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Optimal times to buy and sell a home

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  • Matthew Lorig
  • Natchanon Suaysom

Abstract

We consider a financial market in which the risk-free rate of interest is modeled as a Markov diffusion. We suppose that home prices are set by a representative home-buyer, who can afford to pay only a fixed cash-flow per unit time for housing. The cash-flow is a fraction of the representative home-buyer's salary, which grows at a rate that is proportional to the risk-free rate of interest. As a result, in the long-run, higher interest rates lead to faster growth of home prices. The representative home-buyer finances the purchase of a home by taking out a mortgage. The mortgage rate paid by the home-buyer is fixed at the time of purchase and equal to the risk-free rate of interest plus a positive constant. As the home-buyer can only afford to pay a fixed cash-flow per unit time, a higher mortgage rate limits the size of the loan the home-buyer can take out. As a result, the short-term effect of higher interest rates is to lower the value of homes. In this setting, we consider an investor who wishes to buy and then sell a home in order to maximize his discounted expected profit. This leads to a nested optimal stopping problem. We use a nonnegative concave majorant approach to derive the investor's optimal buying and selling strategies. Additionally, we provide a detailed analytic and numerical study of the case in which the risk-free rate of interest is modeled by a Cox-Ingersoll-Ross (CIR) process. We also examine, in the case of CIR interest rates, the expected time that the investor waits before buying and then selling a home when following the optimal strategies.

Suggested Citation

  • Matthew Lorig & Natchanon Suaysom, 2022. "Optimal times to buy and sell a home," Papers 2203.05545, arXiv.org, revised Mar 2022.
    • Handle: RePEc:arx:papers:2203.05545
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    References listed on IDEAS

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    3. Anglin, Paul M., 2004. "How long does it take to buy one house and sell another?," Journal of Housing Economics, Elsevier, vol. 13(2), pages 87-100, June.
    4. James Albrecht & Pieter Gautier & Susan Vroman, 2016. "Directed Search in the Housing Market," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 19, pages 218-231, January.
    5. Savas Dayanik, 2008. "Optimal Stopping of Linear Diffusions with Random Discounting," Mathematics of Operations Research, INFORMS, vol. 33(3), pages 645-661, August.
    6. Tim Leung & Xin Li, 2015. "Optimal Mean Reversion Trading With Transaction Costs And Stop-Loss Exit," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 18(03), pages 1-31.
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