Pricing of Mortgage Backed Securities
Student Name: Sijo G Raj
Financial institutions offering loans on mortgages to individuals usually
insulate themselves from potential loss due to pre-payments by bundling up
a bunch of mortgage loans and selling them to third parties at a discounted
price, with the understanding that these third parties will absorb whatever
future cash flows are generated from the payment of mortgage loans by
individual borrowers (and hence the need to discount the price of the total
mortgage loan amounts at time 0). These securities sold to third parties by
financial institutions are called Mortgage Backed Securities or MBS. The
problem here is how to appropriately price such MBS in the face of uncertain
nature of pre-payment behavior of individual borrowers. In the begining of
the loan period, the monthly pay-back amount on the loan or the equated
monthly installment (called EMI), say Y is calculated using the
standard annuity formula with the assumption that there will not be any
pre-payment. Now in the i-th month if there is a prepayment of amount
Qi and the total term of the loan is n months, then
again using the annuity formula one can now recalculate the revised EMI for
the remaining n-i months. Let's denote this revised EMI amount after
the i-th month by Yi. Note that Yi<
Y. Thus due to the prepayment at the i-th month, for all the
remaining n-i months there is a loss of amount Y-Yi
and if indeed the loan would have continued for all the remaining n-i
months, the sum of the discounted values of Y-Yi over these
n-i months would have given the net present value of the total loss due
to prepayment at the i-th month. However because of pre-payment in
every month, a loan typically lasts m< n months. Thus the net present
value of the loss due to prepayment at the i-th month is calculated as
the sum of the discounted values of Y-Yi over m-i
months plus the sum of the discounted values of Y for n-m
months, as the loss is total Y in these last n-m months. This
gives the net present value of loss due to pre-payment of amount
Qi in the i-th month, say Li. Now
the net present value of total loss due to pre-payments in all the m
months is given by L= L1+...+ Lm
. Thus if the total mortgage loan amount at the begining is P,
after accounting for the pre-payments, it should be priced as P-L,
yielding the pricing formula for a MBS. Now the formula (based on the basic
annuity formula) is such that one can divide through out by P and
make it free of the actual loan amount and the pricing of the MBS is
expressed as the fraction of P that should be charged as the price
of an MBS. In the process of dividing all the quantities by P,
Qi needs to be interpreted as the fraction of the loan
amount that is pre-paid in the i-th month, and plays a crucial in
determining the price. In practice these Qi's are
random and needs to be modeled using pre-payment data. Towards this end,
monthly payment data are collected longitudinally for 30 years for 1000
MBS from the Bloomberg/Reuters terminal. The price of the MBS, as obtained
above, is calculated for each of these MBS, and then the distribution of
these prices is studied, which is found to be Normal. Based on this the
optimal price of such MBS is found as its mean ± 2 * standard deviation,
which numerically amounted to [0.4313,0.4466]. Thus it is found that the
MBS should be priced between 43.13% to 44.66% of the mortgage amount.