yield curve

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In
finance Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of Production (economics), production, Distribution (economics), distribution, and Consumption (economics) ...

, 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. Typically, the graph's horizontal or x-axis 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 y-axis depicts the annualized yield to maturity. Those who issue and trade in forms of debt, such as loans and bonds, use yield curves to determine their value. 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. For this reason, many traders closely watch the yield curve for U.S. Treasury debt securities, which are considered to be risk-free. Informally called "the Treasury yield curve", it is commonly plotted on a graph such as the one on the right. More formal mathematical descriptions of this relationship are often called the term structure of interest rates.

# Significance of slope and shape

Yield curves are usually upward sloping
asymptotically In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates Limit of a function#Limits at infinity, tends to infinity. In pro ...
: 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 columnist Buttonwood of
The Economist ''The Economist'' is a British weekly newspaper printed in Paper size#Demitab, demitab format and Electronic publishing, published digitally. It focuses on current affairs, international business, politics, technology, and culture. Based in Lo ...
newspaper, the slope of the yield curve can be measured by the difference, or "spread", between the yields on two-year and ten-year U.S. Treasury Notes. A wider spread indicates a steeper slope. There are two common explanations for upward sloping yield curves. First, it may be that the market is anticipating a rise in the
risk-free rate The risk-free rate of return, usually shortened to the risk-free rate, is the rate of return In finance, return is a Profit (accounting), profit on an investment. It comprises any change in value of the investment, and/or cash flows (or securit ...
. If investors hold off investing now, they may receive a better rate in the future. Therefore, under the
arbitrage pricing theory In finance, arbitrage pricing theory (APT) is a multi-factor model for asset pricing In financial economics, asset pricing refers to a formal treatment and development of two main Price, pricing principles, outlined below, together with the resu ...
, 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 long-term investments. Another explanation is that longer maturities entail greater risks for the investor (i.e. the lender). A
risk premium A risk premium is a measure of excess return that is required by an individual to compensate being subjected to an increased level of risk. It is used widely in finance and economics, the general definition being the expected risky Rate of retur ...
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 short-term interest rates higher than long-term. For instance, in November 2004, the yield curve for UK Government bonds was partially inverted. The yield for the 10-year bond stood at 4.68%, but was only 4.45% for the 30-year bond. The market's anticipation of falling interest rates causes such incidents. Negative
liquidity premium In economics Economics () is the social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behavio ...
s can also exist if long-term 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 In microeconomics, supply and demand is an economic model In economics, a model is a theory, theoretical construct representing economic wikt:process, processes by a set of Variable (mathematics), variables and a set of logical and/or q ...
: for instance, if there is a large demand for long bonds, for instance from
pension fund A pension fund, also known as a superannuation fund in some countries, is any plan, fund, or scheme which provides pension, retirement income. Pension funds typically have large amounts of money to invest and are the major investors in listed a ...
s 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 hump-shaped, due to anticipated interest rates being steady, or short-term volatility outweighing long-term 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".

## Types of yield curve

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 The London Inter-Bank Offered Rate is an interest-rate average calculated from estimates submitted by the leading banks in London London is the capital and List of urban areas in the United Kingdom, largest city of England and the Unite ...
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 The London Inter-Bank Offered Rate is an interest-rate average calculated from estimates submitted by the leading banks in London London is the capital and List of urban areas in the United Kingdom, largest city of England and the Unite ...
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 A corporation is an organization—usually a group of people or a company—authorized by the State (polity), state to act as a single entity (a legal entity recognized by private and public law "born out of statute"; a legal person in legal ...
(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 five-year yield curve point for
Vodafone Vodafone Group Public limited company, plc () is a British Multinational corporation, multinational Telephone company, telecommunications company. Its registered office and Headquarters, global headquarters are in Newbury, Berkshire, England. It ...
might be quoted as LIBOR +0.25%, where 0.25% (often written as 25
basis point A basis point (often abbreviated as bp, often pronounced as "bip" or "beep") is one hundredth of 1 percentage point. The related term '' permyriad'' means one hundredth of 1 percent. Changes of interest rates are often stated in basis points. If ...
s or 25) is the credit spread.

### Normal yield curve

From the post-
Great Depression The Great Depression (19291939) was an economic shock that impacted most countries across the world. It was a period of economic depression that became evident after a major fall in stock prices in the United States. The Financial contagion, ...
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 A central bank, reserve bank, or monetary authority is an institution that manages the currency and monetary policy of a country or monetary union, and oversees their commercial bank, commercial banking system. In contrast to a commercial ba ...
will tighten monetary policy by raising short-term 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 fixed-income investing (i.e., buying and selling, not necessarily holding to maturity), particularly if the investing is leveraged. 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 In economics, deflation is a decrease in the general price level of goods and services. Deflation occurs when the inflation rate falls below 0% (a negative inflation rate). Inflation reduces the value of currency over time, but sudden 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.

### Steep yield curve

Historically, the 20-year
Treasury bond United States Treasury securities, also called Treasuries or Treasurys, are government bond, government debt instruments issued by the United States Department of the Treasury to finance government spending as an alternative to taxation. Sin ...
yield has averaged approximately two percentage points above that of three-month Treasury bills. In situations when this gap increases (e.g. 20-year Treasury yield rises much higher than the three-month 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 short-term interest rates; however, rates begin to rise once the demand for capital is re-established by growing economic activity. In January 2010, the gap between yields on two-year Treasury notes and 10-year notes widened to 2.92 percentage points, its highest ever.

### Flat or humped yield curve

A flat yield curve is observed when all maturities have similar yields, whereas a humped curve results when short-term and long-term yields are equal and medium-term yields are higher than those of the short-term and long-term. 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.

### Inverted yield curve

Under unusual circumstances, investors will settle for lower yields associated with low-risk long-term debt if they think the economy will enter a recession in the near future. For example, the
S&P 500 The Standard and Poor's 500, or simply the S&P 500, is a stock market index tracking the stock performance of 500 large companies listed on stock exchanges in the United States. It is one of the most commonly followed equity indices. As of D ...
experienced a dramatic fall in mid 2007, from which it recovered completely by early 2013. Investors who had purchased 10-year 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 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. 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 long-term rates to fall. Falling long-term rates in the presence of rising short-term rates is known as "Greenspan's Conundrum".

# Relationship to the business cycle

The slope of the yield curve is one of the most powerful predictors of future economic growth, inflation, and recessions. One measure of the yield curve slope (i.e. the difference between 10-year Treasury bond rate and the 3-month Treasury bond rate) is included in th
Financial Stress Index
published by the St. Louis Fed. A different measure of the slope (i.e. the difference between 10-year Treasury bond rates and the
federal funds rate In the United States, the federal funds rate is the interest rate at which depository institutions (banks and credit unions) lend reserve balances to other depository institutions overnight on an collateral (finance), uncollateralized basis ...
) is incorporated into the Index of Leading Economic Indicators published by
The Conference Board The Conference Board, Inc. is a 501(c)(3) organization, 501(c)(3) non-profit business membership and research group organization. It counts over 1,000 public and private corporations and other organizations as members, encompassing 60 countries. ...
. An inverted yield curve is often a harbinger of
recession In economics Economics () is the social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the b ...
. A positively sloped yield curve is often a harbinger of
inflation In economics, inflation is an increase in the general price level of goods and services in an economy. When the general price level rises, each unit of currency buys fewer goods and services; consequently, inflation corresponds to a reductio ...
ary 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 short-term interest rates (they use 3-month T-bills) and long-term interest rates (10-year 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. The New York Fed publishes
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 (10-year vs 3-month). 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. The yield curve became inverted in the first half of 2019, for the first time since 2007. Estrella and others have postulated that the yield curve affects the
business cycle Business cycles are intervals of expansion followed by recession in economic activity. These changes have implications for the welfare of the broad population as well as for private institutions. Typically business cycles are measured by exami ...
via the balance sheet of banks (or bank-like financial institutions). When the yield curve is inverted, banks are often caught paying more on short-term deposits (or other forms of short-term wholesale funding) than they are making on new long-term loans leading to a loss of profitability and reluctance to lend resulting in a
credit crunch A credit crunch (also known as a credit squeeze, credit tightening or credit crisis) is a sudden reduction in the general availability of loans (or credit) or a sudden tightening of the conditions required to obtain a loan from banks. A credit cr ...
. When the yield curve is upward sloping, banks can profitably take-in short-term deposits and make new long-term loans so they are eager to supply credit to borrowers. This eventually leads to a
credit bubble An economic bubble (also called a speculative bubble or a financial bubble) is a period when current asset price In financial economics, asset pricing refers to a formal treatment and development of two main Price, pricing principles, outline ...
.

# 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.

## Market expectations (pure expectations) hypothesis

This
hypothesis A hypothesis (plural hypotheses) is a proposed explanation for a phenomenon. For a hypothesis to be a scientific hypothesis, the scientific method requires that one can testable, test it. Scientists generally base scientific hypotheses on prev ...
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 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 rates, along with the assumption that
arbitrage In economics and finance, arbitrage (, ) is the practice of taking advantage of a difference in prices in two or more Market (economics), markets; striking a combination of matching deals to capitalise on the difference, the profit being the di ...
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 In mathematics, the geometric mean is a mean or average which indicates a central tendency of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the Nt ...
of the expected returns on a series of short-term instruments: : $\left(1 + i_\right)^n=\left(1 + i_^\right)\left(1 + i_^\right) \cdots \left(1 + i_^\right),$ where ''i''''st'' and ''i''''lt'' are the expected short-term and actual long-term interest rates (but $i_^$ 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 In finance Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of Production (economics), production, Distribution (economics), distribution, and Consumpt ...
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: : $\left(1 + i_\right)^n=rp_+\left(\left(1 + i_^\right)\left(1 + i_^\right) \cdots \left(1 + i_^\right)\right),$ where $rp_n$ is the risk premium associated with an  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 substitutable. As a result, the
supply and demand In microeconomics, supply and demand is an economic model In economics, a model is a theory, theoretical construct representing economic wikt:process, processes by a set of Variable (mathematics), variables and a set of logical and/or q ...
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 Richard Milhous Nixon (January 9, 1913April 22, 1994) was the 37th president of the United States, serving from 1969 to 1974. A member of the Republican Party (United States), Republican Party, he previously served as a United States House ...
announced that the U.S. dollar would no longer be based on the
gold standard A gold standard is a monetary system in which the standard economic unit of account is based on a fixed quantity of gold. The gold standard was the basis for the international monetary system from the 1870s to the early 1920s, and from th ...
, thereby ending the
Bretton Woods system The Bretton Woods system of Monetary system, monetary management established the rules for commercial and financial relations among the United States, Canada, Western European countries, Australia, and Japan after the 1944 Bretton Woods Agree ...
and initiating the era of
floating exchange rate In macroeconomics and economic policy, a floating exchange rate (also known as a fluctuating or flexible exchange rate) is a type of exchange rate regime in which a currency's value is allowed to fluctuate in response to foreign exchange marke ...
s. Floating exchange rates made life more complicated for bond traders, including those at
Salomon Brothers Salomon Brothers, Inc., was an American multinational bulge bracket investment bank headquartered in New York City, New York. It was one of the five List of investment banks, largest investment banking enterprises in the United States and the mo ...
in
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. 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 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 A short-rate model, in the context of interest rate derivatives, is a mathematical model that describes the future evolution of interest rates by describing the future evolution of the short rate, usually written r_t \,. The short rate Under a sho ...
. 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 A zero coupon bond (also discount bond or deep discount bond) is a bond in which the face value The face value, sometimes called nominal value, is the value of a coin A coin is a small, flat (usually depending on the country or value), ...
s and the creation of long dated mortgages.

# Construction of the full yield curve from market data

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 :$Y\left(t\right) = P\left(t\right)^ -1.$ 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 The money market is a component of the economy that provides short-term funds. The money market deals in short-term loans, generally for a period of a year or less. As Security (finance)#Debt, short-term securities became a commodity, the mone ...
use prices of "cash" from today's LIBOR rates, which determine the "short end" of the curve i.e. for ''t'' ≤ 3m,
interest rate future An interest rate future is a financial derivative In finance, a derivative is a contract that ''derives'' its value from the performance of an underlying entity. This underlying entity can be an asset, Index fund, index, or interest rate, and ...
s which determine the midsection of the curve (3m ≤ ''t'' ≤ 15m) and
interest rate swap In finance, an interest rate swap (finance), swap (IRS) is an interest rate derivative, interest rate derivative (IRD). It involves exchange of interest rates between two parties. In particular it is a Interest_rate_derivative#Linear_and_non-linear ...
s which determine the "long end" (1y ≤ ''t'' ≤ 60y). The example given in the table at the right is known as a
LIBOR The London Inter-Bank Offered Rate is an interest-rate average calculated from estimates submitted by the leading banks in London London is the capital and List of urban areas in the United Kingdom, largest city of England and the Unite ...
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, usually positive, meaning private borrowing is at a premium above government borrowing, of similar maturity is a measure of risk tolerance of the lenders. For the U. S. market, a common benchmark for such a spread is given by the so-called
TED spread The TED spread is the difference between the interest rates on interbank loans and on Treasury security#Treasury bill, short-term U.S. government debt ("T-bills"). TED is an Acronym and initialism, acronym formed from ''T-Bill'' and ''ED'', the ...
. 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 ''i''-th 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 : $AP = F + \varepsilon \,$ where $\varepsilon$ 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 In the mathematics, mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one ...
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 $\epsilon$ 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: #Approximation using Lagrange polynomials #Fitting using parameterised curves (such as splines, the Nelson-Siegel family, the Svensson family, the exponential polynomial family or the Cairns restricted-exponential family of curves). Van Deventer, Imai and Mesler summarize three different techniques for
curve fitting Curve fitting is the process of constructing a curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve ...
that satisfy the maximum smoothness of either forward interest rates, zero coupon bond prices, or zero coupon bond yields #Local regression using kernels #
Linear programming Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear function#As a polynomial function, li ...
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 In general, bootstrapping usually refers to a self-starting process that is supposed to continue or grow without external input. Etymology Tall boot A boot is a type of footwear. Most boots mainly cover the foot and the ankle, while some ...
from one to the next. At the long end, a regression technique with a cost function that values smoothness might be used.

# Effect on bond prices

There is a time dimension to the analysis of bond values. A 10-year bond at purchase becomes a 9-year bond a year later, and the year after it becomes an 8-year 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 Capital gain is an economic concept defined as the profit earned on the sale of an asset In financial accounting, an asset is any resource owned or controlled by a business or an economic entity. It is anything (tangible or intangible) th ...
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 longer-term bonds have a larger duration, a rise in rates will cause a larger capital loss for them, than for short-term 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 long-run 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 long-end does not move quite as much percentage-wise 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.

*
Short-rate model A short-rate model, in the context of interest rate derivatives, is a mathematical model that describes the future evolution of interest rates by describing the future evolution of the short rate, usually written r_t \,. The short rate Under a sho ...
*
Zero interest-rate policy Zero interest-rate policy (ZIRP) is a macroeconomic concept describing conditions with a very low nominal interest rate, such as those in contemporary Bank of Japan, Japan and in the Federal Reserve System, United States from December 2008 th ...
* Multi-curve framework

# Notes

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, intra-day 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. However the 10-year vs 3-month 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). The month average of the 10-year vs 3-month (bond equivalent yield) difference reached zero basis points in May 2019. Both March and April 2019 had month-average spreads greater than zero basis points despite intra-day 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
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declared
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a pandemic.

# References

## Books

* * * * * * * * See in particular the section ''Theories of the term structure'' (section 4.7 in the fourth edition). * *

## Articles

* Ruben D Cohen (2006) "A VaR-Based Model for the Yield Curv