In finance, the yield curve is a graph which depicts how the yields on debt instruments – such as bonds – vary as a function of their years remaining to maturity.^{[1]}^{[2]} Typically, the graph's horizontal or xaxis is a time line of months or years remaining to maturity, with the shortest maturity on the left and progressively longer time periods on the right. The vertical or yaxis depicts the annualized yield to maturity.^{[3]}
Those who issue and trade in forms of debt, such as loans and bonds, use yield curves to determine their value.^{[4]} Shifts in the shape and slope of the yield curve are thought to be related to investor expectations for the economy and interest rates.
Ronald Melicher and Merle Welshans have identified several characteristics of a properly constructed yield curve. It should be based on a set of securities which have differing lengths of time to maturity, and all yields should be calculated as of the same point in time. All securities measured in the yield curve should have similar credit ratings, to screen out the effect of yield differentials caused by credit risk.^{[5]} For this reason, many traders closely watch the yield curve for U.S. Treasury debt securities, which are considered to be riskfree. Informally called "the Treasury yield curve", it is commonly plotted on a graph such as the one on the right.^{[6]} More formal mathematical descriptions of this relationship are often called the term structure of interest rates.
Yield curves are usually upward sloping asymptotically: the longer the maturity, the higher the yield, with diminishing marginal increases (that is, as one moves to the right, the curve flattens out). According to The Economist, the slope of the yield curve can be measured by the difference, or "spread", between the yields on twoyear and tenyear U.S. Treasury Notes. A wider spread indicates a steeper slope.^{[7]}
There are two common explanations for upward sloping yield curves. First, it may be that the market is anticipating a rise in the riskfree rate. If investors hold off investing now, they may receive a better rate in the future. Therefore, under the arbitrage pricing theory, investors who are willing to lock their money in now need to be compensated for the anticipated rise in rates—thus the higher interest rate on longterm investments. Another explanation is that longer maturities entail greater risks for the investor (i.e. the lender). A risk premium is needed by the market, since at longer durations there is more uncertainty and a greater chance of events that impact the investment. This explanation depends on the notion that the economy faces more uncertainties in the distant future than in the near term. This effect is referred to as the liquidity spread. If the market expects more volatility in the future, even if interest rates are anticipated to decline, the increase in the risk premium can influence the spread and cause an increasing yield.
The opposite situation can also occur, in which the yield curve is "inverted", with shortterm interest rates higher than longterm. For instance, in November 2004, the yield curve for UK Government bonds was partially inverted. The yield for the 10year bond stood at 4.68%, but was only 4.45% for the 30year bond. The market's anticipation of falling interest rates causes such incidents. Negative liquidity premiums can also exist if longterm investors dominate the market, but the prevailing view is that a positive liquidity premium dominates, so only the anticipation of falling interest rates will cause an inverted yield curve. Strongly inverted yield curves have historically preceded economic recessions.
The shape of the yield curve is influenced by supply and demand: for instance, if there is a large demand for long bonds, for instance from pension funds to match their fixed liabilities to pensioners, and not enough bonds in existence to meet this demand, then the yields on long bonds can be expected to be low, irrespective of market participants' views about future events.
The yield curve may also be flat or humpshaped, due to anticipated interest rates being steady, or shortterm volatility outweighing longterm volatility.
Yield curves continually move all the time that the markets are open, reflecting the market's reaction to news. A further "stylized fact" is that yield curves tend to move in parallel; i.e.: the yield curve shifts up and down as interest rate levels rise and fall, which is then referred to as a "parallel shift".
There is no single yield curve describing the cost of money for everybody. The most important factor in determining a yield curve is the currency in which the securities are denominated. The economic position of the countries and companies using each currency is a primary factor in determining the yield curve. Different institutions borrow money at different rates, depending on their creditworthiness.
The yield curves corresponding to the bonds issued by governments in their own currency are called the government bond yield curve (government curve). Banks with high credit ratings (Aa/AA or above) borrow money from each other at the LIBOR rates. These yield curves are typically a little higher than government curves. They are the most important and widely used in the financial markets, and are known variously as the LIBOR curve or the swap curve. The construction of the swap curve is described below.
Besides the government curve and the LIBOR curve, there are corporate (company) curves. These are constructed from the yields of bonds issued by corporations. Since corporations have less creditworthiness than most governments and most large banks, these yields are typically higher. Corporate yield curves are often quoted in terms of a "credit spread" over the relevant swap curve. For instance the fiveyear yield curve point for Vodafone might be quoted as LIBOR +0.25%, where 0.25% (often written as 25 basis points or 25bps) is the credit spread.
From the postGreat Depression era to the present, the yield curve has usually been "normal" meaning that yields rise as maturity lengthens (i.e., the slope of the yield curve is positive). This positive slope reflects investor expectations for the economy to grow in the future and, importantly, for this growth to be associated with a greater expectation that inflation will rise in the future rather than fall. This expectation of higher inflation leads to expectations that the central bank will tighten monetary policy by raising shortterm interest rates in the future to slow economic growth and dampen inflationary pressure. It also creates a need for a risk premium associated with the uncertainty about the future rate of inflation and the risk this poses to the future value of cash flows. Investors price these risks into the yield curve by demanding higher yields for maturities further into the future. In a positively sloped yield curve, lenders profit from the passage of time since yields decrease as bonds get closer to maturity (as yield decreases, price increases); this is known as rolldown and is a significant component of profit in fixedincome investing (i.e., buying and selling, not necessarily holding to maturity), particularly if the investing is leveraged.^{[8]}
However, a positively sloped yield curve has not always been the norm. Through much of the 19th century and early 20th century the US economy experienced trend growth with persistent deflation, not inflation. During this period the yield curve was typically inverted, reflecting the fact that deflation made current cash flows less valuable than future cash flows. During this period of persistent deflation, a 'normal' yield curve was negatively sloped.
Historically, the 20year Treasury bond yield has averaged approximately two percentage points above that of threemonth Treasury bills. In situations when this gap increases (e.g. 20year Treasury yield rises much higher than the threemonth Treasury yield), the economy is expected to improve quickly in the future. This type of curve can be seen at the beginning of an economic expansion (or after the end of a recession). Here, economic stagnation will have depressed shortterm interest rates; however, rates begin to rise once the demand for capital is reestablished by growing economic activity.
In January 2010, the gap between yields on twoyear Treasury notes and 10year notes widened to 2.92 percentage points, its highest ever.
A flat yield curve is observed when all maturities have similar yields, whereas a humped curve results when shortterm and longterm yields are equal and mediumterm yields are higher than those of the shortterm and longterm. A flat curve sends signals of uncertainty in the economy. This mixed signal can revert to a normal curve or could later result into an inverted curve. It cannot be explained by the Segmented Market theory discussed below.
Under unusual circumstances, investors will settle for lower yields associated with lowrisk longterm debt if they think the economy will enter a recession in the near future. For example, the S&P 500 experienced a dramatic fall in mid 2007, from which it recovered completely by early 2013. Investors who had purchased 10year Treasuries in 2006 would have received a safe and steady yield until 2015, possibly achieving better returns than those investing in equities during that volatile period.
Economist Campbell Harvey's 1986 dissertation^{[9]} showed that an inverted yield curve accurately forecasts U.S. recessions. An inverted curve has indicated a worsening economic situation in the future eight times since 1970.^{[10]}
In addition to potentially signaling an economic decline, inverted yield curves also imply that the market believes inflation will remain low. This is because, even if there is a recession, a low bond yield will still be offset by low inflation. However, technical factors, such as a flight to quality or global economic or currency situations, may cause an increase in demand for bonds on the long end of the yield curve, causing longterm rates to fall. Falling longterm rates in the presence of rising shortterm rates is known as "Greenspan's Conundrum".^{[11]}
The slope of the yield curve is one of the most powerful predictors of future economic growth, inflation, and recessions. ^{[12]}^{[13]} One measure of the yield curve slope (i.e. the difference between 10year Treasury bond rate and the 3month Treasury bond rate) is included in the Financial Stress Index published by the St. Louis Fed.^{[14]} A different measure of the slope (i.e. the difference between 10year Treasury bond rates and the federal funds rate) is incorporated into the Index of Leading Economic Indicators published by The Conference Board.^{[15]}
An inverted yield curve is often a harbinger of recession. A positively sloped yield curve is often a harbinger of inflationary growth. Work by Arturo Estrella and Tobias Adrian has established the predictive power of an inverted yield curve to signal a recession. Their models show that when the difference between shortterm interest rates (they use 3month Tbills) and longterm interest rates (10year Treasury bonds) at the end of a federal reserve tightening cycle is negative or less than 93 basis points positive, a rise in unemployment usually occurs.^{[16]} The New York Fed publishes a monthly recession probability prediction derived from the yield curve and based on Estrella's work.
All the recessions in the US since 1970 have been preceded by an inverted yield curve (10year vs 3month). Over the same time frame, every occurrence of an inverted yield curve has been followed by recession as declared by the NBER business cycle dating committee.^{[17]} The yield curve became inverted in the first half of 2019, for the first time since 2007.^{[18]}^{[19]}^{[20]}
Recession  Inversion start date  Recession start date  Time between inversion start and beginning of recession  Duration of inversion  Time between start of recession and NBER announcement  Time between disinversion and end of recession  Recession duration  Time between end of recession and NBER announcement  Inversion maximum (basis points) 

1970 recession  Dec68  Jan70  13  15  —  8  11  —  −52 
1974 recession  Jun73  Dec73  6  18  —  3  16  —  −159 
1980 recession  Nov78  Feb80  15  18  4  2  6  12  −328 
1981–1982 recession  Oct80  Aug81  10  12  5  13  16  8  −351 
1990 recession  Jun89  Aug90  14  7  8  14  8  21  −16 
2001 recession  Jul00  Apr01  9  7  7  9  8  20  −70 
2008–2009 recession  Aug06  Jan08  17  10  11  24  18  15  −51 
COVID19 recession  May19  Mar20  10^{[1]}  5  4^{[2]}  TBD  TBD  TBD  −52 
TBD  Oct22  TBD  TBD  TBD  TBD  TBD  TBD  TBD  −182 
Average since 1969  12  12  7  10  12  15  −140.11  
Standard deviation since 1969  3.83  4.72  2.74  7.50  4.78  5.45  138.96 
Estrella and others have postulated that the yield curve affects the business cycle via the balance sheet of banks (or banklike financial institutions).^{[21]} When the yield curve is inverted, banks are often caught paying more on shortterm deposits (or other forms of shortterm wholesale funding) than they are making on new longterm loans leading to a loss of profitability and reluctance to lend resulting in a credit crunch. When the yield curve is upward sloping, banks can profitably takein shortterm deposits and make new longterm loans so they are eager to supply credit to borrowers. This eventually leads to a credit bubble.
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.
This hypothesis assumes that the various maturities are 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 shortterm investments will equal the known final value of a single longterm investment. If this did not hold, the theory assumes that investors would quickly demand more of the current shortterm or longterm bonds (whichever gives the higher expected longterm 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 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 1year interest rates will be next year, the current 2year interest rate can be calculated as the compounding of this year's 1year interest rate by next year's expected 1year interest rate. More generally, returns (1+ yield) on a longterm instrument are assumed to equal the geometric mean of the expected returns on a series of shortterm instruments:
where i_{st} and i_{lt} are the expected shortterm and actual longterm interest rates (but is the actual observed shortterm 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.
The liquidity premium theory is an offshoot of the pure expectations theory. The liquidity premium theory asserts that longterm interest rates not only reflect investors' assumptions about future interest rates but also include a premium for holding longterm bonds (investors prefer shortterm bonds to longterm 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, longterm bond yields tend to be higher than shortterm yields and the yield curve slopes upward. Longterm 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:
where is the risk premium associated with an year bond.
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 shortterm investors are more prevalent in the fixedincome market, and therefore longerterm rates tend to be higher than shortterm rates, for the most part, but shortterm rates can be higher than longterm 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.
This theory is also called the segmented market hypothesis. In this theory, financial instruments of different terms are not substitutable. As a result, the supply and demand in the markets for shortterm and longterm instruments is determined largely independently. Prospective investors decide in advance whether they need shortterm or longterm instruments. If investors prefer their portfolio to be liquid, they will prefer shortterm instruments to longterm instruments. Therefore, the market for shortterm instruments will receive a higher demand. Higher demand for the instrument implies higher prices and lower yield. This explains the stylized fact that shortterm yields are usually lower than longterm 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).
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 rates.
Floating exchange rates made life more complicated for bond traders, including those at Salomon Brothers in New York City. By the middle of the 1970s, encouraged by the head of bond research at Salomon, Marty Liebowitz, traders began thinking about bond yields in new ways. Rather than think of each maturity (a tenyear bond, a fiveyear, 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 riskneutral equivalent martingale measure) and accordingly by shortterm 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 shortrate 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 zerocoupon bonds and the creation of long dated mortgages.
Type  Settlement date  Rate (%) 
Cash  Overnight rate  5.58675 
Cash  Tomorrow next rate  5.59375 
Cash  1m  5.625 
Cash  3m  5.71875 
Future  Dec97  5.76 
Future  Mar98  5.77 
Future  Jun98  5.82 
Future  Sep98  5.88 
Future  Dec98  6.00 
Swap  2y  6.01253 
Swap  3y  6.10823 
Swap  4y  6.16 
Swap  5y  6.22 
Swap  7y  6.32 
Swap  10y  6.42 
Swap  15y  6.56 
Swap  20y  6.56 
Swap  30y  6.56 
A list of standard instruments used to build a money market yield curve.  
The data is for lending in US dollar, taken from October 6, 1997 
The usual representation of the yield curve is in terms of a function P, defined on all future times t, such that P(t) represents the value today of receiving one unit of currency t years in the future. If P is defined for all future t then we can easily recover the yield (i.e. the annualized interest rate) for borrowing money for that period of time via the formula
The significant difficulty in defining a yield curve therefore is to determine the function P(t). P is called the discount factor function or the zero coupon bond.
Yield curves are built from either prices available in the bond market or the money market. Whilst the yield curves built from the bond market use prices only from a specific class of bonds (for instance bonds issued by the UK government) yield curves built from the money market use prices of "cash" from today's LIBOR rates, which determine the "short end" of the curve i.e. for t ≤ 3m, interest rate futures which determine the midsection of the curve (3m ≤ t ≤ 15m) and interest rate swaps which determine the "long end" (1y ≤ t ≤ 60y).
The example given in the table at the right is known as a LIBOR curve because it is constructed using either LIBOR rates or swap rates. A LIBOR curve is the most widely used interest rate curve as it represents the credit worth of private entities at about A+ rating, roughly the equivalent of commercial banks. If one substitutes the LIBOR and swap rates with government bond yields, one arrives at what is known as a government curve, usually considered the risk free interest rate curve for the underlying currency. The spread between the LIBOR (or swap) rate and the government bond yield of similar maturity is usually positive, meaning that private borrowing is at a premium above government borrowing. This spread is a measure of the difference in the risk tolerances of the lenders to the two types of borrowing. For the U. S. market, a common benchmark for such a spread is given by the socalled TED spread.
In either case the available market data provides a matrix A of cash flows, each row representing a particular financial instrument and each column representing a point in time. The (i,j)th element of the matrix represents the amount that instrument i will pay out on day j. Let the vector F represent today's prices of the instrument (so that the ith instrument has value F(i)), then by definition of our discount factor function P we should have that F = AP (this is a matrix multiplication). Actually, noise in the financial markets means it is not possible to find a P that solves this equation exactly, and our goal becomes to find a vector P such that
where is as small a vector as possible (where the size of a vector might be measured by taking its norm, for example).
Even if we can solve this equation, we will only have determined P(t) for those t which have a cash flow from one or more of the original instruments we are creating the curve from. Values for other t are typically determined using some sort of interpolation scheme.
Practitioners and researchers have suggested many ways of solving the A*P = F equation. It transpires that the most natural method – that of minimizing by least squares regression – leads to unsatisfactory results. The large number of zeroes in the matrix A mean that function P turns out to be "bumpy".
In their comprehensive book on interest rate modelling James and Webber note that the following techniques have been suggested to solve the problem of finding P:
In the money market practitioners might use different techniques to solve for different areas of the curve. For example, at the short end of the curve, where there are few cashflows, the first few elements of P may be found by bootstrapping from one to the next. At the long end, a regression technique with a cost function that values smoothness might be used.
There is a time dimension to the analysis of bond values. A 10year bond at purchase becomes a 9year bond a year later, and the year after it becomes an 8year bond, etc. Each year the bond moves incrementally closer to maturity, resulting in lower volatility and shorter duration and demanding a lower interest rate when the yield curve is rising. Since falling rates create increasing prices, the value of a bond initially will rise as the lower rates of the shorter maturity become its new market rate. Because a bond is always anchored by its final maturity, the price at some point must change direction and fall to par value at redemption.
A bond's market value at different times in its life can be calculated. When the yield curve is steep, the bond is predicted to have a large capital gain in the first years before falling in price later. When the yield curve is flat, the capital gain is predicted to be much less, and there is little variability in the bond's total returns over time.
As market rates of interest increase or decrease, the impact is rarely the same at each point along the yield curve, i.e. the curve rarely moves up or down in parallel. Because longerterm bonds have a larger duration, a rise in rates will cause a larger capital loss for them, than for shortterm bonds. But almost always, the long maturity's rate will change much less, flattening the yield curve. The greater change in rates at the short end will offset to some extent the advantage provided by the shorter bond's lower duration.
Long duration bonds tend to be mean reverting, meaning that they readily gravitate to a longrun average. The middle of the curve (5–10 years) will see the greatest percentage gain in yields if there is anticipated inflation even if interest rates have not changed. The longend does not move quite as much percentagewise because of the mean reverting properties.
The yearly 'total return' from the bond is a) the sum of the coupon's yield plus b) the capital gain from the changing valuation as it slides down the yield curve and c) any capital gain or loss from changing interest rates at that point in the yield curve.^{[23]}
1. ^ The New York Federal Reserve recession prediction model uses the month average 10 year yield vs the month average 3 month bond equivalent yield to compute the term spread. Therefore, intraday and daily inversions do not count as inversions unless they lead to an inversion on a monthly average basis. In December 2018, portions of the yield curve inverted for the first time since the 2008–2009 recession.^{[24]} However the 10year vs 3month portion did not invert until March 22, 2019 and it reverted to a positive slope by April 1, 2019 (i.e. only 8 days later).^{[25]}^{[26]} The month average of the 10year vs 3month (bond equivalent yield) difference reached zero basis points in May 2019. Both March and April 2019 had monthaverage spreads greater than zero basis points despite intraday and daily inversions in March and April. Therefore, the table shows the 2019 inversion beginning from May 2019. Likewise, daily inversions in September 1998 did not result in negative term spreads on a month average basis and thus do not constitute a false alarm.
2. ^ The recession prediction model stipulated that the recession began in February 2020, one month before the World Health Organization declared COVID19 a pandemic.
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: CS1 maint: multiple names: authors list (link)Types of bonds by issuer  

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Bond valuation  
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