Average Pricing Error Research Articles

A cross-sectional asset pricing test of model validity

ABSTRACT An out-of-sample cross-sectional regression coefficient test of alpha in asset pricing models is investigated. The test is straightforward, easy to implement, supported by statistical precedent in the literature, applicable to linear and nonlinear models, and supplements in-sample GRS alpha tests of average pricing errors. Using U.S. stock returns, consistent with model misspecification, we find that the CAPM as well as prominent multifactor models have significant alpha coefficients in cross-sectional regressions. We conclude that the cross-sectional distribution of alphas across test assets contains useful information about model validity.

A dynamic conditional regime-switching GARCH CAPM for energy and financial markets

This paper develops a methodology for estimating a time-varying conditional version of the CAPM with regime changes in conditional variance dynamics. Our research goal is related to documenting the power of the beta when it is estimated dynamically. The empirical performance is tested across a sample of 81 financial, energy, and other commodity markets for the period August 1999–January 2018. The conditional regime-switching GARCH CAPM, with time-varying betas explaining both bull and bear markets, outperforms the unconditional (static) CAPM. Among stocks, there are significant time variations in betas across our models and regimes. This empirical feature is even more pronounced in the USA, the UK, Germany, France, China, and Malaysia. Among energy and other commodities, we find similar variations in the market price of risk. The direction of the relation with market returns for Crude Oil, Gold, Copper, Tin, Rubber, Aluminum, and Platinum is the same across our nested models. This result also holds for aggregate markets indices. Secondly, we provide a ranking by mean filtered volatility series where Natural Gas stands out at a high level. Average pricing errors are inferior in the case of the conditional model, and for Crude Oil. Lastly, we demonstrate that the regime-switching model delivers better estimates of one-day-ahead Value-at-Risk than its non-switching counterpart. Our results shed light on the supremacy of the market factor alone associated with time variation in risk premia across the energy and financial markets.

Which Option Pricing Model Is the Best? HF Data for Nikkei 225 Index Options

AbstractIn this study, we analyse the performance of option pricing models using 5-minutes transactional data for the Japanese Nikkei 225 index options. We compare 6 different option pricing models: the Black (1976) model with different assumptions about the volatility process (realized volatility with and without smoothing, historical volatility and implied volatility), the stochastic volatility model of Heston (1993) and the GARCH(1,1) model. To assess the model performance, we use median absolute percentage error based on differences between theoretical and transactional options prices. We present our results with respect to 5 classes of option moneyness, 5 classes of option time to maturity and 2 option types (calls and puts). The Black model with implied volatility (BIV) comes as the best and the GARCH(1,1) as the worst one. For both call and put options, we observe the clear relation between average pricing errors and option moneyness: high error values for deep OTM options and the best fit for deep ITM options. Pricing errors also depend on time to maturity, although this relationship depend on option moneyness. For low value options (deep OTM and OTM), we obtained lower errors for longer maturities. On the other hand, for high value options (ITM and deep ITM) pricing errors are lower for short times to maturity. We obtained similar average pricing errors for call and put options. Moreover, we do not see any advantage of much complex and time-consuming models. Additionally, we describe liquidity of the Nikkei225 option pricing market and try to compare the results we obtain here with a detailed study for Polish emerging option market (Kokoszczyński et al. 2010b).

The “out-of-sample” performance of long run risk models

This paper studies the ability of long-run risk models to explain out-of-sample asset returns during 1931–2009. The long-run risk models perform relatively well on the momentum effect.A cointegrated version of the model outperforms the classical, stationary version. Both the long-run and the short-run consumption shocks in the models are empirically important for the models' performance. The models' average pricing errors are especially small in the decades from the 1950s to the 1990s. When we restrict the risk premiums to identify structural parameters, this results in larger average pricing errors but often smaller error variances. The mean squared errors are not substantially better than those of the classical CAPM, except for Momentum.

Testing linear factor models on individual stocks using the average F-test

In this paper, we propose the average F-statistic for testing linear asset pricing models. The average pricing error, captured in the statistic, is of more interest than the ex post maximum pricing error of the multivariate F-statistic that is associated with extreme long and short positions and excessively sensitive to small perturbations in the estimates of asset means and covariances. The average F-test can be applied to thousands of individual stocks and thus is free from the information loss or the data-snooping biases from grouping. This test is robust to ellipticity, and more importantly, our simulation and bootstrapping results show that the power of the average F-test continues to increase as the number of stocks increases. Empirical tests using individual stocks from 1967 to 2006 demonstrate that the popular four-factor model (i.e. Fama–French three factors and momentum) is rejected in two sub-periods from 1967 to 1971 and from 1982 to 1986.

The 'Out-of-Sample' Performance of Long-Run Risk Models

This paper studies the ability of stationary and cointegrated versions of long-run risk models to explain out-of-sample asset returns during 1931-2009. The models perform similarly overall to the classical Capital Asset Pricing Model in a mean squared error sense, but have smaller average pricing errors. The long-run risk models perform relatively well on the momentum effect. A cointegrated version of the model outperforms a stationary version. A cointegrated model restricting the risk premiums to identify structural parameters has larger average pricing errors and smaller error variances than a reduced-form version of the model.

Long-run cash flow and discount-rate risks in the cross-section of US returns

This paper decomposes the overall market beta of common stocks into four parts reflecting uncertainty related to the long-run dynamics of stock-specific and market-wide cash flows and discount rates. We employ a discrete time version of Merton's intertemporal capital asset pricing model to test whether these four sources of risk command different risk prices. The model performs well in pricing average returns on single- and double-sorted portfolios according to size, book-to-market, dividend-price ratios and past risk. It generates high estimates for the explained cross-sectional variation in average returns, lower average pricing errors than the Fama–French three-factor model and economically and statistically acceptable estimates for the coefficient of relative risk aversion.

The 'Out-of-sample' Performance of Long-Run Risk Models

This paper studies the ability of long-run risk models to explain out-of-sample asset returns during 1931-2009. The long-run risk models perform relatively well on the momentum effect. A cointegrated version of the model outperforms the classical, stationary version. Both the long-run and the short run consumption shocks in the models are empirically important for the models' performance. The models' average pricing errors are especially small in the decades from the 1950s to the 1990s. When we restrict the risk premiums to identify structural parameters, this results in larger average pricing errors but often smaller error variances. The mean squared errors are not substantially better than those of the classical CAPM, except for Momentum.

Asset pricing with a reference level of consumption: New evidence from the cross-section of stock returns

This paper presents an empirical evaluation of recently proposed asset pricing models which extend the standard preference specification by a reference level of consumption. We motivate an alternative model that accounts for the return on human capital as a determinant of the reference level. Our analysis is based on a broad cross-section of test assets, which provides a level playing field for a comparison to established benchmark models. The reference level model extended by human capital does a good job in explaining size and value premia. Estimated on Fama and French's size and book-to-market sorted portfolios, it outperforms Lettau and Ludvigson's scaled CCAPM and delivers average pricing errors comparable to the Fama–French three-factor model.

Distressed debt prices and recovery rate estimation

This paper has two purposes. First, it uses distressed debt prices to estimate recovery rates at default. In this regard, estimates are obtained for three recovery rate models: recovery of face value, recovery of Treasury, and recovery of market value. We show that identifying the “economic” default date, as distinct from the recorded default date, is crucial to obtaining unbiased estimates. The economic default date is defined to be the first date when the market prices the firm’s debt as if it has defaulted. An implication is that the standard industry practice of using 30-day post default prices to compute recovery rate yields biased estimates. Second, we construct and estimate a distressed debt pricing model. We use this model to implicitly estimate the parameters of the embedded recovery rate process and to price distressed debt. Our distressed debt pricing model fits market prices well, with an average pricing error of less than one basis point.

The Canadian macroeconomy and the yield curve: an equilibrium‐based approach

Abstract. The authors develop and estimate an equilibrium‐based model of the Canadian term structure of interest rates. The proposed model incorporates a vector‐autoregression description of key macroeconomic dynamics and links them to those of the term structure, where identifying restrictions are based on the first‐order conditions that describe the representative investor's optimal consumption and portfolio plan. A remarkable result is that the in‐sample average pricing errors obtained with the equilibrium‐based model are only slightly larger than those obtained with a far more flexible no‐arbitrage model. The gains associated with parsimony become obvious out‐of‐sample, where the equilibrium model delivers much more accurate predictions, especially for yields with longer‐term maturities. The preferred equilibrium model has impulse responses that are consistent with long‐term inflation expectations being anchored, so a surprise increase in inflation does not necessarily raise expectations of higher future inflation.

Asset Pricing with a Reference Level of Consumption: New Evidence from the Cross-Section of Stock Returns

This paper presents an empirical evaluation of recently proposed asset pricing models which extend the standard preference specification by a reference level of consumption. We motivate an alternative model that accounts for the return on human capital as a determinant of the reference level. Our analysis is based on a broad cross-section of test assets which provides a level playing field for a comparison to established benchmark models. The human capital extended reference level model does a good job in explaining size and value premia. Estimated on Fama and French's size and book-to-market sorted portfolios it outperforms Lettau and Ludvigson's scaled CCAPM and delivers average pricing errors comparable to the Fama-French three-factor model.

Improving the Asset Pricing Ability of the Consumption-Capital Asset Pricing Model?

This paper compares the asset pricing ability of the traditional consumption-based capital asset pricing model to models from two strands of literature attempting to improve on the poor empirical results of the C-CAPM. One strand is based on the intertemporal asset pricing model of Campbell (1993, 1996) and Campbell and Vuolteenaho (2004). The model takes the traditional C-CAPM as its starting point, but substitutes all references to consumption out, as empirical consumption data is assumed to be error ridden. The other strand to be investigated is based on the premise that the C-CAPM is only able to price assets conditionally as suggested by Cochrane (1996) and Lettau and Ludvigson (2001b). The unconditional C-CAPM is rewritten as a scaled factor model using the approximate log consumptionwealth ratio cay, developed by Lettau and Ludvigson (2001a), as scaling variable. The models are estimated on US data and the resulting pricing errors are compared using average pricing errors and a number of composite pricing error measures. The conditional C-CAPM and the two beta I-CAPM of Campbell and Vuolteenaho (2004) result in pricing errors of approximately the same size, both average and composite. Thus, there is no unambiguous solution to the pricing ability problems of the C-CAPM. Models from both the alternative literature strands are found to outperform the traditional C-CAPM on average pricing errors. However, when weighting pricing errors by the full variance-covariance matrix of returns or the moment matrix of returns, the traditional C-CAPM actually outperforms the models from both the two new litterature strands.

Testing the APT with the Maximum Sharpe Ratio of Extracted Factors

This paper develops a test of the asymptotic arbitrage pricing theory (APT) via the maximum squared Sharpe ratio of the factors extracted from individual stocks using the Connor-Korajczyk method. The test treats the beta pricing relation as approximate without predetermining the systematic factors, unlike the existing tests that take the relationship as exact and systematic factors as given. This paper also examines the magnitude of pricing errors bounded partly by the maximum squared Sharpe ratio. For most 60-month subperiods of the sample, the hypothesis that the maximum squared Sharpe ratio for monthly returns is greater than 0.25 can be rejected. Simulation indicates that the average pricing error in monthly returns is less than 0.001. These results support the asymptotic APT.

On the Tests on Factor Risk Premiums in Beta Estimation Methods

In order to see whether an asset pricing factor is priced, we often test the null hypothesis that a factor risk premium is zero in estimation method. By theoretical analysis and Monte-Carlo study, this paper provides evidence that unless the constant term (average pricing error) is indistinguishable from zero (i.e., unless the model is believed to be correctly specied by other evidence), this test may not provide meaningful information in the sense that even a useless factor (which does not have any cross-sectional information and should be out of the model) may reject this hypothesis just like a true, priced factor. Further, it is shown that a source of Menzly's (2003) is related to this failure to detect useless factors, so infuential observations may signal that the assumed factor is useless. In addition, if assumed factors are excess returns, additional tests of hypotheses related to beta equals one for those factors can be a good remedy to this problem. A new statistic to test this hypothesis is derived using the GMM specification equivalent to the estimation procedure.

Testing Linear Factor Models on Individual Stocks Using the Average F Test

We propose the average F statistic for testing linear asset pricing models. The average pricing error, captured in the the statistic, is of more interest than the ex post maximum pricing error of the multivariate F statistic that is associated with extreme long and short positions and excessively sensitive to small perturbations in the estimates of asset means and covariances. The average F test can be applied to thousands of individual stocks and thus is free from the information loss or the data snooping biases from grouping. This test is robust to ellipticity, and more importantly, our simulation and bootstrapping results show that the power of average F test continues to increase as the number of stocks increases. Empirical tests using individual stocks from 1967 to 2006 demonstrate that the popular four factor model (i.e. Fama-French three factors and momentum) is rejected two sub-periods from from 1967 to 1971 and from 1982 to 1986.

The Pricing of Structured Products in the Swiss Market

Securitized covered call writing strategies play an important role in the Swiss market. The authors analyze the initial pricing of 275 products by comparing them to equivalent strategies in the underlying markets and at EUREX. On average, price distortions in favor of the issuing institutions compensate them for efforts associated with the management of the structured products. There are differences in the pricing behavior of several issuers. The pricing differs in terms of average pricing errors but also in the corresponding dispersions. There are significant pricing differences between products paying a fixed coupon and those without such a coupon. Finally, the authors document the important role of co-lead managers in the case of new issues.

Implied Volatility of Interest Rate Options: An Empirical Application of the Market Model

In this paper we analyze the empirical properties of the volatility implied in options on the 13-week US Treasury bill using both ITM, ATM, and OTM options. This market has not been studied previously. Moreover, these options are interesting because they are identical to options written on zero-coupon bonds which are the simplest interest rate derivate contracts possible. We apply the market model and adapt it to our data by assuming that the 13-week Treasury bill forward rate is log-normally distributed. The use of the market model separates our work from the vast majority of previous studies of implied volatility of interest rate options. We suggest three different volatility specifications: constant, affine, and exponential volatility and run a horse race between them. We document that the constant volatility specification is preferred with respect to parameter stability and the size of the average pricing errors which are rather large, -139% to -87%. Furthermore, we find that the implied volatility of yesterday contains information of the future volatility of the forward interest rates for all three volatility specifications. Finally, we show that moneyness and option type (call/put) explain 28-36% of the pricing errors. Hence, we give a partial explanation of why the market model fails.

Implied Volatility of Interest Rate Options: An Empirical Application of the Market Model

In this paper we analyze the empirical properties of the volatility implied in options on the 13-week US Treasury bill using both ITM, ATM, and OTM options. This market has not been studied previously. Moreover, these options are interesting because they are identical to options written on zero-coupon bonds which are the simplest interest rate derivate contracts possible. We apply the market model and adapt it to our data by assuming that the 13-week Treasury bill forward rate is log-normally distributed. The use of the market model separates our work from the vast majority of previous studies of implied volatility of interest rate options. We suggest three different volatility specifications: constant, affine, and exponential volatility and run a horse race between them. We document that the constant volatility specification is preferred with respect to parameter stability and the size of the average pricing errors which are rather large, -139% to -87%. Furthermore, we find that the implied volatility of yesterday contains information of the future volatility of the forward interest rates for all three volatility specifications. Finally, we show that moneyness and option type (call/put) explain 28-36% of the pricing errors. Hence, we give a partial explanation of why the market model fails.

Asset Pricing Specification Errors and Performance Evaluation

Many evaluation techniques typically measure performance as deviations of average returns on actively managed funds from those predicted by some asset pricing model. Empirical evidence, however, has so far suggested that all asset pricing models lack empirical support, implying that the models contain mis-specification errors to various degrees. Evaluating mutual fund performance relative to any of these models thus becomes problematic. In this paper, we propose an approach to performance measurement that emphasizes minimizing explicitly the pricing error associated with an asset pricing function which is employed to compute performance measures. This approach is henceforth called the minimum specification-error (MSE) method. We also discuss the statistical properties for implementing MSE performance measure. To demonstrate the significance of the pricing error confounded in evaluation measurement, we contrast our methodology with the Grinblatt and Titman (1989) period weighting approach and with the empirical implementation of Chen and Knez (1996). We find that the greater the pricing error of passive assets, the larger the performance measures. Given the average pricing error generated from a collection of 163 diverse passive portfolios used in this analysis the performance values assigned to a large number of the funds become statistically and economically insignificant.