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# Analytical Solution for the Loss Distribution of a Collateralized Loan under a Quadratic Gaussian Default Intensity Process

## Author

Listed:
• Satoshi Yamashita

(Professor, The Institute of Statistical Mathematics (E-mail: yamasita@ism.ac.jp))

• Toshinao Yoshiba

(Director and Senior Economist, Institute for Monetary and Economic Studies, (currently Financial System and Bank Examination Department), Bank of Japan (E-mail: toshinao.yoshiba@boj.or.jp))

## Abstract

In this study, we derive an analytical solution for expected loss and the higher moment of the discounted loss distribution for a collateralized loan. To ensure nonnegative values for intensity and interest rate, we assume a quadratic Gaussian process for default intensity and discount interest rate. Correlations among default intensity, discount interest rate, and collateral value are represented by correlations among Brownian motions driving the movement of the Gaussian state variables. Given these assumptions, the expected loss or the m-th moment of the loss distribution is obtained by a time integral of an exponential quadratic form of the state variables. The coefficients of the form are derived by solving ordinary differential equations. In particular, with no correlation between default intensity and discount interest rate, the coefficients have explicit closed form solutions. We show numerical examples to analyze the effects of the correlation between default intensity and collateral value on expected loss and the standard deviation of the loss distribution.

## Suggested Citation

• Satoshi Yamashita & Toshinao Yoshiba, 2011. "Analytical Solution for the Loss Distribution of a Collateralized Loan under a Quadratic Gaussian Default Intensity Process," IMES Discussion Paper Series 11-E-20, Institute for Monetary and Economic Studies, Bank of Japan.
• Handle: RePEc:ime:imedps:11-e-20
as

File URL: http://www.imes.boj.or.jp/research/papers/english/11-E-20.pdf
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## References listed on IDEAS

as
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3. Li Chen & H. Vincent Poor, 2003. "Markovian Quadratic Term Structure Models For Risk-free And Defaultable Rates," Finance 0303008, University Library of Munich, Germany.
Full references (including those not matched with items on IDEAS)

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### Keywords

default intensity; stochastic recovery; quadratic Gaussian; expected loss; measure change;
All these keywords.

### JEL classification:

• G21 - Financial Economics - - Financial Institutions and Services - - - Banks; Other Depository Institutions; Micro Finance Institutions; Mortgages
• G32 - Financial Economics - - Corporate Finance and Governance - - - Financing Policy; Financial Risk and Risk Management; Capital and Ownership Structure; Value of Firms; Goodwill
• G33 - Financial Economics - - Corporate Finance and Governance - - - Bankruptcy; Liquidation

### NEP fields

This paper has been announced in the following NEP Reports:

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