Bond Math: Duration Risk at the Zero Boundary

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February 9th, 2012 by Econompic Data

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via Econom­pic Data

There seems to be lots of con­fu­sion sur­round­ing Bill Gross' lat­est Invest­ment Out­look, Life and Death Propo­si­tion. First, some back­ground of what Bill Gross stated...

Investors aren't only con­cerned with credit risk (i.e. the abil­ity to get paid back), but also dura­tion risk (the risk of lend­ing for an extended period of time in fear that rates may rise).
In Bill's words:

What per­haps is not so often rec­og­nized is that liq­uid­ity can be trapped by the “price” of credit, in addi­tion to its “risk.” Cap­i­tal­ism depends on risk-taking in sev­eral forms. Devel­op­ers, home­own­ers, entre­pre­neurs of all shapes and sizes epit­o­mize the risk­i­ness of busi­ness build­ing via equity and credit risk exten­sion. But mod­ern cap­i­tal­ism is depen­dent as well on matu­rity exten­sion in credit mar­kets. No ven­ture, aside from one financed with 100% own­ers’ cap­i­tal, could sur­vive on credit or loans that matured or were callable overnight. Build­ings, util­i­ties and homes require 20– and 30-year loan com­mit­ments to smooth and jus­tify their returns.

Investors had been will­ing to take on this dura­tion risk because they would be com­pen­sated with addi­tional yield AND (this is impor­tant) because bonds could appre­ci­ate if rates fell (i.e. when yields fall, bonds rise).

Back to Bill:

Because this is so, lenders require a yield pre­mium, expressed as a pos­i­tively sloped yield curve, to make the extended loan. A flat yield curve, in con­trast, is a dis­in­cen­tive for lenders to lend unless there is suf­fi­cient down­side room for yields to fall and pro­vide bond mar­ket cap­i­tal gains.

And although the yield curve is steep, it is very low in nom­i­nal terms (i.e. there is less room for rates to move down).
Last time to Bill for his main argument:

Even if nod­ding in agree­ment, an observer might imme­di­ately com­ment that today’s yield curve is any­thing but flat and that might be true. Most short to inter­me­di­ate Trea­sury yields, how­ever, are dan­ger­ously close to the zero-bound which imply lit­tle if any room to fall: no mar­gin, no air under­neath those bond yields and there­fore lim­ited, if any, price appre­ci­a­tion. What incen­tive does a bank have to buy two-year Trea­suries at 20 basis points when they can park overnight reserves with the Fed at 25? What incen­tives do invest­ment man­agers or even indi­vid­ual investors have to take price risk with a five-, 10– or 30-year Trea­sury when there are mul­ti­ples of down­side price risk com­pared to appre­ci­a­tion? At 75 basis points, a five-year Trea­sury can only ratio­nally appre­ci­ate by two more points, but the­o­ret­i­cally can go down by an unlim­ited amount. Dura­tion risk and flat­ness at the zero-bound, to make the sim­ple point, can freeze and trap liq­uid­ity by con­vinc­ing investors to hold cash as opposed to extend credit.

Now my over­sim­pli­fied expla­na­tion using two inter­est rate scenarios...

Sce­nario one... bonds yield­ing 5%.
In this sce­nario, bonds with matu­ri­ties 1 year through 5 are yield­ing 5%. Should rates stay at 5%, the bonds are worth PAR (i.e. $100) in all sce­nar­ios. How­ever, the bonds have the poten­tial to appre­ci­ate should yields move lower. In fact, should rates fall all the way to 1% (a huge decline, but this is meant to illus­trate the point), the bonds actu­ally appre­ci­ate almost 20% in the case of the 5 year Trea­sury. Com­pare that to the one year Trea­sury that gained less than 5%.
In other words, in a flight to qual­ity sce­nario there is a HUGE incen­tive to own the longer dura­tion bond when yields have room to com­press.



Sce­nario two... bonds yield­ing 1%.

In this sce­nario, bonds with matu­ri­ties 1 year through 5 are yield­ing 1% (yes the yield curve is upward slop­ing in "real life", but this isn't far off). Should rates stay at 1%, the bonds are again worth PAR (i.e. $100), but in this case they have lim­ited room to move due to the zero bound­ary. Should rates move all the way to 0%, the five year bonds don't appre­ci­ate 20% like in sce­nario 1, they appre­ci­ate only 5%, while the one year Trea­sury appre­ci­ates around 1%.
In other words, in a flight to qual­ity sce­nario the poten­tial ben­e­fit of a longer dura­tion Trea­sury is 75% lower than in sce­nario one and only 4% higher than the one year matu­rity bond.




The exam­ple above is close to cur­rent rates (as of this writ­ing, a five year bond yields 0.82%). The result, as Bill Gross points out, is a lack of incen­tive for a lender to lend and take that risk as they can get roughly the same yield just putting their money in a mat­tress with­out the risk of rates mov­ing higher (0% isn't far from 0.82%). In addi­tion, for an investor that is allo­cat­ing to bonds to diver­sity their equity hold­ings, fixed income will no longer appre­ci­ate in a flight to qual­ity sce­nario to off­set equity losses. As a result, busi­nesses should in the­ory be hav­ing a hard time get­ting money for their busi­nesses out­side of equity financ­ing.
But, the evi­dence doesn't point to any of this being an issue. As far as I know, investors are still will­ing to extend the dura­tion of their invest­ments to pick up this incre­men­tal yield. And why not? The Fed has made it clear there is zero risk that rates will rise going out to at least 2014. So why not pocket that addi­tional 82 bps regard­less of the lack of cap­i­tal appreciation?

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