Hanno Lustig | Macro Paper Warehouse

Common Risk Factors in Currency Markets

The starting claim is that currency risk premia are a robust feature of the data, survive transaction costs, and are determined by exposure to a single global risk factor for which the interest rate itself is the measure of exposure. (The figures below come from the freely available NBER working-paper version of June 2008, which is the full text this summary was built from; the published version appeared in the Review of Financial Studies in 2011.) The empirical design is a cross-sectional sort rather than the usual time-series test: at the end of each month all currencies for which forward contracts trade are allocated to six portfolios by their forward discount – which equals the interest differential when covered interest parity holds – so portfolio 1 holds the lowest interest rate currencies and portfolio 6 the highest, and portfolios are rebalanced monthly. This matters for interpretation, because the classic UIP failure documented by Hansen-Hodrick and Fama concerns currencies whose rates are higher than usual, whereas here “our investment strategy only considers whether the currency’s interest rate is currently high.” The sample is 37 currencies from end-1983 to early 2008, growing from 9 countries to 26 with a maximum of 34, with a 15-country developed subsample as a robustness check; returns are computed from spot and one-month forward contracts net of bid-ask spreads. The basic magnitudes: portfolio 1 currencies trade at an average forward discount of -390 basis points but appreciate by only about 100 basis points, giving a log excess return of -290 basis points; portfolio 6 currencies trade at a discount of 778 basis points but depreciate only 188, giving +590 basis points. Net of transaction costs the spread between the first and last portfolio is 483 basis points a year with a Sharpe ratio of 0.54 – against 7.11 percent and a Sharpe ratio of 0.48 for the Fama-French US market excess return over the same period, which does not net out any transaction costs. In the developed-country subsample the long-short Sharpe ratio is 0.39. A principal component analysis of the six portfolio returns yields two factors explaining more than 80 percent of return variation: a level factor (70 percent of common variation, all portfolios loading equally) that is essentially the average portfolio return, labelled the dollar factor RX, and a slope factor (over 12 percent) with monotonically increasing loadings, essentially portfolio 6 minus portfolio 1, labelled HML_FX. Cross-sectional asset pricing on the six portfolios gives an adjusted R-squared of 69 percent with a root mean squared error around 95 basis points and no rejection of the null that pricing errors are zero. The estimated market price of HML_FX risk is 546 basis points per annum against a sample factor mean of 537 – a nine-basis-point gap, which is what linear factor pricing requires since the factor is itself a traded return. HML_FX betas rise monotonically from -0.39 for portfolio 1 to 0.61 for portfolio 6, so low interest rate currencies insure US investors against this risk while high interest rate currencies expose them to it. The dollar factor’s price (135 basis points, mean 136) prices the average level of returns but none of the cross-section. Three further results tie the pieces together: sorting currencies instead on their estimated HML_FX betas recovers monotonically increasing forward discounts and excess returns, so the interest rate really is measuring exposure; the carry factor also prices momentum portfolios, which supports a risk-based rather than characteristic-based reading; and the average forward discount across portfolios predicts returns better than portfolio-specific discounts, with forecast excess returns on medium-to-high interest portfolios moving counter-cyclically against US industrial production, payrolls and help-wanted indices and positively with term and default premia and the VIX. A no-arbitrage exponentially-affine model with a country-specific and a common SDF factor reproduces this, but only under two conditions: a common factor must exist, since it is “the only source of cross-sectional variation in currency risk premia,” and low interest rate currencies must load more on the common factor when the price of common risk is high. Heterogeneity in country-specific loadings alone cannot do the job.

Does Incomplete Spanning in International Financial Markets Help to Explain Exchange Rates?

Standard international macro-finance models assume complete spanning, in which case the change in the real exchange rate must exactly equal the difference between the foreign and domestic investor’s marginal utility growth. Almost nobody believes markets really are complete, so this paper asks how much of the exchange rate evidence the missing markets could account for. The authors adopt the perspective of an econometrician who has committed to some model for the domestic and foreign log stochastic discount factors and takes observed macroeconomic quantities as given, and then, following Backus, Foresi, and Telmer (2001), insert a stochastic “FX wedge” between the exchange rate change and the difference in discount factors. To make the exercise an upper bound on what incompleteness can do, they allow the most extreme departure they can write down: domestic investors may hold only the foreign risk-free asset (equivalently, only one-period forward currency contracts), with no access to any foreign risky asset, and symmetrically for foreign investors, so that only two Euler equations discipline the wedge. Those two Euler equations turn out to force the wedge to be procyclical, which means it always lowers exchange rate volatility relative to complete markets – one-for-one in variance. Measured on quarterly data for 15 developed countries over 1973.IV-2014.IV, average annualized bilateral exchange rate volatility is 11 percent (11.21 percent, standard error 0.44), the correlation between real exchange rate changes and relative consumption growth is not statistically different from zero, and the carry trade earns an average annualized excess return of 4.4 percent with a Sharpe ratio of about 0.5. Starting from a maximum Sharpe ratio of 0.50 in each country and a generous cross-country stochastic-discount-factor correlation of 0.50, matching the 11 percent volatility requires a wedge with an annualized standard deviation of 49 percent – as large as the maximum Sharpe ratio itself, and larger still (at least 70 percent) if the discount factors are uncorrelated. That same wedge cuts the model’s currency risk premium from about 12 percent to below 6 percent, and to essentially zero or negative for zero or negative wedge drifts; the drift values large enough to preserve an empirically plausible risk premium make the correlation between exchange rates and consumption growth larger in absolute value than under complete markets (above 0.7 with a risk aversion coefficient of 10), which is the opposite of what the cyclicality puzzle requires. The result survives dropping lognormality: in an entropy-based generalization, and in calibrated Merton (1976) jump and Barro-Rietz consumption-disaster models, no admissible wedge delivers both a plausible exchange rate volatility and a significant risk premium. Imposing dynamic no-arbitrage discipline makes things worse rather than better: in a Cox-Ingersoll-Ross specification the wedge’s drift ceases to be a free parameter, a 50 percent reduction in exchange rate volatility implies a 75 percent reduction in the currency risk premium, and the uncovered-interest-parity slope coefficient is always pushed toward one. The authors are careful about what this does and does not show – they hold the projection of the discount factor on domestically traded assets fixed and so do not claim that incomplete-market models are uninteresting, only that incomplete spanning across borders cannot by itself resolve the three puzzles together.

Exorbitant Privilege Gained and Lost: Fiscal Implications

This paper studies three centuries of U.K. fiscal history to understand the fiscal implications of safe asset supplier status — what the authors call “exorbitant privilege” — and how it can be gained and lost. Using the discounted cash flow approach to fiscal capacity developed in Jiang, Lustig, Van Nieuwerburgh, and Xiaolan (2019), the paper measures the present discounted value of expected future primary surpluses (inclusive of convenience yield seigniorage) and compares it to the observed market value of outstanding government debt. The central finding is a sharp historical discontinuity: before World War I, when the U.K. was the world’s dominant safe asset supplier and its gilts served as the global reserve asset, roughly only three-quarters of U.K. debt was backed by future surpluses even after accounting for convenience yields earned from global safe asset demand. After World War II, when the U.K. lost its safe asset supplier status to the U.S., the U.K.’s debt became fully backed by surpluses and fiscal capacity became closely tied to its own macro fundamentals. By contrast, the U.S. after World War II shows a pattern similar to the pre-war U.K. but more extreme: less than one-third of outstanding U.S. Treasury debt is backed by future surpluses according to the paper’s estimates, with the gap between debt and estimated fiscal capacity growing sharply over recent decades.

Fiscal hedging with nominal assets

This paper solves for optimal fiscal and monetary policy in a fully specified general-equilibrium economy in which the government finances distortionary-tax-smoothed spending only with non-contingent nominal bonds of several maturities – no explicit state-contingent debt is available – and both households and the government face a “no lending” constraint that rules out negative bond positions. Two nominal frictions, borrowed from Siu (2004), give the government’s inflation and interest-rate choices real bite: a fraction of firms must set prices before the current shock is known, so inflation surprises misallocate production across sticky- and flexible-price firms, and households face a cash-in-advance constraint on part of their consumption, so positive short-term nominal interest rates misallocate spending across cash and credit goods. Because explicit contingent claims are unavailable, the government can only hedge adverse fiscal shocks (higher spending or lower productivity) indirectly, through contemporaneous inflation surprises and through changes in the price of its outstanding debt at each maturity, both of which are costly. Solving the Ramsey problem recursively and computing calibrated numerical examples, the paper’s central finding is that optimal policy relies almost exclusively on the longest-maturity nominal bond available: long-term debt lets the government postpone the nominal-interest-rate increases used to hedge a shock, paying the associated transaction-cost distortion later and concentrating it in states where it can hedge several past shocks at once, rather than absorbing the full cost immediately. In the calibrated examples, a spell of adverse fiscal shocks produces a gradual rise in short-term nominal rates and a hump-shaped yield curve with the hump at the longest outstanding maturity, reverting to a flatter, lower curve once the shock spell ends or that debt matures; the resulting volatility in long-term bond returns is deliberate policy, not a cost, and functions like an insurance premium the government pays for hedging rather than a reason to shorten the maturity structure, as some earlier literature (e.g., Barro 1997) had argued while treating inflation and the yield curve as exogenous. The welfare gains from allowing longer maximum maturities, while positive, are found to be quantitatively small (roughly 0.02%-0.1% of consumption).