The Term Structure in the Data

Investment Theory

Lorenzo Naranjo

Fall 2026

The Question

  • Forward rates contain an expected future short rate and a term premium: f_{t}(n - 1, n) = \operatorname{E}_{t}\left( y_{t + n - 1}(1) \right) + \mathit{TP}_{t}(n).
  • The term premium is the expected excess return of a long bond over a short one.
  • The expectations hypothesis: the term premium is zero, or constant.
  • This chapter takes the hypothesis to the data:
    • How much have long-term Treasuries earned?
    • Do forward rates forecast short rates or bond returns?
    • How have term premia and the risk of bonds changed?

Data and Notation

  • Fama-Bliss zero-coupon Treasury yields, 1 to 5 years, 1964 to 2025.
  • The Treasury yield curve of Gürkaynak, Sack and Wright for longer maturities.
  • Continuously compounded yields y_{t}(n), log prices p_{t}(n) = -n \, y_{t}(n), and forwards f_{t}(n - 1, n) = p_{t}(n - 1) - p_{t}(n).
  • Hold an n-year bond for a year; its excess log return is \mathit{rx}_{t+1}(n) = p_{t+1}(n - 1) - p_{t}(n) - y_{t}(1).

Average Bond Returns

Mean (%) Volatility (%) Sharpe ratio
2 years 0.36 1.70 0.21
3 years 0.68 3.12 0.22
4 years 0.97 4.34 0.22
5 years 1.04 5.38 0.19
  • One-year excess returns of Fama-Bliss zeros, 1964 to 2024.
  • A 10-year zero earned 2.7% a year over bills since 1971, a Sharpe ratio of 0.25 against 0.48 for stocks.
  • A premium, but a small one per unit of risk, and inflated by four decades of falling rates.

Do Forward Rates Forecast Short Rates?

  • Fama and Bliss (1987): regress the change in the one-year yield on the forward spread, y_{t+n-1}(1) - y_{t}(1) = a + b \left( f_{t}(n - 1, n) - y_{t}(1) \right) + \varepsilon.
    • The expectations hypothesis predicts b = 1.
  • Or regress the bond’s excess return on the same spread, \mathit{rx}_{t+1}(n) = a + b \left( f_{t}(n - 1, n) - y_{t}(1) \right) + \varepsilon.
    • The expectations hypothesis predicts b = 0.

Two Sides of the Same Coin

  • For the 2-year bond, the forward spread splits into the change in the short rate and the excess return: f_{t}(1, 2) - y_{t}(1) = \left( y_{t+1}(1) - y_{t}(1) \right) + \mathit{rx}_{t+1}(2).
  • Regressing both sides on the forward spread, the two slopes add up to exactly one.
  • Whatever the forward spread does not forecast about the short rate, it forecasts about the bond’s return.

Forward Rates Forecast Returns

A grouped bar chart with maturities 2 to 5 years. Slopes for the change in the one-year rate rise from about 0.24 to about 0.74, below the dashed line at one. Slopes for excess returns are about 0.8 to 1.2, far above the dashed line at zero.

Slopes of the Fama-Bliss regressions on the forward spread, 1964 to 2025. The dashed lines are the predictions of the expectations hypothesis.

  • Short-rate slopes rise with the horizon, 0.24 to 0.74; return slopes are 0.8 to 1.2, with t-statistics above 3.1.

Long Yields Move the Wrong Way

  • Campbell and Shiller (1991): under the expectations hypothesis, a steep curve means the long yield should rise over the next year.
  • Regress y_{t+1}(n - 1) - y_{t}(n) on (y_{t}(n) - y_{t}(1))/(n - 1): the hypothesis predicts a slope of one.
  • The slopes are negative, from -0.52 at two years to -1.23 at five.
  • A steep curve forecasts rising short rates over the long term, but a falling long yield over the short term.
  • Both tests point to a time-varying term premium.

Example 1 (An Excess Bond Return) Zero prices are 0.96, 0.92 and 0.88 for one, two and three years, and a year later the two-year zero sells for 0.93. The one-year yield is -\ln 0.96 = 4.08\%, the forward rate f(2, 3) = \ln 0.92 - \ln 0.88 = 4.45\%, and the excess return of the three-year bond is \ln 0.93 - \ln 0.88 - 0.0408 = 1.44\%, well above the forward spread of 0.37%.

One Factor in Bond Risk Premia

A line chart of five coefficients. The 1964 to 2003 line has a tent shape, about minus 2 on the one-year yield, about plus 3 on the forward from two to three years, and about minus 2 on the last forward. The 2004 to 2024 line has a different shape, positive on the one-year yield, about minus 2 on the first forward and positive on the last two.

Coefficients of the regression of the average excess return of the two- to five-year bonds on the one-year yield and four forward rates.

  • Cochrane and Piazzesi (2005): a tent-shaped combination of forwards, R^{2} up to 0.44; here 0.35 to 2003 and 0.21 after.

How Much to Trust the Forecasts?

  • The forecasting variables are persistent and the returns overlap.
  • Bauer and Hamilton (2018): the usual tests overstate the evidence that variables beyond level, slope and curvature forecast bond returns.
  • The broad conclusion survives: forward rates forecast returns better than short rates, so term premia vary.
  • Dai and Singleton (2002): models in which the price of interest rate risk varies with the state of the economy reproduce both puzzles.

Term Premia over Time

A line chart from 1961 to 2025. The term premium rises from about 0.5 percent in the 1960s to above 5 percent in the early 1980s, declines unevenly for three decades, turns negative in the mid-2010s, reaches about minus 1.4 percent in 2020, and rises above zero after 2022.

Estimated term premium in the 10-year Treasury yield, Adrian et al. (2013), published by the New York Fed.

  • Peak 5.2% in May 1984, low -1.4% in July 2020, 0.75% at the end of 2025.

What Moves the Term Premium?

  • Supply: Greenwood and Vayanos (2014): a longer maturity of Treasury debt relative to GDP raises yields and future excess returns.
    • Federal Reserve purchases of long bonds after 2008 worked the other way.
  • Safety and liquidity: Krishnamurthy and Vissing-Jorgensen (2012): Treasuries’ special value lowered their yields by 73 basis points on average, 1926 to 2008.
  • Risk: how bonds do when other investments do badly.

Bonds as Hedges

A line chart from 1976 to 2025. The correlation is positive, between about 0.1 and 0.6, until the late 1990s, first falls below zero in late 2001, stays mostly between minus 0.6 and zero from 2005 to 2021, and rises to between about 0.25 and 0.55 in 2024 and 2025.

Correlation between the monthly excess returns of a 10-year zero and the stock market, rolling 60-month windows.

  • 0.29 over 1971 to 1999 and -0.11 over 2000 to 2025; clearly positive again in 2024 and 2025.

Inflation Bets or Deflation Hedges?

  • Campbell et al. (2017):
    • Inflation risk: stocks and bonds fall together; bonds earn a high premium.
    • Recession risk: bonds rise when stocks fall; bonds hedge and earn a low premium.
  • Campbell et al. (2020): inflation and output became positively correlated in 2001.
  • Pflueger (2025): bonds and stocks co-move with volatile supply shocks and an inflation-focused Fed.
    • After the pandemic: negative betas through 2022, with a Fed that responded late.

Real Rates since 2021

  • Nominal yield = real yield (TIPS) + breakeven inflation.
  • Correlation of stock returns with monthly changes in 10-year yields:
Nominal yield Breakeven Real yield
2010 to 2019 0.49 0.66 0.07
2021 to 2025 -0.54 0.25 -0.69
  • Breakeven: 2.33% in 2025, as in 2003 to 2007. Real yield: -1.06% to 1.93% from 2020 to 2025.
  • Higher real rates raise the discount rate on stocks and bonds alike.

Example 2 (Bonds in a 60/40 Portfolio) With 60% in stocks of volatility 16% and 40% in bonds of volatility 11%, the portfolio’s variance is 0.6^{2} \times 0.16^{2} + 0.4^{2} \times 0.11^{2} + 2 \times 0.6 \times 0.4 \times \rho \times 0.16 \times 0.11. With \rho = -0.3 its volatility is 9.3%; with \rho = +0.5 it is 12.4%.

Takeaways

  • Long-term Treasuries have earned a premium over bills, small per unit of risk.
  • Forward rates forecast bond returns better than short rates: the term premium varies.
    • The forecasts are less reliable than their in-sample fit suggests.
  • Term premia have ranged from about 5% to below zero, moved by supply, the special value of Treasuries and risk.
  • Whether bonds hedge stocks depends on what moves yields in bad times.
    • Since 2021: real yields, not expected inflation.

References

Adrian, Tobias, Richard K. Crump, and Emanuel Moench. 2013. “Pricing the Term Structure with Linear Regressions.” Journal of Financial Economics 110: 110–38.
Bauer, Michael D., and James D. Hamilton. 2018. “Robust Bond Risk Premia.” The Review of Financial Studies 31 (2): 399–448.
Campbell, John Y., Carolin Pflueger, and Luis M. Viceira. 2020. “Macroeconomic Drivers of Bond and Equity Risks.” Journal of Political Economy 128 (8): 3148–85.
Campbell, John Y., and Robert J. Shiller. 1991. “Yield Spreads and Interest Rate Movements: A Bird’s Eye View.” Review of Economic Studies 58 (3): 495–514.
Campbell, John Y., Adi Sunderam, and Luis M. Viceira. 2017. “Inflation Bets or Deflation Hedges? The Changing Risks of Nominal Bonds.” Critical Finance Review 6: 263–301.
Cochrane, John H., and Monika Piazzesi. 2005. “Bond Risk Premia.” American Economic Review 95 (1): 138–60.
Dai, Qiang, and Kenneth J. Singleton. 2002. “Expectation Puzzles, Time-Varying Risk Premia, and Affine Models of the Term Structure.” Journal of Financial Economics 63: 415–41.
Fama, Eugene F., and Robert R. Bliss. 1987. “The Information in Long-Maturity Forward Rates.” American Economic Review 77 (4): 680–92.
Greenwood, Robin, and Dimitri Vayanos. 2014. “Bond Supply and Excess Bond Returns.” The Review of Financial Studies 27 (3): 663–713.
Krishnamurthy, Arvind, and Annette Vissing-Jorgensen. 2012. “The Aggregate Demand for Treasury Debt.” Journal of Political Economy 120 (2): 233–67.
Pflueger, Carolin. 2025. “Back to the 1980s or Not? The Drivers of Inflation and Real Risks in Treasury Bonds.” Journal of Financial Economics 167: 104027.