Abstract
We derive the first closed-form optimal refinancing rule for mortgages: Refinance when the current mortgage interest rate falls below the original mortgage interest rate by at least (1/ψ)[φ W(-exp(-φ))], where W(.) is the principal branch of the Lambert W-function, ψ=((√(2(ρ λ)))/σ), φ=1 ψ(ρ λ)((κ/M)/((1-τ))), ρ is the real discount rate (e.g. ρ= 0.05), λ is the expected real rate of exogenous mortgage repayment, including the effects of moving, principal repayment, and inflation (e.g. λ= 0.15), σ is the annual standard deviation of the mortgage rate (e.g. σ=0.0109), κ/M is the ratio of the refinancing cost and the remaining value of the mortgage (e.g. κ/M= $4,500/$250,000), and τ is the marginal tax rate (e.g. τ= 0.28). This expression is derived by solving a tractable class of stylized mortgage refinancing problems. Our quantitative results closely match those reported by other researchers using numerical methods.
Highlights
Households in the US hold $23 trillion in real estate assets.1 Almost all home buyers obtain mortgages and the total value of these mortgages is $10 trillion, exceeding the value of US government debt
The NPV of the interest saved equals the sum of refinancing costs and the difference between an old ‘in the money’ refinancing option that is given up and a new ‘out of the money’ refinancing option that is acquired
The Model we present a tractable continuous-time model of mortgage refinancing
Summary
Households in the US hold $23 trillion in real estate assets. Almost all home buyers obtain mortgages and the total value of these mortgages is $10 trillion, exceeding the value of US government debt. To eliminate a state variable, we counterfactually assume that mortgage payments are structured so that the real value of the mortgage, M, remains fixed until an exogenous and discrete mortgage repayment event These repayment events follow a Poisson arrival process. Mortgage holders pick the refinancing policy that minimizes the expected NPV of their real interest payments, applying a fixed discount rate, ρ. An optimizing mortgage holder picks a refinancing rule that minimizes the discounted value of her mortgage payments. A second-order Taylor series approximation to f (x∗) at x∗ = 0 is given by: Setting this to zero and solving for x∗ (picking the negative root) yields an approximation that we refer to as the square root rule, to refinance, this differential represents a lower bound to the refinancing decision; they note that calculating the option value is complicated. We evaluate the practical accuracy of this approximation in the calibration section below. We evaluate a third-order approximation, which is given by an implicit cubic equation.
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