Gradient Descent Only Converges to Minimizers: Non-Isolated Critical Points and Invariant Regions

Given a non-convex twice differentiable cost function f, we prove that the set of initial conditions so that gradient descent converges to saddle points where \nabla^2 f has at least one strictly negative eigenvalue has (Lebesgue) measure zero, even for cost functions f with non-isolated critical points, answering an open question in [Lee, Simchowitz, Jordan, Recht, COLT2016]. Moreover, this result extends to forward-invariant convex subspaces, allowing for weak (non-globally Lipschitz) smoothness assumptions. Finally, we produce an upper bound on the allowable step-size.Comment: 2 figure

A Quantitative Model of Sudden Stops and External Liquidity Management

Emerging market economies, which have much of their growth ahead of them, run persistent current account deficits in order to smooth consumption intertemporally. The counterpart of these deficits is their dependence on capital inflows, which can suddenly stop. In this paper we develop and estimate a quantifiable model of sudden stops and use it to study practical mechanisms to insure emerging markets against them. We first assess the standard practice of protecting the current account through the accumulation of international reserves and conclude that, even when optimally managed, this mechanism is expensive and incomplete. External insurance, on the other hand, is hard to obtain because sudden stops often come together with distress in emerging market investors themselves (the most natural insurers). Thus, one needs to find global (non-emerging-market-specific) assets that are correlated to sudden stops. We show an example of such an asset based on the S&P 500's implied volatility index. If added to these countries portfolios, it would significantly enhance their sudden stop risk-management strategies. In our simulations, the median gain in terms of reserves available at the time of sudden stop is around 30 percent. Moreover, in instances where the level of non-contingent reserves is low, the median gain is close to 300 percent. We also find that as countries manage to reduce the size of the sudden stops that afflict them, they should reduce their stock of reserves and significantly increase their share of contingent reserves. The main insights of the paper extend to external liquidity and liability management more generally.

Contingent Reserves Management: An Applied Framework

One of the most serious problems that a central bank in an emerging market economy can face, is the sudden reversal of capital inflows. Hoarding international reserves can be used to smooth the impact of such reversals, but these reserves are seldom sufficient and always expensive to hold. In this paper we argue that adding richer hedging instruments to the portfolios held by central banks can significantly improve the efficiency of the anti-sudden stop mechanism. We illustrate this point with a simple quantitative hedging model, where optimally used options and futures on the S&P100’s implied volatility index (VIX), increases the expected reserves available during sudden stops by as much as 40 percent.

Hedging Sudden Stops and Precautionary Contractions

Even well managed emerging market economies are exposed to significant external risk, the bulk of which is financial. At a moment's notice, these economies may be required to reverse the capital inflows that have supported the preceding boom. While capital flows crises are sudden nonlinear events (sudden stops), their likelihood fluctuates over time. The question we address in the paper is how should a country react to these fluctuations. Depending on the hedging possibilities the country faces, the options range from pure self-insurance to hedging the sudden stop jump itself. In between, there is the more likely possibility to hedge the smoother fluctuations in the likelihood of sudden stops. The main contribution of the paper is to provide an analytically and empirically tractable model that allows us to characterize and quantify optimal contingent liability management in a variety of scenarios. We show, with a concrete example, that the gains from contingent liability management can easily exceed the equivalent of cutting a country's external liabilities by 10 percent of GDP. *This is a revision of the June 2003 version.

Does the European Financial Stability Facility bail out sovereigns or banks?:An event study

On May 9, 2010 euro zone countries announced the creation of the European Financial Stability Facility. This paper investigates the impact of this announcement on bank share prices, bank credit default swap (CDS) spreads, and sovereign CDS spreads. The main private beneficiaries were bank creditors. Furthermore, countries with banking systems heavily exposed to southern Europe and Ireland benefited, as evidenced by lower sovereign CDS spreads. The combined gains of bank debt holders and shareholders exceed the increase in the value of their banks’ sovereign debt exposures, suggesting that banks saw their contingent claim on the financial safety net increase in value

Contingent Reserves Management: An Applied Framework

One of the most serious problems that a central bank in an emerging market economy can face, is the sudden reversal of capital inflows. Hoarding international reserves can be used to smooth the impact of such reversals, but these reserves are seldom sufficient and always expensive to hold. In this paper we argue that adding richer hedging instruments to the portfolios held by central banks can significantly improve the efficiency of the anti-sudden stop mechanism. We illustrate this point with a simple quantitative hedging model, where optimally used options and futures on the S&P100's implied volatility index (VIX), increases the expected reserves available during sudden stops by as much as 40 percent.