# MORAL HAZARD IN HOME EQUITY CONVERSION: A Model of Moral Hazard for These Home Equity Conversion Forms

Let us now suppose that there is a production function/(/) that converts investment expenditure I into home value.2 We interpret the term I very broadly as representing all costly activities that increase home value, not only the maintenance and improvements but also the efforts to get a good price at resale and the moral effort to refrain from under-the-table transactions. The homeowners attach weight a, 0 £ a £ 1 to the resale value, and (1 – a) to the value in use to the homeowners before sale.3 The expected profit for the homeowners in the absence of any risk sharing is given by:

where V is the random (unknown when the investment I is made) value of the home if there is no depreciation in it, random due to changing market conditions, D is the depreciation on the home that would occur if there were no investment made at all in value-preserving or value-increasing activities, and E is the expectations operator. We will assume the usual properties for the production function, that/'(0) = «\ and that/'(7) > 0 and/”(/) < 0 for / > 0. Profit maximizing homeowners will set the derivative of the profit function to zero, and so.

Suppose now that the homeowners enter into a risk-sharing contract so that the homeowners receive on sale instead g(V – D + /(/)) of the resale value of the home, where g is one of the functions shown in Figures 1 through 4. Then the profit function to the homeowners is:

where h(I), the expected homeowner portfolio value on resale as a function of /, equals Eg(V – D + /(/)) and E is the expectations operator. The function h(J) will be central to our analysis below. The differences between the home equity conversion forms, both for homeowner investment incentives and for expected losses to investors, can be summarized in terms of differences in the k(I) functions. The first order condition for maximal profit is then:

We now derive expressions for h(T) to allow us to use equation (4) to gain some perspectives on the effects of the home equity conversion on the investments that homeowners make in their homes. To do this, we make use of the functional forms for g that are shown in Figures 1 through 4. We need also to make some assumptions about the distribution of V and about the production function/(/).

Since our model uses a lognormality assumption for V, it is straightforward to derive expressions for h(F). The function g(-) shown in the figures has the simple form of a straight line or a broken straight line, with a break at V = L = 0.8 (the loan to initial value ratio) for the reverse mortgage (home equity insurance) example (Figure 2) and with a break at V = V0 = 1 (the price from which appreciation is measured as a fraction of initial value) for the shared appreciation mortgage example (Figure 3). We derive that the expression for a conventional mortgage, corresponding to Figure 1, is:

The expression for a reverse mortgage or home insured by home equity insurance, corresponding to Figure 2, is, if f{I) – D is less than L:

where N(-) is the cumulative normal distribution function. If /(7) – D is greater than or equal to L, then expression (5) applies.