PDF Ebook Optimal Mortgage Refinancing with Endogenous Mortgage Rates: an Intensity Based, Equilibrium Approach
While there is widespread agreement that the value of a mortgage contract subject to prepayment but not default risk should be given by an expectation of the present value of the cash flow, the devil is in the details. A wide variety of approaches have been considered, most of which are commonly classified into one of two categories. One kind of approach has been variously called a reduced form approach, an exogenous approach, an empirical approach, and an econometric approach. The basic idea is to build a stochastic model for interest rates and possibly other economic factors, and then add a statistical model describing how the mortgagor’s prepayment behavior depends on the factors. While such an overall model can be quite complicated, it is usually straightforward to use Monte Carlo simulation to estimate the expected value of the discounted cash flow. Some of the many papers in this category are by Schwartz and Torous [17], [18], Deng [1], Deng, Quigley, and Van Order [2], Gorovoi and Linetsky [7], Kariya and Kobayashi [9], Kariya, Pliska, and Ushiyama [10], and Kau, Keenan, and Smurov [11].
Of significance is that in some of this research, dating back at least to Schwartz and Torous [17], [18], it was recognized that the random time when a mortgagor prepays can be described with a hazard rate model, that is, the conditional rate of prepayment given the current state of any factors and no prepayment to date. Perhaps this development was inspired by the engineering literature on reliability theory, as the time of mortgage prepayment is clearly analogous to the failure time of a system. In any event, Schwartz and Torous [17] used this hazard rate viewpoint in conjunction with a two-factor model in order to present a partial differential equation for the value of a mortgage contract. Moreover, as will be seen in this paper, recent developments involving the hazard rate as a model of a default time in the credit risk literature lead to new results, involving intensity processes, for mortgage contract valuation.
The other main kind of approach for the valuation of mortgage contracts is called an option-based or a structural approach. The basic idea is to incorporate some kind of optimal behavior with respect to the mortgagor’s decision about when to refinance. Moreover, the way to do this is to appeal to some intuition based upon the theory of the optimal early exercise decision for American options, usually leading to a recursive valuation procedure that resembles the one used for the binomial option pricing model. For example, Kalotay, Yang, and Fabozzi [8] described the following procedure: first build an interest rate lattice, and then, starting with the final scheduled cash flows of the mortgage, work backwards through the lattice computing the mortgage’s value, comparing the value with no refinancing and the value of a newly refinanced mortgage, with the latter assumed to be par plus the refinancing cost. If the latter is less, then the value of the existing mortgage at the node is replaced by the value of the new mortgage. Some others who took an option-based approach are Dunn and McConnell [3], [4], Stanton [19], Nakagawa and Shouda [13], Stanton and Wallace [20], Dunn and Spatt [5], and Longstaff [12].
The latter three papers are noteworthy because, in contrast to all the other option-based papers which assumed mortgage rates are exogenous, Stanton and Wallace [20], Dunn and Spatt [5], and Longstaff [12] allowed for endogenous values of fixed rate mortgages. These authors studied discrete time, finite horizon models, with time equal to the age of the mortgage contract. The mortgage rates were computed recursively, much like the “binomial option pricing” procedure by Kalotay, Yang, and Fabozzi [8] that was described above. But the model assumptions made by Stanton and Wallace [20], Dunn and Spatt [5], and Longstaff [12] are unclear, due in part to the limited use of mathematics in their expositions. Suffice it to say that their models are significantly different from the one in this paper, as evidenced by the fact that their endogenous mortgage rates seem to depend upon the age of the mortgage contract.
As indicated above, the theory of hazard rates and intensity processes for modelling default times in the credit risk literature has advanced considerably in recent years. Since the time of a default is analogous to the time when a mortgage balance is prepaid, it was natural to translate some of the credit risk developments to mortgage valuation. This was recently accomplished by Goncharov [6], who worked entirely in a continuous time setting. In particular, he showed how to unify the reduced form and the option-based approaches, he derived some explicit formulas for a mortgage’s value, he derived a variety of partial differential equations useful for computing mortgage values, and he used an explicit valuation formula to provide a nonlinear equation for the endogenous mortgage rate.
This paper makes several contributions. First, some intensity based valuation results that Goncharov [6] derived for a continuous time environment are here derived for a discrete time financial market. In particular, with the mortgagor’s prepayment behavior described by a suitable intensity process and with exogenous mortgage rates, in Section 2 the value of the contract is derived in an explicit form that can be interpreted as the principal balance plus the value of a certain swap. This leads in Section 3 to a nonlinear equation for what the mortgage rate must be in a competitive market, and thus mortgage rates are endogenous and depend upon the mortgagor’s prepayment behavior. The complementary problem, where mortgage rates are exogenous and the mortgagor seeks the optimal refinancing strategy, is then solved in Section 4 via a Markov decision chain. Various theoretical results about computational algorithms and existence of solutions are included. The equilibrium problem, where the mortgagor is a representative agent in the economy who seeks the optimal refinancing strategy and where the mortgage rates are endogenous, is developed, solved, and analyzed in Section 5. In particular, the existence of an equilibrium solution is established. Section 6 provides a simple computational example that illustrates various theoretical points, although it is probably not realistic enough to draw conclusions about actual mortgage markets. Section 7 provides some concluding remarks, summarizing our main economic conclusions and suggesting some good directions for future research. Finally, an appendix contains technical derivations and proofs of many of the main results.
Contents
1 Introduction
2 Valuation of Mortgage Contracts
3 Endogenous Mortgage Rates
4 Optimal Refinancing
5 A Mortgage Market Equilibrium Problem
6 A Numerical Example
7 Concluding Remarks
8 Appendix
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