# Intuition behind one factor Merton model for probability of default?

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??

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.