2 Vasicek Model (20 points) Let's get a bit more practice with normal distributions, and use the law of large numbers, all at once.
The Vasicek model is a simple model used for evaluating the credit risk of a loan portfolio consisting of many loans with a particular correlation structure. Suppose your bank has extended n loans, indexed i = 1,...,n. The performance of loan i depends some underlying cash flow $U_i$. We suppose that $U_i$ is given by
$U_i = \sqrt{p}X + \sqrt{1 - p}e_i$
where here $X \sim N(0, 1)$ is a common factor to all of the loans, and $e_i \stackrel{iid}{\sim} N(0, 1)$ for $i = 1,...,n$ is the idiosyncratic/loan-specific component. The X and $e_i$ variables are also independent. The constant p is assumed to be nonnegative, with $p \in [0, 1)$. Note that the coefficients $\sqrt{p}$ and $\sqrt{1 - p}$ give us that $U_i \sim N(0, 1)$. Note also that
$cov(U_i, U_j) = E[(\sqrt{p}X + \sqrt{1 - p}e_i)(\sqrt{p}X + \sqrt{1 - p}e_j)]$
$= E[\rho X^2 + \sqrt{p(1 - p)}e_iX + \sqrt{p(1 - p)}e_jX + (1 - p)e_ie_j]$
$= \rho E[X^2]$
$= \rho$
so this is a model with correlated defaults, and $\rho$ controls the default correlations.
Within the model, we assume that loan i defaults if $U_i < K$ for some constant K (that is the same for all loans). Using that $U_i \sim N(0, 1)$, we know that the default probability for any individual loan is $\Phi(K)$. To make it clearer, we often substitute out K by noting that $K = \Phi^{-1}(PD)$ where PD stands for "probability of default."
We will let $Y_i$ denote the default indicator for loan i: $Y_i = 1$ if loan i defaults, and 0 otherwise. This means that
$Y_i = \begin{cases} 1 & \text{if } U_i \le \Phi^{-1}(PD) \\ 0 & \text{otherwise} \end{cases}$
(a) (5 points) Let's first work conditional on $X = x$. That is, let's analyze the problem after we know the common factor takes on value $X = x$. In that case, $U_i = \sqrt{p}x + \sqrt{1 - p}e_i \sim N(\sqrt{p}x, 1 - p)$ are all iid. Show that the probability of default of loan i (conditional on $X = x$) is
$\Phi(\frac{\Phi^{-1}(PD) - \sqrt{p}x}{\sqrt{1 - p}})$.
Hint: this is really easy. I'm just asking you to calculate the probability that $U_i < K = \Phi^{-1}(PD)$ (or equivalently, $Y_i = 1$) under the conditional distribution.
