Optimal Mortgage Refinancing: A Closed Form Solution

Abstract

We derive the first closed-form optimal refinancing rule: Refinance when the current mortgage interest rate falls below the original rate by at least \(\frac{1}{ψ}\)[φ + W (− exp (−φ))]. In this formula W(.) is the Lambert W-function, ψ = \(\frac{2 (ρ + λ)}{σ}\), φ = 1 + ψ (ρ + λ)\(\frac{κ/M}{(1 − τ )}\), 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, κ/M 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.