Journal of Real Estate Finance and Economics, 19:2, 133±146 (1999) # 1999 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands. A Closed Form Formula for Valuing Mortgages P. COLLIN-DUFRESNE*, JOHN P. HARDING** *Graduate School of Industrial Administration Carnegie Mellon University ** University of Connecticut Abstract We develop a closed form formula for the value of a ®xed-rate residential mortgage that includes the provision that the borrower can prepay at any time with no penalty. The value of the mortgage equals the expectation, under the risk neutral probability measure, of the future cash ¯ows. We model future cash ¯ows by estimating an empirical model of prepayment behavior. A second change of measure leads to a closed form expression for the expectation. The closed form values explain most of the time series variation in MBS prices. The closed form formula signi®cantly shortens the time to calculate mortgage values and durations and can be a useful tool for portfolio management and hedging. Key Words: mortgage valuation, closed form formula, MBS prepayment 1. Introduction The valuation of mortgage-backed securities collateralized by ®xed-rate mortgages is a signi®cant problem. The standard ®xed-rate mortgage is an extremely complex instrument involving American options to prepay and default. An extensive literature has developed dealing with the valuation of FRMs and mortgage-backed securities backed by FRMs. 1 Even when the problem is simpli®ed by restricting consideration to a single state variable (e.g., the interest rate), the resulting partial differential equation and the related boundary conditions can only be solved numerically. The numerical approaches commonly used to solve the partial differential equation (®nite difference methods, lattice methods and Monte Carlo simulation) are computer intensive and the solution time restricts portfolio managers' ability to evaluate alternative funding strategies. Investors such as depository institutions and the government sponsored enterprises (GSEs), Fannie Mae and Freddie Mac, hold mortgages as long-term investments and fund their portfolios using a combination of debt securities with different maturities and call features. These portfolio investors must choose the characteristics (e.g., duration and convexity) of the debt funding the mortgages. The traditional ®xed income portfolio management techniques of duration and convexity matching are dif®cult to apply to mortgage portfolios because of the imbedded options to default and prepay. Fannie Mae and Freddie Mac use simulation techniques extensively in their portfolio management but are limited in their ability to fully evaluate different funding strategies. If one wants to estimate the distribution of returns that can be earned by duration matching a portfolio of mortgages using simulation techniques, the number of required simulation runs grows