Mortgage Rate Insurance Pricing under an Interest Rate Diffusion with Drift
This research was supported by a grant from the Financial Research Foundation of Canada. Computing resources were provided by Digital Equipment Corporation. The author thanks an anonymous referee for helpful comments.
The federal government of Canada has introduced a program whereby a mortgage borrower can purchase insurance which gives protection against interest rate rises at mortgage renewal. A diffusion model of interest rates incorporating a drift term is applied to the valuation of mortgages, and the resulting partial differential equation is solved numerically. The boundary conditions of this problem necessitate a novel solution method which is likely to have applications in other areas. Estimates of the appropriate net premium are given.
As a result of pressure from homeowners renewing mortgages at the high interest rates which prevailed in the early 1980s, the federal government of Canada introduced on March 1, 1984 the Mortgage Rate Protection Program. The Program is administered through Canada Mortgage and Housing Corporation, a federal agency. In Canada, residential mortgages are usually amortized over 25 years, but the mortgage is periodically renewed at prevailing interest rates. The inter-renewal period is usually one, two, three or five years.
Under the Mortgage Rate Protection Program, up to $70,000 of the principal at inception can be insured. For the rate rise a deductible of 2 percent per annum applies, and the portion of renewal mortgage payment resulting from an increase in rates of over 12 percent is not insured. The principal at the end of the insured period is calculated using the market rate rather than the rate reduced by the effect of the insurance. The insurance only pays 75 percent of the increase in monthly payments resulting from a rate rise and only covers payments over a period equal to the original inter-renewal term.
As an example(1) of the operation of the insurance, consider the case of an insured mortgage with an amortization period of 25 years and repayments at the end of each month, when mortgage rates have risen 5 percent in the five year period from the date of mortgage inception and insurance purchase. The situation is illustrated(2) as Table 1. The single premium paid for the insurance is 1.5 percent of insured principal, regardless of the inter-renewal term. Thus the premium structure takes no account of either the longer period of potential payout under the longer inter-renewal periods, or the higher probability that substantial rate increases will have occurred over a longer inter-renewal period. At mortgage renewal at time five years in the example it would be possible to again purchase the insurance to protect against higher monthly payments between times ten years and 15 years. However, the insurance would then be based on the 17.5 percent per annum market rate which prevails at time five years.
It should be noted that the insurance can be transferred to the purchaser if the house changes ownership. Thus, ownership transfer would not affect the value of the insurance. The possibility exists, but is not included in the model below, that a fall in house value could cause the mortgagor to abandon the house and correspondingly to give up any claim to benefits under the insurance. Price volatility extensive enough to lead to house abandonment is unusual in Canada, but to the extent that it occurs the insurance values found below should be regarded as upper limits. This consideration does not affect the result found that the premium charged is too high.
This study presents a method for calculating the value of the insurance. Work on the valuation of other types of mortgages by a different method without insurance has been reported by Dunn and McConnell, while the valuation of bond options has been studied by Dietrich-Campbell and Schwartz. …