A relative-performance mandate can push a manager into far more equity risk. Liang, Xiong and Yang calculate the increase for a mean-variance investor who cares about peers. In their numerical example, one simulated agent raises stock exposure from roughly 60 to roughly 200 dollars as the competition weight moves from 0 to 0.9. Without peer concern, the same agent holds a flat amount. The paper stays entirely theoretical. It makes no empirical claim, estimates nothing from prices, and draws every number from synthetic parameters.
The allocation rule
Each agent holds cash at rate r and owns one stock. Its shocks combine idiosyncratic volatility ξ with a loading σ on the single common Brownian motion shared across stocks. Relative terminal wealth is own wealth minus φ times the population average. The objective multiplies the expectation of that quantity by (μ1x + μ2), then subtracts γ times its variance. Effective risk aversion therefore behaves like γ/(μ1x + μ2), declining as wealth increases when μ1 > 0.
The game ends at an exponential time with hazard λ. This random horizon removes calendar time and reduces the extended HJB (Hamilton-Jacobi-Bellman) system to ODEs with quadratic solutions.
Mean-variance preferences are time-inconsistent. The authors therefore find an equilibrium against each agent's future selves, rather than a pre-commitment optimum, before solving another fixed point across agents. The resulting stock allocation is affine. Dollars invested equal ρpx + ρqy + k1(peers' common-risk exposure) + k2(peers' excess-return exposure) + k3, where x denotes own wealth, y denotes peers' average wealth and ρ = 1/[2(ξ² + σ²)]. The slope p comes from a cubic. As the paper states, "there can be more than one but at most three linear equilibrium feedback strategies, which is consistent with Landriault et al." The selection rule chooses the admissible root with the highest objective. Each peer term, q, k1 and k2, scales with φ.
Two limiting cases connect the result to earlier work. Set μ1 = 0, μ2 = 2 and r = 0, and the allocation becomes a constant matching the CARA (exponential-utility) equilibrium of Lacker and Zariphopoulou. With one agent, ξ = 0, φ = 0, μ1 = 2 and μ2 = 0, the rule recovers the state-dependent equilibrium of Landriault et al. The constant case has a boundary as well. No equilibrium exists once Ψn, the population average of φσ²/((1 − φ/n)ξ² + σ²), reaches 1.
Positions are measured in dollars. When p > 0, exposure increases with wealth, while wealth ranges across the whole real line. A footnote says that restricting wealth or μ1x + μ2 to positive values would make the problem "extremely challenging to solve" with the authors' method. A live implementation would need a wealth floor and position limits absent from the model. Another restriction is quieter: λ > 2r + 2ρp(b − r) + ½ρp². An expected horizon that is too long relative to the wealth slope eliminates the equilibrium.
Does competition increase stock demand?
Inside the model, yes.
The focal agent's position increases monotonically with φ over [0, 0.9]. Across γ from 1 to 8, the competitive curve remains above the φ = 0 curve throughout. The same ordering holds across μ1 on [0, 1] and μ2 on [0, 2]. The authors attribute the added exposure to fear of lagging peers, which partly offsets absolute risk aversion.
A limiting result shows where that exposure enters. For two agents over a short horizon, q and k2 disappear. Only k1, the term loading on peers' common-factor exposure, remains. As λ → ∞ in the two-agent game, k1 approaches 2φρσ/(1 − φ/2), positive whenever φ and σ are.
This is a demand prediction under a particular incentive. The paper never claims that the additional exposure earns more, and its analysis cannot establish that. For an allocator, the practical reading concerns governance: at φ = 0.1465, a peer-benchmarked mandate predicts a larger equity book than an absolute mandate with the same γ.
Preference choices dominate the example
The largest movement comes from γ. Raising it from 1 to 8 lowers the focal position from roughly 300 to roughly 50, about sixfold. Increasing φ from 0 to 0.9 roughly triples the position. Moving μ2 from 0 to 2 changes it from roughly 54 to 66. For μ1 over [0, 1], the paper reports a concave increase and greater sensitivity than for μ2. A mandate designer sets all four preference parameters.
Market inputs are sampled uniformly. Drift b comes from U[0.1, 0.2], while ξ and σ each come from U[0.01, 0.05]. For the focal agent, b = 0.1375, ξ = 0.0480 and σ = 0.0393. Those values imply about 6.2% total volatility, excess drift of 11.75% above r = 0.02, and a Sharpe near 1.9 for a single stock. They also place ρ near 130. At a hypothetical 30% volatility, ρ would be about 5.6, changing the scale of the whole example.
Feasibility determines the horizon choice. The authors search λ over [0.3, 20] using steps of 0.00985, about 2,000 candidates, then select the smallest admissible value. They obtain λ = 0.3, giving an expected horizon of 1/0.3, about 3.3 time units (years if the drifts are interpreted annually). Because 0.3 is the grid's lower endpoint, the admissibility condition did not bind at the selected value.
The reported exercise depends on agent 500 from one random draw, whose wealth is x500 = 919.8344. We did not find results for other draws. There is also a small inconsistency: one sentence says wealth was sampled from U[0.1, 1], while another gives U[100, 1000]. The result reads as a mandate-design model equipped with a closed form.
Our daily implementation on fifty US stocks
We could not reproduce the exact equilibrium setting. The paper assumes continuous time, known and constant drifts, volatilities and common-noise loadings, plus an exogenous exponential stopping time. Our run estimated those quantities from daily US equity bars and simulated the random horizon. It therefore evaluates a discretized, estimated version of the equilibrium. The results measure our construction and bear on none of the paper's theorems.
We fixed the 2020 cohort of the 50 US stocks with the highest dollar volume and assigned each a persistent agent. Starting wealth came from U[100, 1000] and was scaled to the capital base. Each day, we estimated excess drift and a 252-session Ledoit-Wolf covariance matrix using data through the prior session. We projected that matrix onto a nonnegative one-factor fit to separate σ from ξ, then solved the heterogeneous fixed point from the paper's best-response equation. When roots failed the paper's conditions, we rejected them and retained the previous positions.
The preferences matched the paper's focal case: φ = 0.1465, γ = 6.3879, μ1 = μ2 = 0.156, λ = 0.3, with 100 simulated liquidation paths. Trades executed at the closing auction. Commissions were $0.004 a share, with a minimum of $1 and a cap at 1% of trade value.
From 1 January 2020 through 1 July 2024, our construction returned 3.03% in total after commissions. Its Sharpe was 1.68 (Sortino 1.65, Calmar 2.02), with annualised volatility of 0.40% and a maximum drawdown of -0.33%. The paper reports no performance figures of its own, leaving nothing of theirs for comparison.
The book carried almost no risk. A 1.68 Sharpe accompanied only a 3.03% total return over four and a half years. Even after we disabled position and leverage limits, the rule deployed very little risk relative to capital. We have not isolated the reason. The run excludes margin financing, short borrow and market impact.
A convergence result without market evidence
The mean-field section defines average wealth conditional on common noise, then proves that the homogeneous n-agent coefficients converge almost surely to their mean-field counterparts. Its convergence plot covers n of about 500 to 3,000, and every displayed value falls between 64.0 and 64.8. Heterogeneous closed forms are derived only for n = 2. The paper says the general heterogeneous system remains difficult to solve.
The authors write, "We leave the formal proof of convergence under the fully heterogeneous framework to future research," and acknowledge that the best responses take peer averages as given. In their words, this amounts to assessing the convergence of one approximate equilibrium against another.
We did not find historical returns, turnover, drawdowns or cost figures anywhere in the paper, consistent with its stated scope. The peer-risk mechanism works inside the model, although the typesetting contains slips: in (2.71), agent 2's own-wealth coefficient multiplies x1. Evidence that would change my view of the model's practical value is a calibration of φ and γ from actual mandate data that generates stable positions at realistic equity volatility.
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.