Arc Length as a Measure of Risk

• Activity Number: 574
• Type: Contributed
• Date/Time: Wednesday, August 1, 2012 : 2:00 PM to 3:50 PM
• Sponsor: Business and Economic Statistics Section
• Author(s): Tharanga Wickramarachchi*+ and Colin Gallagher and Robert Lund
• Companies: Clemson University and Clemson University and Clemson University
• Keywords: Volatility ; Changepoint ; Central limit theory

Abstract

We introduce arc length methods as a way of quantifying volatility and risk in time series. The idea is that processes with larger sample arc lengths exhibit larger fluctuations, and hence suggest greater risk and volatility. While arc length methods have not been previously proposed to study volatility, measuring volatility via arc lengths (in lieu of squared returns) is less problematic in financial settings, where infinite process fourth moments may be encountered. Here, a Gaussian functional central limit theorem for the sample arc lengths is proven under general finite second moment conditions. The limit theory is shown to apply to most popular financial models of log price ratios, including the autoregressive moving-average, generalized autoregressive conditional heteroscedastic, and stochastic volatility model classes. A cumulative sum statistic based on sample arc lengths is used to identify changepoints in the volatility of the Dow Jones Index. The results are compared to cumulative sum tests based on absolute and squared returns. The risk of financial assets can be compared via arc length. The methods naturally extend to multivariate processes and irregularly-spaced data.