Dynamic estimation of volatility risk premia and investor risk aversion from option-implied and realized volatilities

This paper proposes a method for constructing a volatility risk premium, or investor risk aversion, index. The method is intuitive and simple to implement, relying on the sample moments of the recently popularized model-free realized and option-implied volatility measures. A small-scale Monte Carlo experiment confirms that the procedure works well in practice. Implementing the procedure with actual S&P500 option-implied volatilities and high-frequency five-minute-based realized volatilities indicates significant temporal dependencies in the estimated stochastic volatility risk premium, which we in turn relate to a set of macro-finance state variables. We also find that the extracted volatility risk premium helps predict future stock market returns.     

Modeling common volatility characteristics and dynamic risk premia in European equity markets

The paper applies a Factor-GARCH model to evaluate the impact of the market portfolio, as a single common dynamic risk factor, on conditional volatility and risk premia for the returns on size-based equity portfolios of three major European markets; France, Germany and the United Kingdom. The results show that for the size-based portfolios the factor loading for the dynamic market factor is significant and positive but the association between the risk premia and the conditional market volatility is weak. However, the dynamic market factor is shown to explain common characteristics in the conditional variance such as asymmetry and persistence. This finding is consistent across markets and portfolio sizes.     

The uncertainty in extreme risk forecasts from covariate-augmented volatility models

For a GARCH-type volatility model with covariates, we derive asymptotically valid forecast intervals for risk measures, such as the Value-at-Risk or Expected Shortfall. To forecast these, we use estimators from extreme value theory. In the volatility model, we allow for leverage effects and the inclusion of exogenous variables, e.g., volatility indices or high-frequency volatility measures. In simulations, we find coverage of the forecast intervals to be adequate for sufficiently extreme risk levels and sufficiently large samples, which is consistent with theory. Finally, we investigate if covariate information from volatility indices or high-frequency data improves risk forecasts for major US stock indices. While—in our framework—volatility indices appear to be helpful in this regard, intra-day data are not.     

Probabilistic recalibration of forecasts

We present a scheme by which a probabilistic forecasting system whose predictions have a poor probabilistic calibration may be recalibrated through the incorporation of past performance information in order to produce a new forecasting system that is demonstrably superior to the original, inasmuch as one may use it to win wagers consistently against someone who is using the original system. The scheme utilizes Gaussian process (GP) modeling to estimate a probability distribution over the probability integral transform (PIT) of a scalar predictand. The GP density estimate gives closed-form access to information entropy measures that are associated with the estimated distribution, which allows the prediction of winnings in wagers against the base forecasting system. A separate consequence of the procedure is that the recalibrated forecast has a uniform expected PIT distribution. One distinguishing feature of the procedure is that it is appropriate even if the PIT values are not i.i.d. The recalibration scheme is formulated in a framework that exploits the deep connections among information theory, forecasting, and betting. We demonstrate the effectiveness of the scheme in two case studies: a laboratory experiment with a nonlinear circuit and seasonal forecasts of the intensity of the El Niño-Southern Oscillation phenomenon.     

Multivariate volatility forecasts for stock market indices

Volatility forecasts aim to measure future risk and they are key inputs for financial analysis. In this study, we forecast the realized variance as an observable measure of volatility for several major international stock market indices and accounted for the different predictive information present in jump, continuous, and option-implied variance components. We allowed for volatility spillovers in different stock markets by using a multivariate modeling approach. We used heterogeneous autoregressive (HAR)-type models to obtain the forecasts. Based an out-of-sample forecast study, we show that: (i) including option-implied variances in the HAR model substantially improves the forecast accuracy, (ii) lasso-based lag selection methods do not outperform the parsimonious day-week-month lag structure of the HAR model, and (iii) cross-market spillover effects embedded in the multivariate HAR model have long-term forecasting power.     

Filtering and identification of Heston's stochastic volatility model and its market risk

We study the filtering problem for the stochastic volatility model of Heston by using the nonlinear estimation theory. To solve the estimation problem for the stochastic volatility process, we use the random time change method. The derived basic equation for the filtering is the so-called Zakai equation and its numerically realized algorithm is proposed with the aid of the splitting-up method. Regarding the European call option problem, the identification of the market price of the volatility risk is also studied. Some numerical simulation studies are demonstrated to show the advantage of the proposed method.     

Tests of risk premia in linear factor models

We show that statistical inference on the risk premia in linear factor models that is based on the Fama–MacBeth (FM) and generalized least squares (GLS) two-pass risk premia estimators is misleading when the β$\beta$’s are small and/or the number of assets is large. We propose novel statistics, that are based on the maximum likelihood estimator of Gibbons [Gibbons, M., 1982. Multivariate tests of financial models: A new approach. Journal of Financial Economics 10, 3–27], which remain trustworthy in these cases. The inadequacy of the FM and GLS two-pass t$t$/Wald statistics is highlighted in a power and size comparison using quarterly portfolio returns from Lettau and Ludvigson [Lettau, M., Ludvigson, S., 2001. Resurrecting the (C)CAPM: A cross-sectional test when risk premia are time-varying. Journal of Political Economy 109, 1238–1287]. The power and size comparison shows that the FM and GLS two-pass t$t$/Wald statistics can be severely size distorted. The 95% confidence sets for the risk premia in the above-cited work that result from the novel statistics differ substantially from those that result from the FM and GLS two-pass t$t$-statistics. They show support for the human capital asset pricing model although the 95% confidence set for the risk premia on labor income growth is unbounded. The 95% confidence sets show no support for the (scaled) consumption asset pricing model, since the 95% confidence set of the risk premia on the scaled consumption growth consists of the whole real line, but do not reject it either.     

Debt stabilization games in the presence of risk premia

As a result of the recent financial crisis and the ensuing economic recession, fiscal deficits have soared in many OECD countries. As a consequence, government debt has been on the rise again after a period of stable or declining government debt. In this paper we analyze debt stabilization in a country that features endogenous risk premia, imposed by financial markets that evaluate the probability of debt default by governments. Endogenous risk premia arise by assuming, e.g., simple linear relations between risk premia and the level of debt. As a result the real interest rate on government debt can be written as a constant (measuring the risk-free real interest rate corrected for real output growth) plus an endogenous risk premium that depends on the debt level. We bring such an endogenous risk premium into Tabellini (1986) model and analyze the impact of it. This gives rise to a non-linear differential game. We solve this game for both a cooperative setting and a non-cooperative setting. The non-cooperative game is solved under an open-loop information structure. We present a bifurcation analysis w.r.t. the risk premium parameter.     

New tips from TIPS: Identifying inflation expectations and the risk premia of break-even inflation

This paper decomposes the break-even inflation rates derived from inflation-indexed bonds into inflation risk premia, liquidity risk premia, and inflation expectations. I estimate a common factor model with autoregressive conditionally heteroscedastic (ARCH) errors that extracts co-movements from twenty-two monthly and quarterly indicators to identify these three components. The results indicate that the sharp declines in the 10-year and 5-year break-even inflation rates in 2009 reflect a substantial increase in liquidity risk rather than a decrease in inflation expectations. Break-even inflation rates underestimate inflation expectations over nearly the entire sample due to the liquidity risk premia carried by the inflation indexed bond yields. Also, the model-implied inflation expectations show better forecast performance for the average annual inflation rates than raw break-even inflation rates, the Survey of Professional Forecasters, and the Surveys of Consumers inflation forecasts.     

Priced risk and asymmetric volatility in the cross section of skewness

We investigate the sources of skewness in aggregate risk factors and the cross section of stock returns. In an ICAPM setting with conditional volatility, we find theoretical time series predictions on the relationships among volatility, returns, and skewness for priced risk factors. Market returns resemble these predictions; however, size, book-to-market, and momentum factor returns are not always consistent with our predictions. We find evidence that size and book-to-market may be priced post-crisis but not in the decade before. Momentum does not appear priced by our test. We link aggregate risk and skewness to individual stocks and find empirically that the risk aversion effect manifests in individual stock skewness. Additionally, we find several firm characteristics that explain stock skewness. Smaller firms, value firms, highly levered firms, and firms with poor credit ratings have more positive skewness.     

Saving motives and multivariate precautionary premia

This paper extends the concept of precautionary premium to a multivariate setting so as to measure the intensity of the precautionary saving motive to protect against multivariate risks. This makes it possible to disentangle and to link the various motives of precautionary saving depending on the configuration of the risks the individual wants to protect against.     

Bond risk, bond return volatility, and the term structure of interest rates

This paper explores the time variation in the bond risk, as measured by the covariation of bond returns with stock returns and consumption growth, and in the volatility of bond returns. A robust stylized fact in empirical finance is that the spread between the yields on long- and short-term bonds forecasts future excess returns on bonds at varying horizons positively; in addition, the short-term nominal interest rate forecasts both the stock return volatility and the exchange rate volatility positively. This paper presents evidence that movements in both the short-term nominal interest rate and the yield spread are positively related to changes in the subsequent realized bond risk and bond return volatility. The yield spread appears to proxy for business conditions, while the short rate appears to proxy for inflation and economic uncertainty. A decomposition of bond betas into a real cash flow risk component and a discount rate risk component shows that yield spreads have offsetting effects in each component. A widening yield spread is correlated with a reduced cash-flow (or inflationary) risk for bonds, but it is also correlated with a larger discount rate risk for bonds. The short rate only forecasts the discount rate component of the bond beta.     

Consumer credit and consumption forecasts

Recent advances in the theory of consumer behavior indicate that consumption may exhibit non-linear dynamics characterized by occasional surges. Building upon them, and taking explicitly into account the forward-looking nature of consumption, this paper argues that rising consumer debt can signal such surges, as well as the consumption underprediction which will occur if they are not taken sufficiently into account in forecasting. This insight is tested with and strongly confirmed by the Organization of Economic Cooperation and Developments forecasts for the USA. The results should be of interest not only to professional forecasters and policy-makers, but also to theoretical economists and econometricians who study non-linear dynamic models.     

Scoring rules and survey density forecasts

This article provides a practical evaluation of some leading density forecast scoring rules in the context of forecast surveys. We analyse the density forecasts of UK inflation obtained from the Bank of England’s Survey of External Forecasters, considering both the survey average forecasts published in the Bank’s quarterly Inflation Report, and the individual survey responses recently made available to researchers by the Bank. The density forecasts are collected in histogram format, and the ranked probability score (RPS) is shown to have clear advantages over other scoring rules. Missing observations are a feature of forecast surveys, and we introduce an adjustment to the RPS, based on the Yates decomposition, to improve its comparative measurement of forecaster performance in the face of differential non-response. The new measure, denoted RPS*, is recommended to analysts of forecast surveys.     

Demographic forecasts and fiscal policy rules

All quantitative evaluations of fiscal sustainability that include the effects of population ageing must utilize demographic forecasts. It is well known that such forecasts are uncertain, and some studies have taken that into account by using stochastic population projections jointly with economic models. We develop this approach further by introducing regular demographic forecast revisions that are embedded in stochastic population projections. This allows us to separate, for each demographic outcome and under different policy rules, the expected and realized effects of population ageing on public finances. In our Finnish application, demographic uncertainty produces a considerable sustainability risk. We consider policies that reduce the likelihood of getting highly indebted and demonstrate that, although demographic forecasts are uncertain, they contain enough information to be useful in forward-looking policy rules.     

Evaluating quantile-bounded and expectile-bounded interval forecasts

In many different contexts, decision-making is improved by the availability of probabilistic predictions. The accuracy of probabilistic forecasting methods can be compared using scoring functions and insight provided by calibration tests. These tests evaluate the consistency of predictions with the observations. Our main agenda in this paper is interval forecasts and their evaluation. Such forecasts are usually bounded by two quantile forecasts. However, a limitation of quantiles is that they convey no information regarding the size of potential exceedances. By contrast, the location of an expectile is dictated by the whole distribution. This prompts us to propose expectile-bounded intervals. We provide interpretation, a consistent scoring function and a calibration test. Before doing this, we reflect on the evaluation of forecasts of quantile-bounded intervals and expectiles, and suggest extensions of previously proposed calibration tests in order to guard against strategic forecasting. We illustrate ideas using day-ahead electricity price forecasting.     

Stochastic population forecasts and their uses

"The properties and uses of stochastic forecasts are discussed here. For linear stochastic projections, we show how the computation of forecast moments and the statistical distribution of forecasts depend on the multiplicative and autoregressive structure of the dynamics. Both scalar and vector projection methods are discussed, and their similarities are explored. Next we discuss the uses of stochastic forecasts, arguing that it is important to relate forecasts to the specific decision-making criteria of particular forecast users. The example of [the U.S. system of] Social Security is used to show how a dynamic programming approach may be used to explore alternative decisions in a probabilistic context."