Country allocators should care about the difference between spillovers into profits and wages, rather than the strength of cross-border spillovers alone. Zhang's model makes that distinction cleanly: with equal persistence of capital and labour shocks, balanced spillovers leave foreign holdings unchanged. The abstract credits both the size and the capital-labour mix of spillovers. The paper's equivalence result gives the mix the decisive role when persistence is equal. Its numerical claim is less secure. The simulation and the description of it differ by roughly a factor of ten.
The setup
There are two symmetric countries and one good. Agents have log utility. They trade two equity claims in zero net supply, each paying the capital income of one country. Capital and labour income follow a VAR(1). Own persistence is rho for capital and rho_L for labour; cross-border spillovers are psi' and psi'_L, respectively. Within each country, capital and labour shocks have correlation rho_KL < 0. Domestic equity consequently hedges domestic wages, giving portfolios an initial home bias. Zhang explicitly imposes that negative correlation as a simplification in place of a multi-good production model.
Zhang solves portfolios in closed form using the Devereux-Sutherland and Tille-van Wincoop perturbation method. The external position equals the conventional position multiplied by (1 - Omega). Omega is the portfolio wedge, gathering the effects of spillovers. S measures the domestic-equity share of a country's wealth.
There is no dataset behind the calibration. Zhang sets r = 1.01 and the capital share at 1/3. He chooses rho_KL = -0.35 to deliver S = 0.85 when spillovers are equal, using a target from Corsetti et al. Together, those choices produce an asset stock of 100/3 and an external position of -5. Persistence is 0.91 from Heathcote and Perri (2013). The fixed labour spillover, psi'_L = 0.025, comes from Heathcote and Perri (2002). Zhang varies psi' from 0 to 0.035.
Why can spillovers change nothing?
The optimal position takes the covariance of risk exposure and the excess equity return, then divides by the variance of that return. Spillovers reduce both sides of this calculation. As the dividend streams move together, excess-return volatility falls; relative consumption risk falls as well. At psi' = psi'_L = 0.025, the reductions match exactly. Omega = 0, the external position remains -5, and S remains 0.85. Portfolios look just as they do with no spillovers.
This is the paper's strongest idea.
A desk claim that synchronized climate or geopolitical shocks have killed diversification misses a condition in Zhang's model. Correlation cuts hedging demand and hedging supply together. The portfolio changes when the shocks reach profits and wages differently. If psi' exceeds psi'_L, relative consumption risk declines faster than hedging opportunities, Omega becomes positive, and S rises. Reverse that gap and S falls: labour-heavy spillovers increase integration. Zhang reports a convex, asymmetric response, with the rise above 0.025 steeper than the fall below it.
The mechanism depends on kappa_L = (1 - alpha) rho_KL / phi, which represents labour's share of total-income dynamics. Omega is proportional to kappa_L. Set rho_KL to zero and the wedge disappears at any spillover level. Thus the entire channel relies on the correlation Zhang imposes rather than derives.
The size of the move
The simulation starts at rho = rho_L = 0.91. Raising psi' from 0.025 to 0.035 takes S from 0.85 to 0.93. A one percentage point increase in the spillover produces eight percentage points more home bias.
The introduction describes the same exercise as a 1 percentage point rise in capital-income spillovers relative to labour-income spillovers accompanied by about 0.78 percentage points more home bias. In the simulation section, Zhang calls that move an approximate elasticity of 0.78. Neither label fits the reported endpoints. Their difference is 8 percentage points. Using the benchmark, our own conventional-elasticity calculation is (0.08/0.85)/(0.01/0.025), or about 0.24.
Our arithmetic offers one possible source of the discrepancy: a rise to about 0.928 is 7.8 points and would print as 0.93. A slipped decimal is our inference; the paper gives no such explanation. Read literally, the simulation makes home bias about ten times as sensitive as the introduction says. The discrepancy favours the thesis.
Persistence close to a pole
Zhang acknowledges the source of the shock parameters. The simulation "relies on the estimates for those of conventional TFP and serves as a basis for further research". Accurate estimates for climate, geopolitical and pandemic shocks are lacking; estimating them "is challenging and out of scope of this research". The exercise therefore assigns TFP persistence to climate, war and pandemic shocks, and uses a TFP spillover estimate of 0.025 for labour income.
The coefficients have r minus (rho minus psi') in the denominator, with r at 1.01. Zhang runs rho at 0.91, 0.92 and 0.93; the same spillover gap moves S further as persistence rises. He also cites a Smets-Wouters median of 0.95, calls higher persistence perhaps more realistic, and says it would enhance the effect. The footnote's percentiles appear reversed: it gives the 95th as 0.94 and the 5th as 0.97.
At psi' = 0.025, r minus (rho minus psi') drops from 0.105 when rho = 0.93 to 0.085 when rho = 0.95. The grid ends at 0.93, short of the 0.95 Zhang regards as perhaps more realistic. On his account, the headline 8-point move is a lower bound. No estimate for the shocks in question supports that magnitude.
A result without a backtest
We could not backtest this mechanism. It requires separate country-level series for home and foreign capital income and labour income, with cross-border spillovers estimated for each. We do not have those series. US-listed international ETFs would substitute traded returns for the model's claim on a country's capital income and lose the income-hedging logic responsible for the result.
The paper supplies no empirical test of its own and describes its framework as deliberately stylized to obtain closed forms. Its algebraic equivalence result belongs in discussions of fragmentation. The 0.85-to-0.93 change remains a calibration exercise. I would put more weight on it if psi' and psi'_L were estimated separately for climate or geopolitical shocks and home bias moved in the predicted direction as they diverged.