Rationales of mortgage insurance premium structures

Journal of Real Estate Research, The,  1997  by Barry Dennis,  Chionglong Kuo,  Tyler T Yang

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Comparison of Alternative Premium Structures-Algebraic Approach

Given a premium structure discussed in the above section, the ex-post premium payment made by a borrower depends upon the premium structure and the behavior of the borrower. We first compare the ex-post payments for borrowers under the same premium structure but with different behaviors. Then we compare the ex-post payments for one borrower under different premium structures. The implication of this analysis is that borrowers with different expectations of mobility and default will have different preferences for specific premium structures.

We compare the ex-post payment for three types of borrowers: borrowers who hold the mortgage until maturity (accumulated premium payments denoted by NDNP), borrowers who default at time t and s (accumulated premium payments denoted by DFt and DFs, t > s), and borrowers who prepay at time t and s (accumulated premium payments denoted by PPt and PPs).

Comparison within the Same Premium Structure

Case 1: Annual Premium Only. Since there is no refund, borrowers who default and prepay at the same time pay the same insurance premiums. The earlier borrowers terminate the mortgage, the smaller the premium payments are. Thus the borrowers who hold the mortgage until the maturity date pay the highest premiums. Case 2: Upfront Premium Only, No Refund. Because there is no refund and no annual premiums, the premium payments for all types of borrowers are the same.

Case 3: Upfront Premium with Refund, No Annual Premiums. In the upfront with refund case, defaulters pay the same premiums as borrowers holding the mortgage to maturity because neither receives a refund. The premium payments for prepayers are less than those for the defaulters, and the earlier the borrowers prepay, the lower are the total payments.

Case 4: Upfront Premium with Financing, No Refund, No Annual Premiums. In the upfront with financing case, defaulters pay the least amount of premiums and prepayers pay the same amount of premiums as borrowers holding the mortgages to maturity. The earlier a borrower defaults, the less the total premium payment is. Comparison of Alternative Premium Structures

The comparison of the present value of ex-post premium payments is based upon the premium schedules obtained from a set of default and prepayment probabilities and a fixed gross margin. We separately compare payments for defaulters, prepayers and borrowers holding mortgages to maturity. We use superscripts AN UF, RF, FG to represent Cases 14, respectively. For example, DF,UF represents the ex-post payment for time t defaulters under the upfront premium structure (Case 2). The definitions and abbreviated names for these four cases are summarized as below:

Borrowers Holding Mortgages to Maturity. For borrowers holding mortgages to maturity, the payment in the Upfront case is lower than that in the Annual case because in the Annual case both defaulters and prepayers are paying less than borrowers holding the mortgages to term. In comparison with the Upfront case, under the Annual case borrowers holding the mortgages to maturity, as well as the late defaulters and prepayers, must pay higher premiums to make up for the lower premiums paid by the early defaulters and prepayers. The premium payment in the Refund case is greater than that in the Upfront case because the Refund case prepayers are refunded for the unused premiums while the Upfront case prepayers are not. The premium payments made by early defaulters are lower in the Annual case than those in the Refund case, implying that borrowers holding mortgages to maturity will pay more in the Annual case than under the Refund case. The relationship of the accumulated payments for borrowers holding mortgages to maturity is: