An Actuarial Premium Pricing Model for Nonnormal Insurance and Financial Risks in Incomplete Markets

Abstract

A model for pricing insurance and financial risks, based on recent developments in actuarial premium principles with elliptical distributions, is developed for application to incomplete markets and heavy-tailed distributions. The pricing model involves an application of a generalized variance premium principle from insurance pricing to the pricing of a portfolio of nontraded risks relative to a portfolio of traded risks. This pricing model for a portfolio of insurance or financial risks reflects preferences for features of the distributions other than mean and variance, including kurtosis. The model reduces to the Capital Asset Pricing Model for multinormal portfolios and to a form of the CAPM in the case where the traded and nontraded risks have the same elliptical distribution.     

Modelling implied volatility with OLS and panel data models

The paper performs an empirical estimation of time-varying volatility using OLS regression. Error Components, and Dummy Variable models, by regressing the implied volatility on time to maturity, the strike price and a dummy. Both the daily OLS equations and the panel data model provide more accurate estimates of Black and Scholes option prices than the bench-mark standard deviation of log returns. FT-SE 100 Index European options are used for empirical analysis.     

Systematic risk and timescales

Abstract

In this paper we propose a new approach to estimating the systematic risk (the beta of an asset) in a capital asset pricing model (CAPM). The proposed method is based on a wavelet multiscaling approach that decomposes a given time series on a scale-by-scale basis. At each scale, the wavelet variance of the market return and the wavelet covariance between the market return and a portfolio are calculated to obtain an estimate of the portfolio's beta. The empirical results show that the relationship between the return of a portfolio and its beta becomes stronger as the wavelet scale increases. Therefore, the predictions of the CAPM model are more relevant in the medium long run as compared to short time horizons.     

Comparison of modeling methods for Loss Given Default

We compare six modeling methods for Loss Given Default (LGD). We find that non-parametric methods (regression tree and neural network) perform better than parametric methods both in and out of sample when over-fitting is properly controlled. Among the parametric methods, fractional response regression has a slight edge over OLS regression. Performance of the transformation methods (inverse Gaussian and beta transformation) is very sensitive to ε, a small adjustment made to LGDs of 0 or 1 prior to transformation. Model fit is poor when ε is too small or too large, although the fitted LGDs have strong bi-modal distribution with very small ε. Therefore, models that produce strong bi-model pattern do not necessarily have good model fit and accurate LGD predictions. Even with an optimal ε, the performance of the transformation methods can only match that of the OLS.     

Robust estimation with flexible parametric distributions: estimation of utility stock betas

The distributions of stock returns and capital asset pricing model (CAPM) regression residuals are typically characterized by skewness and kurtosis. We apply four flexible probability density functions (pdfs) to model possible skewness and kurtosis in estimating the parameters of the CAPM and compare the corresponding estimates with ordinary least squares (OLS) and other symmetric distribution estimates. Estimation using the flexible pdfs provides more efficient results than OLS when the errors are non-normal and similar results when the errors are normal. Large estimation differences correspond to clear departures from normality. Our results show that OLS is not the best estimator of betas using this type of data. Our results suggest that the use of OLS CAPM betas may lead to erroneous estimates of the cost of capital for public utility stocks.     

Does the equity premium puzzle persist during financial crisis? The case of the French equity market

This paper examines the effects of the financial crisis that began in 2008 on the equity premium of 6 French sector indices. Since the systematic risk coefficient beta remains the most common explanatory element of risk premium in most asset pricing models, we investigate the impact of the crisis on the time-varying beta of the six sector indices cited. We selected daily data from January 2003 to December 2012 and we applied the bivariate MA-GARCH model (BEKK) to estimate time-varying betas for the sector indices. The crisis was marked by increased volatility of the sector indices and the market. This rise in volatility led to an increase in the systematic risk coefficient during the crisis and first post-crisis period for all the major indices. The results are intuitive and corroborate findings in the empirical literature. The increase of the time-varying beta is considered by investors as an additional risk. Therefore, as expected, investors tend to increase their equity premiums to b ear the impact of financial crisis.     

Hedge fund allocation: Evaluating parametric and nonparametric forecasts using alternative portfolio construction techniques

We propose a model for constructing Asian funds of hedge funds. We compare the accuracy of forecasts of hedge fund returns using an ordinary least squares (OLS) regression model, a nonparametric regression model, and a nonlinear nonparametric model. We backtest to assess these forecasts using three different portfolio construction processes: an “optimized” portfolio, an equally-weighted portfolio, and the Kelly criterion-based portfolio. We find that the Kelly criterion is a reasonable method for constructing a fund of hedge funds, producing better results than a basic optimization or an equally-weighted portfolio construction method. Our backtests also indicate that the nonparametric forecasts and the OLS forecasts produce similar performance at the hedge fund index level. At the individual fund level, our analysis indicates that the OLS forecasts produce higher directional accuracy than the nonparametric methods but the nonparametric methods produce more accurate forecasts than OLS. In backtests, the highest information ratio to predict hedge fund returns is obtained from a combination of the OLS regression with the Fung–Hsieh eight-factor variables as predictors using the Kelly criterion portfolio construction method. Similarly, the highest information ratio using forecasts generated from a combination of the nonparametric regression using the Fung–Hsieh eight-factor model variables is achieved using the Kelly criterion portfolio construction method. Simulations using risk-adjusted total returns indicate that the nonparametric regression model generates superior information ratios than the analogous backtest results using the OLS. However, the benefits of diversification plateau with portfolios of more than 20 hedge funds. These results generally hold with portfolio implementation lags up to 12 months.     

Fuzzy Financial Pricing of Property-Liability Insurance

Abstract

This paper uses fuzzy set theory (FST) to solve a problem in actuarial science, the financial pricing of property-liability insurance contracts. The fundamental concept of FST is the alternative formalization of membership in a set to include the degree or strength of membership. FST provides consistent mathematical rules for incorporating vague, subjective, or judgmental information into complex decision processes. It is potentially important in insurance pricing because much of the information about cash flows, future economic conditions, risk premiums, and other factors affecting the pricing decision is subjective and thus difficult to quantify by using conventional methods. To illustrate the use of FST, we “fuzzify” a well-known insurance financial pricing model, provide numerical examples of fuzzy pricing, and propose rules for project decision-making using FST. The results indicate that FST can lead to significantly different decisions than the conventional approach.     

Estimating the Dynamics of Mutual Fund Alphas and Betas

This article develops a Kalman filter model to track dynamicmutual fund factor loadings. It then uses the estimates to analyzewhether managers with market-timing ability can be identifiedex ante. The primary findings are as follows: (i) Ordinary leastsquares (OLS) timing models produce false positives (nonzeroalphas) at too high a rate with either daily or monthly data.In contrast, the Kalman filter model produces them at approximatelythe correct rate with monthly data; (ii) In monthly data, thoughthe OLS models fail to detect any timing among fund managers,the Kalman filter does; (iii) The alpha and beta forecasts fromthe Kalman model are more accurate than those from the OLS timingmodels; (iv) The Kalman filter model tracks most fund alphasand betas better than OLS models that employ macroeconomic variablesin addition to fund returns.     

Sovereign bond yield spreads: A time-varying coefficient approach

We study the determinants of sovereign bond yield spreads across 10 EMU countries between Q1/1999 and Q1/2010. We apply a semiparametric time-varying coefficient model to identify, to what extent an observed change in the yield spread is due to a shift in macroeconomic fundamentals or due to altering risk pricing. We find that at the beginning of EMU, the government debt level and the general investors’ risk aversion had a significant impact on interest differentials. In the subsequent years, however, financial markets paid less attention to the fiscal position of a country and the safe haven status of Germany diminished in importance. By the end of 2006, two years before the fall of Lehman Brothers, financial markets began to grant Germany safe haven status again. One year later, when financial turmoil began, the market reaction to fiscal loosening increased considerably. The altering in risk pricing over time period confirms the need of time-varying coefficient models in this context.     

Linear beta pricing with inefficient benchmarks in a given factor structure*

ABSTRACT

We show the equivalence between the zero-beta version of a multi-factor arbitrage pricing model and a linear pricing model utilizing undiversified inefficient benchmarks in a given factor structure. The resulting linear model is a two-beta model, with one beta related to the inefficient benchmark and another adjusting for its inefficiency. This linear model shows that there are only two distinctive and computable sources of risk, affecting security expected returns, despite the existence of several risk factors. In a short empirical example we demonstrate that the model can be employed to provide guidance and allow researchers to test for the validity of their selection of the underlying risk factors driving variations in security returns.     

Time-varying beta and the Asian financial crisis: Evidence from Malaysian and Taiwanese firms

This paper empirically investigates the effects of the Asian financial crisis of 1997–1998 on the time-varying beta of 10 firms from each of Malaysia and Taiwan. Daily data from 1990 to 2001 and the bivariate MA-GARCH model (BEKK) are applied to create the time-varying betas for the firms. Results provide ample evidence of the influence of the financial crisis and the period after on the time-varying betas of the twenty firms. Results provided are somewhat mixed, indicating a rise in the beta in some cases and a fall in other cases. Results also show that the 10 Malaysian firms applied were more affected than the Taiwanese firms.     

Regression-Based Tests of the Market Pricing of Accounting Numbers: The Mishkin Test and Ordinary Least Squares

The test developed in Mishkin [1983] (hereafter, MT) is widely used to test the rational pricing of accounting numbers. However, contrary to the perception in the accounting literature, the exclusion of variables from the MT's forecasting and pricing equations leads to an omitted variables problem that affects inferences about the rational pricing of accounting variables. Only if the omitted variables are rationally priced is their exclusion irrelevant. Failure to recognize this issue leads accounting researchers to employ the MT without appreciating how omitted variables affect the inferences they draw. We demonstrate that when additional explanatory variables are included in the MT, the rational pricing of accruals is not rejected. That is, the accrual anomaly documented in Sloan [1996] vanishes when additional explanatory variables are incorporated into the MT. We also show that in accounting research settings, where samples are large, ordinary least squares (OLS) is equivalent to the MT. As a result, accounting researchers should consider using OLS or be more explicit about the exact advantages of the MT over OLS in their research setting.     

Model risk of the implied GARCH-normal model

Model risk causes significant losses in financial derivative pricing and hedging. Investors may undertake relatively risky investments due to insufficient hedging or overpaying implied by flawed models. The GARCH model with normal innovations (GARCH-normal) has been adopted to depict the dynamics of the returns in many applications. The implied GARCH-normal model is the one minimizing the mean square error between the market option values and the GARCH-normal option prices. In this study, we investigate the model risk of the implied GARCH-normal model fitted to conditional leptokurtic returns, an important feature of financial data. The risk-neutral GARCH model with conditional leptokurtic innovations is derived by the extended Girsanov principle. The option prices and hedging positions of the conditional leptokurtic GARCH models are obtained by extending the dynamic semiparametric approach of Huang and Guo [Statist. Sin., 2009, 19, 1037–1054]. In the simulation study we find significant model risk of the implied GARCH-normal model in pricing and hedging barrier and lookback options when the underlying dynamics follow a GARCH-t model.     

On accurate and provably efficient GARCH option pricing algorithms

The GARCH model has been very successful in capturing the serial correlation of asset return volatilities. As a result, applying the model to options pricing attracts a lot of attention. However, previous tree-based GARCH option pricing algorithms suffer from exponential running time, a cut-off maturity, inaccuracy, or some combination thereof. Specifically, this paper proves that the popular trinomial-tree option pricing algorithms of Ritchken and Trevor (Ritchken, P. and Trevor, R., Pricing options under generalized GARCH and stochastic volatility processes. J. Finance, , 54(1), 377–402.) and Cakici and Topyan (Cakici, N. and Topyan, K., The GARCH option pricing model: a lattice approach. J. Comput. Finance, , 3(4), 71–85.) explode exponentially when the number of partitions per day, n, exceeds a threshold determined by the GARCH parameters. Furthermore, when explosion happens, the tree cannot grow beyond a certain maturity date, making it unable to price derivatives with a longer maturity. As a result, the algorithms must be limited to using small n, which may have accuracy problems. The paper presents an alternative trinomial-tree GARCH option pricing algorithm. This algorithm provably does not have the short-maturity problem. Furthermore, the tree-size growth is guaranteed to be quadratic if n is less than a threshold easily determined by the model parameters. This level of efficiency makes the proposed algorithm practical. The surprising finding for the first time places a tree-based GARCH option pricing algorithm in the same complexity class as binomial trees under the Black–Scholes model. Extensive numerical evaluation is conducted to confirm the analytical results and the numerical accuracy of the proposed algorithm. Of independent interest is a simple and efficient technique to calculate the transition probabilities of a multinomial tree using generating functions.     

The Application of Corporate Governance Indicators With XBRL Technology to Financial Crisis Prediction

ABSTRACT

The widespread adoption of eXtensible Business Reporting Language (XBRL) suggests that intelligent software agents can now use financial information disseminated on the Web with high accuracy. Financial data have been widely used by researchers to predict financial crises; however, few studies have considered corporate governance indicators in building prediction models. This article presents a financial crisis prediction model that involves using a genetic algorithm for determining the optimal feature set and support vector machines (SVMs) to be used with XBRL. The experimental results show that the proposed model outperforms models based on only one type of information, either financial or corporate governance. Compared with conventional statistical methods, the proposed SVM model forecasts financial crises more accurately.     

Pricing of credit default index swap tranches with one-factor heavy-tailed copula models

In this paper, we provide two one-factor heavy-tailed copula models for pricing a collateralized debt obligation and credit default index swap tranches: (1) a one-factor double t distribution with fractional degrees of freedom copula model and (2) a one-factor double mixture distribution of t and Gaussian distribution copula model. A time-varying tail-fatness parameter is introduced in each model, allowing one to change the tail-fatness of the copula function continuously. Fitting our model to comprehensive market data, we find that a model with fixed tail-fatness cannot fit market data well over time. The two models that we propose are capable of fitting market data well over time when using a proper time-varying tail-fatness parameter. Moreover, we find that the time-varying tail-fatness parameters change dramatically over a one-year period.     

Diversification evidence from international equity markets using extreme values and stochastic copulas

Tail dependence plays an important role in financial risk management and determination of whether two markets crash or boom together. However, the linear correlation is unable to capture the dependence structure among financial data. Moreover, given the reality of fat-tail or skewed distribution of financial data, normality assumption for risk measure may be misleading in portfolio development. This paper proposes the use of conditional extreme value theory and time-varying copula to capture the tail dependence between the Australian financial market and other selected international stock markets. Conditional extreme value theory enables the model adequacy and the tail behavior of individual financial variable, while the time-varying copula can fully disclose the changes of dependence structure over time. The combination of both proved to be useful in determining the tail dependence. The empirical results show an outperformance of the model in the analysis of tail dependence, which has an important implication in cross-market diversification and asset pricing allocation.     

Return generating process and the determinants of term premiums

This paper examines asset pricing theories for treasury bonds using longer maturities than previous studies and employing a simple multi-factor model. We allow bond factor loadings to vary over time according to term structure variables. The model examines not only the time variation in the expected returns of bonds but also their unexpected returns. This allows us to explicitly test some asset pricing restrictions which are difficult to study under existing frameworks. We confirm that the pure expectation theory of the term structure of interest rates is rejected by the data. Our empirical study of a two-factor model finds substantial evidence of time-varying term-premiums and factor loadings. The fact that factor loadings vary with long-term interest rates and yield spreads suggest that bond return volatilities are sensitive to interest rate levels.     

Time-varying long-run mean of commodity prices and the modeling of futures term structures

The exploration of the mean-reversion of commodity prices is important for inventory management, inflation forecasting and contingent claim pricing. Bessembinder et al. [J. Finance, 1995, 50, 361–375] document the mean-reversion of commodity spot prices using futures term structure data; however, mean-reversion to a constant level is rejected in nearly all studies using historical spot price time series. This indicates that the spot prices revert to a stochastic long-run mean. Recognizing this, I propose a reduced-form model with the stochastic long-run mean as a separate factor. This model fits the futures dynamics better than do classical models such as the Gibson–Schwartz [J. Finance, 1990, 45, 959–976] model and the Casassus–Collin-Dufresne [J. Finance, 2005, 60, 2283–2331] model with a constant interest rate. An application for option pricing is also presented in this paper.