9.1 Introduction

Mortgage backed securities (MBS) are financial assets backed by a pool of mortgages. Investors buy a part of the pool's principal and receive the corresponding mortgages cash flows. The pooled mortgages generally offer the borrower the opportunity to prepay part or all of the remaining principal before maturity. This prepayment policy is the key point for pricing and hedging MBS.

In the existing literature, two broad directions have been explored. On the one hand, the mainstream approach relies on statistical inference. The observed prepayment policy is statistically explained by the level of interest rates and some parameters of the underlying mortgage portfolio, see Schwartz and Torous (1989), Boudhouk et al. (1997). Dedicated to pricing and hedging, these approaches do not address the rationality behind the observed prepayment policy.

On the other hand, authors like Nielsen and Poulsen (2002) directly address the problem of optimal prepayment within consumption based models. This normative approach gives insights into the determinants of prepayments and relies on macro-economic variables. However, this approach appears to be of poor practical use due to the numerous economic variables involved.

In this chapter, we propose a third way. The optimality problem is addressed from an unconstrained, financial point of view. Using arguments similar to those of early exercise of American derivatives, we identify the optimal interest rate level for prepayment. Building on this frontier, we construct a family of prepayment policies based on the spread between interest rates and the optimal prepayment level. The MBS are then priced as the expected value of their forthcoming discounted cash flows, which is in line with classical methodology for flow product valuation.

Mortgages specific characteristics

Mortgage cash flows differ from those of a classical bond since their coupon is partly made of interest and partly of principal refunding. Despite this difference in cash flow structure, the prepayment option enclosed in the mortgage is very similar to the callability feature of a bond. Under classical assumptions on the bond market, an optimal time of early exercise can be exhibited, depending on the term structure and on the volatility of interest rates. Such models predict a rise in exercise probability during low interest rate periods, increasing the value of the callability option attached to the bond. These conclusions are supported by empirical evidence. Historical values of the market price of a non-callable and a callable General Electric bond with the same maturity and coupon are displayed on Figure 9.1.

Figure 9.1: Historical prices of the 10-year US goverment bond (solid line, right axis) and a non-callable (dotted line, left axis) and a callable (dashed line, left axis) General Electric bonds.
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The 10-year US government rate is displayed on the secondary axis. During this period of a sharp decrease of interest rates, the value of the non-callable bond rose much more than the value of the callable one. It may be tempting to adapt callable bonds pricing framework to mortgages. Nevertheless, statistical results prevent such a direct extrapolation. Though most mortgagors prepay for low interest rate levels, a significant percentage chooses to go on refunding their loan, no matter how interesting the refinancing conditions are. This phenomenon is often called burnout, Schwartz and Torous (1989). Conversely, some mortgagors choose to exercise their prepayment right at high interest rate levels. Such observations reveal that mortgagors are individuals whose behavior is in part determined by exogenous factors.

Economic studies suggest that major motivations for early prepayment can be classified within three broad categories, Hayre (1999):

Based on these considerations, the subsequent analysis is divided into three parts. Section 2 is concerned with the determination of the optimal time for prepaying a mortgage in an ideal market where interest rates would be the only variable of decision. This section sheds light on the influence of interest rates on refinancing incentive. In Section 3, the MBS price is expressed as the expected value of its future cash-flows, under some prepayment policy. A numerical procedure based on the resolution of a two-dimensional partial differential equation is put forward. The insights provided by our approach are illustrated through numerical examples.