The peer-wealth term changes the portfolio rule, but the paper offers no market evidence that traders should carry it. Liang, Xiong and Yang start with the classical single-asset Merton allocation, put competitors' average wealth into the objective, and let risk aversion scale with the investor's own wealth. Within the affine class, they derive time-consistent equilibria and an allocation that is affine in wealth. No price series enters the paper. The result is a theorem, untouched by measurement.
The portfolio rule
Each of n agents can hold cash at rate r and one stock. Stock i has drift b_i, idiosyncratic vol xi_i, and loading sigma_i on a single common Brownian factor B. Through that common channel, every book is correlated with every other book. The horizon tau is random, with hazard rate lambda(t).
Agents maximize mean minus variance of relative wealth. Relative wealth is X_i(tau) minus phi_i times the cross-agent average, where phi_i in [0,1] controls the weight placed on the peer group versus the agent's own book. The paper's additional term is the wealth-affine coefficient on expected wealth, mu_i1 * x_i + mu_i2. Implied risk aversion is therefore gamma_i / (mu_i1 x + mu_i2). Björk, Murgoci and Zhou's gamma/x specification appears as the special case, while mu_i1 = 0 returns constant risk aversion.
Mean-variance creates time inconsistency, which rules out the usual dynamic programming solution. The authors instead use an equilibrium formulation. They hold competitors' strategies fixed, establish a verification theorem for the representative agent, and obtain an extended HJB system in three functions: the value function, the conditional mean of relative terminal wealth, and its second moment. Solving a fixed point then closes the full system.
With a constant hazard rate, the PDE system reduces to ODEs. The resulting position is
hat-pi_i = rho_i p_i x_i + rho_i q_i y_-i + k_i1 * (peer common-factor exposure) + k_i2 * (peer excess-return exposure) + k_i3
where rho_i = 1 / [2(xi_i^2 + sigma_i^2)]. The dollar holding in the agent's stock depends linearly on own wealth, the average wealth of everyone else, and two aggregates of competitors' positions. Its own-wealth slope p_i comes from a cubic in z = rho_i (b_i - r) p_i, equation (2.29).
What does peer wealth change?
The decomposition is clean because q_i, c_i and D_i are all proportional to phi_i. The term rho_i p_i x_i + k_i3 gives the position without peer concerns. Everything else, rho_i q_i y_-i and the two peer-exposure terms, scales with the competition parameter and disappears at phi_i = 0. For a benchmark-relative mandate, this split has a direct use: it separates the desired position from the extra position induced by the mandate.
Set mu_i1 = 0, mu_i2 = 2 and r = 0. Risk aversion then becomes state-independent, cash earns nothing, and the equilibrium is constant, with no wealth dependence. The paper says this constant agrees with the equilibrium in Lacker and Zariphopoulou (2019) under exponential preferences. Two remarks also recover the same strategy directly from a CARA objective. The headline wealth dependence therefore comes entirely from mu_i1 > 0 or r > 0. Remove those conditions and the result becomes the exponential-utility answer in different clothing.
The existence boundary is unusually sharp. Psi_n is the cross-agent average, defined as the sum over i of phi_i sigma_i^2 / (n psi_{i,n}), with psi_{i,n} = (1 - phi_i/n) xi_i^2 + sigma_i^2. Its mean-field counterpart is Psi = E[phi sigma^2 / (xi^2 + sigma^2)]. A constant equilibrium exists when Psi_n < 1. Positions scale as 1/(1 - Psi_n), exploding as Psi_n approaches one, and the paper says that no equilibrium exists at exactly one.
In the state-independent, zero-rate setting of Corollary 2.1, the game has no solution once competition intensity multiplied by the common-noise share reaches one. Anyone constructing a peer-relative allocator from this result would need to monitor that quantity first. It depends on parameters that must be estimated.
Three equilibria and the horizon restriction
Multiplicity comes from the cubic (2.29). For the two-agent case, the authors write that "there can be more than one but at most three linear equilibrium feedback strategies." They resolve the choice with a selection rule: take the real root that satisfies the theorem's conditions and produces the maximal mean-variance objective. The convention is defensible, though it makes the choice for the investor.
Admissibility further requires lambda > max{Q, P_i, 2r + 2 rho_i p_i (b_i - r) + (1/2) rho_i p_i^2}. Reverse the reading and the limitation becomes clear. Extend the expected horizon far enough, and the proposed construction leaves the admissible region required by the paper. Horizon length is therefore tied to the leverage sought by the rule. I did not find an economic interpretation of this bound in the paper.
Wealth can take any value on the whole real line.
A footnote explains the choice explicitly. Constraining x_i >= 0 or mu_i1 x_i + mu_i2 > 0 would introduce state constraints that make the extended HJB method, in the authors' words, extremely challenging. The footnote discusses divergence of implied risk aversion only when mu_i1 = 2, mu_i2 = 0, as wealth approaches zero. By construction, risk aversion also changes sign when mu_i1 x + mu_i2 becomes negative. The authors interpret the divergence case as an agent liquidating everything while wealth runs down. That reading is coherent. Implied risk aversion becomes unbounded, yet the equilibrium position remains affine and, with mu_i2 = 0, contracts to zero alongside wealth.
Inputs drawn from uniforms
The numerical section uses n = 1000 homogeneous agents. The type vectors come from U([0.1, 0.2] x [0.01, 0.05]^2 x [0, 1] x [1, 10] x [0, 1]^2), initial wealth is sampled from U[100, 1000], and r = 0.02. There are no prices, sample period, or investment universe. The paper presents itself as theory and makes no claim otherwise.
The treatment of lambda matters. The authors search a uniform grid on [0.3, 20] in increments of 0.00985, giving roughly 2000 candidates. At each point they solve the cubic, retain roots that satisfy admissibility, and choose the smallest admissible value, lambda = 0.3. All later results, sensitivity plots included, use that selection.
The focal agent is also fixed: number 500 from one draw, with x_500 = 919.8344 and parameters (0.1375, 0.0480, 0.0393, 0.1465, 6.3879, 0.1560, 0.1560). I did not find results showing how the comparative statics vary across other draws or across other admissible lambda.
For agent 500, equilibrium investment increases monotonically with phi across [0, 0.9], moving from roughly 60 to 200 in the plotted range. The no-competition position remains flat. Figure 4.2 compares the competitive and non-competitive strategies across gamma in [1, 8]. Both decline convexly, from roughly 300 to 50. Figure 4.3 and Figure 4.4 plot the same pair of strategies against the other preference parameters. The response is concave in mu1 over [0, 1] and linear in mu2 over [0, 2]. The mu2 plot shows a narrow increase from about 54 to 66. Competition raises risk-taking; risk aversion reduces it.
The paper proves and plots convergence from the finite-n equilibrium to the mean-field equilibrium. Across n from about 500 to 3000, values stay within a band of 64.0 to 64.8. The proposition applies only to homogeneous agents. The fully heterogeneous convergence proof is left for future research. The authors also acknowledge that best responses treat the average interaction terms as exogenous, meaning the exercise compares one approximate equilibrium with another.
Heterogeneous closed forms are available only for n = 2.
A historical test would substitute a different estimator
We are building a version of this rule, with strict limits on what that exercise can establish. We have not tested the paper's method. Its equilibrium is a continuous-time construction under exact correlated GBM and an exponential horizon. Any historical run would replace those assumptions with discrete rebalancing, rolling parameter estimates, and simulated stopping times. The resulting object is a different estimator.
Market data contain neither competitors' wealth nor their strategies. Both must therefore come from simulated agents following the paper's feedback rules, so the measured result concerns a model-based implementation. Drifts, idiosyncratic and common volatilities, risk-aversion coefficients, and competition parameters remain structural inputs that require estimation or sensitivity analysis.
The comparison carrying the empirical argument is the identical rule with phi set to zero. Corollary 2.2 supplies that benchmark and isolates the contribution of the peer term. I did not find out-of-sample returns, turnover, drawdown, or transaction-cost figures in the paper. Since the work presents itself as theory, their absence is unsurprising. The mathematics closes. Tradeability depends on structural parameters the paper leaves for the user to supply.
One result could change my view. Psi estimated from a real peer group and real common-factor loadings would need to remain comfortably below 1 through time, while the peer term would need to beat the phi = 0 rule on risk-adjusted return after costs. Corollary 3.1 places the loss of the mean-field equilibrium at Psi = 1 under mu1 = 0, mu2 = 2 and r = 0. That boundary deserves monitoring rather than treatment as a footnote.