For a sufficiently deep loss tail, minimizing VaR on two diversified lognormal assets selects the minimum-variance mix of log-prices. Gach and Hochger obtain that result as α → 0 in their comments on optimization. The sticking point is "far enough". Their explicit test for tail depth is sufficient rather than necessary, and on our reading it does not reach a 5% monthly level. Here diversified means each log-variance exceeds the log-covariance: b1 = A22 − A12 and b2 = A11 − A12 are both positive.
The paper develops theorems, asymptotics and Monte Carlo pictures for its own model. Our backtest figures come from a separate construction. They cannot confirm or contradict the authors' results.
Why the quantile changes shape
The authors study q_α(u), the α-quantile of Σ u_i X_i. Each X_i = exp(Z_i), with Gaussian Z, covariance A and positive coefficients u_i. For elliptical risk factors, the quantile was already known to be concave in u for small α and convex for large α; elliptical VaR is sub-additive in the loss tail for that reason. The corresponding result for lognormal factors was unknown, the authors say. Their starting point is a Hessian formula for the quantile, extending the gradient and Hessian Gourieroux, Laurent and Scaillet derived for linear pay-offs. For a linear portfolio, its sign comes from the y-derivative of the portfolio density multiplied by the conditional variance of positions at the quantile.
The proof turns on where a moving probability measure puts its mass. Rescaling {Σ u_i x_i = y} to a fixed simplex gives a conditional law µ_y. Deep in the left tail, when every b_i is positive, µ_y concentrates at an interior barycenter, x*_i = b_i/(u_i β). Here b_i are the row sums of B = A^{-1}, and β is their total. Otherwise the mass goes to a corner; in the right tail it moves to a corner as well. Figures 2 and 3 show Monte Carlo quantiles from 10^7 trials. The theorems stand independently of those plots.
Theorem 3.6 establishes strict left-tail concavity for any number of assets if every b_i is positive. It also establishes concavity for two assets when b2 < 0. Remark 3.7 leaves b2 = 0 open: it "would require further analysis". On the other side, Theorem 3.12 establishes right-tail convexity for two assets only. The authors keep n = 2 there to avoid boundary case distinctions. Section 4 follows the quantile surface as α moves into the tail and the weights vary. Its limit separates a level-dependent scale from a shape determined by covariance: q_α(u) ∼ k_α q_0(u). The shape q_0 forgets µ entirely. The abstract calls this asymptotic separation of scale and shape. Gulisashvili and Tankov, cited by the authors, located the quantile at fixed weights; the result here concerns how its asymptotic shape changes across weights.
The weight the limit selects
Left-tail concavity gives the maximization of q_α over u1 + u2 = 1 a single answer. As α → 0, the factorization makes that answer independent of α. For b1 = A22 − A12 > 0 and b2 = A11 − A12 > 0, the answer is u_i = (A11 + A22 − A_ii − A12)/(A11 + A22 − 2A12). Remark 5.1 identifies it as the minimum-variance portfolio of log-prices.
With b2 < 0, the variance of the less volatile asset lies below the covariance. The shape curve becomes linear and the optimum moves to u = (1, 0), wholly in that asset. In the right tail, under a dominance condition equivalent to A22 > A11, the profit-maximizing choice is the other corner: u = (0, 1), wholly in the more volatile asset.
Remark 5.2 acknowledges the familiar outcome. The solutions "correspond to intuition and are not surprising per se", the authors write, while calling it "remarkable that these strategies are obtained analytically from a quantile perspective." I agree with both points. The corner appears precisely when the unconstrained minimum-variance weight on the second asset turns negative, so long-only minimum variance already gives the same weights. The practical gain is narrower and useful: deep in the left tail, VaR minimization cannot strand an optimizer at a local solution, and its target does not depend on µ.
Does the proof reach a desk's VaR level?
Theorem 3.6(2), Theorem 3.12 and the Section 4 limits establish the existence of some α0. For the right tail and the shape factorization, we found neither a value nor a convergence rate. Remark 4.1 says the speed of convergence depends on µ. Figure 3 gives a visible warning: with µ = (0, 0), B11 = 0.5, B12 = −0.3 and B22 = 0.25, the right tail "develops convexity only at the last simulated instance."
The left-tail result offers more to check. Theorem 3.6(1) supplies a sufficient condition when every b_i is positive. For all α ≤ α0, strict concavity holds once q_α0(u) < η exp((Σ_j (Bµ)_j + 1)/β), with 0 < η < min u_i. Remark 3.8 guarantees an α0 satisfying it for every u. Figure 1 puts the threshold near 10.75 in its example and describes the condition as non-sharp.
For monthly gross returns normalized to 1, our reading puts the exponential factor close to one. In a two-asset book η cannot exceed 0.5. The sufficient condition therefore calls for a quantile below roughly half the starting value, or a one-month loss greater than 50%. Its failure at 5% proves nothing about whether the quantile is concave there.
The proof has a couple of points to watch. The last paragraph of Section 3.4 concludes about y → ∞ while treating y → 0. The right-tail Laplace expansions retain leading-order terms, and we found no bound on their remainders.
Prices, shares and the holding period
The theorem's u_i are share counts: they multiply lognormal prices, and Section 5 imposes u1 + u2 = 1 on shares. Set each price to 1 at the decision close and those counts become dollar weights for that day. They drift thereafter. The other translation concerns time. A must describe log-prices at the chosen horizon, so a monthly hold calls for a monthly log-return covariance. The weight formula has degree zero in A, which removes the det A = 1 normalization. If daily log returns scale cleanly, the horizon also cancels from the weights. The corner test comes down to A11 < A12.
Our SPY/IWM book stayed at 50/50
We ran a monthly SPY/IWM construction. At each month-end close, we estimated mean and covariance from 252 trailing daily log returns, then drew 100,000 Gaussian scenarios for the next holding period. We selected the weight that maximized the 5% quantile of weighted gross returns from 1,001 possible SPY weights. Trades were placed at the following month's first trading-day close. Every fill incurred $0.004 a share in commissions, subject to a $1 minimum per order.
We also capped each ETF at 50%. In a fully invested two-asset book, that rule fixes both weights at 50/50; the optimizer never affected a trade. The cap was our design error. From 2020-01-01 to 2024-07-01, the book returned 55.11% net of commissions. Its Sharpe was 0.53, annualized volatility 22.96%, and maximum drawdown -40.51%. These are equal-weight SPY/IWM figures for a window starting in January 2020. They tell us nothing about tail-quantile optimization, since our equal-weight and capped minimum-volatility benchmarks have the same target. The authors publish no backtest performance to compare with our figures.
Even without the cap, a monthly 5% quantile remains far from α → 0. The concavity certificate, by the arithmetic above, would not reach it. A computed α0 for the Section 4 shape limit under a realistic monthly covariance would change my view. If it lay near 5%, the log-price minimum-variance weight would become the VaR-optimal weight at a level desks actually report.
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.