We choose a set of parameter values as the benchmark case to make our results in the last section more vivid. We assume that the statutory redemption period lasts for one year, i.e.,
1
T , which is implemented by several states in the U.S. (Baker, Miceli, and Sirmans, 2008).
Both the mortgagor and a buyer at the foreclosure sale expect housing prices to appreciate at 1% per year, i.e., 1%, and this inflation rate evolves stochastically with a volatility equal
to 20% per year, i.e., 20%.14 Both also have a common discount rate equal to 6% per deviation for REITs on commercial property was equal to 15.4%. The volatility of housing price inflation in our benchmark case was slightly higher than it.
20
year, i.e., r6%. The buyer incurs an irreversible transaction cost equal to one unit, i.e., 1
K , and can improve the value of the property by 5% after purchasing, i.e., 0.05. For this benchmark case, the buyer will not purchase the property until its value reaches 64.51 units.15
Figure 1: The timing of purchase and the associated gain for various levels of volatility
This graph shows the timing of purchase and the associated gain for a buyer attending at the foreclosure sale for various levels of volatility of housing price inflation. The solid curve shows the critical level of the value of the property that triggers the buyer to make an immediate purchase, *
Vb defined in Equation (19), and the dotted curve shows *
1( b)
F V (scaled up 100 times) defined in Equation (20), which is the gain at the date of purchasing.
The benchmark parameter values are given by the irreversible transaction cost K1 unit, the redemption period 1
T year, the parameter for the buyer’s managerial ability 0.05, the expected rate of housing price inflation
1% per year, and the discount rate r6% per year.
Figure 1 shows the result for the case where the volatility of housing price inflation, , changes in a region between 0% to 40% per year. We see that there exists a turning point at
3% per year. As the volatility increases, the potential loss incurred by a buyer resulting
15 The transaction cost relative to the purchase price is equal to 1/64.51 = 1.55%, which is plausible as it is smaller than that spent in purchasing a property not subject to the foreclosure sale, i.e., 5% – 6% (Stokey, 2009).
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from the repurchase option owned by the mortgagor also increases. Consequently, the buyer’s gain from an immediate purchase and the option value from waiting will both decrease.
When uncertainty is very insignificant (i.e., the volatility is smaller than 3% per year), the reduction in the former is less than the reduction in the latter such that the buyer will speed up purchasing. This contrasts with the situation when uncertainty is significant (i.e., the volatility is greater than 3%) because the loss resulting from the right for the mortgagor to redeem his/her foreclosed property is sufficiently high such that the buyer will wait longer before purchasing. Given that the volatility of housing price inflation less than 3% per year is uncommon in most real estate markets, we thus predict that a buyer is less likely to bid if housing price inflation fluctuates more severely. Nevertheless, without considering the time value of money, the gain for the buyer at the date of purchasing will unambiguously increase with the volatility.
22 Table 1
Optimal timing of purchase and the associated gain at the date of purchase Benchmark case: 20%, 1%, K1, T1, 0.05, and r6%
redemption (year), the parameter for the buyer’s managerial ability, and the discount rate per year, respectively. The terms *
Vb and *
1( b) F V
are endogenous variables, which denote the critical level of the property value that triggers a buyer to purchase the foreclosed property at
0
t , and the associated gain at the purchasing date, respectively.
The results conform to the theoretical results stated in Propositions 1, 2, and 3. Panel A shows the result for the case where the expected appreciation rate of housing prices, ,
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varies from -1% to 3% per year. It shows that a buyer who expects housing prices to appreciate more rapidly in the future will lose more if making an immediate purchase than if delaying the purchase. Consequently, the buyer will postpone purchasing, thus gaining more at the date of purchasing.
Panel B shows the result for the case where the irreversible transaction cost, K , varies from 0.5 units to 1.5 units. A buyer who incurs larger transaction costs in purchasing a foreclosed property will delay purchasing, and thus will gain more at the date of purchasing.
However, after considering the time value of money, the gain associated with purchasing will decrease with the transaction cost.
Panel C shows the result for the case where the statutory redemption period, T , varies from a half year to one and half years. As expected, a longer statutory redemption period encourages the buyer to delay purchasing, but leaves unchanged the gain at the date of purchasing. However, a longer statutory redemption period will reduce the gain from purchasing after we take the time value of money into account.
Panel D shows the result for the case where a buyer can increase the value of the purchased property by 3% to 7%, i.e., varies from 0.03 to 0.07. As a buyer is more capable in enhancing the value of the bought property, the buyer will speed up purchasing because the gain from purchasing will more than offset the increase of the potential loss incurred by the buyer resulting from the repurchase option owned by the mortgagor. As
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expected, at the purchasing date the buyer’s gain is increasing with the buyer’s managerial ability.
Finally, Panel E shows the result for the case where the buyer’s discount rate, r, varies
from 4% to 8% per year. It shows that a far-sighted buyer (low r) will delay and thus gain more from the purchase.
6. Conclusion
About one third of states in the U.S. offer the right of statutory redemption to a defaulting mortgagor who can reclaim his/her foreclosed property within a certain period of time, usually lasting for one month to one year. We derive a closed-form solution of a buyer’s decision at the foreclosure sale, which predicts that the buyer is less likely to purchase in
states with statutory redemption than in states without it. In states with statutory redemption, a buyer’s likelihood to purchase will decline if the redemption period lasts longer or housing
price inflation fluctuates more severely because the buyer will then be hurt more by the defaulting mortgagor who owns more valuable repurchasing option.
To test our predictions, researchers need to collect empirical data on both the dependent and independent variables. For the former researchers need to use data to proxy a buyer’s likelihood to purchase. We suggest that “the average number of auctions being taken place for settling down a foreclosed property” could be such a proxy. For the independent variables, we
25
suggest that researchers can collect the state-level data on the statutory redemption period from Baker, Miceli, and Sirmans (2008), and the metropolitan-level data on the housing price volatility from Miller and Peng (2004). We, however, leave future work to test our predictions.
Our predictions come from a very simplified model, which can be extended in the following ways. First, we may consider the default decision made by a mortgagor, and thus also take the equitable right of redemption into account. Second, we may allow more buyers to compete at the foreclosure sale rather than consider only one single buyer. The game played by the buyers and the mortgagor will then become a hierarchical one (Jou, 2004;
Krawczyk and Zaccour, 1999). Third, we may allow an auctioneer to lower his/her reservation price each time when resetting the date for the foreclosure sale. Finally, we have not discussed whether it is desirable to eliminate or shorten the period of statutory redemption (see, e.g., Phillips and Rosenblatt, 1997). We may take the objective function of the regulator into account in order to investigate this issue. We leave these extensions to future research.
Acknowledgements
We would like to thank the Editor (Carl R. Chen), two anonymous reviewers, and seminar participants at the 16th New Zealand Finance Colloquium held in Massey University (Albany Campus) in Feb., 2012 and 4th Annual Conference of Global Chinese Real Estate Congress held in Macau in July, 2012.
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References
Asabere, P. K., & Huffman, F. E. (1992). Price determinants of foreclosed urban land. Urban Studies, 29(5), 701-707.
Baker, M. J., Miceli, T. J., & Sirmans, C. F. (2008). An economic theory of mortgage redemption laws, Real Estate Economics, 36(1), 31-45.
Barone-Adesi, G., & Whalley, R. E. (1987). Efficient analytic approximation of American option values, Journal of Finance, 42, 301-320.
Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economic, 3, 637-659.
Brueggeman, W. B., & Fisher, J. D. (2006). Real Estate Finance and Investments (13th edition), McGraw-Hill: New York, NY, 10020.
Carr, P. (1995). The valuation of American exchange options with application to real options.
In Lenos Trigeorgis (Ed.), Real Options in Capital Investment, 109-120, Praeger.
Clauretie, T. M. (1987). The impact of interstate foreclosure cost differences and the value of mortgages on default rates, AREUEA Journal, 15(3), 152-167.
Clauretie, T. M., & Herzog. T. (1990). The effect of state foreclosure laws on loan losses:
Evidence from the mortgage insurance industry, Journal of Money, Credit, and Banking, 22(2), 221-233.
27
Cox, J. C., & Ross, S. A. (1976). The valuation of options for alternative stochastic process.
Journal of Financial Economics 3(1), 145-166.
Dixit, A. K., & Pindyck, R. S. (1994). Investment under Uncertainty, Princeton University Press, New Jersey.
Fischer, E. O. (1993). Analytic approximation for the valuation of American put options on stocks with known dividend. International Review of Economics and Finance, 2(2), 115-127.
Geske, R. L., & Johnson, E. (1984). The American put option values analytically. Journal of Finance, 39, 1511-1524.
Goetzmann, W. N., & Ibbotson, R. G. (1990). The performance of real estate as an asset class.
Journal of Applied Corporate Finance, 3(1), 65-76.
Grenadier, S. R. (2002). Option exercise games: An application to the equilibrium investment strategies of firms. Review of Financial Studies, 15(3), 691-721.
Jaffee, A. (1985). Mortgage foreclosure law and regional disparities in mortgage financing costs. Working paper no. 85-80, Pennsylvania State University.
Jou, J.-B. (2004). Environment, irreversibility and optimal effluent standards. The Australian Journal of Agricultural and Resource Economics, 48(1), 127-158.
Jou, J.-B., & Lee, T. (2008). Irreversible investment, financing, and bankruptcy decisions in
28
an oligopoly. Journal of Financial and Quantitative Analysis, 43(3), 769-786.
Jou, J.-B., & Lee, T. (2009). How does a development moratorium affect development timing choices and land values? Journal of Real Estate Finance and Economics, 39, 301-315.
Kau, J. B., Keenan, D. C., & Kim, T. (1993). Transactions Costs, Suboptimal Termination and Default Probabilities. Journal of the American Real Estate and Urban Economics Association, 24(3), 279-299.
Kau, J. B., Keenan, D. C., & Smurov, A. A. (2011). Leverage and mortgage foreclosures.
Journal of Real Estate Finance and Economics, 42, 393-415.
Krawczyk, J. B., & Zaccour, G. (1999). Management of pollution from decentralized agents by local government. International Journal of Environment and Pollution, 12, 343-357.
Lee, J., & Paxson, D. A. (2003). Confined exponential approximations for the valuation of American options. European Journal of Finance, 9, 449-474.
Meador, M. (1982). The effect of mortgage laws on home mortgage rates. Journal of Economics and Business, 34, 143-148.
Merton, R. C. (1973). The theory of rational option pricing. Bell Journal of Economics and Management Science, 4, 141-83.
Miceli, T. J., & Sirmans, C. F. (2005). Time-limited property rights and investment incentives.
Journal of Real Estate Finance and Economics, 31(4), 405-412.
29
Miller, N. & Peng, L. (2006). Exploring Metropolitan Housing Price Volatility. Journal of Real Estate Finance and Economics, 33(1), 5-18.
Phillips, R. A., & Rosenblatt, E. (1997). The legal environment and the choice of default resolution alternatives: an empirical analysis. Journal of Real Estate Research, 13(2), 145-154.
Phillips, R. A., & VanderHoff, J. H. (2004). The conditional probability of foreclosure: an empirical analysis of conventional mortgage loan defaults. Real Estate Economics, 32(4), 571-587.
Reinsdorf, M. (1994). New Evidence on the Relation Between inflation and Price Dispersion, American Economic Review, 84(3), 720-731.
Stokey, N. L. (2009). The Economics of Inaction, Princeton University Press, New Jersey.
Wong, K. P. (2006). The Effects of Abandonment Options on Operating Leverage and Forward Hedging, International Review of Economics and Finance, 15(1), 72-86.
Wong, K. P. (2009). The Effects of Abandonment Options on Operating Leverage and Investment timing, International Review of Economics and Finance, 18(1), 162-171.
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Table 1 presents the comparative-statics results, in which only one parameter is changed around its benchmark value, while the other parameters are held at their benchmark values.