{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,10]],"date-time":"2026-03-10T00:50:29Z","timestamp":1773103829073,"version":"3.50.1"},"reference-count":50,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2023,12,28]],"date-time":"2023-12-28T00:00:00Z","timestamp":1703721600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"FCT (Portuguese Science Foundation) through national funds under project","award":["CEMAPRE\/REM\u2014UIDB\/05069\/2020"],"award-info":[{"award-number":["CEMAPRE\/REM\u2014UIDB\/05069\/2020"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematics"],"abstract":"<jats:p>The unbiased expectation theory stipulates that long-term interest rates are determined by the market\u2019s expectations of future short-term interest rates. According to this hypothesis, if investors have unbiased expectations about future interest rate movements, the forward interest rates should be good predictors of future spot interest rates. This hypothesis of the term structure of interest rates has long been a subject of debate due to empirical and theoretical challenges. Despite extensive research, a satisfactory explanation for the observed systematic difference between future spot interest rates and forward interest rates has not yet been identified. In this study, we approach this issue from an arbitrage theory perspective, leveraging on the connection between the expectation hypothesis and changes in probability measures. We propose that the observed bias can be explained by two adjustments: a risk premia adjustment, previously considered in the literature, and a stochastic adjustment that has been overlooked until now resulting from two measure changes. We further demonstrate that for specific instances of the Vasicek and Cox, as well as the Ingersoll and Ross, stochastic interest rate models, quantifying these adjustments reveals that the stochastic adjustment plays a significant role in explaining the bias, and ignoring it may lead to an overestimation of the required risk premia\/aversion adjustment. Our findings extend beyond the realm of financial economic theory to have tangible implications for interest rate modelling. The capacity to quantify and distinguish between risk and stochastic adjustments empowers modellers to make more informed decisions, leading to a more accurate understanding of interest rate dynamics over time.<\/jats:p>","DOI":"10.3390\/math12010105","type":"journal-article","created":{"date-parts":[[2023,12,28]],"date-time":"2023-12-28T03:30:34Z","timestamp":1703734234000},"page":"105","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On the Bias of the Unbiased Expectation Theory"],"prefix":"10.3390","volume":"12","author":[{"given":"Renato","family":"Fran\u00e7a","sequence":"first","affiliation":[{"name":"Darras Consulting, 1269-046 Lisbon, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3294-3962","authenticated-orcid":false,"given":"Raquel M.","family":"Gaspar","sequence":"additional","affiliation":[{"name":"REM Cemapre and ISEG, Universidade de Lisboa, 1200-781 Lisbon, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2023,12,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"783","DOI":"10.1090\/S0002-9904-1930-05048-X","article-title":"The Theory of Interest as Determined by Impatience to Spend Income and Opportunity to Invest it","volume":"36","author":"Fisher","year":"1930","journal-title":"Bull. Am. Math."},{"key":"ref_2","unstructured":"Hicks, J.R. (1939). Value and Capital: An Inquiry into Some Fundamental Principles of Economic Theory, Oxford University Press."},{"key":"ref_3","first-page":"19","article-title":"Price-expectations effects on interest rates","volume":"25","author":"Gibson","year":"1970","journal-title":"J. Financ."},{"key":"ref_4","first-page":"854","article-title":"Interest rates and inflationary expectations: New evidence","volume":"62","author":"Gibson","year":"1972","journal-title":"Am. Econ. Rev."},{"key":"ref_5","first-page":"751","article-title":"Price-Expectations Effects on Interest Rates: Reply","volume":"28","author":"Gibson","year":"1973","journal-title":"J. Financ."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"891","DOI":"10.1086\/260614","article-title":"The Gibson paradox and historical movements in real interest rates","volume":"85","author":"Shiller","year":"1977","journal-title":"J. Political Econ."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1016\/0304-405X(84)90013-8","article-title":"The information in the term structure","volume":"13","author":"Fama","year":"1984","journal-title":"J. Financ. Econ."},{"key":"ref_8","first-page":"680","article-title":"The information in long-maturity forward rates","volume":"77","author":"Fama","year":"1987","journal-title":"Am. Econ. Rev."},{"key":"ref_9","first-page":"495","article-title":"Yield Spreads and Interest Rate Movements: A Bird\u2019s Eye View","volume":"58","author":"Campbell","year":"1991","journal-title":"J. Fiancial Econ."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1016\/S0304-405X(97)00007-X","article-title":"On biases in tests of the expectations hypothesis of the term structure of interest rates","volume":"44","author":"Bekaert","year":"1997","journal-title":"J. Financ. Econ."},{"key":"ref_11","unstructured":"Boudoukh, J., Richardson, M., and Whitelaw, R.F. (1995). The Information in Long Maturity Forward Rates: Implications for Exchange Rates and the Forward Premium Anomaly, National Bureau of Economic Research. Working Paper 11840."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1093\/rfs\/hhs108","article-title":"A long-run risks explanation of predictability puzzles in bond and currency markets","volume":"26","author":"Bansal","year":"2013","journal-title":"Rev. Financ. Stud."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1481","DOI":"10.1111\/j.1540-6261.2004.00670.x","article-title":"Risks for the long run: A potential resolution of asset pricing puzzles","volume":"59","author":"Yaron","year":"2004","journal-title":"J. Financ."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2068","DOI":"10.1287\/mnsc.1070.0796","article-title":"Affine general equilibrium models","volume":"54","author":"Eraker","year":"2008","journal-title":"Manag. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1086\/ma.21.25554958","article-title":"Equilibrium yield curves, with comments and discussion","volume":"21","author":"Piazzesi","year":"2006","journal-title":"NBER Macroecon. Annu."},{"key":"ref_16","first-page":"371","article-title":"Risk Premiums in the term structure","volume":"24","author":"Backus","year":"1989","journal-title":"J. Econ."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1269","DOI":"10.2307\/1911873","article-title":"Generalized instrumental variables estimation of Nonlinear rational expectations models","volume":"50","author":"Hansen","year":"1982","journal-title":"Econometrica"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1016\/0304-405X(77)90016-2","article-title":"An Equilibrium Characterization of the Term Structure","volume":"5","author":"Vasicek","year":"1977","journal-title":"J. Financ. Econom."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"385","DOI":"10.2307\/1911242","article-title":"A Theory of the Term Structure of Interest Rates","volume":"53","author":"Cox","year":"1985","journal-title":"Econometrica"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"363","DOI":"10.2307\/1911241","article-title":"An Intertemporal General Equilibrium Model of Asset Prices","volume":"53","author":"Cox","year":"1985","journal-title":"Econometrica"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"631","DOI":"10.1093\/rfs\/hhg010","article-title":"Term structure dynamics in theory and reality","volume":"16","author":"Dai","year":"2003","journal-title":"Rev. Financ. Stud."},{"key":"ref_22","first-page":"1259","article-title":"Interest rate volatility and the term structure: A two-factor general equilibrium model","volume":"47","author":"Longstaff","year":"1992","journal-title":"J. Financ."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"905","DOI":"10.1093\/rfs\/14.4.905","article-title":"Assessing asset pricing anomalies","volume":"14","author":"Brennan","year":"2001","journal-title":"Rev. Financ. Stud."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"138","DOI":"10.1257\/0002828053828581","article-title":"Bond risk premia","volume":"95","author":"Cochrane","year":"2005","journal-title":"Am. Econ. Rev."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1016\/j.jbankfin.2015.03.018","article-title":"Can behavioral biases explain the rejections of the expectation hypothesis of the term 504 structure of interest rates?","volume":"58","author":"Bulkley","year":"2015","journal-title":"J. Bank. Financ."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"473","DOI":"10.1086\/296409","article-title":"Parsimonious modeling of yield curves","volume":"60","author":"Nelson","year":"1987","journal-title":"J. Bus."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Svensson, L.E. (1994). Estimating and Interpreting Forward Interest Rates: Sweden 1992\u20131994, National Bureau of Economic Research. Working Paper 4871.","DOI":"10.3386\/w4871"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1111\/1467-9965.00072","article-title":"Interest rate dynamics and consistent forward rate curves","volume":"9","author":"Christensen","year":"1999","journal-title":"Math. Financ."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"54","DOI":"10.3905\/jfi.1991.692347","article-title":"Common factors affecting bond returns","volume":"1","author":"Litterman","year":"1991","journal-title":"J. Fixed Income"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1943","DOI":"10.1111\/0022-1082.00278","article-title":"Specification analysis of affine term structure models","volume":"55","author":"Dai","year":"2000","journal-title":"J. Financ."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Guidolin, M., and Thornton, D.L. (2008). Predictions of Short-Term Rates and the Expectations Hypothesis of the Term Structure of Interest Rates, European Central Bank.","DOI":"10.2139\/ssrn.1310613"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"77","DOI":"10.3982\/ECTA17440","article-title":"A preferred-habitat model of the term structure of interest rates","volume":"89","author":"Vayanos","year":"2021","journal-title":"Econometrica"},{"key":"ref_33","first-page":"77","article-title":"Bond pricing and the term structure of interest rates: A new methodology for contingent claims valuation","volume":"60","author":"Heath","year":"1992","journal-title":"Econom. J. Econom. Soc."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1111\/1467-9965.00028","article-title":"The market model of interest rate dynamics","volume":"7","author":"Brace","year":"1997","journal-title":"Math. Financ."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1007\/s007800050026","article-title":"LIBOR and swap market models and measures","volume":"1","author":"Jamshidian","year":"1997","journal-title":"Financ. Stochastics"},{"key":"ref_36","first-page":"1209","article-title":"An empirical comparison of alternative models of the short-term interest rate","volume":"47","author":"Chan","year":"1992","journal-title":"J. Financ."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1111\/j.1467-9965.1996.tb00123.x","article-title":"A yield-factor model of interest rates","volume":"6","author":"Duffie","year":"1996","journal-title":"Math. Financ."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"33","DOI":"10.2469\/faj.v46.n1.33","article-title":"A one-factor model of interest rates and its application to treasury bond options","volume":"46","author":"Black","year":"1990","journal-title":"Financ. Anal. J."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"52","DOI":"10.2469\/faj.v47.n4.52","article-title":"Bond and option pricing when short rates are lognormal","volume":"47","author":"Black","year":"1991","journal-title":"Financ. Anal. J."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1080\/13518470122553","article-title":"A family of humped volatility models","volume":"7","author":"Mercurio","year":"2001","journal-title":"Eur. J. Financ."},{"key":"ref_41","unstructured":"Cox, J. (1975). Notes on Option Pricing I: Constant Elasticity of Variance Diffusions, Unpublished Note."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"573","DOI":"10.1093\/rfs\/3.4.573","article-title":"Pricing interest-rate-derivative securities","volume":"3","author":"Hull","year":"1990","journal-title":"Rev. Financ. Stud."},{"key":"ref_43","unstructured":"Von Neumann, J., and Morgenstern, O. (1947). Theory of Games and Economic Behavior, Princeton University Press. [2nd rev ed.]."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"673","DOI":"10.1111\/j.1540-6261.1990.tb03711.x","article-title":"On the Consistency of the Black-Scholes Model with a General Equilibrium Framework","volume":"45","author":"Bick","year":"1990","journal-title":"J. Financ."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1016\/S0304-4076(99)00016-0","article-title":"Nonparametric risk management and implied risk aversion","volume":"94","author":"Lo","year":"2000","journal-title":"J. Econom."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"593","DOI":"10.1093\/rfs\/5.3.593","article-title":"On equilibrium asset price processes","volume":"6","author":"He","year":"1993","journal-title":"Rev. Financ. Stud."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"443","DOI":"10.2307\/3215299","article-title":"Changes of numeraire, changes of probability measure and option pricing","volume":"32","author":"Geman","year":"1995","journal-title":"J. Appl. Probab."},{"key":"ref_48","first-page":"75","article-title":"A short history of stochastic integration and mathematical finance: The early years, 1880\u20131970","volume":"45","author":"Jarrow","year":"2004","journal-title":"Lect. Notes-Monograph Ser."},{"key":"ref_49","unstructured":"Bj\u00f6rk, T. (2009). Arbitrage in Continuous Time, Oxford University Press."},{"key":"ref_50","unstructured":"Zeytun, S., and Gupta, A. (2007). A Comparative Study of the Vasicek and the Cir Model of the Short Rate, Fraunhofer ITWW."}],"container-title":["Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2227-7390\/12\/1\/105\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:43:09Z","timestamp":1760132589000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2227-7390\/12\/1\/105"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,12,28]]},"references-count":50,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2024,1]]}},"alternative-id":["math12010105"],"URL":"https:\/\/doi.org\/10.3390\/math12010105","relation":{},"ISSN":["2227-7390"],"issn-type":[{"value":"2227-7390","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,12,28]]}}}