5.8.1 Constraints & Transaction Costs from Zero

Basic Prerequisites: Mean–Variance from Zero.

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1. Intuition & Practical Objective

This page builds the why of this whole topic with no optimization background beyond the one-line spec of Markowitz. The objective is one idea: the unconstrained mean–variance optimum is a mathematical point, not a tradeable portfolio. It short-sells, it levers, and - crucially - it sits far away from wherever you currently are, so reaching it is itself a large, costly bet. Constraints and transaction costs are not bureaucratic frictions bolted onto the theory; they are the difference between the answer and the implementable answer.

Start with the dumbest question: the optimizer says hold $[-2.26,+3.33,0.28,0.78,0.30,4.96]$ for six assets. What does a portfolio manager actually do with that?

Three “aha”s:

1. The number is not a portfolio. Those weights sum to $1$ but the gross exposure ${\sum }_i|w_i|$ is $11.9$ - you are told to be $12\times$ levered, with a $225\%$ short in asset 1 financed by a $496\%$ long in asset 6. A balance-sheet limit, a long-only mandate, or a $35\%$ position cap kills this answer immediately. The optimizer never knew those rules existed because we never told it.

2. Reaching the optimum is a trade, and trades cost money. From an equal-weight book, hitting those unconstrained weights requires ${\sum }_i|w_i-w_{0,i}|=11.24$ of notional - a $562\%$ one-way turnover. At a 10 bp all-in cost that is $\approx 112$ bp of return gone in one rebalance; at 50 bp it is $\approx 562$ bp. The strategy’s entire theoretical edge can be spent on the arbitrage of getting there.

3. Cost is not a haircut you apply afterwards - it belongs inside the objective. Because the cost term is ${\mathbf{c}}^{\top }|w-w_0|$, it has a kink at $w=w_0$: a non-differentiable cusp. Optimizing a kinked objective creates a no-trade region - a band of current weights in which the optimal action is to do nothing at all, because the alpha gain does not clear the cost hurdle. The optimum of a cost-aware problem is frequently “hold what you own.” That is the single most practically important consequence in this folder.

The one-sentence essence. “A portfolio is the triple (signs you may hold, sizes you may hold, trades you may make); the unconstrained Markowitz optimum ignores all three, and adding them back turns the problem from a matrix inversion into a convex program whose answer is often to not trade.”

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2. Mathematical Ground Truth & Derivations

The unconstrained problem and its closed form. With expected returns $\mu$, covariance $\Sigma$, risk-aversion $\delta >0$ and budget $1^{\top }w=1$,

$$\underset{w\in {\mathbb{R}}^N}{\max }{\mu }^{\top }w-\frac{\delta }{2}{w}^{\top }\Sigma w,w^{\star }=\frac{1}{\delta }{\Sigma }^{-1}\mu .$$

The constrained problem (what you actually solve). Add the feasible set $\mathcal{C}$:

$$\underset{w}{\max }{\mu }^{\top }w-\frac{\delta }{2}{w}^{\top }\Sigma w\text{s.t.}w\in \mathcal{C},\mathcal{C}=\{w:1^{\top }w=1,w_{\min }\le w\le w_{\max },Aw\le b\}.$$

Long-only is $w_{\min }=0$; a position cap is $w_{\max }$; group/sector limits are the rows of $Aw\le b$. $\mathcal{C}$ is a convex polytope and the resulting problem is a quadratic program (QP) - solvable in milliseconds at $N$ in the hundreds.

The turnover definition. With current weights $w_0$, the trade is $\Delta w=w-w_0$ and turnover is the ${\ell }_1$ mass moved:

$$\mathrm{TO}(w)=\frac{1}{2}\Vert w-w_0{\Vert }_1=\frac{1}{2}\sum _{i=1}^N|w_i-w_{0,i}|(\text{one-way; sell-side}=\text{buy-side by the budget constraint}).$$

The cost-augmented objective. Transaction costs come in two families (developed in 03 · Transaction-Cost Models):

$$C(\Delta w)=\underset{\text{linear: spread}+\text{fees}}{\underbrace{{\mathbf{c}}^{\top }|\Delta w|}}+\underset{\text{quadratic market impact}}{\underbrace{\frac{1}{2}\Delta w^{\top }\Lambda \Delta w}},\Lambda =\mathrm{diag}({\eta }_1,\dots ,{\eta }_N),$$

so the net optimization is

$$\boxed{\underset{w\in \mathcal{C}}{\max }{\mu }^{\top }w-\frac{\delta }{2}{w}^{\top }\Sigma w-{\mathbf{c}}^{\top }|w-w_0|-\frac{1}{2}(w-w_0{)}^{\top }\Lambda (w-w_0)}$$

Why the kink matters (first principles). Consider one asset with mean $\mu$, variance ${\sigma }^2$, risk-aversion $\delta$ and a linear cost $c$ per unit traded. The problem ${\max }_w\mu w-\frac{\delta }{2}{\sigma }^2{w}^2-c|w-w_0|$ has first-order conditions that split into three cases:

$$w^{\star }=\{\begin{array}{ll}\frac{\mu -c}{\delta {\sigma }^2}& w_0<\frac{\mu -c}{\delta {\sigma }^2}\\ w_0& |w_0-\frac{\mu }{\delta {\sigma }^2}|\le \frac{c}{\delta {\sigma }^2}\\ \frac{\mu +c}{\delta {\sigma }^2}& w_0>\frac{\mu +c}{\delta {\sigma }^2}\end{array}$$

i.e. a no-trade region of half-width $\theta =\frac{c}{\delta {\sigma }^2}$ around the frictionless target $w^{\ast }=\mu /(\delta {\sigma }^2)$. The width scales linearly in the cost and inversely in risk-aversion and variance: cheap, volatile, high-conviction assets are re-traded; expensive, calm, low-conviction ones are left alone. 03 · Transaction-Cost Models confirms this numerically, and 04 turns the idea into a usable policy.

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3. Computational Implementation - the unimplementable optimum, quantified

One universe, three answers: unconstrained, long-only, long-only capped at $35\%$. For each we report gross exposure, the one-way turnover needed to get there from an equal-weight book, and the net-of-cost return at 10 bp and 50 bp. Runs on numpy + scipy.

Read the second and third blocks together. The unconstrained “optimum” is unattemptable twice over: (i) it is $12\times$ levered and short-sells an asset at $225\%$, so no real mandate can hold it; (ii) even if your mandate allowed it, getting there costs $562\%$ of notional, which at 50 bp is $\approx 562$ bp of return - and note that the naive book’s crude gross-alpha of $1.2552$ is a leveraged $125\%$ that is almost entirely a bet on the sample’s estimation error (see The Estimation-Error Problem). Clipping to the simplex collapses turnover to $59\%$ (one-way) and leaves a portfolio you could actually trade. Constraints here are not conservative - they are the only reason the answer exists.

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4. Failure Modes & First-Principles Breakdowns

1. “Constraints just cost me alpha.” They cost gross alpha and buy implementable alpha. §3 shows the cap-35% book with $0.1932$ gross vs. the naive $1.2552$ - but the naive number is levered $12\times$ and unattainable. Comparing a levered, short-saturated fantasy to a funded portfolio is the most common accounting error in portfolio construction.
2. “I’ll optimize unconstrained, then clip.” Clipping after the fact is not the constrained optimum - the geometry changes what the middle weights should be. The correct procedure solves the QP with the box inside it; see 02 for how much the answer moves.
3. “Costs are a rounding error.” At $562\%$ turnover, a 50 bp cost is $\approx 562$ bp of drag - larger than most strategies’ net alpha. Cost is first-order, not second-order, precisely because unconstrained optima are far away. This is the emotional core of the whole folder.
4. The no-trade region feels like a bug. A beginner sees the optimizer return $w=w_0$ and thinks the code failed. It did not: the kinked cost term makes “do nothing” genuinely optimal below the hurdle $\theta =c/(\delta {\sigma }^2)$. Non-differentiable objectives have interior flat spots; that is the point.

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5. References

• Lobo, Fazel & Boyd (2007), Portfolio Optimization with Linear and Fixed Transaction Costs, Annals of Operations Research 152:341–365
• Grinold & Kahn (2000), Active Portfolio Management, 2nd ed.
• Clarke, de Silva & Thorley (2002), Portfolio Constraints and the Fundamental Law of Active Management, FAJ 58(5):48–66
• Almgren & Chriss (2000/01), Optimal Execution of Portfolio Transactions, Journal of Risk 3(2):5–39
• Hasbrouck (2007), Empirical Market Microstructure

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6. Connected Graph Bridges

• Base: Mean–Variance from Zero · 05 · Estimation-Error Maximizers
• Foundations: Calculus & Optimization (convexity, KKT, the kink) · Statistics & Inference
• Execution sibling: Optimal Execution (Almgren–Chriss) · POV
• Microstructure source: Spread Decomposition & the Roll Model
• Continue: 02 · Weight Constraints · Index Hub