The hedge ratios are the tradeable result in this paper. Total currency demand adds a carry position, sized by an unconstrained mean-variance rule using expected returns estimated over the same window as the reported Sharpe ratios. Viceira and Shen acknowledge most of this. Their candor is one reason the paper deserves a close read.
Two legs inside one optimizer
The framework extends Campbell, Serfaty-de Medeiros and Viceira (2010), Viceira's earlier work on hedging. An investor owns a single-country equity or bond portfolio, then chooses net currency exposure per unit of portfolio value. The paper denotes it Psi. Zero means fully hedged, one means unhedged, and a negative value means selling more foreign currency forward than the asset position itself.
Optimal Psi has two components. The hedging term is the negative slope from a regression of hedged portfolio excess returns on currency excess returns. Covariance alone determines it, regardless of expected currency returns. The speculative term multiplies risk tolerance by expected currency excess return, then divides by its variance. Its economic source is the forward premium puzzle: high-rate currencies tend to depreciate by less than the interest differential, rewarding positions funded in low-rate currencies.
The monthly sample is unbalanced and runs from January 1975 to August 2023. It covers Six developed markets and four emerging ones (Brazil, China, India, Mexico). Equity returns are MSCI local-currency total returns. For bonds, the authors construct 10-year constant-maturity returns with the Campbell (2002) duration approximation. Bloomberg supplies spot and one-month forwards. Forward data start in 1983M12 for the Australian dollar and 1999M2 for the rupee and the yuan, leaving the forward-based analysis with shorter samples than the spot analysis. Expected currency returns come from Lustig, Roussanov and Verdelhan two-factor regressions. One factor is the dollar level factor, the average return from holding all foreign currencies against dollar funding. HML is the second, a carry factor that buys the highest-rate currencies and sells the lowest. Those factor data end in 2021.
The risk-minimizing estimates are unusually clear. For a US equity investor over the full sample, Psi equals -1.03 for the Canadian dollar, -1.85 for the rupee, -1.58 for the peso, -1.46 for the real and -0.45 for the Aussie. Each is significant at 1%. Sterling at 0.01 and the euro at -0.03 imply essentially full equity hedges. The yen stands apart at +0.27, equivalent to a 73% hedge for a US investor holding Japanese equities.
Bond estimates cluster around zero. For the same US investor, Psi is 0.02 on the Canadian dollar, 0.05 on the Aussie and -0.03 on both the yen and sterling. Only the yuan at 0.18 and the real at -0.35 are significant. Changing the base to the Canadian dollar produces the mirror image: optimal USD demand for a Canadian investor holding US equities is exactly 1.00, leaving the dollar exposure untouched.
Does the basis change the hedge?
Using observed forwards instead of currency returns implied by interest differentials barely changes the risk-minimizing equity ratios. The Canadian dollar shifts from -1.03 to -1.05. The Aussie remains -0.44, while the rupee remains -1.79. The real moves from -1.74 to -1.73, and the peso from -1.77 to -1.72.
This near-invariance is the paper's strongest result, and its explanation works. The basis measures the gap between the quoted forward premium and the interest differential, a gap that covered interest parity (CIP) says should equal zero. It changes the levels of hedged and unhedged returns while leaving the covariance behind the regression slope largely unchanged.
The yuan is the conspicuous movement, from -1.94 to -1.13. The paper summarizes the forward-based estimates as qualitatively and quantitatively very similar to the CIP-based estimates over the same window. I see the yuan result as an identification issue rather than an effect of hedging costs. Its forward-based excess return has a standard deviation of only 3.36%, compared with 17.34% for the real, leaving very little variation from which to estimate the slope. Practical consequences are limited: Chinese equity volatility for a US investor stays between 28.31% and 29.83% under every currency policy reported. Viceira and Shen also flag the yuan's effective peg to the dollar and the possibility of sudden devaluation, which sit outside a model based on volatilities and correlations.
Returns are where the basis bites, exactly as the paper's log portfolio return decomposition implies. A positive basis raises the realized return on the hedged asset and reduces the realized currency return. Over the common window beginning in February 1999, Brazilian real excess currency returns average 8.65% a year under CIP and 6.42% using observed forwards. The 2.23-point difference matches the average basis of 2.20%. The yuan loses 1.02 points through the same channel, moving from 2.10% to 1.08%. The rupee and peso gain because their average bases are negative, at -0.58% and -0.75%. Sizing carry from interest differentials instead of dealt forwards overstates Brazilian carry by more than two points annually.
Brazilian equity surrenders the return
For a US investor in Brazilian equities during the forward-data sample, the unhedged portfolio has 51.19% volatility and a 16.19% average excess return, producing a 31.62% Sharpe. Full hedging, with zero currency exposure, cuts volatility to 36.01% and return to 9.81%, for a 27.25% Sharpe. The risk-minimizing overhedge sets Psi at -1.73. Volatility drops to 20.48%, while average excess return becomes minus 1.25% and Sharpe becomes minus 6.09%. The authors label every figure in-sample.
The arithmetic is blunt.
Selling 1.73 units of a currency that returned 6.42% a year costs about 11 points. Subtracting roughly 11 from 9.81% lands near the optimizer's result. Mexico follows the same pattern: the unhedged portfolio returns 9.00% with a 26.58% Sharpe, versus minus 1.91% and minus 10.57% under the risk-minimizing policy.
For Brazilian equity, the paper estimates a volatility reduction of about 60%, from about 51% per annum unhedged to about 20%. Mexican equity volatility declines from 33.88% to 18.06%, nearer 47%. In Brazil, the entire position return pays for that 60% reduction. A liability-driven portfolio could still want the trade. Hedging carries a large price here, plainly visible in the paper's own return and Sharpe tables.
Nine regressions behind the speculative leg
Factor-implied expected returns drive the offsetting demands, yet those estimates diverge sharply from realized sample means. The Canadian dollar's model-implied return is 1.60% a year, against an average of minus 0.62% over the same window. Sterling is similar: 1.53% implied and 0.40% realized. An unconstrained mean-variance rule then converts those estimates into positions.
The table's largest allocation belongs to the rupee, ahead of the real. It reaches 0.598 of portfolio value with risk tolerance at 0.1 and 2.991 at 0.5. The main driver is the rupee's low 6.83% standard deviation of excess currency returns, rather than unusually large carry.
The authors concede this criticism before defending their result. They say the large absolute magnitudes arise from expected-return-driven demands and require caution because unconstrained mean-variance allocations are highly sensitive to expected returns. Their defence shifts attention toward the patterns determining those demands, whose magnitudes constrained optimization can attenuate. The draft reports no constrained version, leaving that defence untestable here. Performance-table volatilities, average returns and Sharpe ratios all use the unconstrained magnitudes rather than the patterns.
The body also qualifies the abstract's claim. According to the abstract, deviations from uncovered interest parity (UIP, the hypothesis that expected currency excess returns are zero) offset and sometimes reverse portfolio-risk minimizing demands, even at low risk tolerance. The discussion of total demands instead says the hedging motive tends to dominate at fairly large levels of risk tolerance. Its figures support the body. With risk tolerance at 0.3, demand for the real remains -0.981 and demand for the peso remains -0.461. The conclusion describes sign changes in total demands as suggesting that UIP deviations are temporary, a tougher trading proposition than the abstract's word persistent implies.
Every volatility, return and Sharpe figure in the performance tables carries the authors' in-sample label. Dynamic policies based on five-year rolling estimates produce the paper's best results: 77.17% Sharpe for Chinese equity and 71.19% for euro-area equity, both at risk tolerance of 0.3 for a US investor. The factor means used in those rolling estimates remain sample objects. The CIP basis is the only cost included anywhere. The paper models no bid-ask spread and no forward roll cost, while risk-minimizing equity positions in the peso, real and rupee range from 1.72 to 1.79 times portfolio value.
We could not test any of this ourselves. The strategy requires a forward-market overlay, and we have no FX spot or one-month forward series for the nine crosses. We also lack the international equity and bond index returns against which the demands are calculated.
Where I draw the line
The two legs deserve separate governance. The covariance hedge barely responds to the basis, and rolling estimates preserve its sign for the Canadian dollar, the Australian dollar, the real and the peso. The analysis now includes emerging markets and extends through August 2023. Equity demands are -1.73 for the real, -1.72 for the peso and -1.79 for the rupee, with all three significant at 1%.
The speculative component is a carry allocation fronted by a factor model. I would constrain it, charge it using dealt forwards instead of interest differentials, and require an out-of-sample test before allowing it to cancel a hedge. The paper reports in-sample results throughout. A single test would change my verdict on this leg: a walk-forward exercise in which rolling factor means are formed strictly before the returns they size, with results reported net of forward bid-ask.