Closed form formula for mortgages
I am looking for a closed form formula to calculate the PV of a
mortgage incorporating constant prepayments. I know how to calculate
the individual cashflows and discount them, but a closed form would be
much easier....
Prepayments are a complicated problem. Prepayments in the US are different
from Prepayments in Canada. In Canada, homeowners may have the right
to partially pay up to 15% of the principal mortgage off in a lumpsum
in any given year, and this is refered to a "partial prepayments". People
may use xmas bonuses etc..to reduce their mortgage from time to time
to pay off the loan faster whether interest rates fall or not. Then,
there are "refinancings", which are prepayments that a pool of mortgages
experience due to falling interest rates, which are really the entire principal
balance of individual mortgages within that pool being paid off in full, but
that only results in a partial reduction of the balance of the pool as a whole.
The refinancing is what is usually treated as CPR in the US. But, then there
are "liquidations" which are mortgages that pay off for reasons not directly
related to interest rates. These are individual mortgages that payoff in full
because homeowner has to move becaue of job changes, for example, so
is forced to sell his home. Then, there are defaults were the homeowner
is unable to pay the mortgage and is foreclosed by the bank,etc. The
"liquidations" and the defaults, although they result in full repayment of
balance of individual mortgages, only have about 3-5% effect on the
balance of a pool of mortgages. So, even for individual mortgages a
prepayment factor related to "liquidations" and another for "defaults"
are included to take into account the probability of such cashflows.
Also, defaults may be treated as loss of principal, because foreclosure
and subsequent forced sale of a home may not result in recovery of principal
balance outstanding etc..