## Danish Mortgages & Negative Equity

In his new book, Richard Posner (HT Tyler Cowen via Steve) notes an interesting aspect of the Danish mortgage system I’ve previously advocated:

The Danish model has another critical and innovative feature. Holders can retire their own mortgages by purchasing the same face amount of mortgage bonds at the prevailing market price. To prepay a mortgage by purchasing bonds, the home owner must give advance notice of several weeks to the MCI [mortgage credit institutions], which designates by lottery the specific bonds to be purchased. Thus, if rising interest rates or other factors cause mortgage bonds to trade at a discount, home owners can reduce the principal or retire the whole mortgage by purchasing an appropriate mortgage bond at a discount.

Cowen terms this “implicit insurance against the prospect of negative equity in the home”, a feature previously noted by Alan Boyce.

When I first read Boyce’s paper & presentation, it took me a little while to explain – to myself – exactly how the Danish system (effectively) ensured against negative equity. This post is my attempt at such an explanation.

1. Recall that, under the Danish mortgage system, a given mortgage is funded by bonds having the same interest rate, face value, & duration as the mortgage itself. These bonds are “callable“, in that a borrower can pay them off at will prior to their maturity date.

2. Note that a mortgage’s principal (P1) is, basically, the Present Value (PV1) of a stream of payments (S1) at the interest rate (I1) & term (T1) of the mortgage in question (*). Of course, if the prevailing interest rate rises to a level (I2) greater than I1, PV1 decreases (and, with it, the value of a given mortgage, and the bonds associated therewith).

3. However, due to the mathematics of amortization calculations, that same higher interest rate (I2) would also decrease the amount of principal (P2) that a notional borrower (having creditworthiness equaling that of the original borrower) could borrow, assuming the same term (T1) & monthly payment stream (S1) as the original borrower. This is the mechanism whereby rising interest rates clobber house prices.

4. Ergo, given #2 & #3: Changes in interest rates affect the values of both mortgage bonds & houses in the same way, via the same basic mechanism.

5. Hence, under a Danish-style system of mortgage financing, you’d never have to worry about rising interest rates leading to underwater homeowners (**). A homeowner’s house value might fall as a result of rising interest rates, but the market value of a quantity of bonds equal to the face value of that homeowner’s mortgage would fall by approximately the same proportion.

(*) This makes sense when you think about it; from the standpoint of a bondholder owning a 100% interest in a given mortgage, the “principal” is basically the amount the bondholder is willing to pay in exchange for receiving a stream of payments – w/ interest – over a certain span of time. That amount, of course, is the present value of such a stream.

(**) Of course, negative equity could still result from other causes – e.g., localized insanity leading to a housing bubble (and subsequent bust) in a given region.

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