The authors' baseline numbers tell a constant relative risk aversion (CRRA) investor to short green and buy brown. A jump that has yet to happen is all that reverses the positions.
Would you short green without a transition?
Cash pays r = 0.02 in the paper's two-stock example. The green stock has μ_g = 0.015 and σ_g = 0.15; brown has μ_b = 0.035 and σ_b = 0.3. Correlation is ρ = 0.7 and risk aversion η = 2. Kroell and Steffensen chose the drifts to reflect evidence that green stocks underperform brown stocks outside crises. The resulting Merton portfolio holds π_g = −0.45 and π_b = 0.24. Green returns less than cash (1.5% against 2%) while sharing a 0.7 correlation with brown. It gets shorted.
Introduce a single shock at an unknown date: green jumps by κ_g and brown by κ_b. Set κ_g = 0.1 and κ_b = −0.2, then raise the arrival rate λ from 0 toward 0.25. Both positions contract before changing sign, leaving the investor long green and short brown. Utility comes from terminal wealth (and the model distortion), without a green-preference term. The expected jump supplies the green tilt.
The continuous-time model contains m geometric Brownian motion stocks, a cash account, and a transition event with constant reference intensity λ. Each stock is multiplied by (1 + κ_i) when the event occurs. Drift and covariance can also change afterwards. The investor doubts λ and considers alternative intensities θ_t·λ. Preferences apply to terminal wealth multiplied by a power of the alternative model's Radon-Nikodým density: U(X_T·Z_T^(1/ε)). This gives a multiplicative ambiguity penalty under power utility. For log utility, it becomes exactly the Hansen and Sargent relative-entropy penalty.
The parameter ε must exceed η − 1; approaching η − 1 makes the problem ambiguity-neutral. If drift and covariance remain unchanged after the transition, letting ε grow without bound leads to the Korn and Leoff worst-case crash portfolio. The authors solve the Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation separately before and after the transition. They also prove that the governing ODE has a solution on [0, T] and supply a verification theorem.
The paper contains no data. Its illustrations use hand-set parameters over a hypothetical 10-year horizon, and the authors make no empirical claim. The backtest below is our own construction from their idea. Its figures say nothing about whether the authors are right.
A fixed hedge direction, a disputed clock
Before the transition, the weight equals the Merton portfolio plus a scalar times Σ0⁻¹κ. Covariance and jump sizes alone determine the direction of that adjustment. Its scale comes from ξ, the post-jump gross return 1 + πᵀκ. A nonlinear equation for ξ* is coupled to the value-function ODE, making the hedge vary with time.
Using the paper's later jumps, κ_g = 0.2 and κ_b = −0.5, our calculation from the base covariance puts Σ0⁻¹κ at roughly (32.7, −17.0). Each unit of hedge therefore buys about two units of green for each unit of brown sold. Correlation determines that direction. For two assets, the authors show that the jump-aware investor holds more green than Merton whenever ρ > −(κ_g/|κ_b|)(σ_b/σ_g). Our arithmetic puts that threshold at −0.8 for these inputs. The corresponding condition for holding less brown is ρ > −1.25 (again our arithmetic), which always holds. At ρ = 0.7, green provides a cheap hedge against the brown loss.
Even setting κ_g = 0 and increasing only |κ_b| produces a green long at λ = 0.1. The authors say it plainly: "it is optimal to be long the green stock only because of an expected loss in the brown stock from the transition".
With low ambiguity and low risk aversion, initial weights at λ = 0.1 reach about π_g ≈ 2 and π_b ≈ −1.2. Increasing either η or ε reduces both positions, making the parameters look interchangeable at first. Their effects diverge. Higher η draws risky weights toward zero. Higher ε takes the portfolio elsewhere: at ε = 20, 50 and in the limit, the base case moves back to long brown and short green. For every plotted ε (1, 2, 5, 20, 50), the distortion θ, which multiplies λ, remains below 1 and declines as ε increases. Swap the drifts and Merton already holds 0.73 green and −0.28 brown. The ambiguity-averse weights then converge quickly toward Merton as θ approaches 0.
The authors spell out the result in their conclusion. Ambiguity-averse investors adjust "towards what would be optimal when the transition is expected to occur on a different timeline". This, they add, "tends to result in the investor moving away from green stocks and towards brown stocks". The useful half is their policy conclusion: confidence in the transition produces a green tilt, giving regulators a reason to send clear signals. That conclusion holds within the model.
Where the authors place the doubt explains the reversal. Timing alone is ambiguous; κ is known with certainty. Nature's available worst case is therefore a transition that never arrives. The authors identify ambiguity about post-transition parameters as an extension and acknowledge that stochastic κ would sacrifice the semi-explicit solution. We regard κ as the larger practical uncertainty. For practical use, that omission matters most.
There is a small slip in Section 6: it states (π^M)ᵀκ > 0 for any κ_g > 0 and κ_b < 0. At weights of −0.45 and 0.24, the product is negative, as the Fig. 4 caption correctly reports.
How large is the assumed event?
Our arithmetic gives a 63% reference probability of a transition within the horizon for λ = 0.1 over T = 10. At λ = 0.25, the top of the Fig. 2a range, that probability reaches 92%. A brown κ of −0.5 halves the stock on the day. These are scenario inputs chosen by the authors.
In the ambiguity-neutral log case, the pre-transition strategy stays constant and is independent of post-transition drift and covariance. Outside the pre-transition market inputs, only λ and κ determine it. Trading is frictionless. Positions of π_g ≈ 2 against π_b ≈ −1.2 amount to gross exposure of about 3.2 times wealth, with the brown short providing funding. Those two inputs set the hedge's size.
Our capped FSLR and XOM run, 2020 to mid-2024
We constructed a daily strategy using First Solar for green and Exxon Mobil for brown. It begins on 2020-01-01 and liquidates at the 2024-07-01 open. We calculate signals after the close and trade at the following open. A trailing 252-day window supplies drift and covariance; cash earns the 3-month Treasury yield, and η = 2. We use the paper's later jump sizes: FSLR +20% and XOM −50%.
The grid combines λ of 0.02, 0.05, 0.1 and 0.2 with ε of 1.5, 2, 5, 20 and 50, for 20 cells. Benchmarks use the ambiguity-neutral ε = 1 and transition-unaware λ = 0 runs. By our arithmetic, those λ values correspond to reference transition probabilities of roughly 9%, 20%, 36% and 59% over the 4.5-year window.
We did not use the continuous-time formula from the paper. Instead, we solved a discrete Bellman problem using 5×5 Gauss-Hermite quadrature, with long-only positions bounded at 0 to 50% per stock and no borrowing. This constraint dominates the comparison. The paper's corner of 2 long and −1.2 short is out of reach. Our implementation can express the hedge only by reducing XOM toward zero and increasing FSLR toward 50%. We used a bounded sigmoid to approximate the timing distortion and have not checked it against the paper's closed-form expression for θ.
No transition appears in the historical prices.
The run can measure the cost of carrying the tilt, never its payoff. The paper supplies no performance results of its own, leaving nothing against which to compare ours. Our headline configuration returned 79.32% in total over 2020-01-01 to 2024-07-01. Sharpe was 0.72 (Sortino 1.01, Calmar 0.56). Volatility was 24.38%, and the worst drawdown reached -24.66%.
With just two names, each capped at 50%, most of that volatility is single-stock risk. Each day, the trailing 252-day drift determines which side of the Merton portfolio gets cut off by the long-only constraint. A weak grid cell tells you about those choices before it tells you anything about Kroell and Steffensen.
The case for a green tilt has limits
The paper delivers a clean, verified answer to a narrow question: how much should an investor hedge a known jump when its date is in doubt? Start with Merton, add a Σ0⁻¹κ hedge, and reduce its scale as doubt about timing increases. This gives no support for a permanent green overweight. Positive correlation makes a green long cheap. When Merton starts long brown, as it does in the base case, doubting the date draws the portfolio back toward brown. Ambiguity about κ would change my view: if the worst case could be a larger brown loss rather than no event, the reversal toward brown could disappear.
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.