Of course, assuming a = 1, that the homeowners care only about resale value and not at all about value in use before sale, was rather extreme. If we instead take a = 0.5, meaning that the homeowners give equal weight to both, then we find that, except for conventional mortgages, they will invest more in the home. For reverse mortgages, our calibrated model shows that the homeowners will invest 18% of the value of the home, and thus, the home will lose 8% of its value on resale. One can visually verify, very roughly, this level of investment for a = 0.5 by using both Figures 1 a and 2a, by finding the level of investment where the average of the slopes of the two h(f) curves equals the slope of the 45 degree line. With this method, one can see visually why lowering a to 0.5 has such an effect on /: while the h(I) curve shown in Figure 2a is fairly linear and with slope less than one except at very low levels of /, the h(I) schedule shown in Figure la has a slope that is much greater than one even at levels of I approaching the optimal level of 0.25. With our assumption of an initial loan to value ratio of 0.8, and an eight-year standard deviation of housing prices of 25.2%, then expression (10) implies that lenders will, in our model, face an expected shortfall per dollar loaned, sfL, due to moral hazard of 2%. This shortfall may seem fairly small, but recall that it is an actual loss, on average, for each mortgage written, and this moral hazard loss is on top of all other losses. This loss will still have to be made up for by a higher mortgage rate, or other charge, if the lender is to break even, but such a higher rate may make the lender uncompetitive.
Whether the a = 1 case or the a = 0.5 case considered above is relevant is a matter of judgment. We think that it is plausible that the true value of a lies between these two extremes, and so our estimated expected shortfall lies between the two cases illustrated.
In the case of a shared appreciation mortgage, Figure 3 a, when a = 1 the homeowners optimally invest 14% of the value of the home, so that the home loses only about 13% of its value. Note that the h(T) curve in Figure 3a resembles that of the conventional mortgage in the low region, Figure la, so there is much less incentive to allow home value to drop sharply than there was in Figure 2a.
Failing to invest in the home will, in the shared appreciation mortgage case, push homeowners into a region of / where they are bearing almost all of the expected losses for failing to invest more. This explains why there is so much more investment in the a = 1 case for shared appreciation mortgages than for reverse mortgages. Lenders face a shortfall s as a fraction of the value of the home, from equation (10), of 4%. When a = .5, then the homeowners invest 18% in the home, just as in the case of reverse mortgages, and from equation (10), the shortfall s as a fraction of the value of the home is 2%. Again, not knowing a, we think it is plausible that it lies somewhere between 0.5 and 1.0, and so the expected shortfall should lie between 2% and 4% of the value of the home.