Correlated Binomial Models and Correlation Structures

We discuss a general method to construct correlated binomial distributions by imposing several consistent relations on the joint probability function. We obtain self-consistency relations for the conditional correlations and conditional probabilities. The beta-binomial distribution is derived by a strong symmetric assumption on the conditional correlations. Our derivation clarifies the 'correlation' structure of the beta-binomial distribution. It is also possible to study the correlation structures of other probability distributions of exchangeable (homogeneous) correlated Bernoulli random variables. We study some distribution functions and discuss their behaviors in terms of their correlation structures.Comment: 12 pages, 7 figure

Evaluation of Tranche in Securitization and Long-range Ising Model

This econophysics work studies the long-range Ising model of a finite system with $N$ spins and the exchange interaction $\frac{J}{N}$ and the external field $H$ as a modely for homogeneous credit portfolio of assets with default probability $P_{d}$ and default correlation $\rho_{d}$. Based on the discussion on the $(J,H)$ phase diagram, we develop a perturbative calculation method for the model and obtain explicit expressions for $P_{d},\rho_{d}$ and the normalization factor $Z$ in terms of the model parameters $N$ and $J,H$. The effect of the default correlation $\rho_{d}$ on the probabilities $P(N_{d},\rho_{d})$ for $N_{d}$ defaults and on the cumulative distribution function $D(i,\rho_{d})$ are discussed. The latter means the average loss rate of the``tranche'' (layered structure) of the securities (e.g. CDO), which are synthesized from a pool of many assets. We show that the expected loss rate of the subordinated tranche decreases with $\rho_{d}$ and that of the senior tranche increases linearly, which are important in their pricing and ratings.Comment: 21 pages, 9 figure

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11 Aspects of Continuity and Transformation From the Research of Four Communities in Tushima Island

報告本稿は，当センター第一期個別共同研究「持続と変容の実態の研究　―対馬60年を事例として」で実施した現地調査の報告である．報告の一部はすでに前年度『年報』第7号に「持続と変容の諸相―　四つの集落の聞き書きから（1）」として発表した．筆者の事情により報告の執筆が遅れたため，「調査報告」として（2）からはじまるという変則的な形になったが，これは，年報全体の編集方針に合わせて形式を変更したためであることをお断りしておきたい