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Say I grant two risky X years loans P(default)=0.1 of 1000 USD. For simplicity let us assume that the interest rate is 0.
I want to sell a X years Senior tranche of 1000 USD and a Junior tranche of USD 1000 backed by these two loans.
How do I calculate the price of these tranches? This exercise seems trivial if the P(default) of the two loans are either completely uncorrelated or perfectly correlated. But how do I do this if: 0< correlation < 1?
Let $D_1 = 1$ be the event that the first loan defaults and let $D_2 = 1$ be the event that the second loan defaults. Assume that we know the correlation $\rho$ between $D_1$ and $D_2$ and we know the marginals $p_1$ and $p_2$.
The aim is to compute the probability that one of the two loans defaults (for the junior tranche) and the probability that both default (senior tranche).
First, notice that: $$ \begin{align*} &\mathbb{E}[D_1] = \Pr(D_1) = p_1\\ &\mathbb{E}[D_2] = \Pr(D_2) = p_2. \end{align*} $$ First, let us try to compute the probability that both loans default. $$ p_{12} = \Pr(D_1 = 1 \text{ and } D_2 = 1) $$ The covariance between $D_1$ and $D_2$ is given by: $$ cov(D_1, D_2) = \mathbb{E}\left[(D_1 - p_1)(D_2 - p_2)\right] = p_{12} - p_1 p_2 $$ Next, the variance of $D_1$ and $D_2$ equals: $$ \begin{align*} var(D_1) &= \mathbb{E}\left[(D_1 - p_1)^2\right] = p_1 - (p_1)^2 = p_1(1-p_1),\\ var(D_2) &= p_2(1- p_2). \end{align*} $$ So the correlation is given by: $$ corr(D_1, D_2) = \rho = \frac{cov(D_1, D_2)}{var(D_1)} = \frac{p_{12} - p_1 p_2}{\sqrt{p_1(1-p_1)} \sqrt{p_2(1-p_2)}} $$ As such, the probability that both default is given by: $$ p_{12} = \rho \sqrt{p_1(1-p_1)} \sqrt{p_2(1-p_2)} + p_1 p_2. $$ Then also: $$ \begin{align*} \Pr(D_1= 1 \text{ and } D_2 = 0) &= \Pr(D_1 = 1)- \Pr(D_1 = 1 \text{ and } D_2 = 1),\\ &= p_1 - \rho \sqrt{p_1 p_2(1-p_1)(1-p_2)} - p_1 p_2\\ \Pr(D_2 = 1 \text{ and } D_2 = 0) &= p_2 - \rho \sqrt{p_1 p_2(1-p_1)(1-p_2)} - p_1 p_2 \end{align*} $$ Then the probability that at least one defaults is given by: $$ \begin{align*} \Pr(D_1 = 1 \text{ or } D_2 = 1) &= \Pr(D_1 = 1 \text{ and } D_2 = 0) + \Pr(D_1 = 0 \text{ and } D_2 = 1) + \Pr(D_1 = 1 \text{ and } D_2 = 1),\\ &= p_1 + p_2 - \rho\sqrt{p_1 p_2(1-p_1) (1-p_2)} - p_1 p_2,\\ \end{align*} $$
