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 the discounted loss distribution for a collateralized loan. To ensure non-negative values for the intensity and interest rate, we assume a quadratic Gaussian process for the 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 <inline-formula id="ILM0001"> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="rquf_a_762459_ilm0001.gif"/> </inline-formula>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 the default intensity and the discount interest rate, the coefficients have explicit closed-form solutions. We use numerical examples to analyse the effects of the correlation between the default intensity and the collateral value on expected loss and the standard deviation of the loss distribution.