We are going to measure risk by duration. The duration of a cash flow is a weighted average of the times at which each component of the cash flow is received. The principal stream has a longer duration than the interest stream of a mortgage since as the age of the mortgage increases, the interest payments will decrease (since they depend on the amount of the mortgage that is yet unpaid) and the principal payments will become a larger part of each monthly payment.
If we let D(p) denote the duration of the principal stream then:
Where we divide by 12 to convert duration in annual units, rather than monthly units. We can rewrite this as:
D(p) = the sum from k=1 to n of w(k)*k where
w(k) = 1/(12*V(0)) * P(k)/(1+r)^k.
In this case is the weight given to each component.
Similarly, we can compute the duration of the interest stream as D(i):
Now we introduce the idea of prepayments.
The interest payment in period k is the same as before:
However, we must reassess the value of M(k) with each prepayment:
where ScheduledPrincipalPayment(k) is now P(k) = B - I(k) for period k.
The risk profiles of interest-only and principal-only MBSs are very different.
Principal-only MBS holders would like prepayments to increase, since they would like their payments earlier (due to the time value of money).
Interest-only MBS holders would like prepayments to decrease, since the slower the mortgage is paid, the more interest will accrue on the mortgage.
In an extreme case where all the mortgage holders of a particular MBS prepay their mortgages immediately after taking them out, the interest-only MBS holder will receive zero, since no interest will have been able to accrue on those mortgages.
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