Intuition behind one factor Merton model for probability of default?

Question

Let

• $Z$ be a standard normal rv,

• $Y_i$ be iid standard normals for $i = 1,\dots, n$,

satisfying the relationship $$ X_i = \sqrt{p} Z + \sqrt{1-p} Y_i $$

In the one factor Merton model, we say that individual $i$ will default with probability $P(X_i < B)$ for some $B$.

I interpret $X_i$ to represent the financial well being of the individual, $Z$ is the well being due to the economy and $Y_i$ is the well being due to idiosyncratic factors.

My question: What is the intuition behind the equation above?

I'm guessing the coefficient of $Z$ is $\sqrt{p}$ because we want the well being between individuals to be linearly correlated with value $p$. I'm also guessing that the other coefficient is $\sqrt{1 - p}$ becuase we want $X_i$ to be standard normal. But why is it important for $X_i$ to have variance = 1??

Answer 1

Like you said the goal of this equation is to describe a simple model where the individual “well being” has correlation $\sqrt{p}$ with some systematic factor and receive also contribution from independent idiosyncratic factor

The fact that $X_i$ has variance 1 is just to simplify the presentation i guess, there is no meaning into it as the equation could be simply adjusted to reflect any variance you want.