Figure 1, showing the case of a conventional mortgage, is very simple since we assume no bankruptcy. The g(V) function is just a straight line with a slope of one, and passes through the horizontal axis at point L If the final value of the home when it is sold equals L, then the homeowners get nothing at time of sale; the sales price is just enough to pay off the mortgage. If the value of the home should become less than the loan balance, then the homeowners have negative equity. In fact, some homeowners will declare bankruptcy under this circumstance, and so one might show the line as having a slope of less than one where V is less than L. Moreover, in nonrecourse states such as California homeowners can walk away from negative equity without bankruptcy. We disregard this complication here to provide a simpler contrast between the different risk management forms.
Figure 2 shows the case of a reverse mortgage in which the homeowners are loaned an amount which will be, when it is sold, 80% of the value of the home today. A reasonable story to tell about this case is that the homeowners are elderly, already own the home, and are using the proceeds of the loan to buy a lifetime annuity which will be consumed. The elderly homeowners will stay in the home until death, and the proceeds of the final sale, if any, will go to heirs. If the home is worth at the time of final sale less than 80% of the initial value of the home, then the heirs get nothing, hence the horizontal slope of the £(V) to the left of L Otherwise, they receive the value of the home minus the loan balance, and so the g(V) curve attains a slope of 1 to the right of L.
Note that Figure 2 is identical to the familiar plot, from elementary finance textbooks, of the payout of a call option on the exercise date as a function of the price of the underlying. Indeed, a home with a reverse mortgage does work out to be essentially such a call option.