Eric Jacquier

Bayesian Analysis of Stochastic Volatility Models

New techniques for the analysis of stochastic volatility models in which the logarithm of conditional variance follows an autoregressive model are developed. A cyclic Metropolis algorithm is used to construct a Markov-chain simulation tool. Simulations from this Markov chain coverage in distribution to draws from the posterior distribution enabling exact finite-sample inference. The exact solution to the filtering/smoothing problem of inferring about the unobserved variance states is a by-product of our Markov-chain method. In addition, multistep-ahead predictive densities can be constructed that reflect both inherent model variability and parameter uncertainty. We illustrate our method by analyzing both daily and weekly data on stock returns and exchange rates. Sampling experiments are conducted to compare the performance of Bayes estimators to method of moments and quasi-maximum likelihood estimators proposed in the literature. In both parameter estimation and filtering, the Bayes estimators outperform these other approaches.

Geometric or Arithmetic Mean: A Reconsideration

An unbiased forecast of the terminal value of a portfolio requires compounding of its initial value at its arithmetic mean return for the length of the investment period. Compounding at the arithmetic average historical return, however, results in an upwardly biased forecast. This bias does not necessarily disappear even if the sample average return is itself an unbiased estimator of the true mean, the average is computed from a long data series, and returns are generated according to a stable distribution. In contrast, forecasts obtained by compounding at the geometric average will generally be biased downward. The biases are empirically significant. For investment horizons of 40 years, the difference in forecasts of cumulative performance can easily exceed a factor of 2. And the percentage difference in forecasts grows with the investment horizon, as well as with the imprecision in the estimate of the mean return. For typical investment horizons, the proper compounding rate is in between the arithmetic and geometric values. An unbiased forecast of the terminal value of a portfolio requires the initial value to be compounded at the arithmetic mean rate of return for the length of the investment period. An upward bias in forecasted values results, however, if one estimates the mean return with the sample average and uses that average to compound forward. This bias arises because cumulative performance is a nonlinear function of average return and the sample average is necessarily a noisy estimate of the population mean. Surprisingly, the bias does not necessarily disappear asymptotically, even if the sample average is computed from long data series and returns come from a stable distribution with no serial correlation. Instead, the bias depends on the ratio of the length of the historical estimation period to that of the forecast period. Forecasts obtained by compounding at the geometric average will generally be downwardly biased. Therefore, for typical investment horizons, the proper compounding rate is in between the arithmetic and geometric rates. Specifically, unbiased estimates of future portfolio value require that the current value be compounded forward at a weighted average of the two rates. The proper weight on the geometric average equals the ratio of the investment horizon to the sample estimation period. Thus, for short investment horizons, the arithmetic average will be close to the “unbiased compounding rate.” As the horizon approaches the length of the estimation period, however, the weight on the geometric average approaches 1. For even longer horizons, both the geometric and arithmetic average forecasts will be upwardly biased. The implications of these findings are sobering. A consensus is already emerging that the 1926–2002 historical average return on broad market indexes, such as the S&P 500 Index, is probably higher than likely future performance. Our results imply that the best forecasts of compound growth rates for future investments are even lower than the estimates emerging from the research behind this consensus. The choice of compounding rate can have a dramatic impact on forecasts of future portfolio value. Compounding at the arithmetic average return calculated from sample periods of either the most recent 77 or 52 years results in forecasts of future value for a sample of countries that are roughly double the corresponding unbiased forecasts based on the same data periods. Indeed, for reasonable risk and return parameters, at investment horizons of 40 years, the differences in forecasts of total return generally exceed a factor of 2. The percentage differences between unbiased forecasts versus forecasts obtained by compounding arithmetic or geometric average returns increase with the ratio of the investment horizon to the sample estimation period as well as with the imprecision in the estimate of the mean return. For this reason, emerging markets present the greatest forecasting problem. These markets have particularly short historical estimation periods and return histories that are particularly noisy. For these markets, therefore, the biases we analyzed can be especially acute. Even for developed economies, however, with their longer histories, bias can be significant if one disregards data from very early periods.

Bayesian analysis of contingent claim model error

Abstract This paper formally incorporates parameter uncertainty and model error into the implementation of contingent claim models. We make hypotheses for the distribution of errors to allow the use of likelihood based estimators consistent with parameter uncertainty and model error. We then write a Bayesian estimator which does not rely on large sample properties but allows exact inference on the relevant functions of the parameters (option value, hedge ratios) and forecasts. This is crucial because the common practice of frequently updating the model parameters leads to small samples. Even for simple error structures and the Black–Scholes model, the Bayesian estimator does not have an analytical solution. Markov chain Monte Carlo estimators help solve this problem. We show how they extend to some generalizations of the error structure. We apply these estimators to the Black–Scholes. Given recent work using non-parametric function to price options, we nest the B–S in a polynomial expansion of its inputs. Despite improved in-sample fit, the expansions do not yield any out-of-sample improvement over the B–S. Also, the out-of-sample errors, though larger than in-sample, are of the same magnitude. This contrasts with the performance of the popular implied tree methods which produce outstanding in-sample but disastrous out-of-sample fit as Dumas, Fleming and Whaley (1997) show. This means that the estimation method is as crucial as the model itself.

Market Beta Dynamics and Portfolio Efficiency

This paper introduces a new estimation for the dynamics of betas. It combines two previously separate approaches in the literature, data-driven filters and parametric methods. Namely, we show how to estimate the parametric beta dynamics by instrumental variables combined with block-sampling - but not overlapping window filters - of data-driven betas. Instrumental variables are needed because of the measurement errorsin empirical betas. We find that, while betas are very strongly autocorrelated, neither aggregate nor firm-specific variables explain much of their quarterly variation. We then compare block-samplers and overlapping window filters using a criterion of economic significance. Namely, we track the out-of-sample performance of portfolios optimized subject to target beta constraints. For target betas of zero, the case of many hedge funds, we show that estimation error results in systematic overshooting of the target beta. These portfolios benefit from the use of medium to long term estimation windows of daily returns.

Asset Allocation Models and Market Volatility

Asset allocation and risk management models assume at least short–term stability of the covariance structure of asset returns, but actual covariance and correlation relationships fluctuate dramatically. Moreover, correlations tend to increase in volatile periods, which reduces the power of diversification when it might most be desired. We propose a framework to both explain these phenomena and to predict changes in correlation structure. We model correlations between assets as resulting from the common dependence of returns on a marketwide factor. Through this link, an increase in market volatility increases the relative importance of systematic risk compared with the unsystematic component of returns. The increase in the importance of systematic risk results, in turn, in an increase in asset correlations. We report that a large portion of the variation in correlation structures can be attributed to variation in market volatility. Moreover, market volatility contains enough predictability to construct useful forecasts of covariance. Asset allocation and risk management models assume at least short–term stability of the covariance structure of asset returns, but actual covariance and correlation relationships fluctuate wildly, even over short horizons. Moreover, correlations increase in volatile periods, which reduces the power of diversification when it might most be desired. This phenomenon, often called “correlation breakdown,” has been widely recognized in the international context, but the pattern is even more characteristic of cross-industry correlations in a domestic context. We attempt to explain correlation breakdown and to present a framework for predicting short–horizon changes in correlation structure. We modeled correlations between assets as resulting from the common dependence of returns on a systematic, marketwide factor. Through this link, an increase in factor volatility increases the importance of systematic risk relative to the unsystematic component of returns. The result is an increase in asset correlations. We found that a simple index model with only one systematic factor can explain a surprisingly large portion of the short–horizon time variation in correlation structure. This finding suggests that univariate models of time variation in volatility, such as the ARCH (autoregressive conditional heteroscedasticity) model and its variants, which are already widely and successfully applied, can be integrated with the index model to form useful short–horizon forecasts of cross-sector correlations. We examined the source of correlation breakdown in the domestic context using returns on 12 industry groups and treating the value–weighted NYSE Index as the systematic factor; in the international context, we used returns on 10 major country indexes and treated the MSCI World Index as the systematic factor. We document that variation in cross-sector correlation is highly associated with market volatility (where “sector” means industry in the U.S. context and country in the international context). Using daily data within quarters to calculate both cross-sector correlations and the volatility of the market index, we measured the tendency for time variation in correlation (across quarters) to track time variation in market volatility. In both the domestic and international contexts, we found that correlation clearly fluctuates in line with market volatility. We found that short–term variation across time in the volatility of the market index can be used to forecast most of the time variation in correlation structure and thus guide managers in updating portfolio positions. The results are qualitatively the same in the international and domestic settings. We found considerably more country-specific volatility, however, than industry-specific volatility, which implies that, although the proposed methodology can be quite effective in the domestic setting, it will be less useful in the international setting. Having established that predictions of market volatility are useful in predicting correlation structure, we next examined the extent to which this methodology can be used in risk management applications. Can predictions of market volatility in conjunction with the index model be used to efficiently diversify portfolio risk? We compared the predictive accuracy of several forecasts of covariance and found that a constrained correlation using a simple autoregressive relationship to forecast next-quarter market variance from current-quarter market variance is highly accurate. In fact, the predictive accuracy of this model is equal to a model of “full-sample constant covariance” (i.e., a covariance estimate obtained by pooling all daily returns and calculating the single full-sample covariance matrix). This latter forecast obviously is not feasible for actual investors because it requires knowledge of returns over the full sample period. It turned out to be the best unconditional covariance estimator, but our constrained index model estimator conditioned on a forecast of market volatility was equally accurate. We conclude that portfolios constructed from covariance matrixes based on an index model and predicted market volatilities will perform substantially better than portfolios that do not account for the impact of time-varying volatility on correlation and covariance structure. The ability of market volatility to explain correlation structure suggests that univariate models of time variation in that volatility can be integrated with the index model to make useful short–horizon forecasts of cross-sector correlations.

A Model of the Convenience Yields in On-the-Run Treasuries

The convenience yield differential between on- and off-the-run Treasury securities with identical maturities has two components. A non-cyclical component may arise due to the higher illiquidity of off-the-run bonds. Also, trading in the market for the next issue often causes cyclical shortages of the on-the-runs. When this occurs, owners of the on-the-run bond can earn riskless profits by borrowing at a special repo rate while lending at the prevailing risk free market rate. This second component of the convenience yield, induced by the auction, is cyclical. We first show that special repo rates and the convenience yield are jointly cyclical over the auction cycle. The patterns are statistically significant and pervasive. Repo specials are highest around the announcement day and disappear by the issue day. The off- minus on-the-run yield spread is highest at the beginning of the cycle and collapses near its end, consistent with a decreasing present value of profits over a decreasing horizon. Second, we develop a first no-arbitrage continuous-time model, with both interest and special repo rates stochastic, that prices the on-the-run bonds that command this convenience yield. A simple implementation of the model can generate yields consistent with the evidence.

Are Underwriting Cycles Real and Forecastable

Speculative efficiency often requires that future changes in a series cannot be forecast. In contrast, series with a cyclical component would seem to be forecastable with decreases, possibly relative to a trend, during the upper part of the cycle and increases during the lower part. On the basis of autoregressive model (AR) estimates, it is considered that there is strong evidence of cycles in insurance underwriting performance as measured by the premium‐to‐loss ratio. Indeed, a large literature attempts to explain this documented cyclicality. First, we show that the parameter estimates from AR models do not lead to any such inference and that in the contrary, the evidence in the data is consistent with no cyclicality at all. Second, we show that a number of different filters lead to the same conclusion: that there is no evidence of in‐sample or out‐of‐sample predictability in annual insurance underwriting performance in the United States.

Credit migration and basket derivatives pricing with copulas

The multivariate modeling of default risk is a crucial aspect of the pricing of credit derivative products referencing a portfolio of underlying assets, and the evaluation of Value at Risk of such portfolios. This paper proposes a model for the joint dynamics of credit ratings of several firms. Namely, individual credit ratings are modeled by univariate continuous time Markov chain, while their joint dynamic is modeled using copulas. A by-product of the method is the joint laws of the default times of all the firms in the portfolio. The use of copulas allows us to incorporate our knowledge of the modeling of univariate processes, into a multivariate framework. The Normal and Student copulas commonly used in the literature as well as by practitioners do not produce very different estimates of default risk prices. We show that this result is restricted to these two two basic copulas. That is, for any other family of copula, the choice of the copula greatly affects the pricing of default risk.

Disentangling Continuous Volatility from Jumps in Long-Run Risk–Return Relationships

Realized variance can be broken down into a continuous volatility and a jump components. We show that these two components have very different degrees of power of prediction on future long-term excess stock market returns. Namely, continuous volatility is a key driver of medium to long-term risk–return relationships. In contrast, jumps do not predict future medium or long-term excess returns. We use inference methods that are robust to persistent predictors in a multi-horizon setup. Specifically, we use a rescaled Student-t to test for significant risk–return relationship and simulate its exact behavior under the null in the case of multiple regressors with different levels of persistence. We also perform tests of equality of the risk–return relationship at multiple horizons. We do not find evidence against a proportional relationship between long-term continuous volatility and future returns.

Horizon effect in the term structure of long-run risk-return trade-offs

The horizon effect in the long-run predictive relationship between market excess return and historical market variance is investigated. To this end, the asymptotic multivariate distribution of the term structure of risk-return trade-offs is derived, accounting for short- and long-memory in the market variance dynamics. A rescaled Wald statistic is used to test whether the term structure of risk-return trade-offs is flat, that is, the risk-return slope coefficients are equal across horizons. When the regression model includes an intercept, the premise of a flat term structure of risk-return relationships is rejected. In contrast, there is no significant statistical evidence against the equality of slope coefficients from constrained risk-return regressions estimated at different horizons. A smoothed cross-horizon estimate is then proposed for the trade-off intensity at the market level. The findings underscore the importance of economically motivated restrictions to improve the estimation of intertemporal asset pricing models.