1
$\begingroup$

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?

$\endgroup$
3
$\begingroup$

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: $$ \begin{align*} \Pr(D_1 = 1 \text{ or } D_2 = 1) &= \Pr(D_1 = 1 \text{ and } D_2 = 0) + \Pr(D_1 = 0 \text{ and } D_2 = 1) + \Pr(D_1 = 1 \text{ and } D_2 = 1),\\ &= p_1 + p_2 - \rho\sqrt{p_1 p_2(1-p_1) (1-p_2)} - p_1 p_2,\\ \end{align*} $$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.