The prepayment behaviour of the wealth maximising borrower
The option theoretic approach to mortgage valuation offers an explanation of prepayment behaviour. This behaviour results from breaching a boundary condition for the value of a risky mortgage involving those values of H and r which induce prepayment. This boundary is known as a free boundary because the borrower can prepay the debt at any time during the life of the current mortgage contract, and will do so depending upon the combination of r and H. This is the main reason why we work backwards to estimate the current mortgage value. The decision to prepay depends upon the future value of the mortgage so that working backwards allows the conditions when prepayment is likely to take place to be determined. This also has the important implication that the decision to prepay a mortgage depends not only upon whether it is currently 'in the money' but also upon the anticipated future values of H and r, and thus the value of the option to prepay.
The analysis in this, and the following section, proceeds on the assumption that prepayment and default are separate decisions. This is particularly useful when presenting diagrammatic explanations of either prepayment or default behaviour. We also retain the focus upon prepayment which takes place for financial reasons only. In other words, the prepayment decision is endogenously determined within the option based model. Such financial prepayment is compatible with the assumption that the borrower aims to maximise his or her wealth and will prepay when such behaviour is consistent with this objective. Other motives for both prepayment and default will be discussed in Chapter 10.
A call option on a mortgage can be said to be 'in the money' when the value of mortgage exceeds the outstanding balance on the debt (book value), as indicated in expression (9.1). The balance can also be considered the price that the borrower pays to exercise the option to prepay. There are two complications to this view of when a mortgage should be prepaid. It may not always maximise expected wealth to exercise the call option when it is 'in the money'. First, once the option is exercised then it cannot be used again, that is the option has value. Interest rates may fall further in the next period increasing the value of the current debt even further. Second, transaction costs may arise when prepaying the mortgage. Both of these features are readily incorporated into the analysis.
Expression (9.8) follows the logic of Follain et al. (1992)5 and indicates the inequality, which would trigger prepayment. This includes the value of the option to prepay C(r, k), that is we now use the arguments of the risky mortgage, and also include transaction costs TC. The outstanding mortgage balance is denoted by OB (or alternatively book value, BOOK). The borrower prepays when the gain on refinancing exceeds the transaction costs of prepayment plus the value of the call option treated as if it is a publicly traded security.
Green & LaCour Little (1999) make the interesting point that the loss of the value of the call option on prepayment is to some extent offset by the option acquired with the new mortgage. Assuming that these two options cancel out can be a useful device for empirical work where the focus can then be on the pure refinancing strategy subject to transaction costs.6 Yang & Maris (1996) also offer an interesting perspective by examining the effect of uncertainty regarding the holding period of the mortgage, that is the T in expression (9.8) is considered stochastic. The results of Yang & Maris suggest that the model with certainty underestimates the interest rate differential required to financially justify prepayment. These considerations will be important when we address the issue of why some households do not prepay their mortgage debt when the financial conditions suggest that it is optimal to do so. The question will be whether premature, delayed or non-occurring prepayments represent sub-optimal behaviour.
A diagrammatic treatment of the wealth maximising prepayment decision
Further insight into the decision rule on when to prepay can be gleaned from a diagrammatic treatment of the issue. Figure 9.1 follows Quigley & Van Order (1990) and illustrates the optimal prepayment rule. The dotted line shows the inverse relationship between the value of the non-callable
- Figure 9.1 Mortgage prepayment and the wealth maximising borrower.
bond component of the mortgage and the current interest rate. This would correspond to the value A(r, T) in expression (9.8). The line labelled Par is the par or book value of the mortgage. The mortgage is assumed to be of a given age to maturity. The curves Z, X and Y represent the relationship between the value of a callable mortgage, that is the risky mortgage, and the interest rate. The three curves represent different solutions to equation (9.7) (using just the one state variable r in this case). Adopting the local expectations hypothesis we can interpret changes in the interest rate as changes in the term structure.
There are an infinite number of valuation curves, satisfying equation (9.7). These possibilities reflect the number of ways that the coupon rate and risk adjusted capital gains can be combined to produce the risk-free rate of return. The conditions of the mortgage contract (e.g. the coupon rate) and the face value fix the valuation curve and provide an interior solution where the par or book value provides the boundary. The optimal interest rate r*, at which prepayment takes place is the point of tangency between the curve representing the value of the callable (risky) mortgage and the par value of the debt. Note that this equality implies that the market has already priced the value of the call option on the mortgage, that is it is presumed to be efficient from the borrower's point of view, no observable surplus accrues to the borrower at this point (Quigley & Van Order 1990).
The curve Z is relevant to the borrower who faces zero transaction costs. The curve X demonstrates the effect of transaction costs. From the borrower's perspective the par value of the debt they face is increased by transactions costs (C). The critical point now is the point of tangency between Y and (Par + C). Introducing transactions costs do not markedly shift the interest rate at which prepayment occurs (r*) and both points are examples of what is termed 'ruthless prepayment'. Quigley & Van Order note that the existence of transactions costs 'drive a wedge' between what the borrower pays and the lender receives. The curve Y takes the lender perspective whereby the lender only ever receives the par value of a loan at r* whereas the mortgage is worth more than par to the borrower.
Transaction costs will feature in the empirical analysis of prepayment behaviour. Bennett et al. (2000) used the idea of the vega threshold in their theoretical work on prepayment. The vega is a measure of the relationship between the option value and changes in interest rate volatility. The theory suggests that volatility has its greatest impact when an option is 'near the money' rather than 'in the money'. Transaction costs, which may in turn be related to individual and household characteristics shift the value of the vega and increase the optimum refinancing threshold.
The preceding discussion has provided the basis of the option theoretic approach to mortgage prepayment in perfectly competitive markets. Mortgage prepayment can be described as 'ruthless' and is determined endogen-ously by purely financial considerations. The contractual terms and the outstanding mortgage balance establish the boundary conditions which along with the valuation of the risky mortgage determine the optimum r at which prepayment will take place. However, prepayment is not automatic when a mortgage is 'in the money' because the option to prepay has value. This option will be correctly priced in a perfectly competitive and efficient capital market. The modelling of prepayment behaviour assumed no default. Default behaviour, however, is not trivial and the mortgage valuation literature has paid increasing theoretical and empirical attention to this phenomenon.
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