# Allowing Underwater Borrowers to Refinance Could Improve Investors’ Sharpe Ratio

Consider borrowers with 6 percent 30-year mortgages that are 20 percent underwater.  Assume that the probability that any one borrower will default in any one month is .2 percent, and that the cost of default to the lender conditional on default is 50 percent.  Assume that at the end of five years, any remaining long balance is paid off).  A security containing such mortgages will have an IRR of 4.83 percent (I am happy to share the spreadsheet for the details.

Now let us convert the borrowers into people with 4 percent mortgages with 20 year terms.  The payment from such mortgages will be essentially the same as before, and the mortgage balance will be paid off more quickly.  The good news for investors is that this lowers the probability of default; the bad news is that it reduces the yield before default.  Assuming default probabilities in any one month go down to .1 percent, the IRR for investors goes down to 3.45 percent.  This seems like a bad deal for investors, except that they will have more certainty about their cash flows; the standard deviation of their investment falls.  Because default is binomial, we can calculate that the variance of returns will be p*(expected loss)*(1-p*expected loss).  The variance of not refinancing is thus .0099 and of refinancing is .004975, which translate into standard deviations of .1 and .07.  Because the riskless rate is currently zero, when we substitute into the Sharpe formula, we find

Sharpe no refinance = .048/.1; Sharpe refinance = .0345/.07.

This is about .5 in both cases, suggesting that investors are getting the same risk adjusted return whether refinancing becomes easy of not, assuming the assumptions are correct.  I am not saying they are; I am saying that in making policy we need to think about these sorts of implications.

Affiliation: University of Southern California

Richard K. Green, Ph.D., is the Director of the USC Lusk Center for Real Estate. He holds the Lusk Chair in Real Estate and is Professor in the School of Policy, Planning, and Development and the Marshall School of Business at the University of Southern California.

Prior to joining the USC faculty, Dr. Green spent four years as the Oliver T. Carr, Jr., Chair of Real Estate Finance at The George Washington University School of Business. He was Director of the Center for Washington Area Studies and the Center for Real Estate and Urban Studies at that institution. Dr. Green also taught real estate finance and economics courses for 12 years at the University of Wisconsin-Madison, where he was Wangard Faculty Scholar and Chair of Real Estate and Urban Land Economics. He also has been principal economist and director of financial strategy and policy analysis at Freddie Mac.

His research addresses housing markets, housing policy, tax policy, transportation, mortgage finance and urban growth. He is a member of two academic journal editorial boards, and a reviewer for several others.

His work is published in a number of journals including the American Economic Review, Journal of Economic Perspectives, Journal of Real Estate Finance and Economics, Journal of Urban Economics, Land Economics, Regional Science and Urban Economics, Real Estate Economics, Housing Policy Debate, Journal of Housing Economics, and Urban Studies.

His book with Stephen Malpezzi, A Primer on U.S. Housing Markets and Housing Policy, is used at universities throughout the country. His work has been cited or he has been quoted in the New York Times, The Wall Street Journal, The Washington Post, the Christian Science Monitor, the Los Angeles Times, Newsweek and the Economist, as well as other outlets.

Dr. Green earned his Ph.D. and M.S. in economics from the University of Wisconsin-Madison. He earned his A.B. in economics from Harvard University.

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