# Mortgage rate insurance pricing under an interest rate diffusion with drift.

Mortgage Rate Insurance Pricing under an Interest Rate Diffusion with
Drift

ACKNOWLEDGEMENTS

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.

ABSTRACT

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.

INTRODUCTION

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[14]. 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[7], while the valuation of bond options has been studied by Dietrich-Campbell and Schwartz[6]. Sharp[15] examines the 1 1/2 percent premium charged under the Mortgage Rate Protection Program in the light of the hypothetical historical payouts under the insurance had the Program been in effect since 1938, and notes that for the first 30 years of the period, no payouts would have occurred because of the relative stability of interest rates.

This study first considers the partial differential equation which results from the application of an interest rate diffusion model. Next the equation is transformed to a form suitable for numerical analysis and estimates of the volatility of rates in various historical periods are presented. Then the method of numerical solution of the partial differential equation is presented and the treatment of the boundary conditions is discussed. The numerical results are presented and some additional comments on the results are then made in the final section.

Diffusion Interest Rate Model Applied to Mortgages

The theoretical premium to be charged under the Mortgage Rate Protection Program depends, among other things, on the probability that a payout under the Program will occur. That is, the premium depends on the probability that interest rates will rise by at least the 2 percent per annum deductible between mortgage inception and renewal. A model of interest rates is required which takes account of the random, unpredictable "noise" fluctuations in rates. Following Brennan and Schwartz[4], this study adopts the Wiener process as a suitable tool, as described by Malliaris and Brock[11].

A Wiener process can be used to describe processes such as the movement of a single molecule under Brownian motion, and can be compared with a random walk where transitions or collisions occur continuously rather than at discrete times. A Wiener process shares with a random walk the property that the standard deviation of the distance from the initial position is proportional to the square root of the time over which the process has been followed. The common analogy to a random walk is the progress from lamp-post to lamp-post of a drunken pedestrian. At each lamp-post collision he or she has probability 0.5 of proceeding further north on the street and probability 0.5 of reversing direction and returning to the lamp-post to the south. Then using the binomial distribution, the standard deviation of the number of lamp-posts of "progress" after n collisions is [(nx0.5x0.5).sup.1/2]. In view of the 0.5 probabilities, the expected progress, taking account of signs, is zero. Correspondingly, for a standardized Wiener process with increase dz over a period dt, one has E(dz) = 0 and var (dz) = E([dz.sup.2]) = dt.

It is required to model [r.sub.i](t), the instantaneous expected rate of return at time t on an investment in a mortgage measured as a continuous rate.(3) Allowing for the possibility of a non-stochastic drift [Beta]([r.sub.i]) the model is adopted: (1) [dr.sub.i] = [Beta]([r.sub.i])dt + [Eta]([r.sub.i])dz Here [r.sub.i] is measured as a force of interest, and [Eta]([r.sub.i]) depends on the volatility assumed for [r.sub.i].

Now denote by V([r.sub.i],t) the market value of a mortgage with principal of $1 at inception. Ito's Lemma(4) can be interpreted as implying that because E([dz.sup.2]) = dt, it is necessary when performing manipulations intended to determine changes dV up to order dt to proceed to the second term in the Taylor series in the [r.sub.i] derivatives: (2) [Mathematical Expression Omitted]

It should be noted that for the purposes of this model it is assumed that the mortgage value V is a function only of the two variables [r.sub.i] and t. Thus other variables such as the long rate of interest and the rate of inflation are assumed to affect V([r.sub.i],t) only to the extent that they are reflected in [r.sub.i](t).

The monthly payments by the mortgage borrower are approximated as being paid continuously at a rate c per annum per dollar of initial principal. Measuring time t from mortgage inception and denoting by [r.sub.c](t) the continuous market rate at inception of a mortgage of inter-renewal period m years and amortization period n years (that is [r.sub.c](t) is the rate used in calculation of the amount of periodic payments) yields:(5) (3) [Mathematical Expression Omitted] Then taking the expectation of equation (2), recalling that E(dz) = 0, adding the correction c for the continuous payments by the borrower and recalling that by definition of [r.sub.i], [Vr.sub.i] must equal the expected instantaneous return it (4) [Mathematical Expression Omitted]

The boundary conditions on (4) are important. As [r.sub.i] approaches infinity the discounted value of the payment streams must approach 0, hence (5) [Mathematical Expression Omitted] In order that the process (1) be constrained to produce only non-negative rates [r.sub.i](t), it is necessary that [Eta](0) = 0. Thus by substituting [r.sub.i] = 0 in equation (4) a natural boundary condition is obtained: (6) [Mathematical Expression Omitted]

The value of the mortgage rate insurance will be obtained as the difference of the time 0 values of two mortgages. Mortgage A is not insured and is renewed at time m, when its market value must equal the outstanding principal: (7) [Mathematical Expression Omitted] Mortgage B is insured, and for the purpose of calculation it is appropriate to assume that the mortgage lender is also the insurer. Thus the market value at inception of mortgage B will be less than that of mortgage A because if rates rise sufficiently between times 0 and m, the periodic mortgage payments between times m and 2m will be lower under mortgage B. Denoting by ([r^.sub.c]) the mortgage rate which under the insurance plan is the maximum that can be charged to the borrower for the period m to 2m, at time m the value of mortgage B is: (8) [Mathematical Expression Omitted] In equation (8), I([r.sub.i](m) denotes the value at time m of continuous payments of $1 per annum between times m and 2m, and it is multiplied by the annual amount [Mathematical Expression Omitted] which would be the borrower's payments per unit time m principal if the insurance became effective. P([r.sub.i](m)) is the value at time m of $1 principal payable at time 2m, and in equation (8) it is multiplied by the outstanding time 2m principal. The use of [r.sub.i](m) in calculating this outstanding principal takes account of the fact that the insurance limits only the periodic payments and not the outstanding principal at time 2m. If the interest rate at time m is insufficiently high to trigger a payout under the insurance, then the "1" in the "minimum" expression of equation (8) becomes the minimum of the pair.

From the mortgage borrower's point of view the purchase of the insurance is somewhat analogous to the purchase of a put option. If under the insurance the principal outstanding at time 2m were limited to that which would be calculated using the inception rate plus the deductible, then the analogy with a put option is straightforward. The security underlying the put option would be a contract to make a series of payments from time m to 2m and to pay the outstanding principal at time 2m, all calculated using the inception rate of mortgage interest plus the deductible. The exercise price would equal the principal outstanding at time m. However, the principal outstanding at time 2m actually depends, under the insurance, on the market mortgage rate at time m, so no such simple analogy with a well-defined put option can be made.

Preliminary Transformation and Data Analysis

In view of the difficulty in handling numerically the semi-infinite range of [r.sub.i], it is useful to use the transformation (9) [Mathematical Expression Omitted] and to define v(u,t) = [Mathematical Expression Omitted] = [Mathematical Expression Omitted] = [Eta]([r.sub.i]). Then the transformed version of the partial differential equation (4) for the mortgage value [Nu] can be verified by use of the chain rule to be (10) [Mathematical Expression Omitted] The boundary conditions (5) and (6) apply to both mortgage A and mortgage B and under the transformation they become respectively (11) v(0,t) = 0, 0 [is not equal to] t [is not equal to] m (12) [Mathematical Expression Omitted] The boundary conditions (7) and (8) remain effectively unchanged if one identifies [V.sub.A]([r.sub.i](m),m) with [V.sub.A](u(m),m) and [V.sub.B]([r.sub.i](m),m) with [Nu.sub.B](u(m),m).

Following Merton [12] the process (1) is specialized by making the assumptions [Beta]([r.sub.i]) = [Alpha] ([Gamma] - [r.sub.i]) and [Eta]([r.sub.i]) = [Sigma][r.sub.i], hence (13) [Mathematical Expression Omitted] The chosen form for [Eta] reflects the intuition, which is supported by the data, that changes in rates will be proportional to the existing rate [r.sub.i] (see e.g. [3]). The term [Alpha]([Gamma] - [r.sub.i]) adds to the model a tendency for the rate to drift towards the target [Gamma] at a rate which depends on [Alpha].

The values found for the insurance depend on the rate volatility parameter [Sigma], and on the drift parameters [Alpha] and [Gamma]. The process (1) is continuous, but Ananthanarayanan [1] has shown that parameter estimates found by directly using the discrete time observations correspond very closely to those found by analytic solution of the continuous stochastic process. Therefore monthly data were used for mortgage rates(6), and denoting by [Delta] [r.sub.c](k) = [r.sub.c](k + 1) - [r.sub.c](k) the month to month change in mortgage rates, an estimate [Sigma] of the volatility was made over the n month period as the sample standard deviation of the proportional rate increases neglecting the effect of the drift term, (14) [Mathematical Expression Omitted] It should be noted that the available data are for mortgage rates [r.sub.c] rather than the expected instantaneous rate of return [r.sub.i], but the difference between the two is relatively slight so the resulting values of [Sigma] are a reasonable guide.

Table 2 displays the calculated values of [Sigma^]. In view of the monthly differencing, the value for 1956-1960 for instance is calculated using data from December 1955 to December 1960 inclusive. As would be expected, the values for the volatility [Sigma^] are particularly high if periods which include the early 1980's are considered.

The estimation of [Alpha] and [Gamma] differs from that for [Sigma] in that several years of data are required if a relatively reliable estimate is to be made. Interest rate behavior differs dramatically between time periods as shown in Table 5. For example during 1977, rates moved within a 0.50 percent range, while during 1981 they moved within a 6.29 percent range. Thus, the estimation of [Alpha] and [Gamma] is very unreliable, and the approach adopted for this study is to present results for a range of values of [Alpha] and [Gamma]. The finding that the Mortgage Rate Protection Program is overpriced is shown to apply to any reasonable choice of parameters.

The parameter [Gamma] represents the drift target, and can be approximated as the long term forward rate on a mortgage investment. Mortgages with a fixed interest term of over five years are uncommon, so it is natural to turn to long bonds for guidance. Mortgages tend to yield more than bonds, and Boyle, Panjer and Sharp [2] indicate that over the period 1962-1986 the yield to maturity of a five year mortgage averaged 2.54 percent more than the yield to maturity on a 3-5 year government bond. Thus, a reasonable estimate of [Gamma] would be the long term forward rate on long bonds plus 2.54 percent. In the results presented in Tables 3 and 4, a range of values of [Gamma] is used which will bracket most such estimates. The parameter [Gamma] can be expected to have, for most historical periods, a value in the range 0.1 to 0.5 (see e.g. [1], [3] and [5]) and results are presented in Tables 3 and 4 for these two values.

Method of Numerical Solution

With the specializations of the stochastic process (1) through the use of equation (13), the partial differential equation (10) becomes (15) [Mathematical Expression Omitted] where c is given by equation (3) and the boundary conditions (7), (8), (11) and (12) apply. Boundary condition (12) simplifies to (16) [Mathematical Expression Omitted] Use is made of an interval h for the u co-ordinate and g for the t co-ordinate and of the notation [Nu](u,t) = [Nu](ih,jg) = [Nu.sub.ij]. The finite difference approximations (17a) [Mathematical Expression Omitted] (17b) [Mathematical Expression Omitted] (18) [Mathematical Expression Omitted] (19) [Mathematical Expression Omitted] are used. Equation (17a) is used wherever the required values of v are available, while (17b) is used at the boundary. It will be noticed that since the calculation proceeds from the known boundary conditions at, for instance, t = m towards the desired value of v at t = 0, the expression (19) leads to an implicit form for the difference equation. The iteration proceeds through equation (20) with the new values being on the left hand side of the equation. The implicit form is preferable to the explicit form in view of its better stability ([10], p. 176).

The finite difference approximation to equation (15) can then be expressed as (20) [Mathematical Expression Omitted] where ph = 1 and (21) [Mathematical Expression Omitted] (22) [Mathematical Expression Omitted] (23) [Mathematical Expression Omitted] At zero interest rate, i = p, one has [Mathematical Expression Omitted], so the pth row of equation (20) corresponds to the equation (16) when use is made of equation (17b). At infinite interest rate, one has by equation (11) a zero value for [Nu] so including a row in equation (20) corresponding to i = 0 is unnecessary.

The system (20) can be converted to tridiagonal form by matrix additions involving the last two rows of the matrix and the column vectors. The tridiagonal system can then be solved by standard techniques ([13], p. 32) to give the value at time 0 of the uninsured mortgage A. The starting point of the iteration is at time m, when for a given mortgage rate [r.sub.c] (0) the boundary condition is given by equation (7). For the insured mortgage B the boundary condition (8) requires special treatment.

Treatment of Boundary Condition

Boundary condition (8) for the insured mortgage A presents novel problems since for a given mortgage rate [r.sub.c] (0) the mortgage rate [r.sub.c] (m) at renewal at time m is not known. Thus the attempt to use the tridiagonal system (20) to iterate back in time from t = m to t = 0 is hindered by lack of a set of values of [Nu] (u,m). This problem is circumvented through consideration of equation (8). For the period m [is less than or equal to] t [is less than or equal to] 2m the tridiagonal system (20) is solved to give values at t = m of I([r.sub.i] (m)) (unit income) and P([r.sub.i] (m)) (unit principal at t = 2m). This is accomplished by suitable modification of the value of c in equations (16) and (20) and by requiring at t = 2m that [Nu.sub.I] (u,2m) = 0 for the values of the unit income and [Nu.sub.p] (u,2m) = 1 for the value of the unit principal. Then the equation (24) [Mathematical Expression Omitted] is for every [r.sub.i] (m) under the finite difference scheme for u solved through the method of successive bisection ([9], p. 258) to give a mortgage rate value [r.sub.c] (m). In other words, for every possible expected instantaneous rate of return at time m, [r.sub.i] (m), there is the corresponding mortgage rate [r.sub.c] (m).

Having thus calculated the values [r.sub.c] (m) corresponding to every [r.sub.i] (m) considered under the differencing scheme, it is possible to use equation (8) to find the value at time m of the insured mortgage B. The value is the minimum of the outstanding principal and the market value at each [r.sub.i] (m) of the future principal at time 2m and the income payments from time m to 2m. The income payments are "capped" by the application of the insurance so that the mortgage rate [r.sub.c] (m) which is used to calculate the income payable under the mortgage is subject to the maximum (25) [Mathematical Expression Omitted] In equation (25) account has been taken of the fact that under the Mortgage Rate Protection Program the 2 percent per annum deductible is calculated on a semi-annually compounded basis, while [r.sub.c] (t) is measured as a continuous rate of interest. The maximum of 12 percent in the covered rise in rates is not taken into account in view of the very low probability for appropriate values of [Sigma] that the process (1) will, over the relevant period, give such an increase in view of the drift term, which corrects large rate deviations.

Thus boundary condition (8) for the insured mortgage B can be put into effect, and for a given mortgage rate at inception [r.sub.c] (0) the market value at inception [Nu.sub.B] (u,0) can be found through solution of the system (20). It is suggested that this method of treatment of the boundary conditions of this problem may have applications in the valuation of other types of bond and mortgage.

Results

Runs of the program were made for an amortization period of 25 years and inter-renewal periods m of 1, 3 and 5 years. The interest rate scale 0 [is less than or equal to] u [is less than or equal to] 1 was discretized into 120 intervals i.e. h = 1/120. After some testing, a time interval was chosen of g = m/1280. Results were produced for mortgage rates at inception [r.sub.c] (0) of 0.05 (i.e. 5% per annum continuous), 0.10 and 0.15 and for a range of parameters [Sigma], [Gamma], and [Alpha]. It should be noted that the standard is adopted here of expressing the parameters on an annual basis.

Table 3 gives the values of the initial instantaneous expected rate [r.sub.i] (0) at which for the given mortgage rate [r.sub.c] (0) the initial market value of the uninsured mortgage A is 1. The values of [r.sub.i] (0) were found from the 120 values of [V.sub.A] ([r.sub.i], (0),0) produced by the program. The solution [r.sub.i] (0) of [V.sub.A] ([r.sub.i] (0),0) = 1 was found by the method of inverse interpolation with unequal intervals ([9], p. 116). It can be seen that the correspondence between [r.sub.i] (0) and [r.sub.c] (0) is, as might be expected, fairly close when [Gamma] is equal to the mortgage rate. The correspondence is less close for large values of m and the volatility [Sigma].

When the drift target [Gamma] and the mortgage rate at inception differ [r.sub.c] (0) are different, then the initial instantaneous expected rate of return [r.sub.i] (0) differs significantly from [r.sub.c] (0). For example, when [r.sub.c] (0) = 0.10, m = 5, [Gamma] = 0.08, [Sigma] = 0.10 and [Alpha] = 0.10 Table 3 shows [r.sub.i] (0) to be 0.10517. In other words, the expected instantaneous rate of return is higher than 0.10 because of the expected future decrease in interest rates towards the 0.08 level.

Table 4 gives the main results of the analysis, i.e. the net single premium value of the insurance at inception. The value of the insurance is calculated for each [r.sub.i] (0) as the difference between the values of the uninsured mortgage A and insured mortgage B. Then using interpolation of the logarithm the insurance value corresponding to the [r.sub.i] (0) of Table 3 was derived. The Mortgage Rate Protection Program single premium of 1.5 percent of principal to cover 75 percent of the mortgage corresponds to a premium per principal insured of 0.015/0.75 = 0.02. This value can be compared with those presented in Table 4. As one might expect, the premium increases with volatility [Sigma] and is larger in situations where the [Gamma] value leads to an expected upward drift in rates. In no case, however, does the value greatly exceed half the amount charged under the Program 0.02.

Conclusions

The Mortgage Rate Protection Program has not proved to be popular with the public. One reason is that lending institutions have no commission incentive to publicise the Program's availability. Another reason is that the memory of the climbing interest rates of the early 1980's may have faded, and house purchasers are reluctant to add to the already high costs involved in a purchase. However, the amount of the premium under the Program is another disincentive to its purchase.

The values of the parameters used in producing Table 4 are intended to represent the range likely to be chosen as representing the most likely future volatility of interest rates. It will be seen that none of the insurance values of Table 4 is as high as the premium (including expenses) of 0.02 charged under the Mortgage Rate Protection Program.

Taking into account the probable risk aversion of the mortgage borrower, the purchase of the insurance could be appropriate in the case of an inter-renewal period of three or five years if a period of high interest rate volatility is anticipated. In general, however, the Mortgage Rate Protection Program appears to be over-priced. [Tabular Data 1 to 5 Omitted]

(1)The example is adapted from the booklet "Mortgage Rate Protection Program", document number NHA 5813 84/08, Canada Mortgage and Housing Corporation, Ottawa. The monthly mortgage payments are calculated by dividing the principal by a monthly annuity factor calculated at the rate of interest compounded semi-annually. See [8]. (2)The monthly mortgage payments are calculated by dividing the principal by a monthly annuity calculated at the rate of interest calculated semi-annually. See [8]. By summing the geometric series one can show that the discounted value of payments for n years of $1/12 at the end of each month at an interest rate of j per annum compounded semi-annually is given by [Mathematical Expression Omitted]. For j = 0.125, [Mathematical Expression Omitted] and we find 50,000/93.71872 = $534. On renewal we use [Mathematical Expression Omitted] at j = 0.175 giving 47887/68.55167 = $699. (3)The annual effective rate of interest corresponding to a continuous rate of interest (or force of interest) [r.sub.i] (t) is given by exp([r.sub.i] (t)). See [8]. The rate of interest compounded semi-annually is 2exp([r.sub.i] (t)/2). (4)For a precise statement of Ito's Lemma and further details of the logic behind equations such as equation (2), see Malliaris and Brock [11]. (5)[Mathematical Expression Omitted] refers to the discounted value of payments of $1 per annum made for n years. Hence, [Mathematical Expression Omitted]. See [8]. (6)The source of the mortgage rate data was the Statistics Canada CANSIM B14024 series - "Conventional Mortgage Lending Rate."

REFERENCES [1]Ananthanarayanan, A.L. Parameter Estimation of Stochastic Interest Rate Models and Applications to Bond Pricing. Unpublished Ph.D. thesis, University of British Columbia (1978). [2]Boyle, P.P., Panjer H.H. and Sharp K.P. Report on Canadian Economic Statistics 1924-1986, (Ottawa: Canadian Institute of Actuaries, 1987). [3]Brennan, A.J. and Schwartz, E.S. "Savings Bonds: Theory and Empirical Evidence", Monograph 1979-4, New York University Salomon Brothers Center, (1979). [4]Brennan, M.J. and Schwartz, E.S., "Savings Bonds, Retractable Bonds and Callable Bonds", Journal of Financial Economics, Vol. 5, (1977), pp. 67-88. [5]Dietrich-Campbell, B. "Bond Option Pricing: Empirical Evidence", unpublished Ph.D. thesis, University of British Columbia (1985). [6]Dietrich-Campbell, B. and Schwartz, E., "Valuing Debt Options: Empirical Evidence", Journal of Financial Economics, Vol. 16, (1986), pp. 321-43. [7]Dunn, K.B. and McConnell, J.J., "Valuation of GNMA Mortgage-Backed Securities", Journal of Finance, Vol. 36, No. 3, (1981), pp. 599-616. [8]Kellison, S.G. The Theory of Interest, (Homewood, IL: Irwin, 1970). [9]Kellison, S.G. Fundamentals of Numerical Analysis, (Homewood, IL: Irwin, 1975). [10]Lapidus, L. and Pinder, G.F. Numerical Solution of Partial Differential Equations in Science and Engineering, (New York: Wiley, 1982). [11]Malliaris, A.G. and Brock, W.A. Stochastic Methods in Economics and Finance, (Amsterdam: North Holland, 1982). [12]Merton, R.C. "Optimum Consumption and Portfolio Rules in a Continuous-Time Model", Journal of Economic Theory, Vol. 3, (1971), pp 373-413. [13]Mitchell, A.R. and Griffiths, D.F. The Finite Difference Method in Partial Differential Equations, (Chichester: Wiley, 1980). [14]Pesando, R.J. and Turnbull, S.M., "Mortgage Rate Insurance and the Canadian Mortgage Market: Some Further Reflections". Canadian Public Policy, Vol. 11, No. 1, (1985), pp. 115-17. [15]Sharp, K.P., "Mortgage Rate Insurance in Canada", Canadian Public Policy, Vol. 12, No. 2, (1986), pp 432-37.

Keith P. Sharp is Assistant Professor at the University of Waterloo in Ontario.

ACKNOWLEDGEMENTS

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.

ABSTRACT

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.

INTRODUCTION

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[14]. 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[7], while the valuation of bond options has been studied by Dietrich-Campbell and Schwartz[6]. Sharp[15] examines the 1 1/2 percent premium charged under the Mortgage Rate Protection Program in the light of the hypothetical historical payouts under the insurance had the Program been in effect since 1938, and notes that for the first 30 years of the period, no payouts would have occurred because of the relative stability of interest rates.

This study first considers the partial differential equation which results from the application of an interest rate diffusion model. Next the equation is transformed to a form suitable for numerical analysis and estimates of the volatility of rates in various historical periods are presented. Then the method of numerical solution of the partial differential equation is presented and the treatment of the boundary conditions is discussed. The numerical results are presented and some additional comments on the results are then made in the final section.

Diffusion Interest Rate Model Applied to Mortgages

The theoretical premium to be charged under the Mortgage Rate Protection Program depends, among other things, on the probability that a payout under the Program will occur. That is, the premium depends on the probability that interest rates will rise by at least the 2 percent per annum deductible between mortgage inception and renewal. A model of interest rates is required which takes account of the random, unpredictable "noise" fluctuations in rates. Following Brennan and Schwartz[4], this study adopts the Wiener process as a suitable tool, as described by Malliaris and Brock[11].

A Wiener process can be used to describe processes such as the movement of a single molecule under Brownian motion, and can be compared with a random walk where transitions or collisions occur continuously rather than at discrete times. A Wiener process shares with a random walk the property that the standard deviation of the distance from the initial position is proportional to the square root of the time over which the process has been followed. The common analogy to a random walk is the progress from lamp-post to lamp-post of a drunken pedestrian. At each lamp-post collision he or she has probability 0.5 of proceeding further north on the street and probability 0.5 of reversing direction and returning to the lamp-post to the south. Then using the binomial distribution, the standard deviation of the number of lamp-posts of "progress" after n collisions is [(nx0.5x0.5).sup.1/2]. In view of the 0.5 probabilities, the expected progress, taking account of signs, is zero. Correspondingly, for a standardized Wiener process with increase dz over a period dt, one has E(dz) = 0 and var (dz) = E([dz.sup.2]) = dt.

It is required to model [r.sub.i](t), the instantaneous expected rate of return at time t on an investment in a mortgage measured as a continuous rate.(3) Allowing for the possibility of a non-stochastic drift [Beta]([r.sub.i]) the model is adopted: (1) [dr.sub.i] = [Beta]([r.sub.i])dt + [Eta]([r.sub.i])dz Here [r.sub.i] is measured as a force of interest, and [Eta]([r.sub.i]) depends on the volatility assumed for [r.sub.i].

Now denote by V([r.sub.i],t) the market value of a mortgage with principal of $1 at inception. Ito's Lemma(4) can be interpreted as implying that because E([dz.sup.2]) = dt, it is necessary when performing manipulations intended to determine changes dV up to order dt to proceed to the second term in the Taylor series in the [r.sub.i] derivatives: (2) [Mathematical Expression Omitted]

It should be noted that for the purposes of this model it is assumed that the mortgage value V is a function only of the two variables [r.sub.i] and t. Thus other variables such as the long rate of interest and the rate of inflation are assumed to affect V([r.sub.i],t) only to the extent that they are reflected in [r.sub.i](t).

The monthly payments by the mortgage borrower are approximated as being paid continuously at a rate c per annum per dollar of initial principal. Measuring time t from mortgage inception and denoting by [r.sub.c](t) the continuous market rate at inception of a mortgage of inter-renewal period m years and amortization period n years (that is [r.sub.c](t) is the rate used in calculation of the amount of periodic payments) yields:(5) (3) [Mathematical Expression Omitted] Then taking the expectation of equation (2), recalling that E(dz) = 0, adding the correction c for the continuous payments by the borrower and recalling that by definition of [r.sub.i], [Vr.sub.i] must equal the expected instantaneous return it (4) [Mathematical Expression Omitted]

The boundary conditions on (4) are important. As [r.sub.i] approaches infinity the discounted value of the payment streams must approach 0, hence (5) [Mathematical Expression Omitted] In order that the process (1) be constrained to produce only non-negative rates [r.sub.i](t), it is necessary that [Eta](0) = 0. Thus by substituting [r.sub.i] = 0 in equation (4) a natural boundary condition is obtained: (6) [Mathematical Expression Omitted]

The value of the mortgage rate insurance will be obtained as the difference of the time 0 values of two mortgages. Mortgage A is not insured and is renewed at time m, when its market value must equal the outstanding principal: (7) [Mathematical Expression Omitted] Mortgage B is insured, and for the purpose of calculation it is appropriate to assume that the mortgage lender is also the insurer. Thus the market value at inception of mortgage B will be less than that of mortgage A because if rates rise sufficiently between times 0 and m, the periodic mortgage payments between times m and 2m will be lower under mortgage B. Denoting by ([r^.sub.c]) the mortgage rate which under the insurance plan is the maximum that can be charged to the borrower for the period m to 2m, at time m the value of mortgage B is: (8) [Mathematical Expression Omitted] In equation (8), I([r.sub.i](m) denotes the value at time m of continuous payments of $1 per annum between times m and 2m, and it is multiplied by the annual amount [Mathematical Expression Omitted] which would be the borrower's payments per unit time m principal if the insurance became effective. P([r.sub.i](m)) is the value at time m of $1 principal payable at time 2m, and in equation (8) it is multiplied by the outstanding time 2m principal. The use of [r.sub.i](m) in calculating this outstanding principal takes account of the fact that the insurance limits only the periodic payments and not the outstanding principal at time 2m. If the interest rate at time m is insufficiently high to trigger a payout under the insurance, then the "1" in the "minimum" expression of equation (8) becomes the minimum of the pair.

From the mortgage borrower's point of view the purchase of the insurance is somewhat analogous to the purchase of a put option. If under the insurance the principal outstanding at time 2m were limited to that which would be calculated using the inception rate plus the deductible, then the analogy with a put option is straightforward. The security underlying the put option would be a contract to make a series of payments from time m to 2m and to pay the outstanding principal at time 2m, all calculated using the inception rate of mortgage interest plus the deductible. The exercise price would equal the principal outstanding at time m. However, the principal outstanding at time 2m actually depends, under the insurance, on the market mortgage rate at time m, so no such simple analogy with a well-defined put option can be made.

Preliminary Transformation and Data Analysis

In view of the difficulty in handling numerically the semi-infinite range of [r.sub.i], it is useful to use the transformation (9) [Mathematical Expression Omitted] and to define v(u,t) = [Mathematical Expression Omitted] = [Mathematical Expression Omitted] = [Eta]([r.sub.i]). Then the transformed version of the partial differential equation (4) for the mortgage value [Nu] can be verified by use of the chain rule to be (10) [Mathematical Expression Omitted] The boundary conditions (5) and (6) apply to both mortgage A and mortgage B and under the transformation they become respectively (11) v(0,t) = 0, 0 [is not equal to] t [is not equal to] m (12) [Mathematical Expression Omitted] The boundary conditions (7) and (8) remain effectively unchanged if one identifies [V.sub.A]([r.sub.i](m),m) with [V.sub.A](u(m),m) and [V.sub.B]([r.sub.i](m),m) with [Nu.sub.B](u(m),m).

Following Merton [12] the process (1) is specialized by making the assumptions [Beta]([r.sub.i]) = [Alpha] ([Gamma] - [r.sub.i]) and [Eta]([r.sub.i]) = [Sigma][r.sub.i], hence (13) [Mathematical Expression Omitted] The chosen form for [Eta] reflects the intuition, which is supported by the data, that changes in rates will be proportional to the existing rate [r.sub.i] (see e.g. [3]). The term [Alpha]([Gamma] - [r.sub.i]) adds to the model a tendency for the rate to drift towards the target [Gamma] at a rate which depends on [Alpha].

The values found for the insurance depend on the rate volatility parameter [Sigma], and on the drift parameters [Alpha] and [Gamma]. The process (1) is continuous, but Ananthanarayanan [1] has shown that parameter estimates found by directly using the discrete time observations correspond very closely to those found by analytic solution of the continuous stochastic process. Therefore monthly data were used for mortgage rates(6), and denoting by [Delta] [r.sub.c](k) = [r.sub.c](k + 1) - [r.sub.c](k) the month to month change in mortgage rates, an estimate [Sigma] of the volatility was made over the n month period as the sample standard deviation of the proportional rate increases neglecting the effect of the drift term, (14) [Mathematical Expression Omitted] It should be noted that the available data are for mortgage rates [r.sub.c] rather than the expected instantaneous rate of return [r.sub.i], but the difference between the two is relatively slight so the resulting values of [Sigma] are a reasonable guide.

Table 2 displays the calculated values of [Sigma^]. In view of the monthly differencing, the value for 1956-1960 for instance is calculated using data from December 1955 to December 1960 inclusive. As would be expected, the values for the volatility [Sigma^] are particularly high if periods which include the early 1980's are considered.

The estimation of [Alpha] and [Gamma] differs from that for [Sigma] in that several years of data are required if a relatively reliable estimate is to be made. Interest rate behavior differs dramatically between time periods as shown in Table 5. For example during 1977, rates moved within a 0.50 percent range, while during 1981 they moved within a 6.29 percent range. Thus, the estimation of [Alpha] and [Gamma] is very unreliable, and the approach adopted for this study is to present results for a range of values of [Alpha] and [Gamma]. The finding that the Mortgage Rate Protection Program is overpriced is shown to apply to any reasonable choice of parameters.

The parameter [Gamma] represents the drift target, and can be approximated as the long term forward rate on a mortgage investment. Mortgages with a fixed interest term of over five years are uncommon, so it is natural to turn to long bonds for guidance. Mortgages tend to yield more than bonds, and Boyle, Panjer and Sharp [2] indicate that over the period 1962-1986 the yield to maturity of a five year mortgage averaged 2.54 percent more than the yield to maturity on a 3-5 year government bond. Thus, a reasonable estimate of [Gamma] would be the long term forward rate on long bonds plus 2.54 percent. In the results presented in Tables 3 and 4, a range of values of [Gamma] is used which will bracket most such estimates. The parameter [Gamma] can be expected to have, for most historical periods, a value in the range 0.1 to 0.5 (see e.g. [1], [3] and [5]) and results are presented in Tables 3 and 4 for these two values.

Method of Numerical Solution

With the specializations of the stochastic process (1) through the use of equation (13), the partial differential equation (10) becomes (15) [Mathematical Expression Omitted] where c is given by equation (3) and the boundary conditions (7), (8), (11) and (12) apply. Boundary condition (12) simplifies to (16) [Mathematical Expression Omitted] Use is made of an interval h for the u co-ordinate and g for the t co-ordinate and of the notation [Nu](u,t) = [Nu](ih,jg) = [Nu.sub.ij]. The finite difference approximations (17a) [Mathematical Expression Omitted] (17b) [Mathematical Expression Omitted] (18) [Mathematical Expression Omitted] (19) [Mathematical Expression Omitted] are used. Equation (17a) is used wherever the required values of v are available, while (17b) is used at the boundary. It will be noticed that since the calculation proceeds from the known boundary conditions at, for instance, t = m towards the desired value of v at t = 0, the expression (19) leads to an implicit form for the difference equation. The iteration proceeds through equation (20) with the new values being on the left hand side of the equation. The implicit form is preferable to the explicit form in view of its better stability ([10], p. 176).

The finite difference approximation to equation (15) can then be expressed as (20) [Mathematical Expression Omitted] where ph = 1 and (21) [Mathematical Expression Omitted] (22) [Mathematical Expression Omitted] (23) [Mathematical Expression Omitted] At zero interest rate, i = p, one has [Mathematical Expression Omitted], so the pth row of equation (20) corresponds to the equation (16) when use is made of equation (17b). At infinite interest rate, one has by equation (11) a zero value for [Nu] so including a row in equation (20) corresponding to i = 0 is unnecessary.

The system (20) can be converted to tridiagonal form by matrix additions involving the last two rows of the matrix and the column vectors. The tridiagonal system can then be solved by standard techniques ([13], p. 32) to give the value at time 0 of the uninsured mortgage A. The starting point of the iteration is at time m, when for a given mortgage rate [r.sub.c] (0) the boundary condition is given by equation (7). For the insured mortgage B the boundary condition (8) requires special treatment.

Treatment of Boundary Condition

Boundary condition (8) for the insured mortgage A presents novel problems since for a given mortgage rate [r.sub.c] (0) the mortgage rate [r.sub.c] (m) at renewal at time m is not known. Thus the attempt to use the tridiagonal system (20) to iterate back in time from t = m to t = 0 is hindered by lack of a set of values of [Nu] (u,m). This problem is circumvented through consideration of equation (8). For the period m [is less than or equal to] t [is less than or equal to] 2m the tridiagonal system (20) is solved to give values at t = m of I([r.sub.i] (m)) (unit income) and P([r.sub.i] (m)) (unit principal at t = 2m). This is accomplished by suitable modification of the value of c in equations (16) and (20) and by requiring at t = 2m that [Nu.sub.I] (u,2m) = 0 for the values of the unit income and [Nu.sub.p] (u,2m) = 1 for the value of the unit principal. Then the equation (24) [Mathematical Expression Omitted] is for every [r.sub.i] (m) under the finite difference scheme for u solved through the method of successive bisection ([9], p. 258) to give a mortgage rate value [r.sub.c] (m). In other words, for every possible expected instantaneous rate of return at time m, [r.sub.i] (m), there is the corresponding mortgage rate [r.sub.c] (m).

Having thus calculated the values [r.sub.c] (m) corresponding to every [r.sub.i] (m) considered under the differencing scheme, it is possible to use equation (8) to find the value at time m of the insured mortgage B. The value is the minimum of the outstanding principal and the market value at each [r.sub.i] (m) of the future principal at time 2m and the income payments from time m to 2m. The income payments are "capped" by the application of the insurance so that the mortgage rate [r.sub.c] (m) which is used to calculate the income payable under the mortgage is subject to the maximum (25) [Mathematical Expression Omitted] In equation (25) account has been taken of the fact that under the Mortgage Rate Protection Program the 2 percent per annum deductible is calculated on a semi-annually compounded basis, while [r.sub.c] (t) is measured as a continuous rate of interest. The maximum of 12 percent in the covered rise in rates is not taken into account in view of the very low probability for appropriate values of [Sigma] that the process (1) will, over the relevant period, give such an increase in view of the drift term, which corrects large rate deviations.

Thus boundary condition (8) for the insured mortgage B can be put into effect, and for a given mortgage rate at inception [r.sub.c] (0) the market value at inception [Nu.sub.B] (u,0) can be found through solution of the system (20). It is suggested that this method of treatment of the boundary conditions of this problem may have applications in the valuation of other types of bond and mortgage.

Results

Runs of the program were made for an amortization period of 25 years and inter-renewal periods m of 1, 3 and 5 years. The interest rate scale 0 [is less than or equal to] u [is less than or equal to] 1 was discretized into 120 intervals i.e. h = 1/120. After some testing, a time interval was chosen of g = m/1280. Results were produced for mortgage rates at inception [r.sub.c] (0) of 0.05 (i.e. 5% per annum continuous), 0.10 and 0.15 and for a range of parameters [Sigma], [Gamma], and [Alpha]. It should be noted that the standard is adopted here of expressing the parameters on an annual basis.

Table 3 gives the values of the initial instantaneous expected rate [r.sub.i] (0) at which for the given mortgage rate [r.sub.c] (0) the initial market value of the uninsured mortgage A is 1. The values of [r.sub.i] (0) were found from the 120 values of [V.sub.A] ([r.sub.i], (0),0) produced by the program. The solution [r.sub.i] (0) of [V.sub.A] ([r.sub.i] (0),0) = 1 was found by the method of inverse interpolation with unequal intervals ([9], p. 116). It can be seen that the correspondence between [r.sub.i] (0) and [r.sub.c] (0) is, as might be expected, fairly close when [Gamma] is equal to the mortgage rate. The correspondence is less close for large values of m and the volatility [Sigma].

When the drift target [Gamma] and the mortgage rate at inception differ [r.sub.c] (0) are different, then the initial instantaneous expected rate of return [r.sub.i] (0) differs significantly from [r.sub.c] (0). For example, when [r.sub.c] (0) = 0.10, m = 5, [Gamma] = 0.08, [Sigma] = 0.10 and [Alpha] = 0.10 Table 3 shows [r.sub.i] (0) to be 0.10517. In other words, the expected instantaneous rate of return is higher than 0.10 because of the expected future decrease in interest rates towards the 0.08 level.

Table 4 gives the main results of the analysis, i.e. the net single premium value of the insurance at inception. The value of the insurance is calculated for each [r.sub.i] (0) as the difference between the values of the uninsured mortgage A and insured mortgage B. Then using interpolation of the logarithm the insurance value corresponding to the [r.sub.i] (0) of Table 3 was derived. The Mortgage Rate Protection Program single premium of 1.5 percent of principal to cover 75 percent of the mortgage corresponds to a premium per principal insured of 0.015/0.75 = 0.02. This value can be compared with those presented in Table 4. As one might expect, the premium increases with volatility [Sigma] and is larger in situations where the [Gamma] value leads to an expected upward drift in rates. In no case, however, does the value greatly exceed half the amount charged under the Program 0.02.

Conclusions

The Mortgage Rate Protection Program has not proved to be popular with the public. One reason is that lending institutions have no commission incentive to publicise the Program's availability. Another reason is that the memory of the climbing interest rates of the early 1980's may have faded, and house purchasers are reluctant to add to the already high costs involved in a purchase. However, the amount of the premium under the Program is another disincentive to its purchase.

The values of the parameters used in producing Table 4 are intended to represent the range likely to be chosen as representing the most likely future volatility of interest rates. It will be seen that none of the insurance values of Table 4 is as high as the premium (including expenses) of 0.02 charged under the Mortgage Rate Protection Program.

Taking into account the probable risk aversion of the mortgage borrower, the purchase of the insurance could be appropriate in the case of an inter-renewal period of three or five years if a period of high interest rate volatility is anticipated. In general, however, the Mortgage Rate Protection Program appears to be over-priced. [Tabular Data 1 to 5 Omitted]

(1)The example is adapted from the booklet "Mortgage Rate Protection Program", document number NHA 5813 84/08, Canada Mortgage and Housing Corporation, Ottawa. The monthly mortgage payments are calculated by dividing the principal by a monthly annuity factor calculated at the rate of interest compounded semi-annually. See [8]. (2)The monthly mortgage payments are calculated by dividing the principal by a monthly annuity calculated at the rate of interest calculated semi-annually. See [8]. By summing the geometric series one can show that the discounted value of payments for n years of $1/12 at the end of each month at an interest rate of j per annum compounded semi-annually is given by [Mathematical Expression Omitted]. For j = 0.125, [Mathematical Expression Omitted] and we find 50,000/93.71872 = $534. On renewal we use [Mathematical Expression Omitted] at j = 0.175 giving 47887/68.55167 = $699. (3)The annual effective rate of interest corresponding to a continuous rate of interest (or force of interest) [r.sub.i] (t) is given by exp([r.sub.i] (t)). See [8]. The rate of interest compounded semi-annually is 2exp([r.sub.i] (t)/2). (4)For a precise statement of Ito's Lemma and further details of the logic behind equations such as equation (2), see Malliaris and Brock [11]. (5)[Mathematical Expression Omitted] refers to the discounted value of payments of $1 per annum made for n years. Hence, [Mathematical Expression Omitted]. See [8]. (6)The source of the mortgage rate data was the Statistics Canada CANSIM B14024 series - "Conventional Mortgage Lending Rate."

REFERENCES [1]Ananthanarayanan, A.L. Parameter Estimation of Stochastic Interest Rate Models and Applications to Bond Pricing. Unpublished Ph.D. thesis, University of British Columbia (1978). [2]Boyle, P.P., Panjer H.H. and Sharp K.P. Report on Canadian Economic Statistics 1924-1986, (Ottawa: Canadian Institute of Actuaries, 1987). [3]Brennan, A.J. and Schwartz, E.S. "Savings Bonds: Theory and Empirical Evidence", Monograph 1979-4, New York University Salomon Brothers Center, (1979). [4]Brennan, M.J. and Schwartz, E.S., "Savings Bonds, Retractable Bonds and Callable Bonds", Journal of Financial Economics, Vol. 5, (1977), pp. 67-88. [5]Dietrich-Campbell, B. "Bond Option Pricing: Empirical Evidence", unpublished Ph.D. thesis, University of British Columbia (1985). [6]Dietrich-Campbell, B. and Schwartz, E., "Valuing Debt Options: Empirical Evidence", Journal of Financial Economics, Vol. 16, (1986), pp. 321-43. [7]Dunn, K.B. and McConnell, J.J., "Valuation of GNMA Mortgage-Backed Securities", Journal of Finance, Vol. 36, No. 3, (1981), pp. 599-616. [8]Kellison, S.G. The Theory of Interest, (Homewood, IL: Irwin, 1970). [9]Kellison, S.G. Fundamentals of Numerical Analysis, (Homewood, IL: Irwin, 1975). [10]Lapidus, L. and Pinder, G.F. Numerical Solution of Partial Differential Equations in Science and Engineering, (New York: Wiley, 1982). [11]Malliaris, A.G. and Brock, W.A. Stochastic Methods in Economics and Finance, (Amsterdam: North Holland, 1982). [12]Merton, R.C. "Optimum Consumption and Portfolio Rules in a Continuous-Time Model", Journal of Economic Theory, Vol. 3, (1971), pp 373-413. [13]Mitchell, A.R. and Griffiths, D.F. The Finite Difference Method in Partial Differential Equations, (Chichester: Wiley, 1980). [14]Pesando, R.J. and Turnbull, S.M., "Mortgage Rate Insurance and the Canadian Mortgage Market: Some Further Reflections". Canadian Public Policy, Vol. 11, No. 1, (1985), pp. 115-17. [15]Sharp, K.P., "Mortgage Rate Insurance in Canada", Canadian Public Policy, Vol. 12, No. 2, (1986), pp 432-37.

Keith P. Sharp is Assistant Professor at the University of Waterloo in Ontario.

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Author: | Sharp, Keith P. |
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Publication: | Journal of Risk and Insurance |

Date: | Mar 1, 1989 |

Words: | 4922 |

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