If a – 1, profit maximization means finding the value / at which the h(I) curve is as high as possible relative to this 45 degree line, that is, where it has the same slope. From Figure la, it is apparent that the point of maximum corresponds to 0.25 in the conventional mortgage case. This is the case where, as shown above, the homeowner will invest a quarter of the value of the home in 8 yean, thereby maintaining the home at its original value. Note that at this point there still appears to be negative profits in this calibration exercise. This fact merely reflects the fact that we have ignored the service benefits that the homeowner obtains from the basic home, before investment, in calculating profit.

For the other three situations, the situations shown in Figures 2 through 4 and for which the h(I) schedules are shown in Figures 2a through 4a, the incentives for the homeowner are very different.

In fact, the maximized profit for Figure 2a if a = 1, the case of a reverse mortgage or home equity insurance, occurs where the homeowners invest only 2% of the value of the home, and the losses to the home will be catastrophic. The reader can check this result visually by noting the level of I in
Figure 2a where the slope of the h(I) schedule matches that of the 45 degree line shown there. This point on the figure is very close to the origin. In this case, the home will lose 36% of its value due to moral hazard. The lender faces an expected shortfall per dollar loaned, sIL, using expression (10), of 20.7% of the amount loaned. In other words, the lender who made a loan so that loan value on the final date was 80% of the initial value of the home would face expected losses due to moral hazard alone of 20.7% of all loans, not just defaulting loans. The figure can be this high because the probability is 70.5% that the home value will be inadequate to cover the loan balance fully, even though the loan to value ratio was 0.8. The reason for this catastrophic shortfall with the reverse mortgage is that if the home value V is sufficiently low the homeowners are protected against losses, and so do not care about them. While the homeowners do not know what V will be, they know at the time the investment I is made that it stands a good chance of being sufficiently low. While the h(I) curve in Figure 2a resembles that of Figure la for high /, it is critically different for the relevant regions, low values of I.