<p>Let <span class="math-container">$D_1 = 1$</span> be the event that the first loan defaults and let <span class="math-container">$D_2 = 1$</span> be the event that the second loan defaults. Assume that we know the correlation <span class="math-container">$\rho$</span> between <span class="math-container">$D_1$</span> and <span class="math-container">$D_2$</span> and we know the marginals <span class="math-container">$p_1$</span> and <span class="math-container">$p_2$</span>.</p>
<p>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).</p>
<p>First, notice that:
<span class="math-container">$$
\begin{align*}
&amp;\mathbb{E}[D_1] = \Pr(D_1) = p_1\\
&amp;\mathbb{E}[D_2] = \Pr(D_2) = p_2.
\end{align*}
$$</span>
First, let us try to compute the probability that both loans default.
<span class="math-container">$$
p_{12} = \Pr(D_1 = 1 \text{ and } D_2 = 1)
$$</span>
The covariance between <span class="math-container">$D_1$</span> and <span class="math-container">$D_2$</span> is given by:
<span class="math-container">$$
cov(D_1, D_2) = \mathbb{E}\left[(D_1 - p_1)(D_2 - p_2)\right] = p_{12} - p_1 p_2
$$</span>
Next, the variance of <span class="math-container">$D_1$</span> and <span class="math-container">$D_2$</span> equals:
<span class="math-container">$$
\begin{align*}
var(D_1) &amp;= \mathbb{E}\left[(D_1 - p_1)^2\right] = p_1 - (p_1)^2 = p_1(1-p_1),\\
var(D_2) &amp;= p_2(1- p_2).
\end{align*}
$$</span>
So the correlation is given by:
<span class="math-container">$$
corr(D_1, D_2) = \rho = \frac{cov(D_1, D_2)}{var(D_1)} = \frac{p_{12} - p_1 p_2}{\sqrt{p_1(1-p_1)} \sqrt{p_2(1-p_2)}}
$$</span>
As such, the probability that both default is given by:
<span class="math-container">$$
p_{12} = \rho \sqrt{p_1(1-p_1)} \sqrt{p_2(1-p_2)} + p_1 p_2.
$$</span>
Then also:
<span class="math-container">$$
\begin{align*}
\Pr(D_1= 1 \text{ and } D_2 = 0) &amp;= \Pr(D_1 = 1)- \Pr(D_1 = 1 \text{ and } D_2 = 1),\\
&amp;= p_1 - \rho \sqrt{p_1 p_2(1-p_1)(1-p_2)} - p_1 p_2\\
\Pr(D_2 = 1 \text{ and } D_2 = 0) &amp;= p_2 - \rho \sqrt{p_1 p_2(1-p_1)(1-p_2)} - p_1 p_2
\end{align*}
$$</span>
Then the probability that at least one defaults is given by:
<span class="math-container">$$
\begin{align*}
\Pr(D_1 = 1 \text{ or } D_2 = 1) &amp;= \Pr(D_1 = 1 \text{ and } D_2 = 0) + \Pr(D_1 = 0 \text{ and } D_2 = 1) + \Pr(D_1 = 1 \text{ and } D_2 = 1),\\
&amp;= p_1 + p_2 -  \rho\sqrt{p_1 p_2(1-p_1) (1-p_2)} - p_1 p_2,\\
\end{align*}
$$</span></p>

CC BY-SA 4.0
2021-05-25T06:25:39.850Z
2021-05-25T17:05:43.980Z
2021-05-25T17:05:43.980Z
tdm
<p>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.</p>
<p>I want to sell a X years Senior tranche of 1000 USD and a Junior tranche of USD 1000 backed by these two loans.</p>
<p>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&lt; correlation &lt; 1?</p>

<finance><financial-economics>
How do I calculate the price and risk of a senior tranche?
Andy
# How do I calculate the price and risk of a senior tranche?

(Asked by: username_0 on 2021-05-24)

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?

**Tags:** finance, financial-economics


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## 1 answers

### Answer by username_1 on 2021-05-25. Score: 3 | Accepted answer

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:
$$
\begin{align\*}
\Pr(D\_1 = 1 \text{ or } D\_2 = 1) &= \Pr(D\_1 = 1 \text{ and } D\_2 = 0) + \Pr(D\_1 = 0 \text{ and } D\_2 = 1) + \Pr(D\_1 = 1 \text{ and } D\_2 = 1),\\
&= p\_1 + p\_2 - \rho\sqrt{p\_1 p\_2(1-p\_1) (1-p\_2)} - p\_1 p\_2,\\
\end{align\*}
$$


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