A linear term in the portfolio objective gives a trader two risky baskets where Markowitz gives one. The portfolio with the highest reported Sharpe ratio generally differs from the portfolio where the capital allocation line touches the risky frontier. Positive homogeneity disappears whenever c or k is nonzero. In the paper's worked example, expected return is 23.38 for the max-Sharpe portfolio and 18.63 for the tangency portfolio. Their risk measure values are 6.14 and 4.09. Gasparavičius and Grigutis call this a new geometric phenomenon in the abstract, and the description fits. Any turnover penalty anchored on current holdings generates the same linear term.
There is no dataset, sample period or backtest. The authors make no empirical claim at all. The paper consists of nine propositions with full Lagrangian proofs and one synthetic four-asset example computed in MATLAB. They describe it as "based on synthetic data chosen solely for illustrative purposes rather than on empirical financial data." This is mathematics about portfolio construction, and it should be judged on those terms.
What the linear term changes
The risk measure is R_P(w) = ½wQw' + cw' + k. Q is symmetric positive definite, while k is at least ½cQ⁻¹c' so that R_P remains nonnegative. Variance returns when c = 0, k = 0 and Q = 2Σ. With c retained, completing the square places the center at -Q⁻¹c. The investor then pays for moving away from a reference portfolio rather than from the origin. That reference point supplies the linear term's economic meaning.
Three instances make the interpretation concrete. A quadratic trading-cost penalty (w_new - w_old)Λ(w_new - w_old)' expands into the same form, with Q = 2Λ, c = -2w_oldΛ and k = w_oldΛw_old'. Benchmark-relative optimization around b places the center at the benchmark. Ridge-style regularization is the c = 0, k = 0 case, with Q = 2(Σ + λI), while a factor covariance gives Q = βΣ_Fβ' + D. The paper also reviews possible sources for Q: Ledoit-Wolf shrinkage toward the identity or a general target, the Minimum Covariance Determinant estimator, and Tyler's and Huber's M-estimators.
The reward for this generality is a closed-form solution. The paper derives minimum-risk weights, the frontier, the max-utility portfolio, max-Sharpe weights and the optimal value sgn(B(1+C)-AD)·sqrt(2Ω/Δ). Each is written through Q⁻¹ contractions with µ, 1 and c. Q, c and k remain fixed inputs throughout. No proposition measures how estimation error in Q or µ passes into the result. Instead, the authors suggest inserting a shrunk or factor-based Q. Section 1.3 explicitly observes that sample covariances become unstable when the number of assets is large relative to the available sample size. This addresses the choice of input, without providing an error analysis.
A curve replaces the capital market line
R_P is strictly convex and fails coherence. Positive homogeneity is the missing axiom: scaling the book ceases to scale risk proportionally. The authors treat this as intentional because proportional scaling can fail under transaction costs, liquidity effects and taxes. That argument is reasonable for the objective they study.
The geometric consequence receives a full treatment. Once a risk-free asset is added, the efficient frontier becomes curved. In the example, it is µ = sqrt(61.53(2R_P - 2.00)) - 0.87 for R_P ≥ 1.00. Without the risk-free asset, the frontier is R_P = 0.02µ² - 0.60µ + 6.68 for µ ≥ 12.16. The authors call the first object a Capital Market Curve. They credit its earlier appearance in a mean-lower partial moment setting instead of claiming the object itself as new. A straight line reappears only when k is set exactly at ½λ²cQ⁻¹c' with no increment.
Tangency and the maximum of (µ - r_f)/sqrt(R_P) therefore occur at different points. A desk can no longer calculate one risky portfolio and move up or down it with cash. The best risky mix changes with the cash allocation because the linear term does not scale with the risky sleeve. Anyone using a cost-penalized optimizer and reporting a Sharpe-maximizing sleeve should keep that distinction in view.
One qualification matters for interpreting the figure. When c or k is nonzero, sqrt(R_P) is not a standard deviation. The paper's "maximum Sharpe ratio" therefore cannot be compared with the Sharpe reported in a monthly letter. The separation between the portfolios remains genuine geometry. Since sqrt(R_P) contains c and k, the optimized quantity differs from a conventional Sharpe ratio.
Fund separation, with moving risk
Fund separation survives in an altered form. Combining frontier portfolios with weights λ₁ = 0.79 and λ₂ = 0.21 reaches a target return of 17.59. The combination lies on a frontier associated with c_new = (1.76, -2.35, 1.17, -0.59). It is efficient because 17.59 exceeds µ_min = 10.71 for that c.
The sharper result sets c₂ = -c₁. The two minimum-risk portfolios have µ = 12.16 with R_P = 3.06 and µ = 5.16 with R_P = 1.06. Their reweighted path remains efficient only for λ in [-0.04, 1.04], corresponding to 14.83 < µ₀ < 26.95.
Two funds survive, while the risk measure governing efficiency changes with the mix.
The example's weights deserve attention before the formulas are treated as a usable portfolio. With a risk-free asset, the max-utility portfolio is (-0.67, 1.00, 0.00, 4.33, -3.67). It produces µ = 60.67 and R_P = 31.77. The setup contains four synthetic assets, permits unconstrained shorting, uses µ = (2, 5, 9, 14) and sets r_f = 1. A long-only constraint, position limit or turnover cap removes the closed forms.
The paper also acknowledges the limitation raised by its own introduction. Variance is criticized for handling tail risk poorly, yet the proposed measure, the authors write, "inherits one important limitation of variance": "it is symmetric, penalizing positive and negative deviations in the same quadratic manner. Consequently, it does not distinguish upside volatility from downside risk." Their response points to uses of quadratic optimization outside portfolio theory, including Lyapunov functions and linear-quadratic regulators in control, Mahalanobis metric learning, and the linear-quadratic model of radiation response. The tail-risk motivation introduced earlier remains unresolved. The measure's flexibility lies in the center and curvature of a symmetric penalty.
Our constrained run
Q, c and k required modeling choices from us before any result could be produced. Real market-impact coefficients cannot be recovered from quotes or order books, so our quadratic anchor examines an assumed cost shape rather than a measured liquidity model. The paper's solutions have no constraints beyond the budget line. We imposed caps and a gross limit, removing the closed form at the center of the paper. Nothing in this run tests the paper's claim because the authors make no empirical claim to test.
We used the top 100 US stocks and ETFs by trailing dollar volume, with monthly rebalancing from 2020-01-01 to 2024-07-01. Expected returns were estimated from 252 daily returns per name and shrunk 50/50 toward the cross-sectional mean. Covariance came from Ledoit-Wolf. We adopted the paper's linear term c = -2w_oldΛ, set Λ = 0.001·I and added the anchor to covariance. Thus Q = 2Σ̂ + 2Λ is our construction rather than theirs, and we dropped the paper's constant k. The portfolio was long/short, with weights capped at ±10%, gross exposure capped at 2.0, and costs of 5bps one-way plus $0.004 a share. We modeled no slippage.
Over that window, the book returned 143.85% in total with a Sharpe of 0.50. Maximum drawdown reached -75.02% on 63.64% annualized volatility. The Sortino was 0.68 and the Calmar 0.29. A 0.50 Sharpe paired with a 75% drawdown is poor. Our anchor calibration produced it.
One choice dominates the outcome: the anchor scale. Λ = 0.001 is expressed in daily variance units. A name with 2% daily volatility contributes 0.0004 to the diagonal of Σ̂. Our anchor term is larger on the diagonal than the risk term, leaving target weights highly sticky and keeping the book close to the previous month's holdings. The 0.001 anchor was our choice.
The universe amplifies the effect. SPY/IVV/VOO, QQQ/TQQQ, EEM/IEMG/VWO and HYG/JNK all appear in the top 100. Regularization determines how weight is divided inside each cluster rather than any forecast.
Evidence that specifying c and Q outperforms writing a benchmark-relative or cost-aware objective directly into a constrained solver would change my view of the framework's usefulness. The paper does not make that comparison. Once a desk adds the constraints it needs, the closed form is unavailable. The geometric result still stands by itself: with a linear term in the objective, tangency and maximum Sharpe identify two portfolios. In the authors' example, their expected returns differ by 4.75 points.
Our backtest stops at 2024-07-01, and everything after that date is deliberately left untouched so the same strategy can be checked out of sample later.