FRM Exam Part II · Portfolio Construction

Portfolio Constraints and Practical Implementation Issues

Updated 11 October 2026 · Fact-checked

Portfolio constraints are limits such as long-only rules, turnover caps, transaction costs and leverage limits that restrict the optimiser. They move the portfolio away from the unconstrained optimum, lower expected utility, and change risk. To solve questions, compare the constrained and unconstrained solutions and interpret the cost and the risk shift.

Understand Portfolio Constraints and Practical Implementation Issues

An unconstrained optimiser picks weights that maximise expected return for a given risk. It can short, lever and trade as much as it likes at no cost. Real portfolios cannot. Mandates, regulation, liquidity and fees all limit what you can hold and how much you can trade.

A constraint can only reduce the best achievable objective, or leave it unchanged if it does not bind. A binding constraint is one the optimal solution would otherwise break. The cost of a binding constraint is called its shadow price: the improvement in the objective if you relaxed the limit slightly.

The long-only constraint is the most common. It forces weights to be at least zero. Mean-variance optimisers tend to give extreme long and short weights from noisy inputs. A long-only limit acts like a shrinkage device, so the portfolio is often more stable and sometimes performs better out of sample, even though in-sample it is worse. It also limits active risk, because you cannot underweight a benchmark stock by more than its benchmark weight.

Turnover and transaction costs link to rebalancing. Each trade costs commissions, bid-ask spread and market impact. The more you trade, the more return you give up. A turnover cap or a cost penalty creates a no-trade zone: you trade only when the weights drift far enough that the gain from rebalancing exceeds the cost. Higher costs mean a wider zone and less frequent trading.

Leverage limits cap gross or net exposure. Leverage scales both return and risk. A leverage cap stops low-risk assets from being scaled up to a target risk level, which is why low-beta strategies are often held back. Other practical issues include position and sector limits, minimum lot sizes, liquidity limits and tracking error budgets. Estimation error in inputs makes all of these more important.

Key formulas to remember

Constrained optimisation: max w'μ − (λ ÷ 2) w'Σw subject to constraints such as w ≥ 0, Σw = 1, Σ|w − w0| ≤ T; w0 is the current portfolio and T the turnover cap. Constraints can only lower the objective or leave it unchanged.
Turnover: Turnover = ½ × Σ|w_new − w_old|; One-way turnover uses the half. Some questions define it as the full sum, so check the definition given.
Transaction cost drag: Net return = Gross return − (Turnover × cost per unit traded); Cost must be applied to the amount actually traded, on both buy and sell sides if stated.
Leverage ratio: Leverage = Gross exposure ÷ Equity capital; Portfolio return = L × asset return − (L − 1) × funding cost, for a simple borrowed position.
Information ratio impact of constraints: IR_constrained ≤ IR_unconstrained; The transfer coefficient measures how much of the unconstrained efficiency is kept after constraints.

How to solve Portfolio Constraints and Practical Implementation Issues questions

Use this approach for any question on constraints and implementation.

1Identify the unconstrained optimum or the starting portfolio and the objective being maximised.
2List every constraint in the question: long-only, turnover, cost, leverage, position limits.
3Decide which constraints bind. A constraint that the unconstrained solution already meets does not change it.
4Compute the trade size: the difference between new and old weights, and the turnover from it.
5Apply the cost to the traded amount and subtract it from expected return.
6Recompute portfolio risk with the constrained weights, using weights and the covariance inputs given.
7Compare constrained and unconstrained results and state the direction: lower expected utility, usually lower or equal information ratio.
8Interpret in words: which constraint costs the most, and what the no-trade zone or shadow price implies.

Quickest way: Shortcut for constraint questions

When to use it: Use when the options differ in direction or size and you need a fast answer.

Remember the direction first: adding a constraint never raises the optimal objective.
Binding or not? If unconstrained weights already satisfy the limit, nothing changes.
For cost questions, compute traded amount × cost rate and subtract from return.
For turnover, sum the absolute weight changes and halve if one-way is asked.
Eliminate options that say costs or constraints improve in-sample performance.

Common mistakes in Portfolio Constraints and Practical Implementation Issues

Saying a constraint always changes the optimal portfolio.
Students forget that non-binding constraints have no effect.
Fix: Check whether the unconstrained solution already satisfies the limit before recomputing.
Applying transaction costs to the whole portfolio value.
The cost rate looks like a flat fee.
Fix: Apply the cost only to the amount traded, which is the weight change times portfolio value.
Confusing one-way and two-way turnover.
Both definitions appear in practice.
Fix: Use the definition in the question. One-way turnover is half the sum of absolute weight changes.
Believing long-only portfolios always perform worse.
Students only think of in-sample optimality.
Fix: In-sample, constraints cost utility. Out of sample, long-only limits can reduce the effect of estimation error and extreme weights.
Forgetting funding cost when leverage is used.
Focus stays on scaling asset return.
Fix: Subtract borrowing cost on the borrowed part: L × R − (L − 1) × funding rate.
Assuming higher costs mean more frequent rebalancing.
Students confuse drift with the cost of correcting it.
Fix: Higher costs widen the no-trade zone, so you rebalance less often.

Worked examples

Example 1

A portfolio manager holds 50% in equities and 50% in bonds on a USD 100 million portfolio. The optimiser's new target is 65% equities and 35% bonds. Trading costs are 0.20% of the amount traded on every trade. Compute one-way turnover and the total cost in USD.

Show the solution

Weight changes: equities +15%, bonds −15%.
Sum of absolute changes = 15% + 15% = 30%.
One-way turnover = ½ × 30% = 15%.
Amount bought = 15% × USD 100 million = USD 15 million. Amount sold = USD 15 million. Total traded = USD 30 million.
Cost = 0.20% × USD 30 million = USD 60,000.

Answer: One-way turnover is 15% and the total transaction cost is USD 60,000, which is 0.06% of the portfolio.

Example 2

An unconstrained optimiser gives weights of 140% in asset A and −40% in asset B. The mandate is long-only and fully invested. Expected returns are A 8% and B 5%. What is the effect on expected return of the constrained portfolio if the optimiser then holds 100% in A, and what does this tell you about the constraint?

Show the solution

Unconstrained expected return = 1.40 × 8% + (−0.40) × 5% = 11.2% − 2.0% = 9.2%.
Constrained expected return = 1.00 × 8% = 8.0%.
Difference = 9.2% − 8.0% = 1.2 percentage points lower.
The constraint is binding because B's weight of −40% breaks the w ≥ 0 limit.
The constrained portfolio also has no exposure to B, so diversification from the combination is lost and risk is that of A alone.

Answer: Expected return falls from 9.2% to 8.0%, a loss of 1.2 percentage points. The long-only constraint binds and removes the short position and the leverage implied by 140% in A.

Exam tips

Start every answer with direction: constraints never improve the in-sample optimum.
Read the turnover definition. Questions often test one-way versus two-way.
Look for the phrase binding. If it is not binding, the answer is unchanged.
Expect interpretation questions: shadow price, transfer coefficient and no-trade zone.
Link constraints to estimation error. Long-only limits can help out of sample.

Practice questions from Portfolio Construction

Portfolio Constraints and Practical Implementation Issues: frequently asked questions

How does a long-only constraint affect the optimal portfolio?

It forces all weights to be zero or positive, so the optimiser cannot short or fund positions with short sales. In-sample, expected utility is lower or equal. It often gives more diversified and stable weights and limits active risk.

How do transaction costs affect portfolio rebalancing?

Each trade costs commissions, spread and market impact, which reduces net return. This creates a no-trade zone where small drifts are left alone. Higher costs widen the zone and reduce rebalancing frequency.

What is a turnover constraint in portfolio optimisation?

It caps how much of the portfolio can be traded in one rebalance, usually as a limit on the sum of absolute weight changes. It keeps costs and market impact down. The portfolio ends up between the old portfolio and the unconstrained optimum.

What is the transfer coefficient?

It measures how much of the unconstrained portfolio's efficiency is preserved once constraints apply. A value below one means constraints have reduced the information ratio. Tighter constraints lower it.