Prof. John Cochrane recently posted a thoughtful piece on the difficulties of extracting the market's expected future interest rates from forward rates. Kamakura Corporation, under the leadership of Managing Director for Research Prof. Robert Jarrow, agrees with Prof. Cochrane that this is a very important topic for central bankers, market participants and the academic community. The graph above plots 3-month U.S. Treasury (NYSEARCA:TLT) (NYSEARCA:TBT) (NYSEARCA:SHY) (NYSEARCA:IEF) forward rates versus the expected future 3-month U.S. Treasury rates. As explained below, the forward rates contain a risk premium and the expected future 3-month U.S. Treasury rates do not. The expected future 3-month U.S. Treasury rates were derived using a 9 factor constant coefficient (affine) Heath, Jarrow and Morton model of U.S. Treasury movements. The coefficients of the model were estimated by Kamakura Risk Information Services using data from the U.S. Department of the Treasury spanning the period from 1962 through September 30, 2014.

The graph below plots the current U.S. Treasury yield curve versus a yield curve constructed from the market's expected future 3-month Treasury bill rates. The difference between the two curves is the "term premium" in the U.S. Treasury yield curve, the reward for accepting the risk of potential interest increases.

**Background on Forward Rates**

The forward rates are derived from the U.S. Treasury zero coupon yield curve produced by Kamakura Risk Information Services using Kamakura Risk Manager, version 8.1. The U.S. Treasury curve is created by the maximum smoothness forward rate method of Adams and van Deventer [1994], which was recently confirmed as consistent with "no arbitrage" standards of Heath, Jarrow and Morton in an important paper by Kamakura Managing Director Prof. Robert A. Jarrow. The issuer's zero coupon yield curve was created by applying the maximum smoothness forward rate approach to zero coupon credit spreads, relative to the base U.S. Treasury curve.

**Forward Rates versus Expected Future Interest Rates**

Many readers who are not familiar with forward rate calculations assume they are a forecast of interest rates with no more credibility than any other forecast. This is an error of understanding that we seek to clarify here. A forecast is a product of judgment, sometimes combined with analytics. Forward rates, which we often label an "implied forecast," involve no judgment. Forward rates are the mechanical calculation of breakeven yields such that investing in any maturity U.S. Treasury and rolling it over for 30 years will yield the same amount of cash in 30 years as investing in any other maturity. For example, if one is given the 4-week U.S. Treasury bill yield and the 13-week Treasury bill yield today, one can calculate how much the 9-week Treasury bill must yield in 4 weeks for a strategy of buying the 3-month Treasury bill or the one month Treasury bill, followed by reinvestment in a 9-week bill, to yield the same amount of cash in 13 weeks. This calculation involves no more judgment than the calculation of the yield to maturity on a bond.

How does one use forward rates to forecast interest rates or to simulate them forward for asset and liability management or other risk management purposes? A series of authors beginning with Ho and Lee in 1985 and then followed by Heath, Jarrow, and Morton in a number of papers beginning in the late 1980s answered this question. The future path of interest rates must bear a specific link to forward rates that depends on the number of random factors driving interest rate movements and the nature of the volatility of those factors. In general, researchers agree that forward rates are biased higher than the average actual rates that will come about because long-term investors earn a risk premium for their long-term commitment of funds. The "expectations hypothesis" taught in economics for at least the last five decades postulated that, contrary to recent findings, forward rates were on average an unbiased forecast of interest rates. This is now understood to be incorrect. Forward rates play an important role, however, in understanding where actual rates will end up.

The empirical relationship between forward rates and actual rates, using actual historical data from multiple countries, is a subject we will return to in the near future. For an introduction to the topic, we encourage readers to review recent Federal Reserve research by Swanson (2007), Rosenberg and Maurer (2008), Kim and Orphanides (2007), and Adrian, Crump, Mills and Moensch (2014). We refer readers with a technical background to Kim and Wright (2005) and the 2005 and 2008 papers by John Cochrane and Monika Piazzesi on this topic. Other excellent papers include Adrian, Crump and Moensch (2013), Durham (2014), Rudebusch, Sack and Swanson (2007), and Cochrane (2007).

**Disclosure:** The author has no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours.

The author wrote this article themselves, and it expresses their own opinions. The author is not receiving compensation for it (other than from Seeking Alpha). The author has no business relationship with any company whose stock is mentioned in this article.