Evaluating the interest-rate risk of adjustable-rate mortgage loans

Journal of Real Estate Research, The,  1997  by Raymond Chiang,  Thomas F Gosnell,  Andrea J Heuson

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Measuring the Interest-Rate Risk of an Adjustable-Rate Mortgage Loan Given a change in the discount rate, the approximate price elasticity of a stream of fixed cash flows is measured by the security's duration (DUR), a statistic first proposed by Macauley (1938). While the original construct has since been enhanced by many authors, Bierwag, Kaufman and Toevs (1983), among others, note that Macauley's measure performs reasonably well when compared with more sophisticated formulations. Furthermore, as noted in Anderson, Barber and Chang (1993), the correlation between the loan rate and the market rate determines the applicability of the conventional (Macauley) duration formula to mortgage cash flows. Given the existence of a marketbased rate adjustment process, it is clear that conventional duration is a more appropriate choice for ARMs than for fixed-rate loans.

It is evident from (1) that duration is calculated by weighting each expected cash flow by the time it occurs, discounting each weighted cash flow to present value at the security's yield, and dividing the sum of the discounted cash flows by the security's price. The beauty of the statistic is that duration multiplied by a given change in yield provides a good approximation for the expected percentage price change or return on a security. As such, it is an absolute measure of interest-rate risk that can be used to compare the price sensitivity of various contracts. The duration measure in (1) cannot be applied directly to adjustable-rate mortgage loans, however, because the C, in equation (1) varies over time in an unknown fashion.

The Cash Flows on an ARM

Consider the rate adjustment process of an ARM with annual and lifetime caps. After the initial year, the mortgage rate can fall into one of three states each year: S1: the lifetime and annual caps are not binding; S2: the lifetime or annual cap is binding on the upside; or S3: the lifetime or annual cap is binding on the downside. The mortgage rate for the coming year is set by the current index level on each adjustment date. For each state, the rate for year t, R^sub t^, is:

where t- 1 refers to the previous time period, I^sub t-1^ is the index at the end of period t- 1, M is the margin, AC is the annual cap, and LC is the lifetime cap.4

Estimating the future values of the R^sub t^'s using equation (2) is not an easy task in a realm of interest-rate uncertainty. The difficulty arises from the recursive nature of the process in that year t's rate is conditional upon the mortgage rate from year t- 1, but the t- 1 rate is conditional upon t-2, etc. As a result, all the rates starting from the inception of the mortgage are relevant in determining Rt. The historical rates and the entire distribution of the index must be taken into consideration to find the anticipated mortgage rate for any given future period.

The terms on the right-hand side of equation (3) are the contribution to the conditional expected mortgage payment made by the different index rates when the caps are nonbinding (top), when the up cap is binding (middle), and when the down cap is binding (bottom). The last two terms are comparable to mortgage payments from fixed-rate mortgages because they represent instances where rate adjustment constraints hold.