As confirmed by empirical evidence, mortgagors do not prepay optimally,
Hayre (1999). Nielsen and Poulsen (2002) provide important insights on the
constraints and information asymmetries faced by mortgagors. Although being
bound by these constraints, individuals aim at minimizing their expected
future cash flows. Thus, it is natural to root their prepayment policy into
the optimal one. Let
be the
distance between the interest rate and the optimal prepayment frontier. The
optimal policy leads to a 100% prepayment of the mortgage if
and
a 0% prepayment if not: it can thus be seen as a Heaviside function of
When determinants of mortgagors' behavior cannot be observed, this behavior
can be modelled as a noisy version of the optimal one. It is thus natural to
look for the effective prepayment policy under the form of a characteristic
distribution function of which introduces dispersion around the
optimal frontier.
A pool of mortgages with similar financial characteristics is now
considered. This homogeneity assumption of the pool is in accordance with
market practice. For the ease of monitoring and investors' analysis,
mortgages with the same coupon rate and the same maturity are chosen for
pooling. Without loss of generality, the MBS can be assimilated into a
single loan with coupon and maturity
issued with principal
normalized at
.
Let be the proportion of unprepaid shares at date
. In the
optimal approach, the prepayment policy follows an ``all or nothing'' type
strategy, with
being worth 0 or 1. When practical policies are
involved,
is a positive process decreasing in time from 1 to 0. One
can look for
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We now come to a particular specification of For simplicity, we
choose an ad hoc parametric form for
in order to analyze its
main sensitivities. In accordance with stylized facts on prepayment, the
prepayment rate
is split in two distinct components
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Structural prepayment can involve many different reasons for prepaying, including:
Such prepayment characteristics appear to be stationary in time, Hayre (1999). Their average effect can be captured reasonably well through a deterministic model.
The Public Securities Association (PSA) recommends the use of a piecewise linear structural prepayment rate:
According to the PSA, the mean annualized values
for and
are
and
respectively. This implies that the
prepayment starts from
at the issuance date of the mortgage, growing
by
per month during the first
months, and being equal to
afterwards. This curve is accepted by the market practice as the benchmark
structural prepayment rate, see Figure 9.7. It is known as the
PSA curve.
The parameter
sets the desired translation level of this benchmark
curve. The PSA regularly publishes statistics on
the level of
according to the geographical region, in the US, of mortgage
issuance.
The refinancing prepayment rate has to account for both the effect of
interest rates and individual characteristics such as burnout. Refinancing
incentives linked to interest rate level can be captured through the optimal
prepayment framework of Section 2. This framework implies a 1 to 0 rule for
MBS principal evolution, depending on the optimal short term interest rate
level for prepaying,
As soon as
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Parameter
is a scale parameter. In this form, being far into
the prepayment zone means that
so that
Parameter
directly
accounts for the magnitude of the burnout effect since it represents the
instantaneous fraction of mortgagors who chose not to prepay even for very
low values of
More precisely, if
was to stay very low
during a time period
and if the refinancing prepayment
was the only prepayment component to be considered, using expression
(9.11), the proportion of unprepaid shares at date
would be equal to
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In order to analyze the main effects of our model, we choose the 100% PSA curve for the structural prepayment rate; the burnout is set equal to 20%.
This means that, whatever the market conditions are, 20% of the mortgagors
will never repay their loan. The time horizon for this burnout effect is
fixed equal to 2 years. Parameters
and
are
calibrated in such a way that when
ten percent of mortgagors
prepay their loan after horizon
, and half of the mortgage is prepaid if
half the distance to optimal prepayment rate is reached.
Market conditions are set as of December 2003 in the EUR zone. The short
rate equals to and the long term rate is
. The volatility of
the short rate
is taken equal to
and
is such
that the volatility of the 10 year forward rate equals to 0.5%. The facial
coupon of the pool of mortgage is
its remaining maturity is set to
years and no prepayment has been made (
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With such parameters, the price of the MBS is displayed in Figure 9.9 as a function of interest rates, together with the optimally
prepaid mortgage (OPM) and the mortgage without callability feature (NPM).
When interest rates go down, the behavior of the MBS is intermediate between
the OPM and the NPM. The value at is controlled by the burnout
level. The transition part is controlled by parameters
and
When interest rates increase, the MBS price is higher than the
NPM's due to the PSA effect. In fact, by prepaying in the optimal region,
mortgagors offer the holder of MBS a positive NPV. This appears clearly when
displaying the value of the option embedded in MBS. Recall that in the case
of the optimally prepaid mortgage, this value was always positive (Figure
9.5). This is no longer the case for MBS as indicated in
Figure 9.10.
As a consequence, the sensitivity of MBS to interest rates moves is reduced.
Duration is computed in Figure 9.11. It is always less than the
underlying pool duration. Its behavior resembles a smoothed version of the
optimally prepaid one.
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Let us now increase the implied volatility of the underlying derivatives market. The embedded option value increases, translating the negative sensitivity of the MBS price to market volatilities, see Figure 9.12. In hedging terms, MBS are ``vega negative''. A long position in MBS is ``short volatility''. This is also well indicated in the variation of duration. Figure 9.13 shows how higher volatility increases the duration when the MBS is ``in the money'' (low interest rates) and decreases for ``out of the money'' MBS. This is not surprising when one thinks of the duration as the ``delta'' of the MBS with respect to interest rates. The effect of volatility on the delta for a standard vanilla put option is known to be opposite, depending on the moneyness of the option.