Finally, to compute the effects of the moral hazard on the investor, note that the sum of h(l) and the expected receipts Щ7) to the investor from loan balance and or sale of (share of) home is always equal to the expected value of the home ехр(ц + о112) +/(/) – D, and so IЦ/) = ехр(ц + o2/2) +/(/) -D-h(T). Thus, the expected shortfall s to the investor due to moral hazard as a function of the amount / that the homeowners choose to invest, as a fraction of the initial value of the home, is:
where /„ is the investment that the homeowner would make in the home if there were no moral hazard. After computing the amount / that the rational homeowner will invest in the home, we can use this expression to compute the expected shortfall. To translate this into a fraction of the amount invested, we must divide s by L for reverse mortgages, by the amounted loaned for shared appreciation mortgages, and by (1 – a) for housing partnerships, shared equity mortgages and sale of remainder.
Calibration of Model
For purpose of simulating our model, we must calibrate the parameters. For the distribution of V, we use the model of home prices presented in Case and Shiller (1987, 1988), and the assumption that the home sale is 8 years after purchase. The Case-Shiller model of home prices was that home prices are driven by three factors, city-wide factors, home specific (including neighborhood factors) and a time of sale noise factor. The first of these two factors were assumed to be lognormal random walks, and the third was assumed to be a serially independent lognormal variable. Averaging over the four cities, we found that average standard deviation of the quarterly change in the city wide variation in log price indices was oc = 2.52%. Again averaging over the four cities, we found that the estimated standard deviation of the quarterly change in the home-specific or neighborhood variation in log price was 3.31% per quarter. The standard deviation of the time of sale noise was oN = 6.37%. For sales 8 years (32 quarters) apart, this implies that the standard deviation о of the change in log price is (32)(ac + oH) + 2oN) = 25.20%. We assume that the mean change in log price \i equals zero for this calibration, reflecting a low inflation environment, though clearly other assumptions could be entertained.