Portfolio Optimisation: When the Best Weights Are Too Fragile to Trade

Portfolio optimisation translates objectives and constraints into weights or trading rules. It does not discover a uniquely correct portfolio independently of its inputs. Expected returns, dependence estimates, risk preferences and implementation limits determine the answer.

This reference combines mean–variance methods, numerical optimisation, tail-risk objectives and continuous-time control. It preserves the teaching methods while distinguishing model-optimal decisions from realised investment outcomes.

Portfolio risk depends on interactions

Let $w$ be the vector of portfolio weights, $\mu$ the vector of expected asset returns and $\Sigma$ their covariance matrix. Expected portfolio return and variance are

$$\mu_p=w^\top\mu,\qquad \sigma_p^2=w^\top\Sigma w.$$

Variance depends on covariance terms as well as individual volatilities. For two assets,

$$\sigma_p^2=w_1^2\sigma_1^2+w_2^2\sigma_2^2 +2w_1w_2\rho\sigma_1\sigma_2.$$

In words: Combine each asset’s weighted variation with their shared movement to obtain total portfolio variation.

Here $\rho$ is return correlation.

Portfolio volatility across the three retained correlation cases

These are model arithmetic, not forecasts of realised diversification.

The covariance matrix should be positive semidefinite so that computed variance cannot be negative. Estimated dependence can change during stress. A low historical correlation is an input estimate, not a contractual guarantee of diversification.

Mean–variance optimisation defines an efficient set

A standard problem minimises variance subject to a target expected return $m$ and full investment:

$$\min_w w^\top\Sigma w \quad\text{subject to}\quad w^\top\mu=m,\quad \mathbf1^\top w=1.$$

In words: Choose weights that minimise portfolio variation while meeting the target expected return and investing the full amount.

Nonnegative weights, concentration limits or leverage bounds can be added. Varying the target traces a feasible risk-return set. The efficient part contains portfolios for which no feasible alternative has both higher expected return and lower risk under the inputs.

Computing mean and variance does not require normally distributed returns. Normality, quadratic utility or other preference assumptions may justify using only those two moments to rank portfolios. Without such assumptions, portfolios with the same mean and variance can have materially different downside behaviour.

For an unconstrained global minimum-variance portfolio with invertible covariance matrix,

$$w_{\mathrm{GMV}}=\frac{\Sigma^{-1}\mathbf1} {\mathbf1^\top\Sigma^{-1}\mathbf1}.$$

In words: The minimum-variation weights come from the covariance structure and are scaled so the weights add to the required total.

Practical implementations should solve the relevant linear system rather than explicitly form an inverse. Singular or nearly singular covariance matrices need additional treatment.

Constraints can improve economic realism

A theoretical optimum may require large short positions, leverage or turnover. These may be unavailable, costly or inconsistent with the investment mandate. Adding constraints changes the feasible set and can materially change the optimum.

A turnover penalty can discourage unnecessary trading. Transaction-cost models may be proportional, nonlinear or state-dependent. Lot sizes and cardinality restrictions can turn a convex problem into a harder discrete or nonconvex optimisation.

An optimiser’s failure to reach a target return may indicate that the target is infeasible under the constraints. Relaxing constraints silently to obtain a result changes the mandate. The report should distinguish infeasibility from numerical failure.

Estimation error is often larger than solver error

Expected returns are difficult to estimate precisely. Optimisation can turn small changes in $\mu$ into large shifts in weights, particularly when assets have similar estimated risk exposures. A highly concentrated solution can be the optimiser exploiting noise.

Three responses to unstable portfolio inputs

These approaches reduce sensitivity but introduce their own assumptions about which deviations are plausible.

Sensitivity analysis should vary expected returns, correlations, volatility levels and transaction costs. A portfolio that changes completely under small plausible input changes should be presented as fragile even if the solver reports an exact optimum.

Tail-risk objectives answer a different question

Value at Risk is a loss quantile, while expected shortfall describes average loss in the relevant tail under an appropriate definition. For a loss function $L(w,\xi)$, random scenario $\xi$ and confidence level $\alpha$, a useful expected-shortfall representation is

$$\min_\eta\left\{\eta+\frac1{1-\alpha} \mathbb E[(L(w,\xi)-\eta)_+]\right\}.$$

In words: Choose the threshold that minimises its value plus the average excess loss divided by the chosen tail probability.

The auxiliary threshold $\eta$ supports scenario-based optimisation. With suitable convex loss functions and linear constraints, the resulting formulation can be computationally tractable. It still depends on the chosen scenario distribution and sufficient representation of tail events.

Stress testing complements probabilistic risk measures by asking what happens under specified shocks. A scenario can be informative without assigning it a precise probability. Combining a stress loss and a statistical quantile into a single number without explaining the weighting obscures their different meanings.

The source teaching materials included a one-million-unit portfolio with 10% annual volatility and a 95% normal quantile of 1.645. The resulting 164,500-unit loss threshold is on the annual horizon under that simplified zero-mean approximation. It was labelled daily in one teaching passage; that horizon label is inconsistent with the stated annual volatility. A daily approximation would require an explicit time conversion and its assumptions.

Dynamic optimisation chooses a policy

For wealth $X_t$, a fraction $\pi_t$ invested in a risky asset with drift $\mu$, volatility $\sigma$, and a risk-free rate $r$ gives

$$dX_t=X_t[r+\pi_t(\mu-r)]dt+X_t\pi_t\sigma dW_t.$$

The fraction $\pi_t$ is different from a currency amount allocated to the risky asset. Mixing these definitions changes the wealth equation and its units.

For terminal utility $U$, define value function

$$J(x,t)=\sup_{\pi}\mathbb E[U(X_T)\mid X_t=x].$$

Under suitable assumptions, its Hamilton–Jacobi–Bellman equation is

$$J_t+\sup_\pi\left\{ x[r+\pi(\mu-r)]J_x+\tfrac12x^2\pi^2\sigma^2J_{xx} \right\}=0.$$

In words: At each time and wealth level, choose the allocation whose expected benefit and uncertainty best support the final objective.

The terminal condition is $J(x,T)=U(x)$. The maximisation expresses the best current decision given the value of future opportunities.

The simple constant-proportion result is conditional

For constant relative risk aversion $\gamma>0$, power utility and constant investment opportunities, the unconstrained Merton fraction is

$$\pi^*=\frac{\mu-r}{\gamma\sigma^2}.$$

With illustrative values $\mu=8\%$, $r=3\%$, $\sigma=20\%$ and $\gamma=4$, the fraction is 31.25%. Higher expected excess return raises the allocation; higher risk aversion or variance lowers it.

This is optimal expected utility within the model. It does not dominate every alternative on every realised path. Changing opportunities, stochastic labour income, consumption, borrowing limits and transaction costs can all change the policy.

When costs make frequent small trades uneconomic, optimal policies can contain no-trade regions rather than continuous adjustment to a target. A control problem with these features may require variational inequalities or other specialised numerical methods.

Judge the process, not just the final weights

Records required across the portfolio-optimisation process

The useful output is an allocation or policy accompanied by its assumptions and fragility. Optimisation becomes a disciplined decision process when it exposes those dependencies instead of concealing them behind a single set of precise weights.

Risk contributions explain concentrated solutions

For portfolio volatility $\sigma_p=\sqrt{w^\top\Sigma w}$, the marginal volatility contribution of weight $w_i$ is $(\Sigma w)_i/\sigma_p$. Multiplying by the weight gives component contribution

$$RC_i=\frac{w_i(\Sigma w)_i}{\sigma_p}.$$

These contributions sum to portfolio volatility under the differentiable homogeneous setup. They show why equal capital weights do not imply equal risk contributions. A low-weight asset with high volatility and strong common exposure can dominate risk.

Negative contributions can occur when a position offsets the rest of the portfolio. They should be interpreted in the context of the full covariance model, not as evidence that the asset is intrinsically riskless. Correlation changes can remove the offset precisely when it is most valuable.

Risk budgeting sets desired contributions or limits rather than directly optimising uncertain expected returns. It reduces dependence on return forecasts but remains dependent on the risk model, chosen assets and constraints. It is therefore another explicit objective, not an assumption-free allocation rule.

A turnover example makes implementation cost visible

Consider an illustrative portfolio moving from weights 60% and 40% to 50% and 50%. It sells 10% of portfolio value in one asset and buys 10% in the other. One-way turnover is commonly half the sum of absolute weight changes, here 10%; gross traded amount is 20% of portfolio value.

If transaction cost is charged on every amount bought and sold, applying the rate to one-way turnover without adjustment understates cost. A 10-basis-point cost on each trade implies two basis points of portfolio cost for this rebalance. The result follows from a stated convention; other turnover definitions should be labelled accordingly.

A strategy that repeatedly moves between similar optimal weights may lose its apparent advantage after these costs. Including costs in optimisation and then evaluating realised turnover provides a more complete test than deducting an arbitrary annual fee from a frictionless backtest.

Stress the inputs and the implementation together

An ordinary covariance stress changes volatilities or correlations. A fuller implementation stress can also reduce available liquidity, widen transaction costs and tighten leverage constraints. These changes may interact: a position that hedges market risk in the model may be expensive or impossible to resize during stress.

A robust portfolio review should examine feasible rebalancing paths rather than only the desired final weights. If every stressed scenario assumes instantaneous movement into the newly optimal portfolio, the analysis may understate the risk of getting there.

Scenario probabilities should not be invented to make the optimisation tractable. Some stresses are better presented as conditional losses or feasibility checks. A worst-case objective over an uncertainty set is a modelling choice whose conservatism depends on how that set is defined.

Validate the optimiser using simpler cases

Before applying a complicated constrained solver, verify a two-asset case that can be checked analytically. Confirm that weights sum to the required amount, constraints are satisfied within tolerance and reported objective value matches an independent calculation.

For convex quadratic problems, first-order optimality conditions and dual information can help identify binding constraints and infeasible targets. Nonconvex problems require greater care: a local optimum is not necessarily the best feasible solution, and results can depend on starting values.

The final decision should distinguish three findings: what is numerically optimal for the stated inputs, how stable that answer is to plausible changes, and whether the resulting trades are operationally feasible. Combining those findings is more informative than presenting an efficient frontier alone.

A feasible benchmark anchors the assessment

Compare the optimised portfolio with a simple feasible alternative, such as existing weights, equal weights or a mandate-consistent low-turnover allocation. Apply the same data, costs, constraints and rebalancing dates to both. An optimiser should earn its additional complexity through a material improvement under that common evaluation.

If the estimated improvement is smaller than plausible input uncertainty or implementation cost, the decision may favour the simpler portfolio even when its model objective is slightly worse. That is a judgement about robustness and economic materiality, not a failure to understand the mathematical optimum.

Record both the expected benefit and the assumptions that would erase it. This gives the eventual allocation a clear review condition: when the assumptions change, the portfolio can be reassessed against its original rationale rather than justified retrospectively by whichever outcome occurred.