Suppose, in \(t=0\), an investment bank is issuing mortgage-backed security (MBS), against an asset pool with two ex ante identical mortgage loans. Each loan will yield a gross return \(R, R>1\) in \(t=1\), with an unconditional probability \(p\) ("repay"), or 0 otherwise ("default"). Assume that the returns of the two mortgage loans are independent and identically distributed (i.i.d.).In \(t=0\), two mortgage-backed securities of identical size are issued:
- A senior tranche that will yield a gross return \(R\) in \(t=1\) if at least one mortgage loan is repaid;
- A junior tranche that will yield a gross return \(R\) in \(t=1\) only if both mortgage loans are repaid.

(a) Compute the expected return of each security. From now on, assume that the returns of the two mortgage loans are correlated in a way that the probability that both loans are repaid is \((1+\epsilon) p^{2}, \epsilon>0\), and the probability that both loans default is \((1+\epsilon)(1-p)^{2}\).

(b) Compute the expected return of each security. How does the correlation in return change your result, compared with that in question (a)?