## Default Distribution Model Truncated by Stochastic Credit Decision

—Application of Skew-normal Distribution—

The univariate skew-normal distribution was introduced by Azzalini in 1985 as a natural extension of the normal density to accommodate asymmetry. In the finance context, a company will be generally deemed to go bankrupt when its credit score exceeds a critical threshold. This paper applies the skew-normal distribution to default company distribution. It assumes that credit decision and corporate credit score are both normally distributed and vary stochastically. The corporate credit distribution is truncated by the stochastic threshold (credit decision) and results in a skew-normal distribution of default companies.

This paper refers to the basic feature of the skew-normal distribution. When the threshold parameters are inferred using empirical default data, the location parameter shows consistency with the historical credit decision trend. Additionally, the skew-normal distribution model can indicate the skewness of the default distribution caused by the parameters, especially the scale parameter of the threshold.

Key words: Skew-normal distribution, hidden truncation, credit risk.

## Binary Prediction for Minimization of Financial Risk: Theory and Applications

Risk involved with financial contracts can often be viewed as uncertainty of binary outcomes. This paper treats risk minimization as a problem of profit maximization, and gives an optimal solution of the cut-off point for binary prediction. This optimality or profit maximization will be asymptotically attained in the sense of convergence in probability. In practice, we have to replace the true parameters inside an indicator function by estimates. Because indicator functions are discontinuous, apparently it looks like a non-standard argument. We show, in spite of this, that we can construct an interval for maximized profit, and even minimize it based on asymptotic theories where the MLE is simply plugged in. Simulation results suggest that the finite sample properties of our asymptotic theories are satisfactory. In an empirical analysis using personal loan data of a south German bank, we show that the total profit realized by our optimal prediction exceeds the actually observed profit regardless of the settings of loan interest.

Key words: Binary prediction, profit maximization, the minimum prospective interval, approximation of an indicator function.

## Regime Switching Factor Analysis and

Its Application to Detection of Risk Factors in J-REIT Market

We study factor analysis subject to regime switching or hidden Markov models (*HMM factor analysis*) and apply the result to conduct an empirical analysis on Japanese REIT (*J-REIT*) markets.

The objective of HMM factor analysis is to detect risk factors solely based on asset prices, where the dimension of risk factors is expected to be far less than that of target assets. The model admits risk factor switching according to hidden economic regimes behind the market. Following the model settings above, we derive an estimation method. The procedure is as follows: (1) introduce a forward-backward algorithm to define the likelihood function and (2) use the Baum-Welch (EM) algorithm to obtain the maximum likelihood estimators. We also mention the variance of derived estimators. Next, we conduct an empirical analysis to apply the model to the J-REIT market to detect common risk factors in the presence of hidden economic regimes. We find that the HMM factor analysis better explains the risk structures in J-REIT prices compared to the standard setting.

While it is common in financial literature to use multi-factor regression analyses to explain the sources of returns in J-REIT, we employ HMM regression for comparison. We find that: (1) in both HMM factor and regression analyses, the estimated smoothers detect essentially the same regimes, (2) regime switching risk factors can be detected by asset prices alone, and (3) the return sources due to regime switching can be better explained by introducing market benchmarks.

Key words: Factor analysis, hidden Markov models, statistical methods, empirical analysis.

## Quote Revisions and Price Discovery before Market Opening

This paper investigates the informational efficiency of preopening quotes of the Tokyo Stock Exchange by unbiasedness regression. As with other exchanges, preopening quotes reflect new information, suggesting that traders learn information from them. The larger the price change of the day, the more frequently preopening quotes are revised. The informational efficiency of opening prices are not deteriorated by the frequency of quote revisions, implying that traders learn information from active revisions of preopening quotes on days with large price movements.

Key words: Informational efficiency, price discovery, preopening quote, unbiasedness regression.

## Backtesting and Studying Risk Measure in Hedge Funds

This paper provides empirical analyses of value at risk and expected shortfall calculated by various models which assume i.i.d. with the hedge fund's losses. We use hedge funds' monthly returns, which include before and after the subprime crisis of 2008.

We use the normal distribution as the hedge fund's loss distribution and also Gaussian mixture, Johnson distribution, which can express negative skewness and large kurtosis of the hedge fund's return.

We conclude that the ratios rejecting risk measure models for the global-macro are low. On the other hand, all risk measures for the relative-value are mostly rejected.

Key words: Hedge fund, risk management, value at risk, expected shortfall.

## Gerber-Shiu Function in Risk Theory and Statistical Inference

Gerber and Shiu introduced the concept of *expected discounted penalty function* in a series of their classical papers. This function is called *the Gerber-Shiu function* in risk theory. Since its introduction, the analysis has been getting a lot of attention in the field of insurance and finance, and many authors have studied it for various risk processes from probabilistic aspects. However, from a practical point of view, statistical inference for the Gerber-Shiu function is also an important issue in actuarial mathematics, since it has some unknown quantities that need to be estimated from real data. This paper briefly reviews some recent results on the Gerber-Shiu function for a class of Lévy insurance risk processes and discuss their statistical inference. The estimation procedure treated in this paper is nonparametric: we consider an empirical type estimator of the Laplace transform of the Gerber-Shiu function, and take a *regularized inverse* to recover the Gerber-Shiu function. We illustrate the procedure for a Wiener-Poisson risk process, and show a kind of consistency and the rate of convergence of a proposed estimator. We also make some remarks on an extension of the method to a generalized risk process, and on another estimation procedure.

Key words: Risk theory, Gerber-Shiu function, generalized risk processes, regularized Laplace inversion, empirical estimation.

## Limit Theorems in Estimation of Diffusion

We discuss limit theorems in estimation problems for diffusion. We treat quasi-likelihood analysis and nonsynchronous covariance estimation and mention new topics in this field.

Key words: Diffusion, volatility, nonsynchronous covariance estimation, mixed normality, martingale expansion.

## Analytical Solution for $m$-th Moment of a Collateralized Loan's Loss under a Quadratic Gaussian Default Intensity Process

In this study, we derive an analytical solution for the expected loss and the higher moment of loss distribution for a collateralized loan, focusing on the correlation between default intensity and collateral value. To ensure non-negativity of intensity, we assume a quadratic Gaussian process for the default intensity. The correlation between the default intensity and the collateral value is expressed by the correlation of the two Brownian motions that drive the movement of the state variables of the default intensity and the collateral value. Given these settings, we show the expected recovery value, which is a component of the expected loss, as given by a Stieltjes integral with a measure-changed survival probability. More generally, we also show that the $m$-th moment of loss distribution can be calculated by a combination of Stieltjes integrals with a measure-changed survival probability. Using numerical examples, we analyze the effect of the correlation on the expected loss and the standard deviation of the loss.

Key words: Default intensity, stochastic recovery, quadratic Gaussian, expected loss, measure change.