## Payment scheduling and mortgage contract heterogeneity

Chapter 4 considered the tilting of mortgage payments towards the early life of the mortgage. In particular the constant payment mortgage would not necessarily reflect the desired or optimum payment schedule for borrowers. This would certainly be true of households who expected their incomes to rise. Thus an optimal mortgage instrument would better align payment profiles with income expectations. The issue of optimum mortgage design also involves interest rate risk, and the sharing of risk between the borrower and the lender. An initially lower payment on a mortgage may better suit those with expectations of growing income, but they may have to bear more, or all, of the risk of adverse change in interest rates (Dokko & Edelstein 1991; Brueckner 1993). A fixed rate constant nominal payment mortgage would impose interest rate risk on the lender, but at the possible expense of a less desirable payment profile for the borrower. These issues offer one explanation for the heterogeneity of mortgage instruments and present a rationale for the ARM (Brueckner 1993).

Lenders are usually large financial institutions with the capacity to diversify, hedge against risk, obtain access to futures and options contracts and interest rate swaps. For these reasons lenders might be expected to be risk-neutral. This means that they will be interested in the size of the cash flows rather than the risk of these flows. Baesel & Biger (1980) suggest that the often limited diversification opportunities open to borrowers makes it more likely that they will be risk averse. This leads to the paradox noted by Brueckner (1993), that theory would indicate borrowers having a preference for the FRM, that is shifting interest rate risk onto lenders, yet we observe consumers choosing ARMs. In theory therefore, the ARM is a suboptimal contract and ought not to be observed. Why then have ARMs been so prevalent in the US mortgage market (or VRMs in the UK)?

Dokko & Edelstein (1991) explored the idea of an optimal mortgage design and presented a single-period theoretical model. The model demonstrated that if both borrowers and lenders were risk averse, and if borrowers behaved according to the precepts of a von-Neumann & Morgernstern utility function, then it would not be optimal for the borrower to take out 100% interest rate protection (that is a fixed rate mortgage). Subsequent research has used two-period frameworks, which allow the simultaneous examination of optimum payment profiles and interest rate risk.

Arvan & Brueckner (1986) develop a two-period model of optimum mortgage design where there is no constraint on the payment profile. Their work assumed risk-neutral borrowers. The Arvan & Brueckner model demonstrated that when the borrower was more impatient than the lender, then the graduated payment mortgage was the optimum mortgage instrument. It is instructive to examine briefly the framework of this model.

The borrower's utility function is given by expression (7.1), with V(-) representing the general form of the utility function. The argument y is income which is constant between the two periods, l is the discount rate and r0 and r1 are the prevailing interest rate (cost of lenders funds) in period 1 and period 2 respectiveley. The argument i0 is the first period payment by the borrower. The borrower's utility is a function of the difference between income and the mortgage payment in the first period, and the discounted value of this difference in the second period. The complication is that the interest rate outcome in period two is stochastic, hence the integral which represents a probability density function. There also has to be some rule for determining the interest charged in the second period, and this is represented by R(r1). The stochastic outcome and the pricing rule determine the second period mortgage payment. Thus utility is derived from the borrower's residual income after recognising the uncertain nature of interest rate outcomes and the lender's pricing policy.

The constraint is the lender's profit function given by expression (7.2). If a lender is risk-neutral then he or she will only be concerned with the cash flow arising from the loan. Of course, cash flows in the second period will still be discounted, hence the lender's discount rate, z. Now the lender's income arises from the difference between the cost of funds and the mortgage payment (we are assuming an interest only mortgage). The payment received by the lender in the second period is a function of the random draw and the pricing rule or loan function applied. Assuming a perfectly competitive market for loans then the lender will operate under the zero profit constraint discussed in Chapter 5. The optimisation problem is to find the first period payment and a loan rate function which maximises the borrower's utility, subject to the zero profit constraint. The solution to this optimisation problem depends upon the borrower's impatience relative to the lender's.2 In the case where the borrower is the more impatient then the optimum mortgage instrument is a graduated payment mortgage.

The problem is that the GPM is not always a prevalent mortgage instrument. The key to the analysis is the loan rate function. Brueckner (1993) argues that borrowers will seek their optimum payment profile by finding mortgage instruments whose loan functions best suit their optimum choice. It is assumed that this loan rate function is both linear, and held to consistantly between the two periods. The borrower chooses the best fitting combination of payment profile and interest rate risk.3 For example, if the term structure of interest rates is upward sloping then the expectation is that the cost of funds will be higher in the second period. This will suit a borrower who is impatient and desires lower initial payments.

However, this payment profile is purchased at the risk of a high draw from the interest rate distribution in the second period. There will be some circumstances where the ARM provides a desireable combination of gradient on the payment profile, and acceptable interest rate risk. Hence the rationale for the existence and sometimes prevalence of the adjustable rate mortgage.

It is useful to look briefly at the loan rate function, as it is suggestive of some other important characteristics of the mortgage market. For example, some empirical work for the United Kingdom has suggested that lags in pricing adjustment can influence borrowers choices. Brueckner (1993) presents a nice example of a linear loan rate function, given in expression (7.3). The term R(r) is the loan rate function where r is the cost of funds, a is a constant indicating a given mark-up and b is a parameter determining the impact of changes in the cost of funds. In the case of the ARM the parameter value is b = 1 while for the FRM b = 0.4 While noting that this linear form is mathematically restrictive, Brueckner argues that it is probably a realistic representation of pricing.

Interestingly, the VRM in the UK has not always responded quickly to changes in the cost of funds (b < 1). The recent introduction of tracker mortgages means that there are now mortgage instruments where b = 1. Lags in interest rate adjustment are not insignificant and mean that variable rate debt can sometimes behave like fixed rate mortgages (see Miles 1992; Leece 2000a). The discussion of credit rationing in Chapter 5 noted how lags in interest rate adjustment reflect the structure of competition in the lending market (Heffernan, 1997). In the US adjustable rate mortgages are also known to have this element of fixity, through their varied speed of response with respect to the interest rate indices used to determine them (see Stanton & Wallace 1999). The possible empirical significance of interest rate adjustment lags for the choice of instrument is returned to in Chapter 8.

Table 7.1 presents the theoretical predictions which follow from Brueck-ner's analysis. The table expresses the decision in terms of the choice of an FRM compared to an ARM. A large variance in the cost of funds represents high interest rate risk and will encourage FRM take up. A normal yield curve will encourage the adoption of an ARM. In fact, a flattening yield curve will progressively discourage ARM take up as the benefits of a graduated payment profile fall. The theoretical predictions of other approaches to modelling the choice of mortgage instrument are also represented in Table 7.1 and are discussed in the appropriate sections below. (Note that some expected signs will differ from those in the published work which have the ARM as the dependent variable.) Though the models which focus upon interest rate risk and payment profiles do not incorporate prepayment or default risk, they offer some key theoretical insights that can be used to guide empirical specifications of choice of mortgage instrument equations.

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