Jacob Boudoukh

Optimal risk management using options

This paper provides an analytical solution to the problem of how an institution might optimally manage the market risk of a given exposure, under the assumption that the institution wishes to minimize its Value at Risk (VaR) using options. The solution specifies the VaR-minimizing level of moneyness of the options as a function of the underlying parameters. We show that the optimal hedge consists of a position in a single option whose strike price is independent of the level of expense the institution is willing to incur for its hedging program. The optimal strike price is increasing in the assets drift, decreasing in its volatility for most reasonable parameter, decreasing in the risk-free interest rate, nonmonotonic in the horizon of the hedge, and increasing in the level of protection desired by the institution (i.e., the percentile of the distribution relevant for the VaR). Finally, we also show that the costs associated with a suboptimal choice of exercise price are economically significant.

Is the ex ante risk premium always positive? A new approach to testing conditional asset pricing models

This paper develops tests of inequality restrictions implied by conditional asset pricing models. The methodology is easy to implement, requires little knowledge of the conditional distribution of asset returns, and is valid under fairly weak assumptions. As an application, we test whether the ex ante risk premium is always positive. We report reliable evidence that the ex ante risk premium is negative in some states of the world; these states are related to periods of high expected inflation and especially to downward-sloping term structures.

Ex Ante Bond Returns and the Liquidity Preference Hypothesis

We provide a formal test of the liquidity preference hypothesis (LPH), that is, the monotonicity of ex ante term premiums, using nonparametric estimates that do not require a structural model for conditional expected returns. Although the point estimates of the term premiums are consistent with previous conclusions in the literature regarding violations of the LPH, the test statistics are generally insignificant, even when powerful conditioning information is used. These results illustrate the importance of correctly accounting for correlations across maturities and of formally testing the inequality restrictions implied by the LPH.

Investigation of a Class of Volatility Estimators

This article examines a class o f volatility estimation models, all ofthem based on a weighted sum ofsquared deviationsjiom the meanfor historical returns. We show how some popular methods, such as RiskMetricsTM, GARCH, and non-parametric density estimation, fa l l into this class. We also conduct a briefempirical comparison ofthese methods. Wefind density estimation and RiskMetricsTMforecasts to be the most accurateforforecasting short-term interest rate Volatility.

Stale Prices and Strategies for Trading Mutual Funds

We demonstrate that an institutional feature of numerous mutual funds—funds managing billions in assets—generates fund net asset values that reflect stale prices. Because investors can trade at these NAVs with limited transaction costs in many cases, obvious trading opportunities exist. These opportunities are especially prevalent in funds that buy Japanese or European equities. Simple, feasible strategies generate Sharpe ratios (excess return divided by standard deviation) that are many times greater than the Sharpe ratio of the underlying fund. We illustrate the potential of the strategy for three Vanguard Group mutual funds. A particular issue to keep in mind is that when the strategies are implemented, the gains from these strategies are matched by offsetting losses incurred by buy-and-hold investors in these funds. In the past few years, the financial press has produced numerous articles about large cash flows into and out of certain mutual funds over short time periods. Most of the funds have had one major identifying characteristic: They invest in international—that is, non-U.S.—assets. We attempted to explain this phenomenon and documented the performance of trading strategies that are consistent with these fund flows. Two key institutional features underlie the trading strategies that lead to the rapid in-and-out trading. First, with the proliferation of mutual funds, a U.S. investor can buy into and exchange out of no-load mutual funds at essentially zero cost. Moreover, the opportunities abound; approximately 700 no-load mutual funds invest in international equities, and a number of them are very large. For example, at least 25 international equity funds have assets under management exceeding

An Equilibrium Model of Nominal Bond Prices with Inflation-Output Correlation and Stochastic Volatility

1 billion. The second institutional feature is that when U.S. investors buy/sell mutual funds during the day, they do so at the prices prevailing at the close of trading in the United States. Those prices are based on the last transaction prices of the stocks in the fund. For Japanese and other Asian equities, the last transaction could have been at the previous 1:00 a.m. (U.S. Eastern Standard Time), and for many European equities, it could have been 12:00 noon. When these markets are closed, information flow does not cease; information relevant for valuation of the securities traded in the closed markets is still being released. For example, the literature contains considerable evidence that international equity returns are correlated at all times, even when one of the markets is closed. Moreover, the magnitude of the correlations may be quite large. This phenomenon induces large correlations between observed security prices during the U.S. trading day and the next days return on these funds. In some cases, derivatives on international markets trading in the United States provide even more informative signals (than U.S. market returns) about the unobserved movements in the prices of securities in the non-U.S. equity funds. This knowledge can be used to generate considerable excess return in the buying and selling of mutual funds. With no transaction costs and perfect liquidity, an investor can purchase funds at stale prices. In the most extreme case, one can use 1:00 a.m. prices to buy a Japan fund while one has information about the “true” price some 15 hours later at 4:00 p.m. Given these facts, it is perhaps no surprise that we document extraordinarily high excess profits and Sharpe ratios for two categories of investment funds: Pacific/Japan equity funds and international/Europe equity funds. Our sample of funds was chosen for the staleness of their underlying prices, the size of the fund, and the ease of implementing the trading strategy. We studied a strategy of switching between a money market account and the underlying fund in response to signals during U.S. market hours. We also studied the effect of the various trading costs from various types of implementation procedures. Because mutual funds do place some limits on the frequency and amount of exchanges between funds, although the limits are not always enforced, we examined strategies with particularly strong signals. We found for both types of fund that, although the strategy recommended active trading only 5–10 percent of the time, the return, on average, substantially exceeded the return to a buy-and-hold strategy during an ex post very good market for equities. More interesting is the fact that for both types of fund, we could predict the next days movement more than 75 percent of the time. Sharpe ratios generally ranged between 5 and 10 on the days the investor was in the market. The range of Sharpe ratios depended on whether the strategy included hedging of equity price movements during non-U.S. trading hours. To illustrate in a detailed manner the mechanics and results of the trading strategy, we provide a case study using three mutual funds from the Vanguard family of funds. This analysis is of special interest to academic readers because these funds are available through the retirement plans of numerous educational institutions and can be easily traded on the Internet or over the phone.

A New Strategy for Dynamically Hedging Mortgage-Backed Securities

A vector autoregressive (VAR) model is used to describe the joint dynamics of consumption growth and inflation. The commonly used homoscedastic VAR is extended to allow for stochastic volatility, driven by an unobservable autoregressive factor. Bond prices, the conditional expectation of a function of these factors, are approximated using Tauchens quadrature method. We show that the mean, variance, and autocorrelation of yields is captured relatively well by the VAR-SV model, calibrated with inflation and consumption data. The co-dependents of consumption and inflation are shown to be important determinants for both real and nominal rates. Time variations in inflation volatility generate realistic variability of risk premia, but unrealistically low average magnitudes. Copyright 1993 by Ohio State University Press.

New Evidence on the Forward Premium Puzzle

This paper develops a new strategy for dynamically hedging mortgage-backed securities (MBSs). The approach involves estimating the joint distribution of returns on MBSs and T-note futures, conditional on current economic conditions. We show that our approach has a simple intuitive interpretation of forming a hedge ratio by differentially weighting past pairs of MBS and T-note futures returns. An out-of-sample hedging exercise is performed for 8%, 9% and 10% GNMAs over the 1990-1994 period for weekly and monthly return horizons. The dynamic approach is very successful at hedging out the interest rate risk inherent in all of the GNMAs. For example, in hedging weekly returns on 10% GNMAs, our dynamic method reduces the volatility of the GNMA return from 41 to 24 basis points, whereas a static method manages only 29 basis points of residual volatility. Moreover, only 1 basis point of the volatility of the dynamically hedged return can be attributed to risk associated with U.S. Treasuries, which is in contrast to 14 basis points of interest rate risk in the statically hedged return.

Long Horizon Predictability: A Cautionary Tale

The forward premium anomaly (exchange rate changes are negatively related to interest rate differentials) is one of the most robust puzzles in financial economics. We recast the underlying parity relation in terms of lagged forward interest rate differentials, documenting a reversal of the anomalous sign on the coefficient in the traditional specification. We show that this novel evidence is consistent with recent empirical models of exchange rates that imply exchange rate changes depend on two key variables: the interest rate differential and the magnitude of the deviation of the current exchange rate from that implied by purchasing power parity.

The Complexity of Liquidity: The Extraordinary Case of Sovereign Bonds

Long-horizon return regressions effectively have small sample sizes. Using overlapping long-horizon returns provides only marginal benefit. Adjustments for overlapping observations have greatly ove...