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Assumption financing and housing prices Hobden, David W. 1991

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A S S U M P T I O N F I N A N C I N G A N D H O U S I N G P R I C E S by David W. Hobden B.Comm., The University of British Columbia, 1988 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF T H E REQUIREMENTS FOR T H E DEGREE OF MASTER OF SCIENCE (BUSINESS ADMINISTRATION) in THE FACULTY OF GRADUATE STUDIES FACULTY OF COMMERCE AND BUSINESS ADMINISTRATION We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA August 1991 David W. Hobden, 1991 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada DE-6 (2/88) Abstract A below market assumable mortgage may imply a discount in the financing costs of prospective property buyers. Sellers may capitalize any discount value into their asking price for the property. Residential properties sold with below market assumption financing may therefore include a capitalized discount in their selling price. This study empirically tests the hypothesis that below market assumption financed residential properties include a capitalized discount in their observed selling price and obtains both a parametric and nonparametric estimate of the average discount capitalization rate. The hypothesis is tested on a large-scale sample of residential condominium sales that occurred in Vancouver over the 13 year period from 1974 to 1986. Both the hedonic pricing and matched pairs methodologies are employed to control for non-financial differences among the sampled transactions. The theoretical value of assumption financing discounts is measured using a modified Cash Equivalent Adjustment model. This study concludes that Vancouver condominiums sold with discount assumption financing do, on average, include a capitalized discount in their selling price. The hedonic price estimate of the average discount capitalization rate is approximately 75 percent of the modified Cash Equivalent discount value; the matched pairs estimate is approximately 50 percent of the modified Cash Equivalent discount value. The average discount capitalization rate appears to be stable across periods of high and moderate inflation. i i Table of Contents Abstract 1 1 List of Tables • iv Acknowledgement v 1. Introduction 1 2. Theory 4 2.1 Discount Valuation 6 2.2 Methodology 14 2.3 Previous Studies 18 3. Experiment • 24 3.1 The Data 24 3.2 Hedonic Price Estimates 35 3.3 Matched Pairs Estimate 48 4. Conclusions and Future Research 52 Notes 55 Bibliography 59 i i i List of Tables Table 1. Sample Selection 26 Table 2. Descriptive Statistics on Assumption and Conventional Financing . . . . 29 Table 3. Sales by Year by Type of Financing 31 Table 4. Sales by Neighbourhood Location 32 Table 5. Definition of Variables 33 Table 6. Descriptive Statistics on Price and Discount 34 Table 7. Hedonic Price Estimates 37 Table 8. Sensitivity Analysis of the Estimated RDISC Coefficient 44 Table 9. Hedonic Price Estimates for Two Time Periods 47 Table 10. Matched Pairs T-Test 50 iv Acknowledgement This thesis is dedicated to Professor Stan Hamilton, without whom it would not have been possible. Professor Hamilton encouraged me to pursue this research topic, provided access to his data base, and offered immediate, lucid advice at critical stages of my research. Even more, Stan convinced me to study Urban Land Economics, provided me with excellent research assistantships, and introduced me to various real estate professionals in business and government. I am beholden to Stan for all his fine assistance. I am also grateful for the timely advice and encouragement provided by Professors Stuart Rosenthal and Larry Jones at various stages of this research project. This research was supported, in part, by a Canada Mortgage and Housing Corporation University Scholarship for Graduate Studies in Housing. v 1. Introduction Conventional wisdom maintains that the buyer of a residential property who assumes an existing mortgage with a below market interest rate obtains a discount in the cost of financing the purchase. If this is so, buyers may be willing to pay a premium for a property that offers a below market assumable mortgage. Realizing this, the property seller offering a below market assumable mortgage may capitalize the value of the discount in the asking price of the property. If this indeed occurs, the observed selling price of a below market assumption financed property would be the sum of the value of the dwelling unit itself plus the amount the buyer is willing to pay for the discount. The purpose of this study is to empirically test the hypothesis that below market assumption financed residential properties do, on average, include a capitalized discount in their selling price and to estimate the average discount capitalization rate, or price per dollar of discount. This study employs a large-scale sample of condominium sales that occurred in selected neighbourhoods of Vancouver and North Vancouver over the 13 year period from 1974 to 1986. Both a parametric and nonparametric estimate of the average discount capitalization rate are obtained by means of the hedonic pricing and matched pairs methodologies, respectively. These methodologies offer different ways of controlling for non-financial differences among the sampled transactions so as to isolate the effect of assumption financing on the selling price of a property. The theoretical 1 value of assumption financing discounts is measured using the Cash Equivalent Adjustment model modified to account for the marginal cost of any subordinate mortgage financing. The sensitivity of the estimated discount capitalization rate to possible errors in measuring the true discount value is tested. The stability of the estimated discount capitalization rate (and other hedonic price estimates) across periods of high and moderate inflation is also examined. Previous studies of the effect of assumption financing on housing prices have generally employed small samples of single detached house sales that occurred in various urban areas in the United States within the period of high mortgage interest rates between 1979 and 1982 (e.g. [2], [4], [16], [24], [28] and [29]). These studies have consistently found a significantly positive capitalization effect but have produced divergent estimates of the average discount capitalization rate ranging in value from 30 to 100 percent. Studying the capitalization of assumption financing discounts is of interest to housing appraisers, property tax assessors, and housing economists. The possibility of inflated sales prices on assumption financed properties introduces problems in appraising the value of residential property. In circumstances where appraised or assessed values tend to match selling prices, the value of assumption financed properties may be overstated. Also, if assumption financed sales are used as market comparables in the appraisal of other properties, the value of those other properties may be overstated. In addition, less than full capitalization of the true value of assumption financing discounts would be consistent with the hypothesis of imperfect competition and inefficiency in the 2 markets for housing and assumption financing discounts. Following this introduction, section 2.1 describes the theoretical value of the discount implied by below market assumption financing. Section 2.2 describes the empirical methodologies employed to measure the effect of below market assumption financing on housing prices. The results obtained in previous studies are reviewed in section 2.3. Section 3.1 provides a detailed description of the data sample employed in this study. Estimates of the average discount capitalization rate using the hedonic pricing and matched pairs methodologies are presented and discussed in sections 3.2 and 3.3. Conclusions and suggestions for future research are presented in section 4. This study concludes that condominiums sold with assumption financing do, on average, include a capitalized discount in their selling price. The hedonic price estimate of the average discount capitalization rate is approximately 75 percent of the modified Cash Equivalent discount value while the matched pairs estimate is approximately 50 percent of the modified Cash Equivalent value. These point estimates, however, lack statistical precision and there may be substantial variation in the rate at which assumption financing discounts are capitalized into selling prices. 3 2. Theory Conventional wisdom maintains that the residential property buyer who assumes an existing mortgage with a below market interest rate obtains a discount in the cost of financing the purchase. The buyer may therefore be willing to pay a premium for a property that offers a below market assumable mortgage. Sirmans, Smith and Sirmans [28] describe the economic rationale that "somewhat validates" this argument. The present value of the outstanding cash flow on a below market assumed mortgage, discounted at the buyer's opportunity cost of financing (the current conventional mortgage rate), is less than the book value of the mortgage. A below market, assumable mortgage may therefore imply a valuable discount in the property buyer's cost of financing. Besides a below market interest rate, other characteristics of an assumable mortgage such as loan-to-value ratio, term to maturity, taxes, and transaction costs may also effect the value of the financing discount. If the assumed loan-to-value ratio is below that available with conventional financing (as is typically the case), and if the buyer's opportunity cost of the incremental financing (either equity or debt) exceeds the conventional mortgage rate, then the net present value of the interest savings implied by a below market interest rate on an assumed mortgage is decreased. A shorter than conventional term to maturity on an assumed mortgage may imply unconventional 4 refinancing risk which may also decrease the net interest savings implied by a below market interest rate. In addition, the interest savings implied by a below market assumed mortgage may only be valued over a buyer's expected holding period for the property since future conventional mortgage rates (and hence future discount values) are uncertain. Tax deductible mortgage interest costs (as in the United States) also decrease the present value of the interest savings implied by a below market, assumed mortgage. Finally, any difference in the buyer's net transaction costs with assumption versus conventional financing directly effects the value of the assumption financing discount. In a competitive market for properties offering below market, assumable financing, buyers may be willing to pay some price for an assumption financing discount. Realizing this, the seller offering below market assumption financing may capitalize the value of the discount in the asking price of the property.1 If this indeed occurs, the observed selling price of an assumption financed property would be the sum of the value of the dwelling unit itself plus the amount the buyer is willing to pay for the discount. The market price of assumption financing discounts (the market capitalization rate) will depend on the degree of competition for properties offering below market, assumable financing. In a perfectly competitive market the capitalization rate would be unity; buyers would pay one dollar for each dollar of discount. Housing markets, however, do not generally function in conditions of perfect competition. Houses are immobile and highly heterogeneous products and price information is discontinuous. Assumable financing only increases the heterogeneity of housing because discount amounts, assumed loan-to-value ratios, and assumed mortgage terms vary widely among 5 such properties. These housing market characteristics potentially restrict the number of buyers competing for a given property. In addition, there may exist uncertainty or incomplete information about the value of an assumption financing discount. Market imperfections could possibly lead to an average discount capitalization rate of less than unity. As Sirmans et. al. [28] note, the negotiating skills of the buyer and seller will, in part, determine the market price of the discount as well as the market price of the property. 2.1 Discount Valuation To estimate the market's average discount capitalization rate, a measure of discount value is required. The discount in a buyer's cost of financing that results from assuming a mortgage with a below market interest rate is not directly observable and must therefore be modelled. A widely used appraisal technique for valuing the discount is known as the Cash Equivalent Adjustment (CEA). The C E A is equal to the net present value of the differences between the buyer's debt repayments with conventional financing and the buyer's debt repayments with assumption financing. The discount rate (the buyer's opportunity cost of financing) is the market rate on conventional financing.2 The C E A formula is D • f C < ( n ~ C ' ( t ) (1) t-i (_+/•)' where D denotes the value of the discount; N is the term to maturity of the assumed mortgage; Ct(i) denotes the repayment at time t on a conventional mortgage at the 6 current market interest rate i*; and C,(i) denotes the repayment at time t on the assumed mortgage with contract interest rate / (/ < i*).3 The C E A model assumes that the buyer's next best alternative to the assumed loan would be to arrange a conventional mortgage with a face value equal to the book value (outstanding balance) of the assumed loan and to hold this conventional mortgage for a term equal to the term to maturity of the assumed loan. Equation 1 can be reformulated as4 D . B - f _ _ _ (2) Sf ( i + r y where B is the book value of the assumed mortgage and the summation term is equal to the market value of the assumed loan under the C E A model assumptions. The strength of the C E A model lies in its ability to provide an objective measure of the benefits of below market assumption financing. One need only know the characteristics of the assumable mortgage and the market rate on conventional financing to compute the CEA. Thus, the C E A model can be applied to all assumable mortgages with below (or above) market interest rates regardless of when or where the property and mortgage are offered for sale. However, previous studies have revealed that the C E A ignores some of the possible costs and benefits associated with assumption financing as compared to conventional financing and may therefore incorrectly measure the true discount implied by an assumed mortgage. Additional factors such as the loan-to-value ratio and term to maturity of assumed and conventional mortgages, as well as income taxes and transaction costs must also be considered. Such factors may introduce idiosyncratic costs and benefits into the value of the discount. In addition, differences in 7 the housing finance and tax environments between Canada and the United States make the potential costs and benefits associated with these factors more significant in the United States. Conventional mortgages with 30 year terms are common in the United States; most Canadian conventional mortgages have terms of five years or less. In addition, interest payments on residential mortgages are income tax deductible in the United States but not tax deductible in Canada. Following the general outline of Jaffee [18], the remainder of section 2.1 examines how these factors effect the value of the assumption financing discount and how the C E A model may be modified to more accurately value the discount. The C E A model does capture the interest savings implied by the below market interest rate on an assumed mortgage. However, it does not account for possible costs resulting from the typically lower than conventional loan-to-value ratio and shorter than conventional term to maturity of assumed mortgages. An example of the typical assumption financed sale, using the approximate median values from the sample of residential property sales employed in this study, will help to illustrate the problem. Suppose a buyer is considering a property valued at $60,000. The buyer could assume an existing, fixed rate, level monthly payment mortgage with a book value of $38,000, term to maturity of 38 months, amortization period of 277 months, and effective annual interest rate of 11 percent. As an alternative to assuming the mortgage, the buyer could originate a conventional mortgage; a fixed rate, level monthly payment mortgage with a four year term to maturity, 25 year amortization period, effective annual interest rate of 13 percent, and loan-to-value ratio of 75 percent for a loan amount of $45,000. With 8 the assumed mortgage, the buyer requires $7,000 in additional cash of financing (plus the value of any capitalized discount) and obtains shorter term financing than with the conventional mortgage. These differences imply costs to the buyer who assumes an existing mortgage — costs that are not captured in the C E A model. A . Additional Financing or Equity. — The additional cash required on the typical assumption sale can be obtained from one or a combination of two general sources: the property buyer's own equity or additional, subordinate financing. The C E A model implicitly assumes the buyer's cost of funding the shortfall is, at most, the market rate on conventional mortgage financing. If the buyer's cost of additional funds exceeds the conventional mortgage rate, the C E A model will overstate the value of the discount; otherwise the C E A model of the value of the discount is unaffected by the additional cash required on the typical assumption sale. If the buyer requires a subordinate mortgage to fund all or part of any cash shortfall on an assumption sale, several sources of such financing may be available. If a second mortgage is arranged with a commercial lender, the interest rate is likely to exceed the market rate on conventional, first mortgage financing. This is due to the inferior collateral priority of a second mortgage. In this case the C E A model will overstate the value of the discount. The appropriate adjustment can be simply obtained, however, by applying equation 2 to the second mortgage. That is, let B2, N2, C2t, and i2 be the book value, term, repayment flow, and interest rate on the second mortgage and /* remains the market rate on conventional financing. On a second mortgage where i2 > /*, equation 2 will yield a negative value which should be deducted from the C E A 9 model of the discount value. This adjustment corrects the C E A valuation of the discount for the cost of any high interest rate, subordinate financing required to fund any cash shortfall caused by a low assumed loan-to-value ratio. Commercial lenders may not be the only source of subordinate financing on assumption sales. The property buyer may be able to arrange a second mortgage, if required, with a government agency, private lender, or the property seller. As already noted, if such financing is obtained at rates below the conventional first mortgage rate, the C E A model remains an appropriate measure of the discount. Seller financing, however, may be provided at a below market rate, implying an additional discount in the property buyer's cost of financing that may also be capitalized into the selling price of the property. The C E A model can be applied to a seller underwritten, subordinate mortgage with a below market interest rate to value the additional discount implied by such financing. Instead of arranging subordinate financing, property buyers may provide additional equity to make up the cash shortfall on the typical assumption financed sale. In this case, the buyer's opportunity cost of such equity funds determines whether or not the C E A model overstates the value of the assumption financing discount. If the buyer's opportunity cost of additional equity exceeds the rate at which subordinate financing could be obtained, the buyer would arrange subordinate financing and the C E A model can be adjusted as previously described. If the buyer's opportunity cost of additional equity is less than the conventional mortgage rate, the C E A model remains an appropriate measure of the discount. Finally, if the buyer's opportunity cost of additional 10 equity is between the rates obtainable on subordinate and conventional financing, the C E A model will overstate the value of the discount. In this case, if the opportunity cost of equity can be observed, an adjustment to the C E A model analogous to that for high interest rate subordinate financing would be appropriate.5 B. Term to Maturity. — Assumed mortgages are typically seasoned conventional mortgages. Thus, the term to maturity on most assumed mortgages is less than the term to maturity on conventional financing. A buyer who assumes a mortgage is therefore generally required to refinance the property earlier than if conventional financing had been used. Since future interest rates are uncertain, the buyer may be expected to adjust his opportunity cost of financing downward from the conventional mortgage rate to compensate for this additional refinancing risk. The C E A model may therefore overstate the value of the discount. A related problem concerns the expected holding period of the assumption financed property. If the buyer sells the property before the assumed financing matures, the value of the remaining discount may not be realized. This would occur if future interest rates were lower at the time of resale, or if the assumed loan-to-value ratio at resale was too low to make passing on the assumable mortgage worthwhile. The C E A model discounts all future cash flows and will therefore overstate the value of the discount to the extent that future cash flows are uncertain. Since this potential error occurs when the buyer's expected holding period is less than the assumed term to maturity, it is most relevant to valuing the discount on fully amortizing assumed mortgages which are relatively rare in Canada. The vast majority of assumed mortgages 11 in Canada have a remaining term to maturity of less than five years (the standard term on new conventional mortgages) and this is unlikely to exceed the expected holding period of most buyers. In the United States, where fully amortizing residential mortgages are relatively common, failure to account for a buyer's expected holding period could result in significant overvaluation of the discount on assumed mortgages. The C E A model can be adjusted to more accurately value the discount by truncating the term over which future interest savings are valued to the estimated average holding period of properties. C. Taxation. — Tax deductible interest payments in the United States are yet another cause for revision of the C E A model. Smith, Sirmans and Sirmans [29] show that the C E A model can be adjusted to capture the effects of tax deductible interest by simply reformulating equation 1 in terms of the after-tax cash flow and opportunity discount rate. A buyer's marginal tax rate must then be known in order to correctly value the discount. Tax deductible interest has the effect of lowering the value of the discount as the buyer's marginal tax rate increases. Therefore, the C E A model unadjusted for tax will overstate the value of the discount in the United States. Mortgage interest payments on principal residences are not tax deductible in Canada and therefore the C E A model need not be adjusted for taxes to value the assumption financing discount in Canada. Two other types of real estate taxes — capital gains tax and capital cost (depreciation) allowances — are effectively not applicable to principal residences in either Canada or the United States and therefore do not effect the value of the discount on the vast majority of assumption financed property sales. 12 D. Transaction Costs. — Transaction costs have largely been ignored in the literature on assumption financing. The small amount of information that is reported, however, suggests that differences in closing costs between assumption and conventionally financed sales may result in significant savings to the buyer who assumes an existing mortgage. As with the other factors examined above, these differences are more significant in the United States than in Canada. This is because lenders in the United States typically charge "points", or a percentage of face value fee, to originate a conventional mortgage. In separate studies of assumption financing in the United States, Sirmans et. al. [28] and Ingram, Gregory and Gaines [16] estimate the difference in closing costs between assumption and conventionally financed sales at 2.5 to 3.0 percent of the loan amount. These estimates imply that the buyer who assumes a $50,000 mortgage may save as much as $1,500 in transaction costs compared to originating a conventional mortgage. In Canada, lenders typically charge a relatively small, flat rate fee to originate a residential mortgage. Discussions with local lenders indicate the transaction cost savings are likely to be only a few hundred dollars in Canada. The C E A model does not account for transaction cost savings and therefore, ceteris paribus, understates the value of the discount implied by assumption financing.6 To briefly summarize the discount valuation issues, the C E A model (equation 4) captures the present value of the interest savings obtained by a property buyer who assumes a below market rate mortgage instead of originating a conventional mortgage. The C E A model tends to overvalue the discount to the extent that it is not adjusted for: potential costs associated with funding the additional equity or financing required on the 13 typical assumption sale; any additional refinancing risk on an assumption sale; a buyer's expected holding period being less than the term to maturity of the assumed mortgage; and the potentially lower tax benefits of assumption financing where interest payments are tax deductible. Various adjustments can be made to the C E A model to limit this potential overvaluation of the discount. In addition, savings in transaction costs would also tend to offset the potential overvaluation. Moreover, differences in the housing finance and tax environments between Canada and the United States significantly reduce the potential overvaluation of the discount using the C E A model in a Canadian setting. 2.2 Methodology Estimating the effect of below market assumption financing on the selling price of dwelling units is essentially an empirical exercise. A study requires a sample of housing sales representing some market, where at least some of the sales are assumption financed. Given the value of the savings in financing costs obtained by buyers who assume below market mortgages (the discount), the major methodological problem is to control for differences in physical and location characteristics among the sampled transactions. Previous studies have almost universally employed the hedonic pricing methodology to study the effect of assumption financing on selling prices (see [2], [15], [16], [24], [28], [29] and [30]). An alternative approach, the paired sample methodology, has been used in relatively few previous studies (see [4], [16] and [26]). A hedonic price equation is an econometric model that "explains" housing prices by regressing a vector of objectively valuable property characteristics on the observed 14 selling prices of those properties. The effect of assumption financing on housing prices is generally studied by including a version of the C E A valued discount as an exogenous variable in a hedonic price equation and estimating the equation on a sample of assumption and conventionally financed property sales. The general model is P. = P(Ht,Dt,et) (3) where i denotes the z'th property sale; P is the observed selling price of the property; H is a vector of variables describing the physical and location characteristics of the property; D is a measure of the present dollar value of the discount implied by any below market, assumption financing associated with the sale; and e is a disturbance term. The variables measuring physical and location characteristics (H) should control for objectively valuable, non-financial differences among properties. If the data sample is not strictly cross-sectional, the model may also include time trend or time dummy variables to control for temporal changes in housing prices. The disturbance term (e) is an unobserved variable included in the econometric model to capture the net effect of any omitted, perhaps subjective, characteristic variables that may influence the observed selling price of a property. The effect of assumption financing on selling prices should therefore be captured independently by the discount variable (D). If the model is correctly specified, dP/dD is an estimate of the average discount capitalization rate, or price per dollar of discount, in the housing market being studied. Previous empirical studies of assumption financing employing the hedonic pricing methodology invariably specify a linear functional form for equation 3.7 The linear hedonic equation is 15 P. = if.p + XD{ + et (4) where /3 is a vector of average dwelling unit attribute prices conditional on the definition of the characteristic variables in H and dP/dD = X is the market's average discount capitalization rate. Assuming the disturbance term has an approximately Normal distribution, with independent and identical variance across property sales, (3 and X can be estimated using an Ordinary Least Squares (OLS) procedure and the usual significance tests can be applied to these estimated factor prices. 8 A variation on equation 4 is to replace the discount quantity variable, D, with a dummy variable indicating assumption financed sales. In this case the coefficient X is interpreted as the dollar value of the average capitalized discount. Dividing the average capitalized discount by the average theoretical discount value implied on below market assumption sales then provides an alternative measure of the average market capitalization rate. A n alternative methodology for measuring the effect of below market assumption financing on selling prices is the paired sample approach. This technique attempts to control for non-financial differences among properties by matching each assumption financed sale with an otherwise equivalent conventionally financed sale. The properties must be closely matched on physical and location characteristics as well as the date of sale. If the match variables indeed control for the effects of non-financial housing characteristics on selling prices, the mean difference in price between the assumption and conventionally financed sales provides an estimate of the average capitalized discount. The significance of this estimate may be tested with a paired sample t-test. Dividing the average capitalized discount by the average theoretical discount value on 1 6 the sample of assumption sales then provides an estimate of the average market capitalization rate. The two methodologies, hedonic pricing and matched pairs, each have a relative advantage for estimating the average capitalized discount (or the average discount capitalization rate). The hedonic price estimate is almost certainly based on more information (larger sample size) because of the relative scarcity of matched pairs of sales that only differ (pair-wise) in whether a sale was assumption financed or not. Therefore, ceteris paribus, the hedonic estimate has a smaller standard error, and allows more powerful inferences, than the matched pairs estimate. On the other hand, the matched pairs estimate, being nonparametric, may be less subject to specification error than the hedonic estimate. If the matched sample is not too small, or the difference in price between assumption and non-assumption sales is distributed approximately Normal, the validity of the matched pairs point estimate and inference statistics follows from the well known central limit theorem. By contrast, the statistical properties of the hedonic estimate depend upon how well the regression model satisfies several assumptions about the relationships among the variables in the equation; i.e. the specified functional form is realistic, the random error term is distributed approximately Normal with zero mean and homoscedastic variance, and the sample matrix of regressors is of full rank and nonstochastic. If the regression model violates these assumptions, the hedonic estimate may be biased or inefficient. Choosing between the estimates obtained with the two methodologies thus requires evaluating the trade-off between sample size and potential specification error. Obtaining separate estimates of the average discount capitalization 17 rate with two methodologies may provide more information than employing either methodology alone.9 In accordance with the theory of discount capitalization, the market capitalization rate is expected to be between zero and unity, depending on the degree of competition that exists for assumption financed properties. However, the "unbiasedness" of the estimated market capitalization rate (X) naturally depends on being able to accurately measure the discount (D) implied on an assumption financed sale. To the extent the discount can be objectively valued, the estimated discount capitalization rate will not be biased by errors in measuring the discount variable. To the extent that idiosyncratic elements or uncertainty surrounding future considerations lead to errors in measuring the value of an assumption financing discount, the estimated capitalization rate may be biased. As described in section 2.1, the C E A model can be modified to provide a fairly accurate value of the discount (D) implied by below market assumption financing, especially in a Canadian setting. In addition, the sensitivity of the estimated market capitalization rate (X) to errors in measuring the discount value (D) can be studied by varying the discount rate (/*) used in the C E A model (equation 2). 2.3 Previous Studies Sirmans, Smith and Sirmans [28] studied the effect of below market assumption financing on selling prices using a sample of single detached houses sold in the Atlanta metropolitan area during the period from July through December of 1980. They assembled a stratified sample from the records of a local computerized listing service. The sample was stratified by two types of financing, assumption and conventional, and 18 by six price level categories of $10,000 ranging from $35,000 through $95,000; an equal number of assumption and conventionally financed sales were selected in each price category. The final sample consisted of 108 observations with a mean sale price of $61,366. Of the 54 assumption sales, 11 had subordinate financing, 10 of which involved seller underwritten financing. Using the C E A model (equation 2), adjusted for secondary financing (total book value minus total present value of debt), the mean discount on the assumption sales was $11,915, approximately 19 percent of the mean selling price. Using this discount variable in a hedonic price equation of the form of equation 4, Sirmans et. al. obtained an estimated discount capitalization rate of 32.2 percent. This estimate was significantly greater than zero at the .05 level and significantly less than unity at the .01 level. With a dummy variable indicating assumption sales in place of the discount variable, the estimated average capitalized discount was $3,361 (significantly greater than zero at the .10 level), or 28.2 percent of the average discount quantity. In a subsequent study using the same data sample, Smith, Sirmans and Sirmans [29] made several further adjustments to the discount valuation model. The first adjustment was to create separate discount variables for the seller underwritten subordinate financing and the assumption financing. The mean seller discount was $2,002 and the mean assumption discount was $11,506. Using both discount variables in their hedonic price equation resulted in an estimated discount capitalization rate of 35 percent on assumption discounts (significantly greater than zero at the .05 level) and an insignificant, negative estimated discount capitalization rate on seller-second discounts. A second further adjustment was to truncate the term to maturity on the assumed 19 mortgages to reflect the average expected holding period of the assumption financed houses. Assuming an average expected holding period of seven years, the estimated discount capitalization rate was 55 percent on assumption discounts. The final further adjustment made by Smith et. al. was to compute the truncated C E A discount on an after-tax basis assuming a market-wide marginal income tax rate of 30 percent. This resulted in an estimated discount capitalization rate on assumption discounts of 64 percent, significantly greater than zero at the .01 level and not significantly less than unity at the .10 level. Rosen [24] studied the combined effects of below market assumption and seller-second financing on selling prices using a sample of houses sold in a Northern California county in June 1981. A sample of house sales that met specific (but unstated) conditions on location, financing terms, and housing characteristics was obtained from the records of a title insurance company. High priced houses were excluded from the sample. The final sample consisted of 57 house sales with a mean selling price of $110,000. Rosen states that the sample represents the middle range of housing transactions in June 1981 in the county being studied. Thirty-eight of the sales involved assumed mortgages, 35 had seller underwritten second mortgages, and 30 had both. Presumably the remainder of the sample were conventionally financed. Using the basic CEA model (equation 2), adjusted for secondary financing, the mean discount was $10,835, approximately 10 percent of the mean selling price. Rosen included this discount variable in a hedonic price equation of the form of equation 4 and obtained an estimated discount capitalization rate of 102 percent, significantly greater than zero at the .05 level. 20 Agarwal and Philips [2] studied the effect of below market assumption financing on selling prices using a sample of houses sold in selected areas of Virginia Beach, Virginia during the period from July 1981 through March 1982. The sample consisted of almost all the houses sold by five real estate sales agencies in 11 tract subdivisions representative of the full range of income classes. The final sample consisted of 97 observations with a mean selling price of $65,431. Ninety-one of the sales had an assumed mortgage and six were conventionally financed. Twenty-nine sales had secondary financing and twenty of these involved seller underwritten secondary financing at below market interest rates. Using the basic C E A model (equation 2), adjusted for secondary financing, the mean discount was $16,067, approximately 25 percent of the mean selling price. Agarwal and Philips divided their sample in half based on the median selling price. They included the discount variable in a hedonic price equation of the form of equation 4 and obtained an estimated discount capitalization rate of 36.4 percent on the lower priced houses. This estimate was significantly greater than zero at the .01 level and significantly less than unity at the .01 level. The estimated discount capitalization rate on the higher priced half of the sample was not significantly greater than zero. Focusing on the lower priced half of the sample, Agarwal and Philips calculated separate discount variables for the assumption and seller underwritten financing. Re-estimating their hedonic price equation with both discount variables resulted in an estimated discount capitalization rate of 37 percent on assumption discounts and an insignificant capitalization of seller-second discounts. Finally, the assumption financing discount was calculated for a truncated holding period of seven years, resulting in an 21 estimated discount capitalization rate of 48 percent on assumption discounts. This estimate was significantly greater than zero at the .01 level and significantly less than unity at the .01 level. Ingram, Gregory and Gaines [16] studied the effect of assumption financing on selling prices using a sample of single detached houses that sold in a selected area of Columbia, South Carolina during the period from January 1978 through March 1982. The sample was selected from Multiple Listing Service records of sales in four subdivisions within approximately one mile of each other. Only sales involving either conventional or assumption financing were included in the sample. Sales involving seller underwritten financing were excluded from the sample. The final sample consisted of 455 observations with a mean selling price of $59,625. No discount quantity was computed for the assumption sales and no distinction was made between below and above market assumed mortgages. Instead, a dummy variable indicating assumption sales was included in a hedonic price equation of the form of equation 4. The estimated average capitalized discount was $1,379 (significantly greater than zero at the .05 level), approximately 2.3 percent of the mean selling price. Ingram et. al. further tested the discount capitalization hypothesis using the alternative, paired sample methodology. From their sample they obtained 58 matched pairs of house sales, each pair consisting of an assumption and a conventionally financed sale. The sales were matched on sales date (± 3 months), age of the houses (± 2 years), size of the houses (± 50 square feet), time on the market ( ± 3 months), and by subdivision. The mean difference in selling prices between the assumption and 22 conventionally financed sales was $670. A paired sample t-test indicated that this estimate of the average capitalized discount, though small, was significantly greater than zero at the .05 level. The previously published studies described above appear to unanimously support the hypothesis that properties sold with below market assumption financing do, on average, contain a capitalized discount in their selling prices. A l l of these studies found a significant, positive capitalization effect on below market assumption financed sales. The results are much less clear on the degree to which assumption discounts are capitalized into selling prices. The studies by Sirmans et. al. [28], Smith et. al. [29], and Agarwal and Philips [2] appear to show that the market capitalization rate (price per dollar) on assumption discounts is less than unity, consistent with imperfect housing markets. However, these studies also appear to show that the basic C E A model (equation 2) overvalues the discount and downward biases the estimated market capitalization rate. As the discount valuation model is adjusted to reflect subordinate financing, expected holding periods, and taxes, the estimated capitalization rate increases. Even so, the results of most previous studies still appear to support less than full capitalization. 23 3. Experiment 3.1 The Data The key to undertaking this study was the existence of a one-fourteenth scale random sample of residential condominium units in Vancouver and North Vancouver.10 This data set includes the histories of title transfers and financing charges, as well as a set of physical and location characteristics, for each sampled property. The sample frame is the set of all residential condominium units existing in Vancouver and North Vancouver as of mid-1987. A one-twelfth scale random sample of units was obtained by selecting the first and every third strata lot (i.e. lots 1,4,7, etc.) in the first and every fourth strata plan (project). A detailed set of physical characteristics as well as the neighbourhood location of each sampled unit was obtained from the records of the British Columbia Assessment Authority (BCAA). The complete histories of title transfers and financing charges for approximately 85 percent of the sampled units were obtained from the records of the Vancouver Land Titles Office (LTO). LTO records for the remaining 15 percent of sampled units were in use and unavailable at the time of data acquisition.11 The sample consists of nested records describing 1,485 residential strata lot properties, 3,495 title transfers, and 5,854 financing charges. The initial random sample was tailored to suit the objective and methodology of the present study by applying several selection criteria. First, the sample was restricted 24 to apartment and townhouse condominiums. This entailed the exclusion of duplex strata lot properties which accounted for only 1.2 percent of the strata lot units represented in the sample. Second, the sample was restricted to neighbourhoods with more than three assumption financed sales in order to reduce the number of neighbourhoods represented in the sample while retaining most of the sampled transactions.12 The number of neighbourhoods was thus reduced from twenty-four to nine while only about seven percent of total title transfers were excluded. Third, the sample was restricted to title transfers occurring from 1974 through 1986. Although the initial sample represented title transfers occurring from 1970 through mid-1987, less than three assumption financed sales were sampled in any year prior to 1974 or in the first half of 1987.13 After applying these three selection criteria, one temporal, one spatial, and one with respect to structure type, the remaining sample consisted of 2,523 observations representing the history of title transfers from 1974 through 1986 on a one-fourteenth scale sample of residential condominiums in selected neighbourhoods of Vancouver and North Vancouver. The sample required further refinement as outlined in Table 1. Part A shows the distribution of title transfers among single property sales transactions, non-sale transfers, and multi-property transfers. Non-sale transfers include transactions between otherwise legally related parties (non-arm's length transactions) and any other transfers deemed unsuitable for sales analysis by the BCAA. Multi-property transfers involve the sale of two or more condominiums in a single transaction. Since the fair market price cannot be readily ascertained for non-sale and multi-property transfers, only single property sales are included in the final sample. 25 Table 1. Sample Selection Single Property Sales A: Initial Sample Non-Sale Transfers Multi-Property Transfers Total Transfers 1863 467 193 2523 B: Single Property Sales by Type of Financing Assumed Originated All-equity Other Mortgage Mortgage 276 986 565 36 Total Sales 1863 Type of Financing: Selection Criteria C: Final Sample Selection Sales Count Final Sample Total Sales Assumed Mortgage Sales: Below Market, No Vendor Financing Vendor Underwritten Subordinate Financing At or Above Market Rate Assumption Financing Missing Key Financial Data Originated Mortgage Sales: Conventional Lenders Other Lenders All-Equity Financed Sales: Other Sales: Total Sales 206 743 565 1514 206 6 59 5 743 243 565 36 1863 Part B of Table 1 shows the distribution of condominium sales among three 26 mutually exclusive methods of financing the property purchase. The buyer either obtained one or more mortgages to finance the purchase or financed the purchase entirely with equity. If the purchase was mortgage financed, it either involved an assumed mortgage or involved only new mortgage financing. In other words, the "Assumed Mortgage" category only includes mortgage financed purchases that involved an assumed mortgage while the "Originated Mortgage" category only includes mortgage financed purchases that did not involve an assumed mortgage; the "All-Equity" category only includes purchases involving no mortgage financing. The sales were classified in this way by comparing the type, original registration date, cancelation date, term, and loan amount of each financing charge registered against each title with the date and sale price of each title transfer. Thirty-six sales, or 1.9 percent of the sample, could not be classified by the type of purchase financing due to insufficient or missing data. Part C of Table 1 shows the final criteria applied in selecting the sample employed in this study. Of 276 assumption financed sales, 206 were included in the final sample. The financing on these 206 sales suggest a positive discount in the property buyer's financing cost when the discount is calculated as per equation 2 and adjusted for the effect of any above market rate subordinate financing as per the discussion in subsection 2.1, part A. In addition, the 206 below market assumption financed sales did not involve any vendor underwritten subordinate financing.14 Six assumption sales involving vendor underwritten financing were excluded from the sample because the present study focuses on the capitalization of assumption financing discounts. Fifty-nine assumption sales involved an assumed mortgage with an interest rate at or above the 27 contemporaneous conventional mortgage rate.3 Since these sales do not imply a financing discount they were excluded from the sample. Missing interest rate data excluded a further five assumption sales from the sample. Among the non-assumption, mortgage financed sales, only the conventionally financed sales were included in the final sample. Conventionally financed sales are those with first mortgages originated by an institutional lender (bank, trust company, or credit union) and with any second mortgages originated by an institutional lender, mortgage broker, or the provincial government. The excluded non-assumption, mortgage financed sales involved mortgages originated by a vendor, private investment company, private individual, or an unspecified lender. Of 986 non-assumption, mortgage financed sales, 743 were conventionally financed. A l l of the all-equity financed sales were included in the sample. The final sample employed in this study thus consists of 1514 condominium sales; 206 below market assumption financed, 743 conventionally financed, and 565 all-equity financed. Table 2 provides statistics that describe the key mortgage characteristics of assumption and conventionally financed sales. On below market assumed mortgages, the median difference between the assumed interest rate and the contemporaneous conventional rate is 2.17 percent per annum. The median book value to sales price ratio on assumed mortgages is 66 percent compared with a median loan-to-price ratio of 75 percent on conventional first mortgages. The median remaining term to maturity on assumed mortgages is 38.5 months compared with a median term of 48 months on conventional mortgages. Thus, on average, below market assumed mortgages provide 28 Table 2. Descriptive Statistics on Assumption and Conventional Financing Type of Financing: Med- Mean Std. Min. Max. Sales . . ian Dev. Count Characteristics Assumption Sales: Assumed Mortgages: _ _ _ _ _ 206 Rate Difference3 2.17 2.65 1.84 0.14 8.99 -(percent per annum) Loan-to-Price Ratio 5 0.66 0.63 0.19 0.06 .1 .13 -Remaining Term 38.5 48.8 58.8 5.0 348.0 -(months) Seasoned Term 23.0 27.7 21.6 3.0 104.0 -(months) Subordinate Mortgages: _ _ _ _ _ 30 Rate Difference3 0.23 0.48 2.07 -4.08 3.38 -(percent per annum) Loan-to-Price Ratio" 0.15 0.23 0.22 0.06 0.86 -Term (months) 89.0 140.5 122.0 9.0 351.0 -Conventional Sales: First Mortgages: _ _ _ _ _ 743 Loan-to-Price Ratio" 0.75 0.71 0.16 0.04 1.00 -Term (months) 48.0 60.1 74.7 6.0 360.0 -Subordinate Mortgages: _ _ _ _ _ 125 Loan-to-Price Ratio" 0.13 0.17 0.14 0.04 0.79 -Term (months) 290.0 251.3 96.1 6.0 361.0 -3 Conventional mortgage interest rate minus assumed (or subordinate) mortgage interest rate. " Mortgage book value divided by observed property selling price. lower loan-to-price ratios and shorter terms to maturity than conventional mortgages. This is as expected since an assumed mortgage is partially amortized and, in general, the 29 nominal price of a condominium has tended to increase as the mortgage financing seasons. The statistics on subordinate financing reveal several interesting facts about assumption financed condominium sales. First, only 30 of 206 below market assumption sales involved any subordinate financing at all. This implies that most buyers who assume an existing, below market mortgage make up the typical financing shortfall (compared to conventional financing) with equity rather than subordinate debt. Second, the median interest rate on subordinate mortgages on assumption sales is approximately equal to the prevailing conventional mortgage rate. In fact, 63 percent of these subordinate mortgages have interest rates below the contemporaneous conventional first mortgage rate. Most of these below market subordinate mortgages are small, fully amortizing loans provided by the Provincial government (at subsidized rates).15 Thirty seven percent of the subordinate mortgages on assumption sales do carry an interest rate premium over the conventional first mortgage rate; typically these mortgages are provided by institutional lenders. Thus, it appears that most buyers who assume an existing mortgage do not obtain subordinate mortgage financing. In addition, most buyers who do obtain subordinate mortgage financing have traditionally obtained subsidized government loans. In only 11 of 206 sales involving a below market assumed mortgage does a subordinate mortgage lower the value of the assumption financing discount. Table 3 provides a crosstabulation of the final sample by year and by type of financing. Also included in the table, by year, are the mean conventional mortgage rate (percent per annum) and the mean condominium sales price for the sample (nominal 30 Table 3. Sales by Year by Type of Financing Year Type of Financing Total Sales Mean Conventional Mean Sales Below Market Assumption Conventional Al l -Equity Mortgage Rate3 Priceb 1974 12 68 31 111 11.24 34,527 1975 15 89 43 147 11.43 40,689 1976 22 48 37 107 11.78 46,254 1977 4 62 47 113 10.35 45,125 1978 3 19 23 45 10.59 41,817 1979 27 31 46 104 11.98 48,071 1980 59 82 70 211 14.32 73,090 1981 30 36 40 106 18.23 106,251 1982 16 34 57 107 17.89 97,462 1983 8 53 53 114 13.29 92,440 1984 5 58 42 105 13.61 95,414 1985 2 72 34 108 12.18 86,364 1986 _3 _91 _42 136 11.22 100,341 Total 206 743 565 1514 3 Annual average of the monthly series of average rates charged by large institutional lenders for five year residential mortgages as recorded by the Bank of Canada (percent per annum, compounded semi-annually). b Measured in nominal dollars. dollars). Although a formal analysis of the determinants of assumption financing is beyond the scope of this study, casual inspection of Table 2 shows a definite positive correlation between conventional mortgage rates and the frequency of assumption sales. As conventional mortgage rates rise, the number of seasoned mortgages with below market interest rates must necessarily increase and apparently this implies a greater 31 frequency of assumption financed sales. This observation is consistent with the empirical results obtained by Jaffee [17] who found that conventional mortgage rates were positively and significantly related to the volume of alternative housing finance in the United States. A formal study of the determinants of assumption financing may be a worthwhile area for future research. Table 4 shows the distribution of sampled sales by neighbourhood location. The nine neighbourhoods represented in the sample were grouped into six relatively homogeneous locations by combining contiguous neighbourhoods. Most of the condominiums in Vancouver are located in and around the downtown peninsula. Most of the condominiums in North Vancouver are located in the downtown Lonsdale area. Table 4. Sales by Neighbourhood Location Neighbourhood Location Total Sales 1. Lower and Central Lonsdale 516 2. Blueridge 29 3. West End 101 4. Kitsilano and Fairview 555 5. Mt. Pleasant and Grandview 277 6. Marpole 36 Total 1514 The variables used in this study for hedonic price modelling are listed and defined in Table 5. DISC is a measure of the assumption financing discount, or the savings in financing costs obtained by buyers who assume an existing mortgage; the discount is 32 Table 5. Definition of Variables Name Definition Sample Mean PRICE Observed sale price of property (nominal $). 70,929 RPRICE Real observed sale price of property (1981 $). 75,897 A S A L E 0-1 variable indicating assumption financed sales. 0.136 DISC Assumption financing discount as defined in the text (nominal $). 2,101.4a RDISC Real assumption financing discount (1981 $). 2,370.0a SQFT Floor area of unit (square feet). 863.2 B E D Number of bedrooms in unit. 1.57 B A T H Number of bathrooms in unit. 1.32 FIRE 0-1 variable indicating unit has a fireplace. 0.517 FLOOR Floor level of unit. 2.11 A G E Age of unit (months). 43.87 THOUSE 0-1 variable indicating townhouse units. 0.082 CSALE 0-1 variable indicating all-equity financed sales. 0.373 L2 - L6 0-1 variables indicating neighbourhood locations. _b T75 - T86 0-1 variables indicating year of sale. _b Sample mean for assumption financed sales only. b See Tables 3 and 4 for temporal and spatial frequency distributions of the sample. calculated as per equation 2 and adjusted for the effect of any above market rate subordinate financing as per the discussion in section 2.1, part A. The discount has zero value on all non-assumption sales. The conventional mortgage rate series used in valuing the discount is the monthly average of rates charged by large institutional lenders for 5-year residential mortgages as recorded by the Bank of Canada. RPRICE and RDISC are the deflated versions of the PRICE and DISC variables. They are adjusted for inflation, 33 and measured in real 1981 dollars, by dividing the nominal dollar series by the monthly Consumer Price Index (CPI) for Metropolitan Vancouver as recorded by Statistics Canada. Studio condominiums are counted as having zero bedrooms. Ensuite bathrooms are counted as full bathrooms. The remaining variables used for hedonic price modelling are defined in Table 5. Table 6 provides statistics that describe condominium sales prices and the theoretical value of assumption financing discounts over the sample. In real 1981 dollars, condominium prices ranged from $17,308 to $488,599. The theoretical discount value on below market assumption financed sales ranged from $118 to $16,008 in real 1981 dollars. The final row in Table 6 describes the value of the theoretical discount as a percentage of condominium selling price for the sample of assumption financed sales. On average, the theoretical value of an assumption financing discount represents approximately 3.5 percent of the selling price of a condominium. Table 6. Descriptive Statistics on Price and Discount Variable Median Mean Std. Dev. Min. Max. Sales Count RPRICE (real 1981 $) 67,011 75,897 36,085 17,308 488,599 1,515 RDISC (real 1981 $) 1,807 2,370 2,301 118 16,008 206 RDISC/ RPRICE 2.77 3.56 3.36 0.20 17.26 206 (percent) 34 3.2 Hedonic Price Estimates Price estimates of the financial, physical, location, and temporal characteristics of condominium sales were obtained by including the variables listed in Table 5 in a linear hedonic price equation of the form of equation 4. Altogether, four specifications of the model were estimated using different combinations of the sales price and assumption sales variables; the model was estimated in both nominal and real dollars using either a dummy variable or a discount quantity variable to measure the assumption sales. The regressions were estimated by means of a stepwise, ordinary least squares (OLS) procedure. Except for the variables of primary interest in this study (ASALE, DISC, or RDISC), explanatory variables were excluded from the regression if the absolute value of their f-ratio did not exceed unity (equivalent to a one-sided significance level of approximately .159 with 1,500 degrees of freedom).16 The regression results are presented in Table 7. Regression I uses the nominal dollar measure of sales price (PRICE) as a dependent variable and the dummy variable (ASALE) to indicate assumption financed sales. A l l of the variables in the equation have the expected signs. A l l of the physical characteristic variables are significant at the .01 level or better. A l l of the location variables are significant at the .01 level or better except for L6 (Marpole). A l l of the sale year dummy variables are significant except for T75 and T78. L6, T75, and T78 were excluded from the estimation by the stepwise exit criteria; the correct interpretation of this is, ceteris paribus, the Marpole area appears to bear a price close to that of the Lonsdale area (the omitted location variable) and nominal condominium price levels in 35 1975 and 1978 were close to the nominal condominium price level in 1974 (the omitted sale year variable). The variable of primary interest in regression I is A S A L E . The estimated coefficient of $1,986 may be interpreted as an estimate of the average capitalized discount on assumption financed sales relative to conventionally financed sales. The standard error of the estimated average capitalized discount is $1,925 and the estimate is only significantly greater than zero at the .15 level. Dividing this estimate by the average theoretical discount value ($2,101) yields an estimate of the average market capitalization rate equal to 94.5 percent with an approximate standard error of 91.6 percent.17 Since a 10 percent confidence interval around the point estimate of the average capitalization rate includes both zero and unity, regression I provides inconclusive evidence on whether, and to what extent, assumption financing discounts are capitalized in selling prices. Nevertheless, the estimated average capitalization rate is not small and appears to suggest the true average capitalization rate may be closer to unity than to zero. The relatively large standard errors of the estimated average capitalized discount and the estimated average capitalization rate from regression I may indicate substantial market-wide variation in the rate at which assumption financing discounts are capitalized into sales prices. Previous studies have suggested possible systematic determinants of the discount capitalization rate. Ferreira and Sirmans [8] suggest that sellers may trade off discount value to reduce the time required to sell a property. Haurin and Hendershott [15] suggest the capitalization rate may depend on the level of housing affordability. 36 Table 7. Hedonic Price Estimates (f-ratios in parentheses) Variable Regression Nominal Dollars Real 1981 Dollars I II III IV CONSTANT -33,095 -32,773 -1,124.1 -1,106.7 (-12.95)* (-12.97)* (-0.44) (-0.44) A S A L E 1,986.5 a 1,273.1 a (1.03) (0.74) DISC a 0.539 a a (0.94) RDISC a a a 0.751 (1.55)*** SQFT 25.65 25.63 22.78 22.69 (10.07)* (10.05)* (10.00)* (9.97)* BED 12,599 12,562 14,503 14,458 (9.64)* (9.62)* (12.41)* (12.37)* B A T H 15,268 15,296 12,630 12,686 (9.94)* (9.95)* (9.23)* (9.28)* FIRE 7,472.8 7,449.2 8,910.7 8,920.8 (5.58)* (5.56)* (7.43)* (7.45)* FLOOR 4,304.7 4,297.3 4,669.7 4,676.7 (9.36)* (9.35)* (11.36)* (11.39)* A G E -109.12 -109.28 -76.43 -77.46 (-5.62)* (-5.62)* (-4.59)* (-4.65)* THOUSE 7,280.2 7,182.8 8,671.3 8,465.0 (2.71)* (2.67)* (3.61)* (3.52)* CSALE 2,719.6 2,550.4 3,282.1 3,410.5 (2.07)** (2.00)** (2.79)* (2.98)* L2 -28,852 -29,001 -25,778 -26,108 (-5.66)* (-5.68)' (-5.66)* (-5.73)* (continued) 37 Table 7. (continued) Variable Regression - Nominal Dollars Real 1981 Dollars I II III IV L3 16,298 16,127 17,331 17,123 (5.91)* (5.85)* (7.09)* (7.00)* L4 20,576 20,396 21,885 21,799 (12.21)* (12.19)* (14.77)* (14.80)* L5 -7,473.5 -7,595.9 -6,733.4 -6,826.7 (-4.15)* (-4.22)* (-4.19)* (-4.26)* L6 _ b _ b _ b _ b T75 b _ b -3,188.5 -3,118.1 (-1.45)*** (-1.41)*** T76 -3,822.0 -3,633.1 -10,476 -10,391 (-1.47)*** (-1.40)*** (-4.30)* (-4-28)* T77 7,274.8 7,252.0 -4,682.9 -4,584.0 (2.83)* (2.82)* (-1.93)** (-1.89)** T78 _ b __b -16,665 -16,532 (-4.95)* • (-4-91)* T79 4,747.3 4,896.8 -16,907 -16,948 (1.72)** (1.79)** (-6.78)* (-6.82)* T80 28,102 28,217 4,421.5 4,363.7 (12.00)* (12.11)* (2.10)** (2.08)** T81 62,823 62,736 31,793 31,475 (22.49)* (22.30)* (12.66)* (12.50)* T82 42,485 42,481 b _ b (15.21)* (15.20)* T83 38,494 38,551 -8,225.3 -8,038.8 (13.22)* (13.24)* (-3.22)* (-3.14)* T84 47,576 47,582 -3,018.3 -2,832.1 (15.71)* (15.71)* (-1.13) (-1.06) (continued) 38 Table 7. (continued) Variable Regression Nominal Dollars Real 1981 Dollars I II III IV T85 41,547 41,502 -10.097 -9,904.5 (13.50)* (13.49)* (-3.74)* (-3.67)* T86 46,071 46,034 -10,630 -10,477 (16.01)* (16.00)* (-4.20)* (-4.14)* n 1,514 1,514 1,514 1,514 R2 .7367 .7367 .6868 .6872 S E E 22,763 22,764 20,357 20,344 ****** Indicates one-sided significance at the 0.01, 0.05, and 0.10 levels respectively. a Indicates the variable was not part of the specification. b Indicates the variable was excluded because the absolute value of its .-ratio was less than unity. Variable rates of discount capitalization may also be consistent with imperfect competition in the market for condominiums or assumption financing discounts. It is not difficult to imagine that the balance of bargaining power between buyer and seller, as well as the amount of information available concerning the value of a below market assumable mortgage, shifts and varies from sale to sale. However, part of the difficulty in obtaining a statistically precise estimate of the average capitalization rate may simply result from the small magnitude of assumption financing discounts relative to the unexplained variance in condominium prices. Even assuming full capitalization, the average theoretical discount is less than one-tenth the magnitude of the standard error of the estimated regression. Regression II has the same specification as regression I except the dummy 39 variable A S A L E is replaced by the theoretical discount variable DISC. The estimated average capitalization rate of assumption financing discounts is 53.9 percent. As in regression I, the standard error of this estimate is relatively high; the point estimate is only significantly greater than zero at the .17 level and only significantly less than unity at the .21 level. A 10 percent confidence interval around the estimated average capitalization rate includes both zero and unity and therefore the estimate provides inconclusive evidence on whether, and to what extent, assumption financing discounts are capitalized into selling prices. Nevertheless, the point estimate of a 53.9 percent average capitalization rate is similar to the better estimates obtained in the studies by Smith et. al. [29] and Agarwal and Philips [2]. A l l of the other hedonic price estimates obtained in regression II are very similar to those obtained in regression I. Regressions III and IV have the same specifications as regressions I and II except the dependent variable is the inflation adjusted sales price, RPRICE. The deflated version of the theoretical discount value, RDISC, is employed in regression IV. The estimated hedonic prices in regressions III and IV are thus measured in real 1981 dollars. The inflation adjusted models are preferable to the nominal dollar models because they contain more relevant information. A single variable OLS regression of the monthly CPI for metropolitan Vancouver on the sample of condominium sales prices indicated that general consumer price levels explained 28.4 percent of the variance in condominium prices from 1974 through 1986.18 Either inflation adjusted model appears to fit the data better than the nominal dollar models judging by the lower standard error of the estimated regression. In the inflation adjusted regressions, explanatory power is 40 shifted from the annual dummy variables to the physical characteristic variables. In regression III, assumption sales are measured with the dummy variable A S A L E . The estimated average capitalized discount on assumption sales is $1,273. Dividing by the average real discount value ($2,370) yields an estimated average market-wide capitalization rate of 53.7 percent with an approximate standard error of 72.6 percent.17 Once again, the estimate provides inconclusive evidence on whether, and to what extent, assumption financing discounts are capitalized into selling prices. Regression IV is the preferred specification for several reasons: it is adjusted for inflation and therefore includes more relevant information than regressions I or II; it has a slightly higher R 2 than regression III, indicating that RDISC provides more relevant information than A S A L E ; and it yields the smallest standard error of the estimated capitalization rate of any of the four regressions. The estimated average capitalization rate is 75.1 percent with a standard error of 48.3 percent. This estimate is significantly greater than zero at the .06 level but only significantly less than unity at the .30 level. Thus, regression IV appears to indicate that assumption financing has a significant positive effect on the selling price of condominiums. In addition, the estimated average market-wide capitalization rate appears to be closer to unity than to zero. The relatively large standard error of the estimate may indicate substantial market-wide variation in the rate at which assumption financing discounts are capitalized into selling prices. Aside from the effect of assumption financing on sales price, regression IV appears to reveal two somewhat surprising results about condominium prices. The first is the positive and significant estimated coefficient on the CSALE variable. This 41 coefficient measures the price of all-equity financed condominiums relative to conventionally financed condominiums holding all other factors constant. In the appraisal literature, a cash sale is considered one in which the seller receives all cash (and settles any outstanding debt) and, for purposes of valuation, no distinction is made between conventionally financed and all-equity financed purchases. The results of this study appear to indicate that, ceteris paribus, all-equity financed purchases imply a premium in sales price compared with conventionally financed purchases. This apparent all-equity financing premium has at least two interpretations, both of which depend on the assumption that buyers who pay all cash are, on average, less wealth constrained than buyers who obtain conventional financing. Assuming this is so, perhaps wealthier buyers shop less, preferring instead to purchase a property they like sooner rather than later. This could be a rational strategy if the buyer's search or negotiating costs are sufficiently reduced. Another explanation for the apparent all-equity financing premium may be that wealthier buyers, on average, purchase more luxurious condominiums than conventional buyers and the variables that explicitly quantify these marginal amenities are omitted from the model. A second result from regression IV worth noting involves the estimated coefficients on the sale year dummy variables, T75 through T86. These estimated coefficients constitute an annual index of real condominium price levels within the sampled neighbourhoods. The estimates appear to indicate that condominium price levels did not keep pace with inflation throughout most of the 13 year period from 1974 to 1986. Real condominium price levels appear to have only exceeded their 1974 level 42 during 1980 and 1981, a period surrounding an unusually large rise and fall in Vancouver real estate prices. If the price index is accurate, the result clarifies the conventional belief that real estate ownership provides a solid hedge against the erosion of wealth by inflation. It may be that land (real estate) ownership has historically provided a solid hedge against inflation, but condominium (real property) ownership, which does not typically include a high land-to-improvement interest, appears to have been a poor hedge against inflation over the 1974 to 1986 period (at least in the neighbourhoods of Vancouver included in this study). Hamilton and Dale-Johnson [14] provide annual series of the deflated mean prices of both single detached houses and residential condominiums sold through the Multiple Listing Services in metropolitan Vancouver over the 1978 through 1989 period. These series appear to indicate that the real price levels of single detached houses and residential condominiums follow very similar trends and appear to have been no higher in 1986 than in 1978. Returning to the capitalization of assumption financing discounts, Table 8 shows the sensitivity of the estimated average capitalization rate obtained in regression IV to possible errors in measuring the discount variable, RDISC. Recall that RDISC may overvalue the true discount implied by assumption financing if, for instance, buyers adjust their opportunity cost of financing downward from the conventional mortgage rate. Such an adjustment might reflect the additional refinancing risk implied by the typically shorter term to maturity of assumed mortgages or the marginal cost of supplying the additional equity typically used in assumption financed purchases. RDISC may also overvalue the true discount if the remaining term on the assumed mortgage exceeds the 43 Table 8. Sensitivity Analysis of the Estimated R D I S C Coefficient Cell Contents: Estimate (f-Ratio) [Mean RDISC] {n} Change in Discount Rate (basis points) Maximum Term to Maturity Valued (months) Full 84 60 0.621 0.463 0.373 -50 (1.11) (0.79) (0.62) [2,173] [2,108] [2,079] {1,487} {1,487} {1,487} 0.751 0.633 0.559 0 (1.55)" (1.24) (1.07) [2,370] [2,295] [2,259] {1,514} {1,514} {1,514} 0.778 0.694 0.634 + 50 (1.84)* (1.54)" (1.37)** [2,742] [2,654] [2,611] {1,523} {1,523} {1,523} *,** Indicates one-sided significance at the 0.05 and 0.10 levels respectively. buyer's expected holding period of the property. This may occur because the present value of any remaining interest savings at some future resale date depends on the uncertain conventional mortgage rate at that time. Consequently, buyers may not value the interest savings implied by a below market assumed mortgage beyond their expected property holding periods. The first column in Table 8 shows the estimates of the average capitalization rate (k in the notation of equation 4) assuming the buyer's opportunity cost of financing (i* in the notation of equation 2) is 50 basis points above or below the prevailing conventional mortgage rate. In either case the estimated capitalization rate is positive and significantly greater than zero at the .13 level or better. However, these estimates 44 also indicate that dk/di*>0, which is inconsistent with the mathematically derived sign of this derivative.19 This inconsistency may indicate that the model is somehow mis-specified. The centre row of Table 8 shows the estimates of the average capitalization rate assuming buyers only value the interest savings of below market assumed mortgages over average expected holding periods of seven or five years. Only 13 of 206 assumption sales involved an assumed mortgage with a remaining term exceeding five years and only 15 of 206 assumption sales involved a subordinate mortgage with a term exceeding five years. For truncated terms of seven and five years the estimated average capitalization rate is positive and significantly greater than zero at the .14 level or better. However, once again the estimates indicate that dX/dN>0, which is inconsistent with the mathematically derived sign of this derivative.19 Most previous studies of the capitalization of assumption financing discounts have employed samples of housing sales that occurred within the period of high interest rates between 1979 and 1982 (e.g. [2], [4], [16], [24], [28] and [29]). The data sample employed in this study allows a test of the stability of the estimated average capitalization rate (and other parameters) across periods of high and moderate inflation. Vancouver experienced a period of unusually high inflation during 1980 and 1981. Annualized, monthly changes in the metropolitan Vancouver Consumer Price Index were between 10.0 and 28.5 percent in 20 of the 24 months of 1980 and 1981. Inflation in the average price of housing during this period was even higher than inflation in the consumer price index (see Table 3). To test the stability of the hedonic price estimates across periods of high and moderate inflation, the sample was divided into two parts; sales occurring in 1980 45 or 1981 (high inflation) and sales occurring in the remainder of the sample (moderate inflation). Regression IV was re-estimated on each of the sub-samples. The results are shown in Table 9. The first column of Table 9 shows the hedonic price estimates for the high inflation period of 1980 and 1981. Quarterly dummy variables are used in this regression to control for the more rapid temporal changes in housing price levels during this period; otherwise the model has the same specification as regression IV. The second column of Table 9 shows the hedonic price estimates over the moderate inflation periods of 1974 through 1979 and 1982 through 1986. The last column of Table 9 shows the original hedonic price estimates obtained using the whole sample (regression IV). Recall that the dependent variable in these equations is RPRICE and the parameter estimates are thus measured in real 1981 dollars. The estimated average discount capitalization rate is relatively stable across all three regressions shown in Table 9. This result suggests that the market is equally efficient in capitalizing assumption financing discounts across periods of high and moderate inflation. Some of the estimated hedonic prices of the physical and location characteristics of condominiums are somewhat less stable over the two sub-periods being analyzed. For instance, the estimated average marginal price per square foot of living area is $38 during the high inflation period and $22 during the moderate inflation period. Such changes in hedonic price estimates over the two sub-periods may reflect changes in consumer preferences. Unstable parameter estimates over sub-samples may also be indicative of multicollinearity among the regressors in the model. 46 Table 9. Hedonic Price Estimates for Two Time Periods (r-ratios in parentheses) Variable Regression 1980 - 1981 1974 - 1979 and 1982 - 1986 IV (From Table 7) CONSTANT -21,148 (-3.24)* -1,120.1 (-0.43) -1,106.7 (-0.44) RDISC 0.703 (1.12) 0.793 (1.03) 0.751 (1.55)*** SQFT 38.08 (3.35)* 22.48 (9.88)* 22.69 (9.97)* B E D 12,801 (3.96)* 14,077 (10.98)* 14,458 (12.37)* B A T H 7,005.5 (1.99)** 13,442 (9.16)* 12,686 (9.28), FIRE 12,589 (4.87)* 8,057.9 (6.09)* 8,920.8 (7.45)* FLOOR 4,715.8 (5.50)* 4,658.1 (10.14)* 4,676.7 (11.39)* A G E _ b -76.12 (-4.39)* -77.46 (-4.65)* THOUSE 9,002.8 (1.89)** 6,760.9 (2.36)* 8,465.0 (3.52)* CSALE 4,340.0 (1.69)** 3,392.2 (2.72)* 3,410.5 (2.98)* L2 -20,225 (-2.46)* -27,033 (-5.07)* -26,108 (-5.73)* L3 23,714 (4.30)* 16,604 (6.17)* 17,123 (7.00)* L4 25,163 (8.45)* 20,846 (12.27)* 21,799 (14.80)* L5 -16,998 (-4.86)* -3,511.6 (-1.94)** -6,826.7 (-4.26)* L6 b 5,385.7 (1.15) b Q2 7,776.6 (1.69)** a a Q3 15,979 (3.74)* a a Q4 28,553 (6.49)* a a Q5 39,658 (8.76)' a a Q6 40,870 (7.30)* a a Q7 55,976 (10.40)* a a Q8 24,183 (3.50)* a a (continued) 47 Table 9. (continued) Variable Regression 1980 - 1981 1974 - 1979 and IV 1982 - 1986 (From Table 7) T75 a -3,453.1 (-1.60)*** -3,118.1 (-1.41)*** T76 a -10,151 (-4.27)* -10,391 (-4.28)* T77 a -5,409.8 (-2.27)** -4,584.0 (-1.89)** T78 a -16,832 (-5.08)* -16,532 (-4.91)* T79 a -16,765 (-6.85)" -16,948 (-6.82)* T80 a a 4,363.7 (2.08)** T81 a a 31,475 (12.50)* T82 a _ b _ b T83 a -7,609.0 (-2.98)* -8,038.8 (-3.14)* T84 a -2,757.9 (-1.03) -2,832.1 (-1.06) T85 a -9,723.7 (-3.58)* -9,904.5 (-3.67)* T86 a -9,757.1 (-3.84)* -10,477 (-4.14)* n 317 1,197 1,514 R2 .7830 .6617 .6872 SEE 19,260 19,876 20,344 Mean RDISC 3,028.8 1,868.8 2,370.0 ****** indicates one-•sided significance at the 0.01, 0.05, and 0.10 levels respectively. a Indicates the variable was not part of the specification. b Indicates the variable was excluded because the absolute value of its r-ratio was less than unity. c Sample mean for assumption financed sales only. 3.3 Matched Pairs Estimate An alternative method of measuring the effect of below market assumption financing on selling prices is the paired sample approach. This method attempts to control for non-financial differences among sampled properties by matching each assumption financed 48 sale with an otherwise equivalent non-assumption financed sale. Each pair of sales must therefore be closely matched on sale date, physical characteristics, and location. If the match variables indeed control for non-financial differences between each pair of sales, the mean of the differences in price between the assumption and non-assumption financed sales provides an estimate of the average capitalized discount on below market assumption financed sales. Dividing this estimate by the average theoretical discount value provides an estimate of the average market-wide capitalization rate of assumption financing discounts. The standard deviation of the differences in prices between the assumption and non-assumption sales provides the basis for calculating f-ratios with which the statistical precision of these estimates may be gauged. From the sample of 1,514 condominium sales, 48 matched pairs of assumption and non-assumption sales were obtained. Each sale pair was rigorously matched on location, physical characteristics, and sale date according to the following criteria: 1. Same strata plan. 2. Floor area within ±50 square feet. 3. Floor level within ±2 floors. 4. Sale date within ±3 months. Each matched pair thus involves condominiums that sold within three months of each other in the same building complex with virtually the same physical characteristics. In addition, the sample distributions of the differences in the values of the match variables between assumption and non-assumption sales were all highly symmetric around zero indicating no systematic bias in the matching criteria. 49 The results of the matched pairs t-test are shown in Table 10. ARPRICE and RPRICE are the selling prices of the assumption and non-assumption financed sales, respectively. These price variables are measured in real 1981 dollars having been deflated by the monthly CPI for metropolitan Vancouver. The estimated average capitalized discount is $1,385 with a standard error of $2,469. Dividing this estimate by the mean value of RDISC on the 48 assumption financed sales ($2,760) provides an estimate of the average market-wide capitalization rate equal to 50.2 percent with an approximate standard error of 89.5 percent.17 Thus, a 10 percent confidence interval around the estimated average capitalized discount includes both zero and unity. The matched pairs t-test provides inconclusive evidence on whether and to what extent assumption financing discounts are capitalized into selling prices. However, the point estimate of a 50.2 percent average discount capitalization rate is consistent with the hedonic price estimates reported in section 3.2. T a b l e 10. M a t c h e d P a i r s T-Test Variable Sample Size Mean Standard Deviation Difference f-Ratio Mean Std. Dev. RAPRICE 75,767 32,613 48 1,385 17,103 0.56 RPRICE 74,382 33,288 The matched pairs analysis appears to demonstrate that the relatively low significance of the estimated average capitalized discount results, in part, from the small 50 magnitude of the average assumption financing discount relative to the unexplained variance in condominium selling prices. To see this, consider the distribution of the differences in selling prices between the assumption and non-assumption sales which has a mean of $1,385 and a standard deviation of $17,103 (see Table 10). Compare this with the distribution of the theoretical discount value over the sample which has a mean of $2,760 and a standard deviation of $1,910. Despite a rigorous matching of the assumption and non-assumption sales to control for non-financial differences in property characteristics, the variation in assumption financing discounts could at most (assuming full capitalization) explain only a small fraction of the unexplained variation in the sales prices of assumption and non-assumption financed condominium sales. In other words, unexplained differences in selling prices are partly responsible for the relatively large standard errors of the estimates of the average capitalized discount and average discount capitalization rate obtained using the matched pairs methodology. The matched pairs point estimate of the average capitalization rate is 50.2 percent with a standard error of approximately 89.5 percent; the most reliable hedonic price point estimate is 75.1 percent with a standard error of 48.3 percent (regression IV). As discussed in section 2.2, each of these estimates has a relative advantage over the other. The hedonic price estimate is based on much more information (larger sample size) and hence the smaller standard error of the estimate. However, the hedonic price estimate may be more sensitive to specification error than the nonparametric matched pairs estimate. In either case, the point estimates suggest that, on average, a substantial portion of the discount implied by below market assumption financing is capitalized in the selling price of assumption financed residential properties in Vancouver. 51 4. Conclusions and Future Research This study has focused on measuring the capitalization of the net interest savings implied by below market assumption financing in the selling prices of Vancouver condominiums over the 13 year period from 1974 to 1986. The most reliable hedonic price point estimate obtained of the average capitalization rate is 75 percent of the modified Cash Equivalent discount value. This estimate is significant at the .06 level allowing the conclusion that condominiums sold with below market assumption financing do, on average, include a capitalized discount in their selling price. This estimate appears to be stable across periods of high and moderate inflation. The matched pairs point estimate of the average capitalization rate is 50 percent of the modified Cash Equivalent discount value. The results of this study are thus consistent with those obtained by Smith, Sirmans and Sirmans [29] in finding significant but apparently less than full discount capitalization. However, the statistical precision of the estimates of the average capitalization rate obtained in the present study (as in Smith et. al. [29]) do not allow the hypothesis of full discount capitalization to be conclusively rejected. The point estimates of the average discount capitalization rate obtained in this study must be interpreted with caution. The estimates have low statistical significance and therefore do not allow conclusive rejection of the hypotheses of either a very low or a very high average discount capitalization rate. In addition, the point estimates may 52 be biased by errors in measuring the discount quantity variable. Recall that several non-interest rate characteristics can effect discount value and are not accounted for in the basic C E A model. In addition, a property seller may avoid a prepayment penalty by allowing a mortgage assumption in which case the discount may be valued as the difference between savings of the buyer and savings of the seller. Many questions concerning the capitalization of assumption financing discounts remain to be investigated. For instance, the relatively large standard errors of the estimates of the average capitalization rate obtained in this study could possibly indicate substantial variation in the capitalization rate over the sample. A more precise estimate is naturally more desirable and future research could investigate whether the capitalization rate has systematic determinants. Ferreira and Sirmans [8] have suggested that sellers may trade off discount value to reduce the time required to sell a property. Haurin and Hendershott [15] have suggested the capitalization rate may depend on the level of housing affordability. Characteristics of buyers, sellers, and their agents such as wealth, portfolio composition, and education, may also be potential determinants of the discount capitalization rate. Several other areas closely related to this study may be worthy of further investigation. For instance, future research could investigate the determinants of the volume of assumption financing. Jaffee [17] has suggested that conventional mortgage rates and the credit rationing standards of institutional lenders are significant determinants of the volume of unconventional housing finance. Future research could also focus on investigating the volume or capitalization of other types of unconventional, 53 below market financing such as seller underwritten mortgages and interest rate buy-downs by builders of new housing. 54 Notes 1. The discount appears to be a potential windfall gain to the property seller. However, the seller retains liability for the mortgage if the buyer defaults unless the mortgagee releases this contingent claim (for consideration). The rational seller must therefore receive a price for a mortgage assumption at least equal to the expected cost of this contingent liability. 2. In this study, the discount rate used in valuing the discount is the market rate on 5-year, conventional, fixed rate mortgages. Although buyers may obtain short term or variable rate conventional mortgages, under a pure expectations theory of yield-term structure, the long rate is equivalent to a series of expected short rates. Thus, holding constant the term to maturity of the assumed mortgage, either the long rate or a series of expected short rates provide equivalent discount values. 3. Some assumption financed property sales involve an assumed mortgage with an interest rate at or above the prevailing conventional mortgage rate (i > /*). An above market rate mortgage assumption may be explained by a high pre-payment penalty or a non-prepayable mortgage, a saving in the transaction costs of originating conventional financing, or a buyer who is excluded from.conventional financing by institutional credit standards. 4. From equation 1: N C N C D = £ _ _ _ ± _ Y-^L; t-i (i+r)r »-i (i+ry D = B - £ — V L L , M (i+ry where B is the book value of the assumed mortgage. 5. Findlay and Fischer [9] suggest an alternative model of the value of an assumption financing discount that attempts to account for different loan-to-value ratios on assumed and conventional mortgages. Based on the Modigliani-Miller theorem, Findlay and Fischer argue that, in perfect capital markets, the average interest rate on any combination of a first and second mortgage, weighted by loan-to-value ratio, must equal the interest rate on a single first mortgage with the same overall loan-to-value ratio. If this condition did not hold, arbitrage opportunities would return the market to 55 this equilibrium. However, as Peach and Crellin [23] note, the Findlay-Fischer model values the assumption financing discount from the perspective of a mortgage investor rather than a house buyer. For the original theorem see Modigliani, F. and M . Miller, "The Cost of Capital, Corporate Finance, and the Theory of Investment", American  Economic Review (June 1958) pp. 261-297. 6. If a property seller avoids a prepayment penalty by allowing a mortgage assumption, the seller may not capitalize the full discount into the asking price for the property. If this indeed occurs, the C E A model may significantly overvalue the capitalized discount. The prepayment penalties on conventional mortgages range from nothing to six months interest on the outstanding balance of the mortgage. 7. It may seem odd that previous studies have invariably specified the linear functional form when studies of other housing market issues have found that double-log or semi-log functional forms often fit housing sales data better. These alternative specifications allow for non-linearities in the relationship between the characteristics and selling prices of properties. However, these alternative functional forms may be inconsistent with modelling the capitalization of assumption financing discounts. The double log relationship is a constant elasticity function where the elasticity is given by the coefficient. Therefore, the double log model is consistent with, on a $100,000 house, the marginal capitalization rate on a $10,000 discount being twice the marginal capitalization rate on a $5,000 discount. The semi-log model is a constant proportional change function where the marginal proportional change is given by the coefficient. Therefore, the semi-log model is consistent with the marginal capitalization rate on a $100,000 house being twice the marginal capitalization rate on a $50,000 house. By contrast, the linear model is consistent with a constant marginal capitalization rate. The marginal discount capitalization rate is probably closer to constant than either the constant elasticity or constant proportional change functions would imply. This may explain why previous studies unanimously specify the linear form of equation 3. 8. Ph and hence et, cannot truly be distributed Normal since Pt > 0. In practice, the necessary condition for applying standard inference tests to the OLS regression estimates is that Pi be distributed approximately Normal over the relevant range of property prices. 9. The estimates obtained with either methodology are subject to potential omitted variable, error in variable, or sample selection biases. 10. This data set was assembled by Dr. S.W. Hamilton and his research assistants at the University of British Columbia. 11. Thus, the one-twelfth scale sample of residential condominiums became a one-fourteenth scale sample (i.e. 1/12 x 85/100 « 1/14). 56 12. The sample was concentrated on neighbourhoods representing most of the sampled transactions because of the difficulty in measuring location factors and in identifying location factor prices given the limited data available on location characteristics and the generally small number of transactions in the excluded neighbourhoods. 13. Condominiums first appeared in British Columbia in 1967 and hence few resale transactions, and fewer assumption financed sales, occurred prior to 1974. 14. A vendor underwritten mortgage or agreement for sale may imply an additional discount in the property buyer's financing cost. Although the theoretical value of a vendor discount is analogous to that of an assumption discount, studies by Smith et. al. [29] and Agarwal and Philips [2] indicate the capitalization rates may differ. The two types of discounts must therefore be studied independently. 15. From 1969 to 1989 the B.C. Provincial government offered second mortgages to qualified buyers of moderately priced residential properties. Qualified buyers were B.C. residents with at least a five percent down payment and first time users of the Second Mortgage Program. Qualified residential properties were priced below a specified limit ($85,000 in 1989). The second mortgages were typically 25 year term, fully amortizing, fixed rate, level monthly payment loans with a face value of up to $5,000 prior to November 1982 and up to $10,000 thereafter. The interest rate was set at the first mortgage rate, as specified under the National Housing Act (NHA), on the date of application. 16. The regressions were estimated by means of a stepwise OLS procedure. This procedure steps the explanatory variables into the equation one at a time and includes or excludes variables according to pre-specified significance criteria. The entry and exit criteria were set at an F statistic of 1.0 (equivalent to a one-sided significance level of approximately .159 with 1,500 degrees of freedom). If the absolute value of the r-ratio on a variable exceeded unity the variable was included in the equation; if at any step the absolute value of the f-ratio on an included variable dropped below unity the variable was excluded from the equation. Stepwise regression is useful when theoretical grounds for specifying particular explanatory variables are exiguous. The relatively low significance criteria was chosen over the conventional .05 level because multi-collinearity can inflate estimated standard errors. In addition, the adjusted R2 on a regression increases with the addition of a regressor if the F statistic for that variable exceeds unity. 17. The standard error of this estimate of the capitalization rate is based on the simplifying assumption that the denominator in the estimate (the average theoretical discount value) is non-stochastic. In fact, the average theoretical discount value is a random variable and therefore the standard error of the estimated capitalization rate is not precisely as stated. Nevertheless, the simplifying assumption is unlikely to have materially altered the interpretation of the estimate. The asymptotic distribution of the 57 estimated capitalization rate can be estimated using the Delta method. Stated simply, the Delta method provides the distribution of an estimator that is a differentiable function of other estimators provided that these other estimators converge to a Normal distribution with finite variance. For a description and proof of the Delta method see Billingsley, P., Probability and Measure (1979, John Wiley and Sons, New York, NY). 18. PRICE = -11,825 + 90,219(CPI) + e ; n= 1,514 i?* = .2843 (-3.37) (24.51) 19. From equation 4: P-//p -e D ' k = -(?-H$-e)— -X 3D z 0 . D dX_ dN -(P-i/p -e) BD <L 0 . 58 B i b l i o g r a p h y [1] Agarwal, V . B . and R . A . Philips, "The Effect of Mortgage Rate Buydowns on Housing Prices: Recent Evidence from F H A - V A Transactions", Journal  of American Real Estate and Urban Economics Association 11:4 (1983) pp. 491-503. [2] Agarwal, V . B . and R . A . Philips, "The Effects of Assumption Financing Across Housing Price Categories", Journal of American Real Estate and Urban  Economics Association 13:1 (Spring 1985) pp. 48-57. [3] Clauretie, T . M . and D.S. Bible, "Cash Equivalency: Appraisers' Views and Applications", The Appraisal Journal (January 1987) pp. 25-31. [4] Dale-Johnson, D., M . C . Findlay, A . L . Schwartz and S.D. Kapplin, "Valuation and Efficiency in the Market for Creatively Financed Houses", Journal of American  Real Estate and Urban Economics Association 13:4 (1985) pp. 388-403. [5] D'Ardenne, R . H . , "Valuation Subject to Existing Financing", The Appraisal  Journal (October 1980) pp. 512-520. [6] Edgren, J .A. and S.C. Hayworth, "The Implications of Land Contracts for Property Tax Assessment Practices", Housing Finance Review 3:2 (April 1984) pp. 177-189. [7] Ferreira, E.J . and G.S. Sirmans, "Assumable Loan Value in Creative Financing", Housing Finance Review 3:2 (April 1984) pp. 139-147. [8] Ferreira, E.J . and G.S. Sirmans, "Selling Price, Financing Premiums, and Days on the Market", Journal of Real Estate Finance and Economics 2 (1989) pp. 209-222. [9] Findlay, M . C . and F .E . Fischer, "On Adjusting The Price of Creatively Financed Residential Sales: Cash Equivalence vs. F F V A " , Housing Finance Review 2:1 (January 1983) pp. 63-80. [10] Findlay, M . C , F .E . Fischer and D . A . Hester, "Valuing the Right of Mortgage Assumptions: A n F F V A Approach", The Real Estate Appraiser and Analyst (Winter 1985) pp. 23-28. 59 [11] Friedman, J.P. and B. Lindeman, "Cash Equivalent Analysis", The Appraisal  Journal (January 1979) pp. 35-43. [12] Gau, G.W., "Impact of Creative Financing on the Market Prices of Apartment and Commercial Properties", Property Tax Journal 5 pp.203-218. [13] Guntermann, K.L., "Financing and Selling Price of Single-Family Houses", Research in Real Estate 1 (1982) pp. 255-273. [14] Hamilton, S.W. and D. Dale-Johnson, "Multiple Listing Sales Data as an Indicator of Market Behaviour: Prices, Volume and Submarket Activity", Canadian Real Estate Research Bureau, University of British Columbia, Working Paper #90-ULE-011. [15] Haurin, D.R. and P.H. Hendershott, "Affordability and the Value of Creative Finance: An Application to Seller Financed Transactions", Housing Finance  Review 5 (1986) pp. 189-206. [16] Ingram, F.J., C.W. Gregory and J.P. Gaines, "The Impact of Assumption Financing on Housing Prices", The Appraisal Journal (October 1985) pp. 585-596. [17] Jaffee, D.M., "Creative Finance: Measures, Sources, and Tests", Housing  Finance Review 2:1 (January 1984) pp. 1-17. [18] Jaffee, D.M., "House Price Capitalization of Creative Finance: An Introduction", Housing Finance Review 3:2 (April 1984) pp. 107-17. [19] Koch, D.L., D.W. Steinhauser and K.R. Ihlanfelt, "The Risks of Creative Financing", Economic Review (Federal Reserve Bank of Atlanta, December 1982) pp. 4-13. [20] Lipscomb, J.B., "Discount Rates for Cash Equivalent Analysis", The  Appraisal Journal (January 1981) pp. 23-33. [21] Nothaft, F.E. "Survey of Home-Seller Finance, 1983", Federal Reserve  Bulletin (October 1985) pp. 767-775. [22] Patchin, P.J., "Common Sense About Cash Equivalency", The Appraisal  Journal (July 1985) pp. 340-346. [23] Peach, R.W. and G.E. Crellin, "The Below-Market Financing Premium: The Buyer's Viewpoint", The Appraisal Journal (October 1985) pp. 562-573. 60 [24] Rosen, K.T., "Creative Financing and Housing Prices: A Study of Capitalization Effects", Housing Finance Review 3:2 (April 1984) pp. 119-27. [25] Rosen, S., "Hedonic Prices and Implicit Markets: Product Differentiation in Pure Competition", Journal of Political Economy 82 pp. 34-55. [26] Schwartz, A.L . and S.D. Kapplin, "Economic Implications of Alternative Home Financing", Housing Finance Review 3:2 (April 1984) pp. 165-175. [27] Sirmans, G.S., C.F. Sirmans and S.D. Smith, "Adjusting Comparable Sales for Assumption Financing", The Appraisal Journal (January 1984) pp. 84-91. [28] Sirmans, G.S., S.D. Smith and C.F. Sirmans, "Assumption Financing and Selling Price of Single-Family Homes", Journal of Financial and Quantitative Analysis 18:3 (September 1983) pp. 307-17. [29] Smith, S.D., G.S. Sirmans and C.F. Sirmans, "The Valuation of Creative Financing in Housing", Housing Finance Review 3:2 (April 1984) pp. 129-138. [30] Strathman, J.G., P.B. DeLacy and K.J. Dueker, "Creative Financing Concessions in Residential Sales: Effects and Implications", Housing Finance Review 3:2 (April 1984) pp. 149-163. 61 

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