In the Longstaff and Rajan top-down correlated default model, assume that losses $L _ { 4 }$ in a credit portfolio are given by the following dynamic process in a one-factor setting: $\frac { d L _ { t } } { 1 - L _ { t } } = \gamma d N ( \lambda )$ where $\gamma$ is a fractional loss (of the current portfolio value) that occurs every time there is a default, assumed to be generated by a Poisson process $N$ with loss arrival rate $\lambda$ (a constant). What is the expected loss of a $100 portfolio in a year if $\gamma = 0.01$ and $\lambda = 2$ ?
A) $1.5
B) $2.0
C) $2.5
D) $3.0

Question

In the Longstaff and Rajan top-down correlated default model, assume that losses  $L _ { 4 }$ in a credit portfolio are given by the following dynamic process in a one-factor setting: $\frac { d L _ { t } } { 1 - L _ { t } } = \gamma d N ( \lambda )$ where $\gamma$ is a fractional loss (of the current portfolio value) that occurs every time there is a default, assumed to be generated by a Poisson process $N$ with loss arrival rate $\lambda$ (a constant). What is the expected loss of a $100 portfolio in a year if $\gamma = 0.01$ and $\lambda = 2$ ?
A) $1.5
B) $2.0
C) $2.5
D) $3.0

Quizplus · Accepted Answer

The expected loss in a year for a portfolio can be calculated using the formula for the expected value of a Poisson distribution, which is simply the mean of the distribution. The mean of a Poisson distribution is given by its rate parameter, $\lambda$. In this case, $\lambda = 2$, indicating that on average, there are 2 defaults per year. Since each default causes a loss of $\gamma = 0.01$ (or 1% of the portfolio value), and the portfolio value is $100, the expected loss per default is $1. Therefore, the expected loss in a year is $2 \times $1 = $2.0$.

In the Longstaff and Rajan Top-Down Correlated Default Model, Assume L4L _ { 4 }L4​

In the Longstaff and Rajan Top-Down Correlated Default Model, Assume $L _ { 4 }$