Note on the Tail Behavior of Random Walk Maxima with Heavy Tails and Negative Drift

Abstract

This paper investigates the asymptotic tail behavior of maxima of a random walk with negative mean and heavy-tailed increment distribution. A simple proof is given to improve the related result in Ng et al. (2002).     

On time-heterogeneous credibility estimation

Abstract

Let χ_i be the total claim amount of an insurance policy in calendar year i. We assume that the χ_i's are conditionally independent given an unknown random parameter ø, and that for all i. In the present paper it is under these assumptions shown how to calculate the credibility estimator of m(ø) by recursive updating. We also give estimators for the unknown parameters α_i, β_i, and ?_i based on portfolio data. Finally we mention some related models.     

Analyst Earnings Forecasts for Publicly Traded Insurance Companies

Several trends in the insurance and financial services industry, including demutualizationconsolidation, and deregulation, have attracted increasing attention from investors and financial analysts. This paper investigates the accuracy of the earnings forecasts of financial analysts for insurance companies. Our empirical results indicate that analyst forecasts outperform random walk time-series forecasts. Furthermore, we find that both disagreement over earnings forecasts among analysts and the relative forecasting error in the mean forecasts is smaller for life insurers than for property-casualty insurers, whereas the relative errors for forecasts for multiple-line insurers are in between the two. Forecasting error is a negative function of firm size and the number of analysts who are following a company, and is a positive function of the disagreement among analysts.Analyst forecasts have a timing advantage over the random walk model. Our results also suggest that the fair value reporting requirement (SFAS 115), which has been in effect since 1994, has enhanced the accuracy of analyst forecasts. The SFAS 115 has improved the superiority of analyst forecasts over the random walk forecasts for life insurers, but not for property-casualty insurers, and there is a weak improvement for multiple-line insurers. JEL Classification: G15     

Nils Ekholm

Abstract

The problem of χ² tests of a linear hypothesis H₀ for ‘matched samples’ in attribute data has been discussed earlier by the author (Bennett, 1967, 1968). This note presents corresponding results for the hypothesis that the multinomial probabilities p satisfy (c ?1) functional restrictions: F ₁(p) = 0, ... , F _C?1(p) = 0. An explicit relationship between the usual ‘goodness-of-fit’ χ² and the modified minimum χ² (=χ*²) of Jeffreys (1938) and Neyman (1949) is demonstrated for this situation. An example of the test for the 2 × 2 × 2 contingency table is given and compared with the solution of Bartlett (1935).     

Stochastic simulations of time series within Weierstrass–Mandelbrot walks

Abstract

In the present work we extend Lévy walks to allow the velocity of the walker to vary. We call these extended Lévy walks Weierstrass-Mandelbrot walks. This is a generalized model of the Lévy walk type which is still able to describe both stationary and non-stationary stochastic time series by treating the initial step of the walker differently. The model was partly motivated by the properties of financial time series and tested on empirical data extracted from the Warsaw stock exchange since it offers an opportunity to study in an unbiased way several features of the stock exchange in its early stages. We extended the continuous-time random walk formalism but the (generalized) waiting-time distribution (WTD) and sojourn probability density still play a fundamental role. We considered a one-dimensional, non-Brownian random walk where the walker moves, in general, with a velocity that assumes a different constant value between the successive turning points, i.e. the velocity is a piecewise constant function. So far the models which have been developed take only one chosen value of this velocity into account and therefore are unable to consider more realistic stochastic time series. Moreover, our model is a kind of Lévy walk where we assume a hierarchical, self-similar in the stochastic sense, spatio-temporal representation of WTD and sojourn probability density. The Weierstrass-Mandelbrot walk model makes it possible to analyse both the structure of the Hurst exponent and the power-law behaviour of kurtosis. This structure results from the hierarchical, spatio-temporal coupling between the walker displacement and the corresponding time of the walks. The analysis makes use of both the fractional diffusion and the super-Burnett coefficients. We constructed the diffusion phase diagram which distinguishes regions occupied by classes of different universality. We study only such classes which are characteristic of stationary situations. We proved that even after taking a moving averaging of the stochastic time series which makes results stationary in the sense that they are independent of the beginning moment of the random walk, it is still possible to see the non-Gaussian features of the basic stochastic process. We thus have a model ready for describing data presented, e.g., in the form of moving averages. This operation is often used for stochastic time series, especially financial ones. Based on the hierarchical representation of WTD we introduce an efficient Monte Carlo algorithm which makes a numerical simulation of individual runs of stochastic time series possible; this facilitates the study of empirical stochastic time series.     

Forecasting stock market volatility: Further international evidence

Abstract

This paper evaluates the out-of-sample forecasting accuracy of eleven models for monthly volatility in fifteen stock markets. Volatility is defined as within-month standard deviation of continuously compounded daily returns on the stock market index of each country for the ten-year period 1988 to 1997. The first half of the sample is retained for the estimation of parameters while the second half is for the forecast period. The following models are employed: a random walk model, a historical mean model, moving average models, weighted moving average models, exponentially weighted moving average models, an exponential smoothing model, a regression model, an ARCH model, a GARCH model, a GJR-GARCH model, and an EGARCH model. First, standard (symmetric) loss functions are used to evaluate the performance of the competing models: mean absolute error, root mean squared error, and mean absolute percentage error. According to all of these standard loss functions, the exponential smoothing model provides superior forecasts of volatility. On the other hand, ARCH-based models generally prove to be the worst forecasting models. Asymmetric loss functions are employed to penalize under-/over-prediction. When under-predictions are penalized more heavily, ARCH-type models provide the best forecasts while the random walk is worst. However, when over-predictions of volatility are penalized more heavily, the exponential smoothing model performs best while the ARCH-type models are now universally found to be inferior forecasters.     

Exchange rates,innovations and forecasting

In this paper an ex-post forecasting experiment is performed on the basis of a version of the ‘news’ model of exchange rate determination. For several exchange rates the ‘news’ formulation of monetary exchange rate models leads to relatively accurate ex-post exchange rate forecasts at a number of forecasting horizons. For a majority of the exchange rates studied, however, the results do not compare favorably with those obtained from the naive random walk forecasting rule. Thus, the findings in this article provide mixed evidence with regard to a suggestion in the literature that the finding by Meese and Rogoff that structural models do not even outperform the random walk in an ex-post forecasting experiment, may be due to the fact that these models were not properly tested in a ‘news’ framework.     

On stationary point processes and Markov chains

Abstract

Introductory. In the theory of random processes we may distinguish between ordinary processes and point processes. The former are concerned with a quantity, say x (t), which varies with time t, the latter with events, incidences, which may be represented as points along the time axis. For both categories, the stationary process is of great importance, i. e., the special case in which the probability structure is independent of absolute time. Several examples of stationary processes of the ordinary type have been examined in detail (see e. g. H. Wold 1). The literature on stationary point processes, on the other hand, has exclusively been concerned with the two simplest cases, viz. the Poisson process and the slightly more general process arising in renewal theory (see e. g. J. Doob 3).     

A note on cointegration of international stock market indices

Cointegration is frequently used to assess the degree of interdependence of financial markets. We show that if a stock's price follows a stock specific random walk, market indices cannot be cointegrated. Indices are a mere combination of n different random walks which itself is non-stationary by construction. We substantiate the theoretical propositions using a sample of 28 stock indices as well as a simulation study. In the latter we simulate stock prices, construct indices and test whether these indices are cointegrated. We show that while heteroscedasticity misleads cointegration tests, it is not sufficient to explain the high correlation between stock market index returns. A common random walk component and correlated price innovations are necessary to reproduce this feature.     

Dynamic Efficiency in the East European Emerging Markets

The paper re-investigates the efficiency of the East European emerging markets of the Czech Republic, Hungary, Poland and Russia analyzed by Rockinger and Urga (2000, 2001) based on the data from September, 1995 through December, 2004. We propose a first-order autoregressive (AR (1)) type time varying parameter model with a non-stochastic linear time trend including the random walk (RW) type model as a special case. The observed data rejects the RW type model for the AR (1) type one. The markets exhibit dynamic efficiency for all the four countries in the sense that the linear time trend approaches to zero over time. The empirical result for the Russian markets differs from that of Rockinger and Urga (2000). JEL Classifications: G14, G15     

A diffusive wander through human life

Moshe Milevsk takes a random walk through the human life cycle and reflects on how quantitative finance is transforming thefield of personal wealth management.     

The skewed multifractal random walk with applications to option smiles

Abstract

We generalize the construction of the multifractal random walk (MRW) due to Bacry et al (Bacry E, Delour J and Muzy J-F 2001 Modelling financial time series using multifractal random walks Physica A 299 84) to take into account the asymmetric character of financial returns. We show how one can include in this class of models the observed correlation between past returns and future volatilities, in such a way that the scale invariance properties of the MRW are preserved. We compute the leading behaviour of q-moments of the process, which behave as power laws of the time lag with an exponent ζ _q =p?2p(p?1)λ² for even q=2p, as in the symmetric MRW, and as ζ _q =p + 1?2p ²λ²?α (q=2p + 1), where λ and α are parameters. We show that this extended model reproduces the ‘HARCH’ effect or ‘causal cascade’ reported by some authors. We illustrate the usefulness of this ‘skewed’ MRW by computing the resulting shape of the volatility smiles generated by such a process, which we compare with approximate cumulant expansion formulae for the implied volatility. A large variety of smile surfaces can be reproduced.     

Measuring informational efficiency of the European carbon market — A quantitative evaluation of higher order dependence

This paper introduces a new method for measuring nonlinear predictability in financial price changes: the so-called intermittency coefficient, a parameter of the multifractal random walk model by Bacry et al. (2001). As the intermittency coefficient can quantify the degree of nonlinear deviation from a random walk, we employ its estimates from financial data as a proxy for the loss of financial market efficiency. In addition, we propose a new statistical test of the random walk hypothesis. In an empirical application using data from the largest currently existing market for tradable pollution permits, the European Union Emissions Trading Scheme (EU ETS), we show that the degree of efficiency of this market remains largely unchanged over the period of observation 2008–2019. This suggests that the market has reached a mature state: informational efficiency in Phase III remains at a level comparable to Phase II. What is more, the EU ETS is found to be more efficient than the US stock market. This result, surprising as such, is largely attributable to the lower exposure to global economic shocks of the EU ETS.     

On Testing the Random-Walk Hypothesis: A Model-Comparison Approach

The main intention of this paper is to investigate, with new daily data, whether prices in the two Chinese stock exchanges (Shanghai and Shenzhen) follow a random‐walk process as required by market efficiency. We use two different approaches, the standard variance‐ratio test of Lo and MacKinlay (1988) and a model‐comparison test that compares the ex post forecasts from a NAÏVE model with those obtained from several alternative models: ARIMA, GARCH and the Artificial Neural Network (ANN). To evaluate ex post forecasts, we utilize several procedures including RMSE, MAE, Theil's U, and encompassing tests. In contrast to the variance‐ratio test, results from the model‐comparison approach are quite decisive in rejecting the random‐walk hypothesis in both Chinese stock markets. Moreover, our results provide strong support for the ANN as a potentially useful device for predicting stock prices in emerging markets.     

On certain compound frequency distributions

Abstract

Let ?(χ/y, z, …) be an ordinary frequency distribution where χ is the variate and y, z, … are parameters characterising the function. If then χ is a graduated variate fd 32_1 represents the probability of an observation drawn from this universe, falling between χ and χ + dχ. This probability is a function of χ. It is, however, also a function of the parameters. If one or more parameters are changed, this probability also changes. d f is a relative probability dependent on the values of y, z, …     

Using Daily High/Low Time to Test for Intraday Random Walk in Two Index Futures Markets

This paper investigates the time of the daily high/low price in the Hang Seng and S&P 500 index futures and uses it to test for deviation from the predictive behavior of an intraday random walk model. Theoretical distributions of the daily high/low time under the random walk model are derived assuming either uniform or time-varying intraday trading speed. We show that under a random walk model, daily high/low time is more likely to occur near market open/close than in the middle of the trading day. Empirical distributions of the daily high/low time are compared with its theoretical distributions to test for the random walk model. It is found that for the intraday movement of the S&P 500 futures, the random walk hypothesis cannot be rejected. However, it is discovered that in the Hong Kong market, daily high/low time tends to appear significantly more often than is predicted by the random walk model in the first 15-minute time interval when the market opens in the morning or in the afternoon. The results remain valid even after we have taken the time-varying trading speed into account. By comparing the price behavior across markets, we can better understand the microstructure of the futures market.     

Does the exchange rate follow a random walk? A Monte Carlo study of four tests for a random walk

Several authors have urged that real and nominal exchange rates follow a random walk. Conditions under which the exchange rate follows a random walk are reviewed in this paper. Many of these do not appear to hold. Therefore, the empirical evidence presents a puzzle : some evidence suggets the exchange rate follows a random walk, while other evidence suggests the opposite should be true. This paper attempts to reconcile the evidence. It is argued that the tests for a random walk have low power. Monte Carlo evidence on four tests for a random walk, is presented that suppors this claim.     

Investment intensity of currencies and the random walk hypothesis: Cross-currency evidence

This paper studies the cross-currency and temporal variations in the random walk behavior in exchange rates. We characterize currencies with relatively large investment flows as investment intensive and conjecture that the more investment intensive a currency is, the closer its exchange rate adheres to random walk. Using 29 floating bilateral USD exchange rates, we find that the higher the investment intensity, the less likely it is to reject random walk and the smaller the deviation from random walk is. However, the effect of investment intensity is non-monotonic. Application of threshold models shows that after investment intensity reaches the estimated thresholds, the level of investment intensity has no further effect on the deviation from random walk. These findings help reconcile the previous conflicting results on the random walk in exchange rates by focusing on the effect of cross-currency and temporal variations in investment intensity.     

On distributions of functions of samples from a normally distributed infinite population

Abstract

7. The joint distribution of the moments a ₁₁, a ₂₂,…, a_nn and a ₁₂, …, a _1n may be deduced explicitly in the case, in which the variates χ₁, …, χ _n in (1) are mutually uncorrelated. In this case we have for the population values of the moments: α_μv = 0 for μ ═ v and, consequently, A_μv = 0 for μ ═ v, so that according to (6) λ_μv = 0 for μ ═ v; the distribution (5) of the moments α_μv is then      

A stronger version of Gautschi's inequality satisfied by the gamma function

Abstract

Let X be a random variable defined on a probability space, and E denote the expectation operator.