Statistical Tools for Finance and Insurance

Many of the concepts in theoretical and empirical finance developed over the past decades – including the classical portfolio theory, the Black-Scholes-Merton option pricing model or the RiskMetrics variance-covariance approach to Value at Risk (VaR) – rest upon the assumption that asset returns follow a normal distribution. But this assumption is not justified by empirical data! Rather, the empirical observations exhibit excess kurtosis, more colloquially known as fat tails or heavy tails (Guillaume et al., 1997; Rachev and Mittnik, 2000). The contrast with the Gaussian law can be striking, as in Figure 1.1 where we illustrate this phenomenon using a ten-year history of the Dow Jones Industrial Average (DJIA) index.

It has been nearly 50 years since the appearance of the pioneering paper of Mandelbrot (1963) on the non-Gaussianity of financial asset returns, and their highly fat-tailed nature is now one of the most prominent and accepted stylized facts. The recent book by Jondeau et al. (2007) is dedicated to the topic, while other chapters and books discussing the variety of non-Gaussian distributions of use in empirical finance include McDonald (1997), Knight and Satchell (2001), and Paolella (2007).

A vast amount of econometrical and statistical research deals with modeling financial time series and their volatility, which measures the dispersion of a series at a point in time (i.e., conditional variance). Although financial markets have been experiencing many shorter and longer periods of instability or uncertainty in last decades such as Asian crisis (1997), start of the European currency (1999), the “dot-Com” technology-bubble crash (2000–2002) or the terrorist attacks (September, 2001), the war in Iraq (2003) and the current global recession (2008–2009), mostly used econometric models are based on the assumption of stationarity and time homogeneity; in other words, structure and parameters of a model are supposed to be constant over time. This includes linear and nonlinear autoregressive (AR) and moving-average models and conditional heteroscedasticity (CH) models such as ARCH (Engel, 1982) and GARCH (Bollerslev, 1986), stochastic volatility models (Taylor, 1986), as well as their combinations.

The universal benchmark for option pricing is flawed. The Black-Scholes formula is based on the assumption of a geometric Brownian motion (GBM) dynamics with constant volatility. Yet, the model-implied volatilities for different strikes and maturities of options are not constant and tend to be smile shaped (or in some markets skewed). Over the last three decades researchers have tried to find extensions of the model in order to explain this empirical fact.

Global warming increases weather risk by rising temperatures and increasing between weather patterns. PricewaterhouseCoopers (2005) releases the top 5 sectors in need of financial instruments to hedge weather risk. An increasing number of business hedge risks with weather derivatives (WD): financial contracts whose payments are dependent on weather-related measurements.

Traditionally volatility is viewed as a measure of variability, or risk, of an underlying asset. However recently investors have begun to look at volatility from a different angle, variance swaps have been created.

This work presents one of the more recent and efficient learning systems – support vector machines (SVMs). SVMs are mainly used to classify various specialized categories such as object recognition (Schölkopf (1997)), optical character recognition (Vapnik (1995)), electric load prediction (Eunite (2001)), management fraud detection (Rätsch and Müller (2004)), and early medical diagnostics. It is also used to predict the solvency or insolvency of companies or banks, which is the focus of this work. In other words, SVMs are capable of extracting useful information from financial data and then label companies by giving them score values. Furthermore, probability of default (PD) values for companies can be calculated from those score values. The method is explained later.

The distance matrix method goes in line with the network analysis of market structure such as clustering analysis (Focardi and Fabozzi, 2004), geometry of crashes (Araujo and Louca, 2007), degree distribution of nodes (Boss et al., 2004; Liu and He, 2009) etc. These methods allow, among other things, to investigate the time evolution of correlations in time series such as stocks. The analysis of such time series correlations has emerged from investigations of portfolio optimization. The standard approach is based on the cross-correlation matrix analysis and optimizations of share proportions (see e.g. Adams et al., 2003; Cuthberson and Nitzsche, 2001). The basic question regarding what the most desirable proportion are among different shares in the portfolio lead to the introduction of a distance between time series, and in particular, of the ultrametric distance, which has become a classical method of correlation analysis between stocks (Bonanno et al., 2001; Mantegna and Stanley, 2000). The method allows to analyze the structure of the market, and therefore, simplifies the choice of shares. In fact, this question about structure of the stock market should be tackled before portfolio optimization.

A loss model or actuarial risk model is a parsimonious mathematical description of the behavior of a collection of risks constituting an insurance portfolio. It is not intended to replace sound actuarial judgment. In fact, according to Willmot (2001), a well formulated model is consistent with and adds to intuition, but cannot and should not replace experience and insight. Moreover, a properly constructed loss model should reflect a balance between simplicity and conformity to the data since overly complex models may be too complicated to be useful.

In examining the nature of the risk associated with a portfolio of business, it is often of interest to assess how the portfolio may be expected to perform over an extended period of time. One approach involves the use of ruin theory (Panjer and Willmot, 1992). Ruin theory is concerned with the excess of the income (with respect to a portfolio of business) over the outgo, or claims paid. This quantity, referred to as insurer’s surplus, varies in time. Specifically, ruin is said to occur if the insurer’s surplus reaches a specified lower bound, e.g. minus the initial capital. One measure of risk is the probability of such an event, clearly reflecting the volatility inherent in the business. In addition, it can serve as a useful tool in long range planning for the use of insurer’s funds.

The purpose of insurance rate making is to ensure that the book of business generates enough revenue to pay claims, expenses, and make profit. Apart from managing the overall level the actuary also needs to design a rating segmentation structure. The insurance company must understand which policies are more and which are less likely to generate claims. The price of insurance should reflect these differences.

Catastrophe (CAT) bonds are one of the more recent financial derivatives to be traded on the world markets. In the mid-1990s a market in catastrophe insurance risk emerged in order to facilitate the direct transfer of re-insurance risk associated with natural catastrophes from corporations, insurers and reinsurers to capital market investors. The primary instrument developed for this purpose was the CAT bond.

In the recent years an increasing demand for capital guaranteed equity-linked life insurance products and retirement plans has emerged. In Germany, a retirement plan, called Riester-Rente, is supported by the state with cash payments and tax benefits. Those retirement plans have to preserve the invested capital. The company offering a Riester-Rente has to ensure that at the end of the saving period at least all cash inflows are available. Due to the investors demand for high returns, banks and insurance companies are not only offering saving plans investing in riskless bonds but also in products with a high equity proportion. For companies offering an equity-linked Riester-Rente the guarantee to pay out at least the invested capital is a big challenge. Due to the long maturities of the contracts of more than 30 years it is not possible to just buy a protective put. Many different concepts are used by banks and insurance companies to generate this guarantee or to reduce the remaining risk for the company. They vary from simple Stop Loss strategies to complex dynamic hedging strategies. In our work we analyze the return distribution generated by some of these strategies.