Minimizing range value-at-risk with a positive lower edge can leave you holding more stock than Kelly and accepting zero wealth in bad states. Mi, Pan and Xu spell out that bargain for a continuous-time investor. In their one-asset example, pure minimum-RVaR at α = 0.2 puts 3.5974 times initial wealth into the risky asset at inception. The growth-optimal (Kelly) portfolio holds 2.6667.
The paper is theoretical. It offers no empirical claim or backtest. The authors calculate their examples in a synthetic Black-Scholes market with r = 0.05, σ = 0.15, θ = 0.4, x = 1 and T = 1. All figures involving real data below are ours.
What does the RVaR envelope buy?
The investor maximizes λE[R_T] − (1−λ)RVaR_{α,β}(R_T), scoring the annualized log return R_T = (1/T)log(X_T/x). RVaR averages the return quantile over [α, β]. Set α = 0 and it becomes expected shortfall at β; shrink β toward α and it becomes VaR at α. The endpoints of the growth weight have familiar meanings: λ = 1 gives the Kelly problem, while λ = 0 gives pure risk minimization. Using log returns rather than terminal wealth makes the objective concave. That concavity prevents the mean-risk problem from collapsing into extreme leverage, the failure He, Jin and Zhou documented for weighted VaR on terminal wealth.
Wei and Xu (2025) supplied the machinery by solving the general weighted-VaR problem on log returns. First reduce the portfolio problem to a static one with a budget constraint, and couple wealth anti-monotonically to the state-price density ξ_T. A change of variables removes the pricing kernel from the budget. The convex envelope of a transformed weighting function φ_λ then gives optimal wealth through its right derivative. Xu appears on both papers. The contribution here is the geometry specific to RVaR.
Because RVaR weights an interior interval, φ_λ consists of three convex pieces. Its slope jumps up at w(α) and down at w(β). The downward jump breaks global convexity, and the envelope has two ways to bridge it.
Scenario 1 starts the supporting line at w(α). With λ = 0, the resulting digital payoff is a constant x/(E[ξ_T][1−w(α)]) when ξ_T ≤ G_ξ(1−α), and zero otherwise. With 0 < λ < 1, the payoff is GOP-constant-GOP: λx/ξ_T in both good and bad states, separated by a flat segment. In Scenario 2, the supporting line touches inside [w(α), w(β)]. At λ = 0, wealth is zero in the worst states, x/((β−α)ξ_T) in the middle, and constant at the top. With 0 < λ < 1, the payoff has four regions; one pays A_λx/ξ_T, where A_λ = λ + (1−λ)/(β−α).
The authors prove that Scenario 1 applies whenever β is close enough to α. Its contact point then does not depend on β, making the RVaR payoff exactly equal to the VaR payoff. At α = 0, Scenario 2 applies and recovers the ES solution.
Exact recovery of VaR for β close to α and ES at α = 0 is the theoretical result. Table 1 carries the practical argument.
The worst α of states get nothing
Both λ = 0 payoffs leave zero wealth when ξ_T > G_ξ(1−α), the worst α fraction of states. These states fall below the window. Allocating wealth to them cannot reduce RVaR, so the investor shifts the budget above the α quantile. The authors describe the sacrifice directly: pure minimum-RVaR "may improve the targeted RVaR criterion by sacrificing states outside the penalized quantile range," and P(X_T = 0) > 0 forces E[R_T] = −∞.
Their response is to motivate λ > 0. For terminal wealth, that response works: every mean-RVaR payoff delivers at least λx/ξ_T in the worst states, giving the investor a fraction λ of the Kelly payoff.
Exposure still exceeds Kelly at λ = 0.4
The position held at inception gives a less reassuring answer. For α = 0.2, β = 0.21, initial exposure over λ = 0, 0.2, 0.4, 0.6, 0.8, 1 is 3.5974, 2.6455, 2.7909, 2.4185, 2.2801, 2.6667. The risk-controlled portfolio remains above Kelly at λ = 0.4. Widening the window to β = 0.7 produces exposure of 3.1902 at λ = 0.2, again above 2.6667.
A positive growth weight restores positive terminal wealth in the excluded tail. It gives no guarantee of below-Kelly exposure. At α = 0.2 and λ = 0.4, both widths still produce positions above 2.6667: 2.7909 and 2.8233. Both fall below it only at λ = 0.6.
The ES end of the table also resists reading λ as a risk-aversion dial. For α = 0, β = 0.01, exposure runs 0, 0.5333, 1.0667, 1.5971, 1.1780, 2.6667. It falls between λ = 0.6 and 0.8. Changing the width gives no consistent direction either. At λ = 0.2 with α = 0, increasing β from 0.01 to 0.5 cuts exposure from 0.5333 to 0.2509. With α = 0.05, increasing β from 0.06 to 0.55 raises it from 1.2002 to 1.2698. The authors treat this as a result in the abstract and return to it in the conclusion, writing that the flexibility "does not imply a uniform ordering of portfolio risk, return, or leverage across parameter configurations."
One pattern is clear. For λ = 0 and α of 0.05 or more, β makes no difference to the initial position. Both widths give 1.3717 at α = 0.05, 2.2236 at α = 0.1, and 3.5974 at α = 0.2. This matches the Scenario 1 digital, whose payoff depends only on α. Minimum-RVaR exposure is then determined entirely by the tail the investor agrees to forfeit. At α = 0.01, however, exposure changes: 0.4275 for β = 0.02 versus 0.2939 for β = 0.51. That suggests a different envelope, though the paper does not label the cells.
We did not find a statement identifying the scenario for each cell or explaining how the contact points were computed. Those details determine the hedge when payoffs contain digital and kinked joins.
Our SPY adaptation
We built a construction from the authors' idea. It is an adaptation, and it does not test the paper, which makes no claim about real markets. SPY replaces the generic risky asset. Cash earns the three-month Treasury yield, DGS3MO, lagged at least one business day. Because this is a quoted yield, the cash account itself is modeled.
SPY is no complete Brownian market, and its coefficients change. At each annual inception we estimate drift and volatility using three trailing years, requiring at least 700 daily log returns, then hold those estimates fixed. The paper hedges continuously. We rebalance daily at the close with a one-session lag. Discretization error is worst around the digital and kinked boundaries, where the continuous delta spikes. We kept λ = 0 out of trading because its zero-wealth region makes log returns singular; an insolvent ledger stops rather than being floored.
The primary run uses RVaR on [0.01, 0.05] with λ = 0.5 and starting capital of $1,000,000, covering 2020-01-01 to 2024-07-01. It includes four complete annual horizons, with the March 2020 crash in the first. The 2024 horizon is censored. The figures include commissions of $0.004 a share. They exclude short borrow, margin financing and market impact.
Our construction returned 15.43% in total over that window. Sharpe was 0.18 (Sortino 0.19), volatility was 19.28%, maximum drawdown was -38.16%, and Calmar was 0.09. The result is weak. It comes from one automated pass on a substitute market and says nothing either way about the paper's theorems. Four horizons, using frozen estimates of θ, are far too few to judge a window covering the 1% to 5% band of annual outcomes. The paper reports no performance of its own to put alongside these figures.
Our own constraint deserves attention first. We cap SPY notional at $1,000,000, one times starting capital; the paper's Kelly example holds 2.6667 times. When the target hedge calls for more than one times, that cap binds ahead of the 4.0 leverage ceiling. The executed portfolio then departs from the paper's payoff.
A payoff map worth reading
The theory earns its place: it makes explicit the payoff map for RVaR budgets left implicit by Wei and Xu's general result. The addition to Wei and Xu is that map and a numerical warning. In the authors' example, α = 0.2 produces a minimum-risk portfolio holding about 35% more stock than Kelly, 3.5974 against 2.6667, while sacrificing the worst fifth of states. Anyone approving RVaR limits with a positive lower edge should read Table 1 first. Evidence that a daily hedge of the GOP-constant-GOP payoff, using estimated θ, tracks its target within costs would change my view on implementation.
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.