $2c$, Glosten–Milgrom’s $(V_H-V_L)\mu$), quadratic ($\frac{1}{2}\Delta w^{\top }\Lambda \Delta w$, impact à la Almgren–Chriss and Kyle’s $\lambda$), or concave (the empirical $|\Delta w{|}^{3/2}$ square-root law, generalized-Roll spread $2(c+\lambda )$).
• A trade-off. Turnover on one axis, net alpha on the other; the optimum sits where marginal alpha equals marginal cost - exactly the true cost, verified numerically below.
• A dynamic. Across periods you trade partially toward a long-horizon aim, not to the current target: the Gârleanu–Pedersen policy $x_t=x_{t-1}+(I+\kappa \Sigma {)}^{-1}(\text{aim}-x_{t-1})$.

This page is the folder’s hub: it gives the fast formula lookup below (job #1) and routes you to six sub-pages - intuition → constraints → cost models → turnover & multi-period → failure modes → extensions.

The one-sentence essence. “The implementable portfolio solves ${\max }_{w\in \mathcal{C}}{\mu }^{\top }w-\frac{\delta }{2}{w}^{\top }\Sigma w-{\mathbf{c}}^{\top }|w-w_0|-\frac{1}{2}(w-w_0{)}^{\top }\Lambda (w-w_0)$ - and because the ${\ell }_1$ cost term has a kink at $w=w_0$, the optimal answer is frequently to not trade at all.”

2. Mathematical Ground Truth & Derivations

Quick-Reference Lookup (job #1). Formulas are transcribed from Lobo–Fazel–Boyd (2007), Gârleanu–Pedersen (2013), Almgren–Chriss (2000) and Hasbrouck (2007, Ch 3/5/7/8); the check-column numbers were re-executed and reproduced exactly on the folder’s shared universe ($N=6$ assets, $T=60$ monthly returns, seed fixed; risk aversion $\delta =3$; starting portfolio $w_0=1/N$; assumed linear cost $c=10$ bp; quadratic impact $\Lambda =\mathrm{diag}(0.020,0.015,0.030,0.025,0.010,0.008)$).

Notation: $w$ weights, $w_0$ current weights, $\Delta w=w-w_0$, $\Sigma$ covariance, $\delta$ risk aversion, $\mathbf{c}$ linear costs, $\Lambda =\mathrm{diag}({\eta }_i)$ impact coefficients, $c$ half-spread, $\lambda$ adverse-selection/impact cost, $\mathcal{C}$ the feasible set.

The critical interpretation caveat. The static, diagonal-$\Lambda$, one-period problem above is an idealization. Three corrections matter in production: (i) the trades are coupled through $\Sigma$ - the fraction you close is a matrix $F$, not a scalar; (ii) impact is not additive across correlated names (use a full $\Lambda$ or the factor form $q{q}^{\top }D$); (iii) the empirical impact law is concave, so it must be convexified by successive convex approximation. All three are developed in 06 · Advanced Extensions.

3. Computational Implementation - the cost-aware engine

One universe, one solver: unconstrained, cost-blind long-only, linear-cost, and linear-plus-impact. Runs on numpy + scipy and reproduces every number in the table’s check column.

The engine makes the folder’s central claim numerically: pricing the trade turns a $0.1562$ strategy into a $0.1708$ one while cutting turnover by $72\%$ - and in the linear-cost solution two weights sit exactly at their current $1/6$, the signature of the no-trade region.

4. Failure Modes & First-Principles Breakdowns

Hub signposts - the folder’s failure-mode analysis lives in 05 · Failure Modes & Practice. In one line each:

1. Cost underestimation. Optimizing with $\hat{c}<c_{\text{true}}$ overtrades; assuming zero cost costs $12.3\%$ of achievable certainty equivalent. Assuming double the cost is also wrong but paralyzes instead ($\mathrm{TO}=0.0000$).
2. Turnover explosion. A cost-blind optimizer on a $10$ bp/month-noisy alpha turns over $3.83\times$/yr; cost-awareness cuts it to $0.91\times$/yr and raises net alpha ($0.1958\to 0.2034$).
3. Constraint interaction & tracking drift. A turnover budget leaves a permanent gap ($\tau =0.25\Rightarrow$ gap $0.3517$; $\tau =0.05\Rightarrow 0.8348$); a group cap that duplicates a per-name cap is free but invisible without its multiplier.

5. References

• Lobo, Fazel & Boyd (2007). Portfolio Optimization with Linear and Fixed Transaction Costs. Annals of Operations Research 152:341–365. The modeling backbone: convex cost-aware QPs, the ${\ell }_1$ no-trade region, the fixed-cost relaxation. ★ MUST-HAVE
• Gârleanu & Pedersen (2013). Dynamic Trading with Predictable Returns and Transaction Costs. Journal of Finance 68(6):2309–2340. The aim portfolio and the closed-form partial-adjustment policy. ★ STRONG
• Almgren & Chriss (2000/01). Optimal Execution of Portfolio Transactions. Journal of Risk 3(2):5–39. Permanent + temporary impact and the trading frontier. Cross-pillar: owned by Pillar 2 (optimal execution). ★ STRONG
• Grinold & Kahn (2000). Active Portfolio Management, 2nd ed., McGraw-Hill. The practitioner’s bible for constrained active portfolios, tracking error and cost-adjusted rebalancing. ★ MUST-HAVE
• Hasbrouck (2007). Empirical Market Microstructure, OUP. *Ch 3 (Roll: $c=\sqrt{-{\gamma }_1}$, spread $2c$)
• Clarke, de Silva & Thorley (2002). Portfolio Constraints and the Fundamental Law of Active Management. FAJ 58(5):48–66. The transfer coefficient $=$ the price of constraints. ★ STRONG
• Boyd, Busseti, Diamond, Kahn, Koh, Nystrup & Speth (2017). Multi-Period Trading via Convex Optimization. FnT in Optimization 3(1):1–72. The multi-period convex formulation and its implementations.

6. Connected Graph Bridges

• Foundational base: Mean–Variance & Efficient Frontier · Calculus & Optimization · Statistics & Inference
• Sibling topics: Robust Optimization · Covariance Shrinkage & RMT · Black–Litterman · Risk Parity & ERC · HRP
• Execution (Pillar 2): Optimal Execution (Almgren–Chriss) · POV
• Market making (Pillar 6): Spread Decomposition & the Roll Model · Adverse Selection & Glosten–Milgrom · Avellaneda–Stoikov · Inventory Management & Quote Skewing
• Legacy flat page (superseded by this folder): Transaction Costs & Turnover Constraints
• Sub-pages (in-folder): 01 From Zero · 02 Weight Constraints · 03 Transaction-Cost Models · 04 Turnover & Multi-Period · 05 Failure Modes · 06 Advanced Extensions

Beginner: start at 01 · Practitioner: start at 05

• Absolute beginner: 01 · From Zero - no optimization background needed.
• Math + code (undergrad/job-seeking): 02 · Weight Constraints → 03 · Transaction-Cost Models → 04 · Turnover & Multi-Period.
• Robustness (practitioner/graduate): 05 · Failure Modes → 06 · Advanced Extensions.
• Forward links: Optimal Execution · Avellaneda–Stoikov · Robust Optimization