For an allocator, the missing number is the risky share. Ha, Jeon and Oh derive a three-term formula for it in a household model with sticky spending and a mean-reverting risk premium. As far as we can find, their numerical section never evaluates the share. The paper uses no data, only illustrative parameters.
Two barriers that move
Ha, Jeon and Oh charge the household α utility units to raise consumption and λ to cut it. They put it in the Kim-Omberg market, where expected excess return b mean-reverts and one Brownian motion drives both the risky asset and the premium. With a constant premium, the earlier Choi-Jeon-Koo solution has two fixed wealth-to-consumption ratios. Spending rises above the upper ratio and falls below the lower one. Between them, it stays put. Ha, Jeon and Oh make both ratios functions of the current premium.
The portfolio absorbs shocks while spending stays inside its band. To solve for that policy, the authors use duality in a normalized shadow price z and the premium b. The marginal value of consumption solves a double obstacle problem. Its contact sets determine an increase threshold a_λ(b) and a reduction threshold d_λ(b).
A sharp occupation estimate controls how quickly those thresholds move. Under the illustrative parameters, a change of one unit in b changes a threshold's log by at most L = 12.4417. A 0.01 premium move therefore shifts either threshold by at most 0.124 in log terms. The smoothness proof covers only |b| above B = 0.0672, away from 0.0990, the zero of the transport coefficient G outside the band. Its other zero, -0.0240, is inside the band already. The authors acknowledge that the excluded central interval contains the long-run mean of 0.05 and holds 0.6544 of the stationary distribution. Verification, however, needs only the Lipschitz bound and smooth fit along the diffusion. Their optimal policy is solved on the whole state space; for a trader, the remaining regularity gap is a mathematician's problem.
The paper uses no data. It sets r = 0.03, δ = 0.04, σ = 0.18, κ = 0.25, b̄ = 0.05, η = 0.03, α = 2 and γ = 2, borrowing the market coefficients from Ferrari and Schütz. At λ = 40 and b̄, the reported wealth-to-consumption thresholds are 27.8767 for a raise and 12.7979 for a cut. Together, the two costs produce an intertemporal loss aversion coefficient of 2.8261, forcing the barriers apart by a ratio of at least 1.6811. On one simulated 35-year path, consumption fell from 1 to 0.9452 across five days around t = 3.4. It stayed flat afterward until t = 32.6746.
What happens to the SPY weight?
We cannot trade the paper's asset. We substitute SPY for its stylized risky asset and cash for the money market.
The portfolio formula has three terms. Its myopic CRRA share, b/(γσ²), is about 0.77 at b = 0.05 with the paper's parameters. An adjustment-cost term follows; for positive b, it is nonpositive. A hedging term proportional to η/σ completes the share, with no prescribed sign.
That hedging term changes the constant-premium result. Choi, Jeon and Koo found that the risky share equals the myopic share at an adjustment threshold. Here the hedging term remains at the threshold, so the equality need not survive. The paper establishes the point from the formula's structure. We did not find a risky share reported in its figures or tables, leaving its size unreported. The authors also distinguish the normalized barrier problem, which does not depend on risk aversion, from its translation into consumption and wealth, which does.
From payout bands to daily trading
For a daily rule, treat "consumption" as a payout stream, such as an endowment draw or a retiree's withdrawal. Estimate b_t each day, then use that premium to find the two wealth thresholds. Wealth over payout above the upper threshold calls for a raise to wealth divided by that threshold. Below the lower threshold, the same calculation sets a cut. Inside the band, leave the payout alone. Set the SPY weight with the three-term formula.
The costs α and λ apply to payout changes. Trading SPY incurs no cost in the model, and its weight changes whenever b changes, even if the payout has been frozen for decades. Leverage is unlimited there too: at b = 0.15, the myopic term alone is about 2.3. A live version needs trading costs and a leverage cap.
The substitution depends on the estimated SPY premium behaving as the model requires. Premium shocks must load on the same Brownian motion as returns, with the premium falling when SPY rallies. They must also satisfy κσ > η; the example has 0.045 against 0.03. Our construction fits κ, b̄ and η to a premium estimate and first checks how its innovations correlate with SPY returns. The benchmark freezes the premium at b̄ and uses the paper's explicit closed form for that rule. Signals form at the close and execute at the next close, with cash financing and a leverage cap. That strategy is being evaluated now. We have no results to report.
Solver accuracy, and the missing share
On the main grid, discrete Bellman residuals are below 10^-13 for λ = 40 and for the ratcheting put. Against the explicit constant-premium solution, the main scheme's relative threshold errors at λ = 40 are 0.0256 and -0.0271. The primal scheme returns 38.1909 and 16.1373, versus exact values of 38.1827 and 16.0613.
Errors grow in the Kim-Omberg model. The main grid exceeds the extrapolated finite-difference values by 0.0581 for a_40(b̄) and 0.0844 for the ratcheting threshold. Extending the log z truncation from the main domain to the largest one reduces the upper wealth threshold by 0.0406 in relative terms. The authors warn against reading small differences across premia from the figures. They also identify the increase threshold's peak near b = 0 as numerical and outside their theory.
A practitioner could be caught by the result proved only for a constant premium. As λ rises, the cut threshold in wealth units tends to 15.8614, below half the ratcheting solvency floor of 1/r = 33.3333. At λ of 40, 160 and 2560, the sequence is 16.0613, 15.8907 and 15.8617. Substituting λ = ∞ directly into the wealth trigger instead gives 33.3333, while the finite-cost triggers tend to 15.8614. The consumption policy itself still converges to ratcheting. The paper proves that convergence from a fixed initial wealth above the floor and states the distinction plainly: "the limit of the thresholds differs from the threshold of the limit."
The paper reports no return, Sharpe or drawdown, and no estimated premium. If SPY premium innovations are far from perfectly negatively correlated with returns, its complete-market setup no longer applies. The authors leave that incomplete market as an open problem.