Sunday, August 4, 2013

A Tutorial on Bond Yields and Returns

Articles like this one can be confusing to someone not familiar with how fixed-income securities work. Bond yields have gone up, and so bond prices have gone down.  Because the Federal Reserve holds a significant position in different types of bonds, some note that it has "caused a mark-to-market loss of $192 billion on the Fed's holding assets, equivalent to approximately all of the unrealized gains that the Fed had accumulated [emphasis mine]."

Mark-to-market can be dramatically misleading when dealing with fixed-income securities.  They are, after all, fixed income.  Bonds are initially priced with one future set of paths of interest rates in mind.  When new information comes, and interest rates change, their current price naturally changes to fit this new interest rate:  otherwise, there would be an arbitrage opportunity.  However, absent any default risk, the end point of bonds remains fixed:  their price path swivels.  Below, Corrections plots the interest rate path:  in blue, the expected interest rate path, a constant 5%, with a "surprise" in the second period (4 periods before bond maturity) in which interest rates rise to 12% (click to enlarge).
The consequence of these interest rate changes is given simply by calculating the bond prices backwards from maturity (click to enlarge).  Notice that while the surprised bond path went down, the slope increased, as it must catch up to its initial path by maturity.
This idea can be seen most easily by taking the log of these two price paths (click to enlarge).  When the shock hits, the bond price is 0.194 log points lower than it would have been otherwise (the vertical distance between the initial shocked path and the unshocked path).  Not-coincidentally, our slope changed from 0.0488 to 0.113:  a change of 0.0645 log points.  Added together three times for the three remaining periods, they exactly make up the difference, summing to 0.194 again.  
The idea, therefore, is that while the price of bonds has declined and the Fed could lose money in a mark-to-market sense, it now holds bonds that have higher yields, and will make money faster.  Consequently, they only have to hold to maturity to not lose money.  This is especially true the change was due to mid-term rates, as it is empirically true that interest rates are long-term mean reverting.  Such a temporary, mid-course change in interest rates (click to enlarge) yields a temporary change in bond prices (click to enlarge).  Such mark-to-market accounting can be particularly misleading when dealing with price changes in fixed-income securities due to interest rate changes, rather than default risk, as is the case currently.   

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