Research using numerical simulation focuses upon the effects of changes in the state variables, and their volatilities, upon the value of the risky mortgage debt (Kau & Keenan 1995; Pereira et al. 2002). This involves analysis of the different components of the risky mortgage, that is the embedded options and the value of the cash flows on the debt. Default and prepayment are seen as a joint probability of mortgage termination. A related consideration, and the main focus of empirical work, is the likely rates of default and prepayment. Numerical analysis offers insights into interactions between prepayment and default value, and thus mortgage termination behaviour (Schwartz & Torous 1992).

It is necessary to examine the relationship between each of the components of mortgage value, including the value of the non-callable bond (discounted payments). These in turn determine the overall value of the risky debt which is a key element in the option to prepay or default (see Figure 9.1 and Figure 9.2). Thus the interactions can be quite complex and ultimately have a bearing upon the fair pricing of a mortgage and the valuation of insurance; default insurance in the US (Kau et al. 1993) and mortgage indemnity guarantees in the UK (Pereira et al. 2002, 2003).

It is useful to think in terms of different regions, delineated by combinations of the interest rate and house price, were the values of one or the other of the options to terminate dominates. Correspondingly, the likelihood of observing prepayment or default behaviour will vary in interest rate/house price space (Deng et al. 2000). For example, numerical analysis reveals that at low interest rates and high house prices the value of the prepayment option is high (Pereira et al. 2002). This is because when house prices are high the value of the default option is low and default is less likely.

Schwartz & Torous (1992) show that at low interest rates it is possible to have a mortgage value which exceeds the value of the property, which suggests default, and a house value which is greater than the mortgage balance which would argue against default. So the household would not be facing negative equity but the interest rate charge on alternative debt would be so low that it raises the value of the current mortgage above the value of the property. In theory both the value of the mortgage and the balance must be lower than the value of the property (see Figure 9.2) before default occurs. When the constraints are inconsistent in this way prepayment will dominate. Schwartz & Torous also note that at low house prices default is likely to dominate prepayment, reducing the likelihood of prepayment to zero.

Changes in the economic environment are represented by changes in interest rate and house price volatility. Increased interest rate volatility, ceteris paribus, raises the value of the option to prepay. Increased house price volatility, ceteris paribus, raises the value of the option to default. Thus the effects on the value of the risky mortgage depend upon the impact of the respective volatilities on the value of the components of the joint option to terminate. The volatility of interest rates also effects the value of the cash flows attached to the debt. Higher volatility of interest rates should increase the value of the cash flows to the mortgage. This effect occurs because the relationship between the net present value of the fixed cash flows and the discount rate is convex, so that lower discount rate outcomes have a comparatively high positive impact on NPV. However, greater interest rate volatility reduces the value of the risky mortgage via the option values, reducing the likelihood of default and prepayment.7

Increased house price volatility has no impact upon the value of cash flows attributable to the mortgage. However, higher house price volatility increases the value of the option to default at the expense of the option to prepay. With greater volatility of house prices it is more likely that a change in the house price would reach the default region (Pereira et al. 2002). Consequently, the value of the risky mortgage will fall with increases in house price volatility. This in turn makes prepayment less likely. These effects and the other possibilities discussed above demonstrate the complexity and competing nature of the interactions between default and prepayment behaviour. However, the analysis so far has been in terms of a fixed rate mortgage, a key question is how far option theoretic arguments apply to mortgage designs other than the FRM?

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