Editing Yield curve (section)

==Theory==
There are three main economic theories attempting to explain how yields vary with maturity. Two of the theories are extreme positions, while the third attempts to find a middle ground between the former two.{{Citations needed|reason=section should be supported by citations to reliable sources. See https://en.wikipedia.org/wiki/Wikipedia:Reliable_sources| date=July 2021}} 

===Market expectations (pure expectations) hypothesis===
{{main|Expectations hypothesis}}

This [[hypothesis]] assumes that the various maturities are [[Substitute good|perfect substitutes]] and suggests that the shape of the yield curve depends on market participants' expectations of future interest rates. It assumes that market forces will cause the interest rates on various terms of bonds to be such that the expected final value of a sequence of short-term investments will equal the known final value of a single long-term investment. If this did not hold, the theory assumes that investors would quickly demand more of the current short-term or long-term bonds (whichever gives the higher expected long-term yield), and this would drive down the return on current bonds of that term and drive up the yield on current bonds of the other term, so as to quickly make the assumed equality of expected returns of the two investment approaches hold.

Using this, [[futures contract|futures rates]], along with the assumption that [[arbitrage]] opportunities will be minimal in future markets, and that futures rates are unbiased estimates of forthcoming spot rates, provide enough information to construct a complete expected yield curve. For example, if investors have an expectation of what 1-year interest rates will be next year, the current 2-year interest rate can be calculated as the compounding of this year's 1-year interest rate by next year's expected 1-year interest rate. More generally, returns (1+ yield) on a long-term instrument are assumed to equal the [[geometric mean]] of the expected returns on a series of short-term instruments:

: <math>(1 + i_{lt})^n=(1 + i_{st}^{\text{year }1})(1 + i_{st}^{\text{year }2}) \cdots (1 + i_{st}^{\text{year }n}),</math>

where ''i''<sub>''st''</sub> and ''i''<sub>''lt''</sub> are the expected short-term and actual long-term interest rates (but <math>i_{st}^{\text{year}1}</math> is the actual observed short-term rate for the first year).

This theory is consistent with the observation that yields usually move together. However, it fails to explain the persistence in the shape of the yield curve.

Shortcomings of expectations theory include that it neglects the [[interest rate risk]] inherent in investing in bonds.

===Liquidity premium theory===
The liquidity premium theory is an offshoot of the pure expectations theory. The liquidity premium theory asserts that long-term interest rates not only reflect investors' assumptions about future interest rates but also include a premium for holding long-term bonds (investors prefer short-term bonds to long-term bonds), called the term premium or the liquidity premium. This premium compensates investors for the added risk of having their money tied up for a longer period, including the greater price uncertainty. Because of the term premium, long-term bond yields tend to be higher than short-term yields and the yield curve slopes upward. Long-term yields are also higher not just because of the liquidity premium, but also because of the risk premium added by the risk of default from holding a security over the long term. The market expectations hypothesis is combined with the liquidity premium theory:

: <math>(1 + i_{lt})^n=rp_{n}+((1 + i_{st}^{\mathrm{year }1})(1 + i_{st}^{\mathrm{year }2}) \cdots (1 + i_{st}^{\mathrm{year }n})),</math>

where <math>rp_n</math> is the risk premium associated with an <math>{n}</math> year bond.

===Preferred habitat theory===
The preferred habitat theory is a variant of the liquidity premium theory, and states that in addition to interest rate expectations, investors have distinct investment horizons and require a meaningful premium to buy bonds with maturities outside their "preferred" maturity, or habitat. Proponents of this theory believe that short-term investors are more prevalent in the fixed-income market, and therefore longer-term rates tend to be higher than short-term rates, for the most part, but short-term rates can be higher than long-term rates occasionally. This theory is consistent with both the persistence of the normal yield curve shape and the tendency of the yield curve to shift up and down while retaining its shape.

===Market segmentation theory===
This theory is also called the '''segmented market hypothesis'''. In this theory, financial instruments of different terms are not [[substitute good|substitutable]]. As a result, the [[supply and demand]] in the markets for short-term and long-term instruments is determined largely independently. Prospective investors decide in advance whether they need short-term or long-term instruments. If investors prefer their portfolio to be liquid, they will prefer short-term instruments to long-term instruments. Therefore, the market for short-term instruments will receive a higher demand. Higher demand for the instrument implies higher prices and lower yield. This explains the [[stylized fact]] that short-term yields are usually lower than long-term yields. This theory explains the predominance of the normal yield curve shape. However, because the supply and demand of the two markets are independent, this theory fails to explain the observed fact that yields tend to move together (i.e., upward and downward shifts in the curve).

===Historical development of yield curve theory===
On August 15, 1971, U.S. President [[Richard Nixon]] announced that the U.S. dollar would no longer be based on the [[gold standard]], thereby ending the [[Bretton Woods system]] and initiating the era of [[floating exchange rate]]s.

Floating exchange rates made life more complicated for bond traders, including those at [[Salomon Brothers]] in [[New York City]]. Encouraged by [[Martin L. Leibowitz|Marty Liebowitz]], traders began thinking about bond yields in new ways by the middle of the 1970s. Rather than think of each maturity (a ten-year bond, a five-year, etc.) as a separate marketplace, they began drawing a curve through all their yields. The bit nearest the present time became known as the ''short end''—yields of bonds further out became, naturally, the ''long end''.

Academics had to play catch up with practitioners in this matter. One important theoretic development came from a Czech mathematician, [[Oldrich Vasicek]], who argued in a 1977 paper that bond prices all along the curve are driven by the short end (under risk-neutral equivalent martingale measure) and accordingly by short-term interest rates. The mathematical model for Vasicek's work was given by an [[Ornstein–Uhlenbeck process]], but has since been discredited because the model predicts a positive probability that the short rate becomes negative and is inflexible in creating yield curves of different shapes. Vasicek's model has been superseded by many different models including the [[Hull–White model]] (which allows for time varying parameters in the Ornstein–Uhlenbeck process), the [[Cox–Ingersoll–Ross model]], which is a modified [[Bessel process]], and the [[Heath–Jarrow–Morton framework]]. There are also many modifications to each of these models, but see the article on [[short-rate model]]. Another modern approach is the [[LIBOR market model]], introduced by Brace, Gatarek and Musiela in 1997 and advanced by others later. In 1996, a group of derivatives traders led by Olivier Doria (then head of swaps at Deutsche Bank) and Michele Faissola, contributed to an extension of the swap yield curves in all the major European currencies. Until then the market would give prices until 15 years maturities. The team extended the maturity of European yield curves up to 50 years (for the lira, French franc, Deutsche mark, Danish krone and many other currencies including the ecu). This innovation was a major contribution towards the issuance of long dated [[zero-coupon bond]]s and the creation of long dated mortgages.