A New Exploring on Definition and Measuring of Financial Ris.doc

A New Exploring on Definition and Measuring of Financial Risk -Based on Wavelet AnalysisHan HaiboSchool of Statistics,Lanzhou Commercial College, 730020Abstract There are tow important concept in traditional field of financial risk defining and measuring. That is the degree of loss or volatility of return and the occurred probability of them. However we ignore one fact. Usually we think that two assets have same risk if their variance of return were same. On the other hand, does this mean that an asset has more risk whose frequency of return volatility is higher? We should research this problem practically. Traditional measuring method of financial risk focus on the timely changing characteristics of variance or the calculating of volatility amplitude. This method cant estimate the frequency of volatility. This paper tries to analyze the timely changing characteristics of frequency using wavelet analysis and give a new concept of financial risk.Key words Financial risk, Wavelet analysis, Timely changing characteristics, Measurement1. IntroductionAt present，there is no uniform definition of risk in academia , because scholars who study risk from different angles, has different interpretations on the concept,. We can conclude some representative viewpoint as follows:(1) Risk is the uncertainty of the event possible result in the future. ( A. H. Mowbray ,1995)(2)Risk is the uncertainty of loss. (F. G. Crane, 1984)(3)Risk is the damage degree of possible loss. (Hu Yida, 2001)(4)Risk is the size of loss and the possibility of taking place. (Zhu Shuzhen, 2002)(5)Risk is the interacted result of factor which composed of risk. (Yi Danhui, 2000)(6)Define risk as volatility with standard statistics method. (P. Jorion, 1997)(7) Define risk as stochastic characteristics of uncertainty2. Exploring of financial riskFrom the concept of risk above we can find that each definition and measurement of risk is inseparable from the two key concepts, the volatility of return and the probability of it taking place. Because these two core concept were foundations, risk measurement also cannot be inseparable from variance which describe separate degree of data, as well as its probability distribution function. At present most method of risk measurement is based on these two key factors. The method is more and more complex and the result is more and more precise. But its result only describes the volatility amplitude, whatever how precise is the results and how complex are the models. (such as ARCH model, SV model, etc.) Therefore, we think the risk is the uncertainty degree, the variance large the possible of loss is bigger, vice versa.However we ignore one fact. Usually we think that two assets have same risk if their variance of return were same. On the other hand, does this mean that an asset has more risk whose frequency of return volatility is higher? We should research this problem practically.To a rational investor, return of assets was not influenced even it changed frequently if he was a long term investor. However frequent changes of return should change the expectation of investor, and change his decision of investment. On the other hand, frequent changes of return will increase the difficulty which select the time of exchange to a short term investor. From this analysis we believe that frequent changes of return will change investment behavior. So frequency of volatility is also the resource of risk.Author: Associate professor Han Haibo. Economics master of Wuhan University. Main study field is analysis of financial data. Tel:13919235333 E-mail:dickkkyTraditional method of financial risk focus on the timely changing characteristics of variance and the calculation of volatility amplitude. This method cant estimate the frequency of volatility. So we need establish a new method to discover and measure the timely changing character of volatility frequency. Because of its amelioration of tradition Fourier transformation, wavelet analysis method is the best one what we need. 3. Brief introduction of wavelet analysisThe fundamental idea behind wavelets is to analyze according to scale. Indeed, some researchers in the wavelet field feel that, by using wavelets, one is adopting a whole new mindset or perspective in processing data. Wavelets are functions that satisfy certain mathematical requirements and are used in representing data or other functions. This idea is not new. Approximation using superposition of functions has existed since the early 1800's, when Joseph Fourier discovered that he could superpose sins and cosines to represent other functions. However, in wavelet analysis, the scale that one uses in looking at data plays a special role. Wavelet algorithms process data at different scales or resolutions. If we look at a signal with a large "window," we would notice gross features. Similarly, if we look at a signal with a small "window," we would notice small discontinuities. The result in wavelet analysis is to "see the forest and the trees."Can you see why these features make wavelets interesting and useful? For many decades, scientists have wanted more appropriate functions than the sins and cosines which comprise the bases of Fourier analysis, to approximate choppy signals. By their definition, these functions are non-local (and stretch out to infinity), and therefore do a very poor job in approximating sharp spikes. But with wavelet analysis, we can use approximating functions that are contained neatly in finite domains. Wavelets are well-suited for approximating data with sharp discontinuities. The wavelet analysis procedure is to adopt a wavelet prototype function, called an "analyzing wavelet" or "mother wavelet." Temporal analysis is performed with a contracted, high-frequency version of the prototype wavelet, while frequency analysis is performed with a dilated, low-frequency version of the prototype wavelet. Because the original signal or function can be represented in terms of a wavelet expansion (using coefficients in a linear combination of the wavelet functions), data operations can be performed using just the corresponding wavelet coefficients. And if you further choose the best wavelets adapted to your data, or truncate the coefficients below a threshold, your data is sparsely represented. This "sparse coding" makes wavelets an excellent tool in the field of data compression. Other applied fields that are making use of wavelets are: astronomy, acoustics, nuclear engineering, sub-band coding, signal and image processing, neurophysiology, music, magnetic resonance imaging, speech discrimination, optics, fractals, turbulence, earthquake-prediction, radar, human vision, and pure mathematics applications such as solving partial differential equations.4. An application of wavelet analysis on volatility frequency identifying.Wavelet analysis can locate the frequency shift accurately. So we can provide material of decision through it. We also can analyze the efficiency and liquidity of securities market via this method.Now, well explain how wavelet analysis identifies volatility frequency by MATLAB wavelet tool box.Suppose signal y generate by follow process:, Looking into figure 1, we can find frequency changed hardly at Figure 1 Raw signalNow, decompose raw signal by wavelet db5Figure 2 Decomposed signal by wavelet db5We can find frequency changed easily at. So wavelet analysis can find frequency change point accurately5. An empirical analysis on timely changing characteristic of volatility frequency of Chinas stock market5.1 Choosing dataUse close index of SSE as the analysis object from Dec 26th 2003 to Sep 18th 2007. And calculate logarithm return.5.2 decomposed by wavelet sym3Figure3 Logarithm returns of SSE indexdecompose raw signal by wavelet sym3：Figure4 Decomposed Logarithm returns of SSE index by wavelet db56. Conclusion(1) Looking into d4, we can see the volatility frequency changed in Oct 2004, Jun 2005, Jun2006, Jan 2007 and May 2007. Combine with stock index of SSE, we find:(2) In these five periods, the government declared some important policy related with stock market. For example, treasury department decide to raise stock exchange stamp tax form 1 to 3 at May 30th 2007. So this policy makes change of volatility frequency and wavelet analysis finds this change.In brief, we can say volatility frequency is also the financial risk and it can be measured by wavelet analysis.References1 Gao Zhi & Yu Xiaohai. Principle & Application of Matlab wavelet tool box. National Defense Industry Publishing Company,2004:105-1082 Yuan Y, Wavelet analysis and its applications. Springer, 2001: 200-2023 Stephane Mallat, A Wavelet Tour of Signal Processing. Academic, 2002:112-115