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Optimal Mortgage Refinancing: A Closed-Form Solution





  • We thank Michael Blank, Lauren Gaudino, Emir Kamenica, Nikolai Roussanov, Jann Spiess, Dan Tortorice, Tim Murphy, Kenneth Weinstein, and Eric Zwick for excellent research assistance. We are particularly grateful to Fan Zhang who introduced us to Lambert's W-function, which is needed to express our implicit solution for the refinancing differential as a closed form equation. We also thank Brent Ambrose, Ronel Elul, Xavier Gabaix, Bert Higgins, Erik Hurst, Michael LaCour-Little, Jim Papadonis, Sheridan Titman, David Weil, participants at seminars at the NBER Summer Institute and Johns Hopkins, the editor, and two anonymous referees for helpful comments. Laibson acknowledges support from the NIA (P01 AG005842) and the NSF (0527516). Earlier versions of this paper with additional results circulated under the titles “When Should Borrowers Refinance Their Mortgages?”, and “Mortgage Refinancing for Distracted Consumers.” The views expressed in this paper do not necessarily reflect the views of the Federal Reserve Board.


We derive the first closed-form optimal refinancing rule: refinance when the current mortgage interest rate falls below the original rate by at least

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In this formula W(.) is (the principal branch of) the Lambert W-function,

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where ρ is the real discount rate, λ is the expected real rate of exogenous mortgage repayment, σ is the standard deviation of the mortgage rate, inline image is the ratio of the tax-adjusted refinancing cost and the remaining mortgage value, and τ is the marginal tax rate. This expression is derived by solving a tractable class of refinancing problems. Our quantitative results closely match those reported by researchers using numerical methods.