Bond Math: Duration Risk at the Zero Boundary

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?