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Don founded Kamakura Corporation in April 1990 and currently serves as its chairman and chief executive officer where he focuses on enterprise wide risk management and modern credit risk technology. His primary financial consulting and research interests involve the practical application of leading edge financial theory to solve critical financial risk management problems. Don was elected to the 50 member RISK Magazine Hall of Fame in 2002 for his work at Kamakura. Read More

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kamakura blog
Oct 4

Written by: Donald van Deventer
10/4/2017 12:11 AM 

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The magnitude of a “term premium” or risk premium in long term U.S. Treasury yields is a major focus of research by economists in the Federal Reserve System. A recent paper by Canlin Li, Andrew Meldrum, and Marius Rodriguez summarizes two important papers on this topic and reviews their methodologies. Adrian, Crump, Mills and Moench (2014) summarize their term premium findings as follows: “The evolution of term premia has been of particular interest since the Federal Reserve began large-scale asset purchases. Over this time, short-term interest rates have been close to zero, and our estimates show that the term premium has been compressed and has at times even been negative.” Estimates of the term premium are a function of the data used, the modeling approach taken, and market expectations. The focus of this note is a simple one: the calculation of historical realized term premiums and “in progress” term premiums. A historical perspective on actual realized term premiums is a potentially important contributor to market expectations, subject to the caveat stated by Robert A. Jarrow, “History is just one draw from a Monte Carlo simulation.”

We seek to answer this question: “Which investment has provided the best total dollar return to investors, a U.S. Treasury bond maturing in X years or a money market fund that invests only in Treasury bills?” We address the answer to that question with respect to U.S. Treasury fixed rate bonds with maturities of 1, 2, 3, 5, 7, 10, 20 and 30 years.


We use the time series of U.S. Treasury yields maintained by the U.S. Department of the Treasury and distributed by the Board of Governors of the Federal Reserve.2 We assume that on each day for which data is available, an investor invests $1000 in the U.S. Treasury bond and $1000 in 6-month Treasury bills. Every six months, the investor will receive a coupon on the bond. We assume that cash from the coupon payment is invested in 6-month Treasury bills and that this investment is rolled over in new 6-month Treasury bills until the underlying bond matures. The investment in the “money fund” starts with an investment in 6-month Treasury bills and all cash thrown off is reinvested in new 6-month Treasury bills until the underlying investment in the fixed rate bond matures. Interest on the six-month bills is calculated on an actual/365 day basis because the U.S. Treasury data series for short term rates is on an “investment” basis.

The payment dates, 6-month bill rates, and the value of the “money fund” are given in this spreadsheet.

The total dollar returns on 1, 2, 3, 5, 7, 10, 20, and 30-year Treasury bonds, both realized and “in progress,” are given in a separate spreadsheet:

Summary Results

The results of the term premium analysis are summarized in the following table:

The results show that realized total dollar excess returns have never been negative for maturities of 10, 20, and 30 years. Moreover, pending “in progress” excess returns have never been negative (so far) for maturities of 5, 7, 10, 20 and 30 years. At the shorter maturities, results are mixed. For 1 year, 74.10% of realized dollar excess returns have been positive and 56.35% of pending dollar excess returns are positive at this point in time. For the 2-year maturity, the same probabilities of positive excess dollar returns are 81.96% (realized) and 88.49% (pending). At three years, the results are 84.14% (realized) and 99.34% (pending).

Graphical Analysis

Appendix A shows the evolution of total dollar return (in red) versus the 6-month T-bill money fund total dollar return (in blue) for completed 1, 2, 3, 5, 7, 10, 20 and 30-year terms beginning January 4, 1982. For most holding periods, interest rates have declined substantially over the investment periods that have been completed. The graphs in Appendix A also include the evolution of the initial six-month Treasury bill yields (in light blue) and the relevant fixed rate Treasury bond yields (in pink) on the origination dates.

Appendix B shows the histogram of realized dollar excess returns for each of the 8 fixed rate maturities. Appendix C reports on the histograms of “in progress” pending dollar excess returns for each maturity. For these incomplete periods, we define the “pending excess return” as the known dollar advantage of the fixed-rate Treasury bond over the 6-month T-bill investment as of the observation date. For example, on August 28, 2017, the amount of interest that will be paid on a 6-month Treasury bill equivalent that matures in 6 months (not necessarily 182 days) on February 28, 2018 is known with certainty. We tally all of the pending dollar term premiums, all of which have a differing time to maturity, in the histograms in Appendix C.

One of the reasons for these surprisingly positive term premium results is the large size of the interest earned on coupon payments, which are assumed to be invested in 6-month Treasury bills. The highest total realized dollar return for all of the completed fixed-rate periods are shown in Appendix D. The lowest total realized dollar returns for each of the fixed-rate holding periods are shown in Appendix E.


The total dollar returns for the 10, 20 and 30-year Treasury bond have exceeded the total dollar returns for the 6-month T-bill money fund on every completed and every partially completed holding period for which data is available. This is due in part because interest earned on coupon payments rises when rates rise over the holding period, and, at least so far, this has been enough to generate strictly positive excess returns for 30-year bonds. Even at maturities of 5 and 7 years, 100% of pending excess dollar returns are positive as of today.

It goes without saying that there is no guarantee that the future will be like the past. At the same time, expectations for the future should be set with the past in mind.


Adrian, T., R. K. Crump, B. Mills and E. Moench (2014), 'Treasury Term Premia: 1961-Present', Liberty Street Economics, available at:

Adrian, T., R. K. Crump and E. Moench (2013), 'Pricing the Term Structure with Linear Regressions', Journal of Financial Economics 110(1), pp. 110-138.

Li, Canlin, Andrew Meldrum, and Marius Rodriguez (2017). "Robustness of long-maturity term premium estimates," FEDS Notes. Washington: Board of Governors of the Federal Reserve System, April 3, 2017,

Heath, David, Robert A. Jarrow and Andrew Morton (1992), ”Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claim Valuation,” Econometrica, 60(1), pp. 77-105.

Kim, D. H. and J. H. Wright (2005), 'An Arbitrage-Free Three-Factor Term Structure Model and the Recent Behavior of Long-Term Yields and Distant-Horizon Forward Rates', Federal Reserve Board Finance and Economics Discussion Series 2005-33.

Appendix A: Time Series of Period End Bond and Money Market Values

Appendix B: Histogram of Historical Realized Excess Returns

Appendix C: Histogram of Pending Realized Excess Returns

Appendix D: Highest Realized Total Dollar Returns

Appendix E: Lowest Realized Total Dollar Return