9.3 Valuation of Mortgage Backed Securities

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 $ d_{t}\stackrel{{\mathrm{def}}}{=}r_{t}^{opt}-r_{t}$ be the distance between the interest rate and the optimal prepayment frontier. The optimal policy leads to a 100% prepayment of the mortgage if $ d_{t}>0$ and a 0% prepayment if not: it can thus be seen as a Heaviside function of $ d_{t}.$

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 $ d_{t},$ which introduces dispersion around the optimal frontier.


9.3.1 Generic Framework

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 $ c$ and maturity $ T,$ issued with principal $ N$ normalized at $ 1$.

Let $ F_{t}$ be the proportion of unprepaid shares at date $ t$. In the optimal approach, the prepayment policy follows an ``all or nothing'' type strategy, with $ F_{t}$ being worth 0 or 1. When practical policies are involved, $ F_{t}$ is a positive process decreasing in time from 1 to 0. One can look for

$\displaystyle \left. \begin{tabular}{l} $F_{t}\stackrel{{\mathrm{def}}}{=}\exp \left( -\Pi _{t}\right) $\ \\ $F_{0}=1,$\end{tabular} \right.$    

where, in probabilistic terms, $ \Pi _{t}$ is the hazard process associated with the refunding dynamics. The size of the underlying mortgage gives incentives to model $ \Pi _{t}$ as an absolutely continuous process. In mathematical terms, this amounts to assuming the existence of an intensity process $ \pi _{t}$ such that

$\displaystyle d\Pi _{t}=\pi _{t}dt,$    

or equivalently

$\displaystyle F_{t}=\exp \left( -\int_{0}^{t}\pi _{u}du\right) .$ (9.11)

In this framework, the main point lies in the functional form of the refunding intensity $ \pi _{t}.$ As it will be precised in the next subsection, $ \pi _{t}$ must be seen as a function of $ d_{t}$ instead of directly $ r_{t}.$ The valuation consists in discounting a continuous sequence of cash flows. Given the prepayment policy $ \pi _{t},$ the MBS cashflows during $ \left[ t,T\right] $ can be divided in two parts. Firstly,

$\displaystyle \int\limits_{t}^{T}\exp \left( -\int_{t}^{s}r_{u}du\right) F_{s}\phi ds$    

is the discounted value of the continuous flows $ \phi$ refunded on the outstanding MBS principal $ F_{s}$. Secondly,

$\displaystyle \int\limits_{t}^{T}\exp \left( -\int_{t}^{s}r_{u}du\right) \pi _{s}K\left( s\right) dF_{s}$    

is the discounted value of the principal prepaid at time $ s.$ The MBS value equals the risk neutral expectation of these cash flows

$\displaystyle P\left( t,r_{t},F_{t}\right) =\mathrm{E}\left[ \int\limits_{t}^{T...
...ht) \cdot \left\{ F_{s}\phi +\pi _{s}F_{s}K\left( s\right) \right\} ds\right] .$ (9.12)

Because $ \pi _{t}$ is chosen as a function of $ d_{t},$ the explicit computation of $ P$ involves the knowledge of $ r_{t}^{opt}$. As opposed to the classical approach, a simple Monte Carlo technique cannot do the job. $ P$ can be characterized as a solution of a standard two dimensional partial differential equation. In our one dimensional framework, this means that:

PROPOSITION 9.2   Under Assumption A, the MBS price $ P\left( t,r_{t},F_{t}\right) $ solves the partial differential equation
$\displaystyle \frac{\partial P\left( t,r,F\right) }{\partial t}+\mu \left( t,r\...
...al r}-\pi \left( t,r\right)
F \frac{\partial P\left( t,r,F\right) }{\partial F}$      
$\displaystyle +\frac{1}{2}\sigma^{2}\frac{\partial ^{2}P\left( t,r,F\right)}{\p...
... r^{2}}
+F(\phi +\pi \left( t,r\right) K\left( t\right) )
-rP\left(t,r,F\right)$ $\displaystyle =$ $\displaystyle 0.$ (9.13)
$\displaystyle %%
P\left( T,r,F\right)$ $\displaystyle =$ 0  

where $ \mu \left( t,r\right) \stackrel{{\mathrm{def}}}{=}\lambda \left\{ \theta \left(
t\right) -r\right\} $ and $ \sigma$ are fixed by assumption A and $ \pi _{t}$ has to be properly determined.


9.3.2 A Parametric Specification of the Prepayment Rate

We now come to a particular specification of $ \pi _{t}.$ For simplicity, we choose an ad hoc parametric form for $ \pi $ in order to analyze its main sensitivities. In accordance with stylized facts on prepayment, the prepayment rate $ \pi _{t}$ is split in two distinct components

$\displaystyle \pi _{t}=\pi _{t}^{S}+\pi _{t}^{R},$    

where $ \pi _{t}^{S}$ represents the structural component of prepayment and $ \pi _{t}^{R},$ as a function of $ d_{t},$ accounts for both the refunding decision and burnout.


9.3.2.1 Structural prepayment

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.

Figure 9.7: The 100% PSA curve.
\includegraphics[width=0.91\defpicwidth]{psacurve.ps}

The Public Securities Association (PSA) recommends the use of a piecewise linear structural prepayment rate:

$\displaystyle \pi _{t}^{S}=k\left( atI{\left( 0\leq t\leq 30\text{ months} \right) }+bI{\left( 30\text{ months}\leq t\right) }\right) .$ (9.14)

This piecewise linear specification takes into account the influence of the age of the mortgage on prepayment.

According to the PSA, the mean annualized values for $ a$ and $ b$ are $ 2\%$ and $ 6\%,$ respectively. This implies that the prepayment starts from $ 0\%$ at the issuance date of the mortgage, growing by $ 0.2\%$ per month during the first $ 30$ months, and being equal to $ 6\%$ afterwards. This curve is accepted by the market practice as the benchmark structural prepayment rate, see Figure 9.7. It is known as the $ 100\%$ PSA curve. The parameter $ k$ sets the desired translation level of this benchmark curve. The PSA regularly publishes statistics on the level of $ k$ according to the geographical region, in the US, of mortgage issuance.

Figure 9.8: Prepayment policy.
\includegraphics[width=0.98\defpicwidth]{prepaymentpolicy.ps}


9.3.2.2 Refinancing prepayment

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, $ r_{t}^{opt}.$ As soon as

$\displaystyle d_{t}>0,$    

if the mortgagors were optimal, the whole MBS principal would be prepaid. In order to reflect the effect of individual characteristics on prepayment rate causing dispersion around the optimal level $ d_{t}=0$, we introduce the standard Weibull cumulative distribution function

$\displaystyle \pi _{t}^{R}=\overline{\pi }\cdot \left[ 1-\exp \left\{ -\left( \dfrac{d_{t} }{\overline{d}}\right) ^{\alpha }\right\} \right] .$ (9.15)

We do not claim that this parametric form is better than other found in the literature. Its main advantage comes from the easy determination of parameters thanks to an analytic inversion of its quantile function. In fact, as suggested by Figure 9.8, the determination of quantile ensures that this parametric specification can easily be interpreted.

Parameter $ \overline{d}$ is a scale parameter. In this form, being far into the prepayment zone means that $ d_{t} / \overline{d} \gg 0$ so that $ \pi _{t}^{R}\sim \overline{\pi }.$ Parameter $ \overline{\pi }$ 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 $ r_{t}.$ More precisely, if $ r_{t}$ was to stay very low during a time period $ \left[ 0,h\right] $ and if the refinancing prepayment was the only prepayment component to be considered, using expression (9.11), the proportion of unprepaid shares at date $ h$ would be equal to

$\displaystyle F_{h}=\exp \left( -\bar{\pi}h\right) .$    

This proportion is the burnout rate during the time horizon $ h.$ Parameter $ \alpha $ controls the speed at which prepayment is made, linking the PSA regime to the burnout regime.

Figure 9.9: The relation between MBS and prepayment policy: MBS without prepayment (solid line), mortgage with prepayment (dashed line), and MBS (dotted line).
\includegraphics[width=0.89\defpicwidth]{priceMBS.ps}


9.3.3 Sensitivity Analysis

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 $ h$ for this burnout effect is fixed equal to 2 years. Parameters $ \overline{d}$ and $ \alpha $ are calibrated in such a way that when $ d_{t}=0,$ ten percent of mortgagors prepay their loan after horizon $ h$, 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 $ 2.3\%$ and the long term rate is $ 5\%$. The volatility of the short rate $ \sigma$ is taken equal to $ 0.8\%$ and $ \lambda $ is such that the volatility of the 10 year forward rate equals to 0.5%. The facial coupon of the pool of mortgage is $ c=5\%,$ its remaining maturity is set to $ T=15$ years and no prepayment has been made ($ F_{0}=1).$

Figure 9.10: Embedded option price in MBS for a steeper forward-rate curve (dotted line) and a less steep forward-rate curve (solid line).
\includegraphics[width=0.91\defpicwidth]{optionpriceMBS.ps}

Figure 9.11: Duration of the MBS: MBS without prepayment (solid line), mortgage with prepayment (dashed line), and MBS (dotted line).
\includegraphics[width=0.91\defpicwidth]{durationMBS.ps}

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 $ r_{t}=0$ is controlled by the burnout level. The transition part is controlled by parameters $ \overline{d}$ and $ \alpha.$ 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.

Figure 9.12: The sensitivity of the MBS price to interest-rates volatility: volatilities of the 1-year and 10-year bonds are 90 bps and 37 bps (solid line) and 135 bps and 55 bps (dotted line), respectively.
\includegraphics[width=0.91\defpicwidth]{priceoptionMBSsensivol.ps}

Figure 9.13: The sensitivity of the MBS duration to interest-rates volatility: volatilities of the 1-year and 10-year bonds are 90 bps and 37 bps (solid line) and 135 bps and 55 bps (dotted line), respectively.
\includegraphics[width=0.91\defpicwidth]{durationMBSsensiv.ps}

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.