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 currrently 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.

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 currrently 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.

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