Below a drift threshold, a goal-seeking investor buys more stock as expected returns fall. That is the striking result in the model solved by Liang, Strub, Wang and Yang. The authors put its scope carefully in their abstract and conclusion: optimal policies "can be decreasing in the drift of risky assets". Their result is exact. At the paper's calibration, though, the position is about 2.85 times wealth (our arithmetic).
The paper develops theory, makes no empirical claim and runs no backtest. Its figures come from analytical examples or numerical illustrations in a simulated market. The backtest figures later in this review are ours, from a strategy we built around the idea.
The payoff arrives on a date
The investor chooses a trading strategy to reach a wealth goal sooner. In the authors' stochastic control problem, the outcome under control is the hitting time. The preference over arrival dates, and the policies it produces, supply the novelty; there is no alpha source.
Let Z be wealth divided by the value of the goal. It starts at 0.6 in the authors' numerics. A hit at 1 at time τ, before deadline T, pays ρ(τ), where ρ discounts the arrival date. If T comes first, the payoff is ρ(T)u(Z_T), discounted utility of the fraction funded. Following Ebert's theory of time-risk preferences, the discount function governs risk over dates as utility governs risk over wealth. With convex ρ, the investor prefers a gamble over arrival dates to their certain average.
The authors examine hyperbolic ρ(s)=1/(1+ks) and exponential discounting at rate log(1+k). Each discounts year one by the same amount, with k in {0.05, 0.1, 0.2}. Non-exponential discounting usually creates time inconsistency. The clock here starts at inception and keeps running in calendar time, so the inconsistency does not arise.
Their proofs establish continuity and Bellman's principle, then give a Hamilton-Jacobi-Bellman (HJB) equation, a verification theorem and concavity of the value function in Z. The verification theorem and concavity result require smoothness; concavity also assumes an optimizer exists. The viscosity characterization is stated, while its proof is deferred to Wang's 2024 PhD thesis. With no deadline, a constant goal, zero rate and deterministic coefficients, the HJB becomes the backward heat equation. Widder's theorem then supplies a complete catalogue of smooth solutions. For finite deadlines, the authors use Howard's policy iteration on an implicit finite-difference grid with Δz=Δt=5×10^-4. Their Black-Scholes illustrations simulate 5×10^5 Monte Carlo paths with μ=0.09, σ=0.35 and T=10.
Why does a lower drift raise the position?
Example 4 gives the clearest answer. It assumes one stock, an infinite horizon, a constant market price of risk and exponential discount rate γ. The optimal fraction in stock is π* = μ/Σ² + 2γ/μ. Merton's term is μ/Σ². Impatience contributes 2γ/μ, which rises as μ falls; the whole allocation decreases in drift when μ < √(2γ)Σ.
Using the paper's calibration, the turning point is μ ≈ 0.15, with γ = log(1.1) ≈ 0.095 and Σ = 0.35. At benchmark μ = 0.09, the falling branch calls for about 2.85 times wealth in stock. As volatility diverges, Merton's term goes to zero and the position approaches 2γ/μ, about 2.12. Those calculations are ours, and Example 4 makes stricter assumptions than the numerical section.
The authors describe the trap in cash: it "preserves wealth but guaranties that the goal is never attained." For an investor with no deadline and no reward for preserving wealth, safety makes failure certain. The rule expresses a preference for reaching the goal, rather than a favourable view of expected returns. The finite-deadline numerics show the same turn. At time 5 and funding ratio 0.6, the policy falls and then rises as μ runs from roughly 0.06 to 0.3 with σ = 0.35. At that state, it falls as σ runs from 0.15 to 0.5, though the slope flattens at higher volatility.
A deadline weakens the timing motive. Near T = 10, the gap between ρ(τ) and ρ(T) shrinks to zero, and the policy approaches the Merton allocation set by terminal utility. At Z = 0.6, that allocation is about 0.61 under CARA and 1.47 under CRRA u(z)=√z. With hyperbolic k = 0.1, the CARA limits are about 1.22, 0.73 and 0.46 at starting ratios of 0.3, 0.5 and 0.8. Starting nearer the goal delays the switch.
Impatience and the tails
Raise k from 0.1 to 0.2 and the hitting-time distribution gains mass at both extremes: more paths hit within two years, and more remain short at year 10. Among the paths that miss, more finish below a funding ratio of 0.2. Terminal risk aversion works the other way. Under hyperbolic discounting, increasing CARA γ from 0.5 to 2 reduces both the chance of a hit before the deadline and the chance of Z_T landing in [0, 0.2]. Its effect is smaller at k = 0.2 than at 0.1. As the authors put it, "a more time-risk-seeking investor behaves as if they were less outcome-risk averse."
The ordering changes in the authors' OU factor model, which lets shocks affect wealth and future returns together. With ξ in {0.4, 0.8}, a = 2 and σY = 0.03, smaller k and hyperbolic discounting produce the higher risky allocation at (s, z) = (5, 0.6). The Black-Scholes ranking runs the other way. The authors attribute the reversal to precautionary hedging and confine it to "the present calibration". It uses a single setting of a and σY.
Six times wealth, without a financing charge
The Black-Scholes benchmark plots positions reaching about 6 times wealth. We did not find a leverage limit, borrowing spread or trading cost in the setup. Square-integrability is the admissibility condition, and the bond pays the same rate r on borrowing and lending. Even before the plotted extreme, the falling branch is levered: our arithmetic gives 2.85 times wealth at μ=0.09.
The numerical scheme sets w(t,0)=0. Remark 2 calls this boundary condition redundant for the PDE.
Our ETF-and-cash version
The authors simulate one risky asset and a bond. We used IWM, QQQ and SPY, with cash earning the 3-month Treasury rate. A cohort is a sleeve of capital begun on a given date with its own goal. Its goal stays fixed at starting capital divided by 0.6. Our window starts 1 January 2020, and the deadline is 1 July 2024.
Returns and covariance use up to 253 trailing sessions. We solve the value equation by policy iteration on a 0.01 funding grid, with hyperbolic k = 0.1 and CARA γ = 2. Weights change in steps of 1/12. Each ETF is capped at 1/3, total risky weight at 1, with the balance in cash. Trading occurs daily at the next close; each cohort stops at its first observed hit. Costs are $0.004 per share, subject to a $1 minimum per order, plus 5 basis points per dollar traded.
We had no data for the paper's simulated stock-and-bond market, so IWM, QQQ, SPY and cash are a substitute. This construction draws on the idea; it does not replicate or test the paper's results. From January 2020 to July 2024, our capped ETF-and-cash book returned 44.30%. Sharpe was 0.63 (Sortino 0.68) on 15.60% volatility. Maximum drawdown was -26.31% (Calmar 0.32).
A 26% drawdown sits badly with a goal-reaching rule.
The paper reports no performance of its own, leaving no authors' return series for comparison. Our return figures also leave the central goal-reaching questions unanswered. We lack per-cohort hit rates, time to target and shortfall severity against fixed ETF-and-cash mixes. A win would require that comparison after the same costs.
Our cap of 1 lies below the 2.12 floor from our Example 4 arithmetic. Positions on the falling-in-drift branch were therefore unavailable to us: 2.85 times wealth at μ=0.09, or 2.12 as volatility diverges, in that same arithmetic. The caps replace the paper's unconstrained optimizer. Our goal is fixed, and our run has no stochastic factor. Our figures describe this capped construction.
The trade that could settle it
The theoretical result earns attention. The authors give a time-consistent control problem under non-exponential discounting and characterize its smooth solutions with no deadline, a constant goal, r=0 and deterministic coefficients. They also derive a clean reversal of Merton's drift relation. The trading rule from Example 4 has a narrower setting: an infinite horizon, exponential discounting and a constant market price of risk. There, when drift is poor, the position has a 2γ/μ stock floor. At the paper's calibration it is 2.12 times wealth (our arithmetic), so borrowing is required.
A book permitted 2 to 3 times wealth, paying a realistic financing spread, would change our mind if it beat a fixed equity-and-cash mix on hit rate without worsening the tail of terminal shortfalls.
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.