Application of Cox Proportional Hazard Model to the Stock Exchange Market

Survival analysis is widely used in mechanical research, engineering and many other fields. This paper introduces the properties and modeling methods for survival data, then fits a Cox Proportional Hazards Model for stock data in the Shanghai Security Market. Introduction By September 17th, the Shanghai stock exchange’s benchmark index had plunged 64% since the start of the year, reaching a 52-week low and crashing past the 2000 points barrier to close at 1986.64. The market is thus worth less than a third of its value at its peak in mid-October 2007, when it reached 6395.76. This marks the most rapid decline of any major market, even in such an internationally gloomy year. Though it is an overall crash of a stock market, differences still exist among individual stocks. Some experienced wild ups and downs in price, while others rapidly fell down, almost straight down to half of their highest prices. This paper aims to find out what the main factors are that influence price performances of quoted companies, and what kind of companies are more likely to survive this meltdown. Dismissing the macro factors, such as a change of the stamp tax on stock trading and macro economy regulation and control, my study focuses on the financial data of each individual stock. The data come from the SSE 50 index of the Shanghai Security Market. The basics of survival analysis and the definition of survival time of stocks is are introduced first. This is followed by an introduction of theories of modeling survival data and the necessary foundations for Cox Proportional Hazards 12 B.S. Undergraduate Mathematics Exchange, Vol. 6, No. 1 (Spring 2009) Model. The estimation parameters using a Cox Proportional Hazards Model for the stock data in Shanghai Security Market are given later. The paper ends with a summary section. Introduction of Survival Data and Definition of Stock Survival Time The object of survival analysis is data in the form of times from a well-defined time origin to an end point. The end point could be the occurrence of some particular events or a particular time point. In medical research the time origin will often correspond to the recruitment of an individual in an experimental study, such as a clinical trial to compare two or more treatments. The outcome of interest is the duration until an event occurs; that is, an analysis of the time until an event occurs. Such events include the time to respond to treatment, relapse-free survival time, time to death, time to device failure, and time to regain mobility. More generally, survival times can also be observed in other application areas, such as the time taken by an individual to complete a task in a psychological experiment, the storage times of seeds held in a seed bank, or the lifetimes of industrial or electronic components. One reason that survival data are not amenable to standard statistical procedures used in data analysis is that it generally is not symmetrically distributed. Typically, a histogram constructed from the survival times of a group of similar individuals will have a longer “tail” to the right of the interval that contains the largest number of observations. Also, survival time is positive while a normal distribution is defined on the entire real line. Thus, the normal distribution assumption is not valid. This difficulty could be resolved by first transforming the data to give a positive and more symmetric distribution, however, a more satisfactory approach is to adopt an alternative distributional model for the original data. The main feature of survival data that renders standard methods inappropriate is that survival times are frequently censored. An individual survival time is said to be censored when the end-point of interest has not been observed. Censoring is classified into two types. In Type I censoring, the number of uncensored observations is a random variable, while in Type II censoring, the number of uncensored observations is fixed in advance. In this study, I am interested in the survival time of stocks. The time origin is defined as the date when a stock’s price reached its highest point in this year. The end-point is the date its price dropped to below 40% of that price for the first time. The number of days between these two dates is then the survival time of a stock. As an example, the study times of eight stocks are shown in Figure 1. The length of my study is 8 months, from Jan 1 to Aug 31 in 2008. The time of entry to study is represented by a dot. Stocks 1, 4, 5 and 8 died (D) during the study period, because their prices had dropped below 40% of their highest prices prior to the end of study. Stocks 3 and 6 were alive (A), which means their prices were still above 40% of their highest when the study period ended. Stocks 2 and 7 had stopped trading for some reason at the time points labeled “L”. They were lost-to-follow-up. The survival times B.S. Undergraduate Mathematics Exchange, Vol. 6, No. 1 (Spring 2009) 13