Decision making with Expected Shortfall and spectral risk measures: The problem of comparative risk aversion

We analyze spectral risk measures with respect to comparative risk aversion following Arrow (1965) and Pratt (1964) for deterministic wealth, and Ross (1981) for stochastic wealth. We argue that the Arrow–Pratt-concept per se well matches with economic intuition in standard financial decision problems, such as willingness to pay for insurance and simple portfolio problems. Different from the literature, we find that the widely-applied spectral Arrow–Pratt-measure is not a consistent measure of Arrow–Pratt-risk aversion. Instead, the difference between the antiderivatives of the corresponding risk spectra is valid. Within the framework of Ross, we show that the popular subclasses of Expected Shortfall, and exponential and power spectral risk measures cannot be completely ordered with respect to Ross-risk aversion. Thus, for all these subclasses, the concept of Ross-risk aversion is not generally compatible with Arrow–Pratt-risk aversion, but induces counter-intuitive comparative statics of its own. Compatibility can be achieved if asset returns are jointly normally distributed. The general lesson is that these restrictions have to be considered before spectral risk measures can be applied for the purpose of optimal decision making and regulatory issues.     

The analysis of foreign exchange data using waveform dictionaries

This paper uses waveform dictionaries to decompose the signals contained within three foreign exchange rates using tick-by-tick observations obtained world wide. The three exchange rates examined are the Japanese Yen and the German Deutsche Mark against the U.S. dollar and the Deutsche Mark against the Yen. The data were provided by Olsen Associates.A wabeform dictionary is a class of transforms that generalizes both windowed Fourier transforms and wavelets. Each waveform is parameterized by location, frequency, and scale. Such transforms can analyze signals that have highly localized structures in either time or frequency space as well as broad band structures; that is, waveforms can, in principle, detect everything from shocks represented by Dirac Delta functions, to short bursts of energy within a narrow band of frequencies, to the presence of frequencies that occur sporadically, and finally to the presence of frequencies that hold over the entire observed period. Waveform dictionaries are most useful in analyzing data that are not stationary and non-stationarity up to second order is well recognized in the context of foreign exchange rates.     

On asymptotic properties of Bonus–Malus systems based on the number and on the size of the claims

A general framework for a Bonus–Malus system (BMS) based on the number and the size of the claims is presented, the set of the bonus classes being an interval [a,?b], say 0<a<1<b. The BMS is interpreted as a general Markov chain with state space [a,?b]. It turns out that, under certain assumptions, the Markov chain possesses an invariant limit distribution to which it converges with a geometric rate. We show how the invariant distribution can be evaluated by means of simulation. We also deal with the best possible convergence rate and show how it can be presented by means of the spectral theory of Banach spaces.     

Real earnings and business cycles: new evidence

In the time domain, the observed cyclical behavior of the real wage hides a range of economic influences that give rise to cycles of differing lengths and strengths. This may serve to produce a distorted picture of wage cyclicality. Here, we employ frequency domain methods that allow us to assess the relative contribution of cyclical frequency bands on real wage earnings. Earnings are decomposed into standard and overtime components. We also distinguish between consumption and production wages. Frequency domain analysis is carried out in relation to wages alone and to wages in relation to output and employment cycles. Our univariate analysis suggests that, in general, the dominant cycle followed by output, employment, real consumer and producer wages and their components is 5–7 years. Consistent with previous findings reported in the macro-level literature, our bi-variate results show that the various measures of the wage are generally not linked to the employment cycle. However, and in sharp contrast with previous macro-level studies we find strong procyclical links between the consumer wage and its overtime components and the output cycle, especially at the 5–7 years frequency.

Observed real wages are not constant over the cycle, but neither do they exhibit consistent pro- or counter-cyclical movements. This suggests that any attempt to assign systematic real wage movements a central role in an explanation of business cycles is doomed to failure. (lucas 1977)

     

A Frequency-domain Based Test for Non-correlation between Stationary Time Series

A one-sided asymptotically normal test for non-correlation between two stationary time series is proposed based on the spectral coherence function. The test statistic is a properly standardized version of the integrated spectral coherency and has similar asymptotic properties as a previously introduced time domain based test for non-correlation. Unlike its time domain counterpart, the proposed test does not require prewhitening of the time series and, thus, is a truly nonparametric test for non-correlation. In a simulation study, we evaluate the small sample performance of the proposed test in comparison with the time domain test and address the problem of bandwidth selection. Furthermore, we present a modification of the test statistic that allows to test for non-correlation over frequency bands. This version shows higher power of detecting interrelationships restricted to the frequency band of interest. This work has been carried out at the Institute of Applied Mathematics at the University of Heidelberg and partly while the author was visiting the Department of Statistics at the University of Chicago.     

Spectral calibration of exponential Lévy models

We investigate the problem of calibrating an exponential Lévy model based on market prices of vanilla options. We show that this inverse problem is in general severely ill-posed and we derive exact minimax rates of convergence. The estimation procedure we propose is based on the explicit inversion of the option price formula in the spectral domain and a cut-off scheme for high frequencies as regularisation.     

A New Measure of Multicollinearity in Linear Regression Models

Databases with a lot of data very often mean little information. It is because of the collinearity of variables which consist of the data of the database. This collinearity is in fact a kind of redundancy of the database.
In the study a new indicator is given. With this indicator, which contains the eigenvalues of the variables' correlation matrix, it is possible to quantify the percentage of collinearity: from 0% (all the eigenvalues are equal to 1) to 100% (all the eigenvalues, except the first, are equal to 0).     

Reconsidering the macroeconomics of the oil price in Germany: testing for causality in the frequency domain

This paper reconsiders the macroeconomics of the oil price for Germany. It investigates whether causality between the oil price and a selection of both macroeconomic and financial market variables differs between frequency bands. Both a bivariate frequency-wise causality measure and its higher-dimensional extension are applied. The main findings are that short-run causality exists between the oil price and variables such as short-term interest rates and the German share price index, while long-run causality is found between the oil price and long-term interest rates. Moreover, the oil price predicts the consumer price index at a high number of different frequencies, while no significant causality is found to run from the oil price to industrial production and the unemployment rate.      

机制纸与手抄纸及其印品老化过程研究

探索纸张本身颜色和印刷品颜色随环境条件变化而变化的规律,对于纸张的使用和保存甚至文物的保存都有一定的意义。实验中选取三种纸张和三原色油墨,通过IGT印刷适性仪进行打样,分别将样条置入氙弧灯老化仪中在一定条件下进行老化后,使用分光光度计对老化前后样条的L＊、a＊、b＊值进行测量,计算得出色差ΔE,并绘出老化时间与色差关系曲线。同时,观察光谱反射率曲线变化,再利用拉曼光谱仪测量其谱图变化。研究结果表明,纸张本身和印品样条在老化前后产生色差,纸张本身的光谱反射率也产生一定变化,而印刷品光谱反射率曲线无明显变化,且纸张分子结构未发生变化,但拉曼强度增强。     

The financial cycles in four East Asian economies

This paper characterizes proxy measures of financial cycles using available data on four East Asian economies, viz., Hong Kong, Malaysia, the Philippines and Thailand. Spectral analysis is adopted to characterize the financial cycles and these cycles are compared with the business cycles of the four East Asian economies. The empirical findings indicated that with the exception of the equity price growth in Hong Kong, the period of the proxy measures for financial cycles is slightly longer than the period of the business cycle. More to the point, there is no evidence to show that the period of the proxy measures for financial cycles in these economies are operating at low frequencies similar to the period of the cycles of between 8 to 32 years observed for advanced economies such as the US, UK and Germany. Taking one step further, the paper finds that the financial cycles of these four economies are better captured by a band-pass filter estimated using the periods obtained in the paper as opposed to using long period cycles of between 8 to 32 years. These findings imply that one needs to be careful in making an a priori assumption on the frequency range the financial cycle is believed to operate.