The valuation of a mortgage contract can be seen as the value of three different forms of security. The actual contract terms, and the discounted cash flows to the lender (using the current market rate of interest) can be represented as a non-callable bond. However, the borrower has both the option to prepay or to default on the mortgage. The option to default is a put option involving the possibility of selling the property back to the lender to repay the outstanding debt. The option to prepay is a call option involving the possibility of buying back the outstanding mortgage balance by prepaying the debt early. Consequently, the value of a mortgage is a composite of the non-callable bond, the call and the put option. An example of a mortgage option having value is a fixed rate mortgage which looks expensive at current interest rates. This makes prepayment look desirable, but if this was done then the option to default and the chance of refinancing under even more favourable terms would be lost.

The modelling presented in this chapter is based upon the idea that the household maximises its wealth by minimising the value of its mortgage debt. So what is meant by the value of a mortgage? Take the example of a household with a fixed rate contract. The cash flows on this contract can be discounted at the current market rate of interest. Assume that the current mortgage rate is lower than the contract rate. Discounting the mortgage payments at the current mortgage rate, over its expected life, will increase the present value of the cash flows. A wealth maximising mortgage holder will be tempted to quit this contract and adopt an alternative lower value mortgage. Remember that prepayment might not take place depending upon the values of the embedded call and put options. The value of these options reduces the value of the mortgage to the borrower. That is, they are benefits that reduce the liability.

If the fixed rate on the mortgage is equal to the current market rate of interest then the discounted present value of future cash flows on that contract will equal the amount borrowed, or book value. Thus mortgage valuation also offers an implicit pricing formulae for a mortgage, that is setting a price where the value of the debt equals the current loan balance.

This equality requires that the term structure of interest rates is flat, that is there is no expectation of increases in interest rates and no premium for this in the price. Given this, then equation (9.1) represents the net value of a mortgage NV(r, k). This is defined for a current market rate of interest r, and the time to maturity k. The net value is the difference between the cash flows due on the current contract discounted at the market rate of interest A(r, k) and the book value (BOOK). The wealth maximising household will wish to minimise this difference. The net value in equation (9.1) can be alternatively described as the extent to which the option to prepay is 'in the money', or the intrinsic value of the debt. However, we also need to consider the value of the options to prepay or default, together with transaction costs arising from mortgage termination.

Kau et al. (1992) present a nice general form for the effects of prepayment and default on gross mortgage value. Expression (9.2) represents the value of a mortgage in terms of the noted present value of the contractual payments on the mortgage A(r, k) discounted at the current spot rate of interest r, less a joint possibility of default or prepayment J(r, H, k). Equation (9.3) shows the joint probability to consist of the put option to default D(r, H, k) and the call option to prepay C(r, H, k). The arguments in parenthesis are the variables that determine the value of the non-callable bond and the options. These are the interest rate r, the time to maturity k, and an additional factor influencing the value of both the call and the put option, the house price H. Note that time is represented in a variety of ways in the literature, being the age of the mortgage, current point of time, time to maturity and time of maturity. For purposes of consistency the discussion in this chapter generally uses the time to maturity, k, to represent time, the other indicators being implicit in this measure.

The value of the mortgage given by V(r, H, k) in expression (9.2) can be described as the value of the risky mortgage, or a callable (and with Federal guarantees default free) bond. Clearly the value of the risky mortgage is a function of the stochastic behaviour of interest rates and house prices, that is, it is the stochastic behaviour of these variables that effects size of pay offs to the options. Having examined the components of the value of the risky mortgage, the discussion of prepayment and default, later in this chapter, will return to the kind of comparison made in expression (9.1). The contractual terms of the mortgage contract determine the boundary conditions for any option valuation model.

There are two important things to note regarding the general expression for the mortgage value given in expression (9.2). First, prepayment and default decisions are determined endogenously by the variables r, k and H.

This means that prepayment and default are treated as functions of these financial variables. However, some prepayment and default can arise out of exogenous effects, for example, moving house when the mortgage is not assumable will generate a prepayment. Equally, adverse life events such as unemployment may induce default. For the moment such factors are ignored and the focus is upon the financial calculations implied by the option theoretic approach.

The second point to note is that the prepayment and default decisions are interdependent. For example, exercising the default option precludes exercising the prepayment option. This is the reason for including house price behaviour in the prepayment function, that is at very low house prices default will dominate. In empirical work this interdependence has been recognised by the estimation of so called competing risk models, a topic to be considered in Chapter 10. The links between the prepayment and default decision are discussed below.

Of course, the valuation of mortgage-backed securities depends upon the valuation of the underlying risky asset (the mortgage). The value of this pass through security is modelled in the same way as the underlying mortgage. However, the cash flows to the pass through security holder are different. Pay outs on the MBS must reflect the fact that mortgage terminations lead to a return of the outstanding balance. It is worth noting that from the MBS security holders' perspective default and prepayment has the same effect, that is they are both terminations of cash flows. The main point is that the valuation of the MBS depends upon the prepayment and default behaviour of the underlying mortgage holders (Schwartz & Torous 1992). There are a number of alternative contingent claims models for valuing residential mortgages and pass through securities. These mainly differ in the number of variables used. Chatterjee et al. (1998) find models with the two variables, short rate of interest and building value, to be the most efficient. The discussion which follows uses these two state variables.

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