The Term Structure of Mortgage Rates

Abstract

M ortgage valuations require the calculation of projected prepayment speeds along given interest rate paths. These paths are typically of benchmark rates such as swaps or Treasuries. However, prepayments are determined by primary market mortgage rates, as these are what the borrower sees. Given a certain path for swap or Treasury rates, how do we obtain a corresponding path for mortgage rates? Constant Spread to the 10-Year Mortgage Citigroup MOATS ModelTo overcome the limitations of these traditional approaches, Citigroup developed a new methodology called the Mortgage Option-Adjusted Term Structure (MOATS) model. 2 The MOATS model assumes a constant OAS for the current-coupon mortgagebacked security (MBS) at all times and across all interest rate paths. From this assumption, it calculates the current-coupon yield at all nodes.

Computing the endogenous mortgage rate without iterations

“…Finding a level curve requires computation of only two conditional expectations, and the number of level curves equals the number of points on the mortgage rate grid, which is one-dimensional. This is in sharp contrast to the existing algorithm proposed in [3] and [6] where the number of expectations to be computed is the number of points needed to reconstruct the mortgage rate function (surface) over the domain of state factors and is a number growing exponentially with the dimensionality of the problem (the complexity of the Citigroup's MOATS methodology [1] is of the same order).…”

“…The solution of this equation is a manifold l 1 of A [1](m 1 , x) as function of x. We will refer to this solution as a "level curve", i.e., as if p = 2.…”

“…In the current set-up m 0 is a solution to the scalar equation m 0 = lim r→0+ A [1](m 0 , r), where "1" stands for "no refinancing" condition. 7 Now, for every m n we find an appropriate interest rate value r n in turn.…”

“…To illustrate our basic idea in the simplest case possible, in this section we assume that the only stochastic process in the model is the instantaneous interest rate r t which is a one-factor diffusion process with r t ∈ D = (0, ∞). In this simple interest rate environment the whole 6 A generalized (with the interest rate tree propagated to 30 × (n + 1) years) version of MOATS mortgage rate model in [1] can be written as iterations mn+1(x) = A[mn(·)](mn+1(x), x), where m0(x) is taken as a solution to an auxilary mortgage rate problem over a [30 × n, 30 × (n + 1)]-year interval where mortgages originated at time t ∈ [30 × n, 30 × (n + 1)] have maturity of 30 × (n + 1) − t years. yield curve is determined by the value of r t , and thus the mortgage rate process m t is merely a function of the interest rate r t , i.e., m t = m(r t ).…”

“…Goncharov, Okten, and Shah [6] applied randomized quasi Monte Carlo method for the mortgage rate computation. Bhattacharjee and Hayre [1] presented a mortgage rate model (called MOATS model) which in fact is a version of the iteration procedure of the fixed-point mortgage rate problem in Goncharov [4], [3] given a certain "educated" initial guess. This guess comes from solving an auxiliary mortgage problem over a [30,60]-year interval where mortgages originated at time t ∈ [30, 60] have maturity of 60 − t years.…”