An Expected Shortfall limit on terminal wealth is cheap in this calibration. In the complete-market case, the limit binds at c = -0.94. Average dollar exposure drops 25.5%, from the Merton 0.4000 to 0.2981. Expected terminal wealth moves from 1.0322 to 1.0242, a decline of about 0.8%.
The policy changes shape as well as size. Exposure responds to the investor's current wealth. Along the selected favorable path, it rises back toward 0.400 and wealth finishes near 1.311. Along the selected unfavorable path, exposure contracts to roughly 0.068 and wealth finishes near 0.947.
The abstract gives the paper's central result directly: the investor "reduces risky exposure following adverse outcomes but preserves, and near maturity may increase, exposure following favorable outcomes."
What Hu, Pesenti and Shi prove
The authors study a continuous-time problem. An investor minimizes a convex running-plus-terminal cost over adapted dollar exposures while keeping CVaR of terminal loss at the 95% level below a fixed limit. The Rockafellar-Uryasev representation expresses CVaR as an infimum over an auxiliary scalar threshold eta. The constraint then becomes jointly convex in eta and the trading strategy.
The authors restrict eta to a compact interval independent of the limit. They establish existence of a primal optimizer and, under strict convexity, uniqueness of the optimal control. A Slater condition yields strong duality and an explicit bound for the optimal multiplier.
The numerical method follows the proof. Once the multiplier and threshold are fixed, the inner step is an unconstrained stochastic control problem with a modified terminal cost. Any existing dynamic-programming or PDE solver can act as the oracle. A golden-section search chooses eta, while an outer bisection adjusts the multiplier until the constraint residual reaches zero. Theorem 4.4 proves convergence when inner accuracy increases quickly enough relative to outer bisection depth.
There is no market data in the paper. It reports no Sharpe ratio or other return-risk metric. Every numerical result is a Monte Carlo moment from one scalar Black-Scholes stock, calibrated with T = 1, r = 0.00, mu = 0.08, sigma = 0.20, gamma = 5, w0 = 1. Merton exposure is 0.40, and alpha = 0.95.
The paper uses exactly two constraint levels: c = -0.86, which does not bind, and c = -0.94, which does. In the complete market, the nonbinding case reproduces the Merton policy exactly and assigns it a multiplier of zero. Terminal loss CVaR is -0.8671 against the -0.8600 limit. The corresponding incomplete-market CVaR is -0.8620, while the quadratic-regularization case gives -0.8971. These are numerical illustrations rather than empirical claims. Nothing from our own run below can support or undermine them.
A disclosure is necessary before presenting our figures. We could not run the continuous-time control solution as written, so we used daily rebalancing and scenario-based CVaR optimization as an approximation. We also left out the endogenous square-root price-impact extension. OHLCV-only data provides no bid-ask, trade-level or order-book information.
Where the paper advances continuous-time work
One branch of the continuous-time literature treats CVaR and quantile constraints under completeness, turning the problem into a choice among replicable terminal payoffs. Another retains self-financing dynamics with risk limits that are re-evaluated over time, relying on growth-optimal or CRRA structure. Hu, Pesenti and Shi dispense with both assumptions. Nontraded endowment risk can remain unreplicable, and general running costs can make the entire state-control path matter. Once frictions enter, the current position becomes another state variable. Existence and strong duality still hold at that level of generality. This is the paper's real contribution.
A quieter change matters for implementation. Complete-market CVaR and quantile problems often generate digital, lottery-like terminal payoffs. The benchmark objective in Section 5 adds a quadratic variation penalty, the gamma/2 sigma^2 phi^2 running term. It smooths away the digital payoff and produces state-dependent de-risking.
Unhedgeable risk makes the mandate dearer
With nontraded endowment exposure set to beta_perp = 0.02, the unconstrained solution changes little. Merton exposure remains 0.4000, and terminal loss CVaR shifts from -0.8671 to -0.8620. The binding case is different. Its multiplier nearly doubles from 0.0720 to 0.1400. Average exposure reaches 0.2549, sitting 14.5% below the complete-market binding value and 36.3% below Merton. Expected terminal wealth is 1.0206, about 1.1% below the nonbinding policy. A small amount of untradeable basis risk roughly doubles the shadow cost of the same mandate.
The friction cases arrive at similar outcomes through different mechanisms. Under a quadratic penalty on the trading rate, Lambda = 0.01, average binding exposure is 0.2415 versus 0.3104 without the binding constraint. The calibrated threshold is -0.947499 and the multiplier is 0.06001. Expected terminal wealth declines from 1.0247 to 1.0192.
Square-root price impact uses Lambda_{3/2} = 0.0045 and deducts execution costs from wealth. Binding exposure averages 0.2422 against 0.3500, with a multiplier of 0.07818. Expected wealth falls from 1.0260 to 1.0180. The authors state plainly that these square-root wealth dynamics lie outside their affine-state convex framework. Their convergence results therefore do not cover this case. They offer it as a sensitivity experiment, leaving the table as numerical evidence without theoretical support.
The cheapness claim belongs to the authors. Their conclusion says that "across the environments considered, stronger lower-tail protection is achieved at a comparatively modest cost in expected terminal wealth." The scope of that qualifier is narrow: one scalar Black-Scholes stock, two constraint levels (c = -0.86 and c = -0.94), and a single nontraded exposure beta_perp = 0.02.
Missing constants in the numerical recipe
The compact interval for the threshold is E = [-M_X/sqrt(alpha), M_X/sqrt(1-alpha)]. The multiplier lies in [0, Lambda], where Lambda = (J(phi_bar) - J_low)/delta. These bounds establish existence. An actual search needs four numerical inputs: M_X, a feasible strictly-slack strategy phi_bar, a lower bound J_low, and the Slater margin delta. For its own calibration, the paper's text supplies no numerical M_X, delta or J_low. The linked repository is left to provide the implementation details.
The inner error estimate contains the Lipschitz constant kappa_alpha = max{1, alpha/(1-alpha)}. With alpha = 0.95, it equals 19. Convergence asks 2^{N_lambda} rho^{N_eta} to vanish, where rho = (sqrt5 - 1)/2. Inner accuracy must therefore increase with outer depth. Theorem 4.4 establishes convergence without supplying a rate. Its analysis also treats the control oracle as exact, excluding grid error and Monte Carlo error from the guarantee. We found no wall-clock time or oracle-call count in the paper, and no comparison with a constant scaled-down Merton exposure.
Reported feasibility deserves a close look. Realized CVaR is -0.9406 in the complete market, -0.9405 in the incomplete market, -0.9413 with quadratic regularization and -0.9409 under square-root impact. The limit is -0.9400. Only the two friction cases are labeled out-of-sample, which makes the complete- and incomplete-market values in-sample relative to that limit. Slack at -0.9406 amounts to 0.0006 of initial wealth.
The authors also delimit what can be inferred from interim risk. In the reported zero-rate, no-inflow experiments, the loss-CVaR diagnostic remains below the terminal limit at every reported intermediate date in all four environments. Their complete-market discussion immediately cautions that "this observation is not a dynamic constraint and need not persist under other calibrations." Remark 3.7 adds that eta* is not unique in general.
Our book diverges from the paper
Our run spans 2020-01-02 to 2024-07-01. It uses the annual top 100 US stocks by capitalization and stays long-only. Each stock is capped at 10%, with total risky weight capped at 100%. We set a terminal horizon of 63 sessions, alpha = 0.95 and c = -0.94, then solved again each day using up to 1,260 sessions of history.
Costs were exogenous: $0.004 per share, a $1 minimum per order and zero modelled slippage. The strategy returned 55.16% cumulatively, with 10.28% CAGR and Sharpe 0.75. Maximum drawdown reached -26.81%, while beta to SPY was 0.65 across 12,245 trades. These figures come from our implementation over that window.
The nearest paper figure in spirit is expected terminal wealth of 1.0242 over one year, or about 2.4%. Our result is 10.28% CAGR. Those quantities differ. The paper reports a simulated moment for one stock at mu = 0.08, without deducting costs. We report realized net P&L from a 100-name book during a window containing the post-COVID rebound and the 2023-24 mega-cap rally.
Diversification further separates the levels. Spreading a terminal-CVaR budget across up to 100 capped long positions allows much greater total risky exposure than the paper's 0.24 to 0.40 dollar exposure in one stock at sigma = 0.20. The result is higher return and higher beta in our book.
The drawdown is the honest indictment of our construction.
The authors define c = -0.94 as a bound on the average of the worst 5% of terminal outcomes. It does not impose a pathwise floor. Our -26.81% drawdown, alongside 16.39% realized volatility and beta 0.65, therefore conflicts with none of their claims. It exposes the weakness of our rolling implementation. Calmar was 0.38 against that -26.81%.
Re-solving a 63-session terminal constraint each day resets the downside budget from current wealth, leaving the cumulative multi-year drawdown uncapped. Our search creates another problem. We ran the 80-reduction eta search and 30-iteration bisection without a defensible initial bracket because the paper gives no numerical M_X, delta or J_low from which to construct one. The traded policy may therefore differ from the constrained optimum reported in the paper's tables.
We recorded no episode-level terminal CVaR, multiplier or state-conditioned exposure path. The paper's central mechanism, asymmetric de-risking, consequently went untested in our run. Its reported binding multipliers range from 0.06001 for quadratic regularization to 0.1400 for the incomplete market. The available evidence does not fully explain the gap between our results and the paper's. Universe and period account for the return side; the risk side remains unresolved. Comparison is limited further because the paper provides no tradable P&L series. It reports simulated one-year moments for a single stock at mu = 0.08 and sigma = 0.20.
The transferable results are the feedback shape and the incomplete-market theory. The tables are calibration-specific. Cost will decide whether the idea survives in practice, as it did in our trend-following note (break-even cost remains the constraint). We charged four tenths of a cent a share plus a $1 minimum per order across 12,245 trades. I would change my mind after seeing a run that records realized terminal-horizon CVaR and the multiplier path against a constant scaled-down Merton book, net of costs, and shows that the state-dependent policy delivers tail relief beyond the flat cut.
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.