Proceedings of the Institute of Statistical Mathematics Vol. 50, No. 2, 119-132(2002)

Markov Switching Stochastic Trend Model

Morikazu Hakamata

(Department of Statistical Science,

The Graduate University for Advanced Studies)

The Graduate University for Advanced Studies)

This paper focuses on two characteristics of financial asset price movement: trend slope change and heteroscedasticity. To capture these characteristics, we propose a stochastic trend model with Markov switching slope change and ARCH (MS-SC/ARCH), and evaluate the usefulness of the MS-SC/ARCH model for the trading strategy. This model consists of a no-slope change and low volatility regime, and a slope change and high volatility regime. The time series shifts between two regimes according to the first-order Markov switching process. In the empirical analysis using TOPIX, we estimate of the effective trend slope for trading, and obtain superior performance over some other trading strategies by using the trading strategy based on the MS-SC/ARCH model.

**Key words: ARCH, heteroscedasticity, Markov switching, TOPIX, trading strategy, trend slope.**

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Proceedings of the Institute of Statistical Mathematics Vol. 50, No. 2, 133-147(2002)

the Term Structure of Interest Rates

Akihiko Takahashi

(Department of Mathematical Science, University of Tokyo)

Seisho Sato

(The Institute of Statistical Mathematics)

We have developed a new methodology for estimating the general class of term structure models based on a Monte Carlo filtering approach. We utilize the generalized state space model, which can be naturally applied to the estimation of term structure models based on Markovian processes. It is also possible to introduce measurement errors in a general way without any bias. Moreover, we illustrate our method using an affine term structure model and JGB data.

**Key words: Generalized state space model, Monte Carlo filter, interest rate model, affine term structure model.
**

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Proceedings of the Institute of Statistical Mathematics Vol. 50, No. 2, 149-164(2002)

and Their Application to Yield Curve Estimation

Yoshinori Kawasaki

(The Institute of Statistical Mathematics)

Tomohiro Ando

(Graduate School of Mathematics, Kyushu University)

There have been a considerable number of researches on estimation of yield curves. Since McCulloch (1971), utilizing cross-sectional data of coupon bonds, a method for regressing cash flows on a set of basis functions to estimate discount factors has been widely used and discussed by both academic researchers and practitioners. Attention has focused on the stability of the implied forward rate curve, and also on the optimal choice of the number of basis functions. However, there seems to be no definitive method for overcoming these shortcomings. Based on the observation that the least squares approach tends to result in a so-called \lq improper' problem, this article presents a regularization method or penalized maximum likelihood approach to stabilize the shapes of yield curves. It also proposes a Gaussian radial basis function as an alternative to commonly adopted basis functions such as McCulloch's natural cubic spline. In model determination, what is essential is the choice of regularization parameter, the number of basis and the dispersion parameter in the Gaussian radial basis function. To determine these quantities, we propose a tailor-made version of generalized information criteria (GIC) constructed in the same manner as in Konishi and Kitagawa (1996). In the final section, we show worked examples with Japanese governmental bond data. By the use of bootstrapping, we demonstrate that the forward rate curve estimated by our method is much more stable than the one derived by the cubic spline model with cross validation. It is also shown that even in a small sample case our model with Gaussian basis still gives rise to a considerably stable forward rate compared to those obtained by natural cubic spline.

**Key words: Yield curve, penalized likelihood, Gaussian radial basis function, generalized information criteria.
**

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Proceedings of the Institute of Statistical Mathematics Vol. 50, No. 2, 165-199(2002)

Hiroki Masuda

(Graduate School of Mathematical Sciences, University of Tokyo)

Several analytical facts relating to *GIG* (generalized inverse Gaussian) distribution on *R*_{+}and *GH* (generalized hyperbolic) distribution on *R* are reviewed. The latter is defined as a normal variance-mean mixture where the mixing distribution is the former. Recently these distributions have been applied to mathematical finance in various ways, and their great potential for modelling certain non-Gaussian variations is also presented. In particular, we refer to the fact that *GIG* and *GH* distributions possess selfdecomposability. This concept is closely related to the invariant distribution of the one-dimensional Ornstein-Uhlenbeck type process, which has recently attracted attention as a possible model of the latent volatility process in financial literature. We also briefly review a multi-dimensional version of the *GH* distribution as an appendix.

**Key words: Generalized hyperbolic distribution, generalized inverse Gaussian distribution, infinitely divisibility, normal variance-mean mixture, selfdecomposability.
**

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Proceedings of the Institute of Statistical Mathematics Vol. 50, No. 2, 201-216(2002)

Analysis of Using Ordered Logit Model and

Continuation Ratio Model

Takehiko Yasukawa

(ChuoAoyama PricewaterhouseCoopers Financial & Risk Management, Ltd.)

Since bond rating is typical ordinal data, ordered logit models or ordered probit models are being widely used for credit analysis. However, these models require the equal slope assumption, which has not been verified in previous research in this field. This paper tests this assumption using the ordered logit model and continuous ratio model using Japanese bond ratings data. As a result, the equal slope assumption was rejected and extended continuation ratio models relaxing this assumption were found to be a useful approach for analyzing bond rating.

**Key words: Bond ratings, ordered logit model, continuation ratio model, equal slope assumption.**

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Proceedings of the Institute of Statistical Mathematics Vol. 50, No. 2, 217-240(2002)

on a Corporate Bond Pricing Model

with a Correlation Structure

among Individual Bond Prices

Hiroshi Tsuda

(The NLI Research Institute)

The recent increase in the number of failures among listed companies has drawn attention to the problem of credit risk. For corporate bonds, credit risk is reflected in market prices. This paper proposes a new pricing model for straight coupon bonds among corporate bonds, which enables us to obtain information on credit risk such as default probability and principal recovery rate. This model has the following features: 1) stochastic treatment of cash-flow discount functions of each individual bond, 2) evaluation for default probability and principal recovery rate for individual companies implied in the corporate bond market, and 3) correlation structure among individual bond prices by assuming a variance-covariance structure for the random part of the stochastic discount function. We obtained significant information on the term structure of default probability by testing our model empirically with Japanese corporate bond price data.

**Key words: Straight coupon bond among corporate bonds, random cash-flow discount function, term structure of default probability, expected value of loss, generalized least squares.
**

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Proceedings of the Institute of Statistical Mathematics Vol. 50, No. 2, 241-258(2002)

Using Credit Risk Database

Hisanao Takahashi

(Department of Statistical Science, The Graduate University for Advanced Studies;

Financial Services Agency)

Financial Services Agency)

Satoshi Yamashita

(The Institute of Statistical Mathematics; CRD)

It is important in credit risk management to determine the probability of bankruptcy. Few reliable analyses of bankruptcy have been developed for small and medium-sized enterprises because of the delay in developing of databases to capture credit risks for the enterprises. Recently, a large-scale database for estimating credit risks for such enterprises has become available as ``Credit Risk Database". In this paper, we estimate the probability of bankruptcy by applying the logit model to the data from this database. We use the * t * value to evaluate the significance of the model's parameters. We discuss the differences in explanatory factors of credit risk depending on the enterprise scale.

**Key words: Credit risk, logit model, failure probability, small and medium-sized enterprises.
**

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Proceedings of the Institute of Statistical Mathematics Vol. 50, No. 2, 259-278(2002)

Harumi Yashiro

(Asian Disaster Reduction Center;

The Tokio Marine and Fire Insurance Co., Ltd.)

The Tokio Marine and Fire Insurance Co., Ltd.)

Sei'ichiro Fukushima

(Tokyo Electric Power Services Co., Ltd.)

In recent years, some risk analysis methods have been proposed for quantitative evaluations of risk. However although it is important for carrying out effective risk management, the treatment of quantified risk has not been considered sufficiently. This paper proposes a seismic portfolio analysis method considering the effect of risk transfer such as securitization, which has been adopted to the portfolio of 73 buildings in Tokyo. Through its application, the following findings are obtained: the grid has a relatively large effect on the PML (probable maximum loss) although the forfeiture rate of principal does not, risk transfer is more effective from the viewpoint of investment for risk with a smaller grid and a higher trigger level that is the minimum magnitude for forfeiture, and some combinations of grid and forfeiture rate can be identical to each other regardless of principal.

**Key words: Portfolio of buildings, risk management, seismic risk, risk transfer, securitization of risk.**

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Proceedings of the Institute of Statistical Mathematics Vol. 50, No. 2, 279-301(2002)

for Pension Fund

Satoshi Yamashita

(The Institute of Statistical Mathematics)

Tomoo Yato

(Nomura Research Institute, Ltd.)

Recently, there has been extensive study on asset and liability management (ALM) as a multi-period portfolio selection problem. However, most studies have assumed finite investment periods and consequently cannot be implemented in actual asset management. This paper takes the pension fund ALM as an example with an infinite investment horizon and proposes a method for constructing an optimum multi-period portfolio. In the process of deriving the main result, problems in dynamic programming and scenario analysis are pointed out and we compare their pros and cons. We apply the Markov decision process (MDP) to solve the pension fund ALM problems. The method's validity is examined based on a criterion that focuses on the changes in the rate of pension premium.

**Key words: Markov decision process, pension fund, asset liability management, multiperiod optimization, rate of pension premium.**

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