# Yield curve

The US dollar yield curve as of 9 February 2005. The curve has a typical upward sloping shape.

In finance, the yield curve is the relation between the interest rate (or cost of borrowing) and the time to maturity of the debt for a given borrower in a given currency. For example, the current U.S. dollar interest rates paid on U.S. Treasury securities for various maturities are closely watched by many traders, and are commonly plotted on a graph such as the one on the right which is informally called "the yield curve." More formal mathematical descriptions of this relation are often called the term structure of interest rates.

The yield of a debt instrument is the annualized percentage increase in the value of the investment. For instance, a bank account that pays an interest rate of 4% per year has a 4% yield. In general the percentage per year that can be earned is dependent on the length of time that the money is invested. For example, a bank may offer a "savings rate" higher than the normal checking account rate if the customer is prepared to leave money untouched for five years. Investing for a period of time t gives a yield Y(t).

This function Y is called the yield curve, and it is often, but not always, an increasing function of t. Yield curves are used by fixed income analysts, who analyze bonds and related securities, to understand conditions in financial markets and to seek trading opportunities. Economists use the curves to understand economic conditions.

The yield curve function Y is actually only known with certainty for a few specific maturity dates, while the other maturities are calculated by interpolation (see Construction of the full yield curve from market data below).

## The typical shape of the yield curve

The British pound yield curve as of 9 February 2005. This curve is unusual in that long-term rates are lower than short-term ones.

Yield curves are usually upward sloping asymptotically; the longer the maturity, the higher the yield, with diminishing marginal growth. There are two common explanations for this phenomenon. First, it may be that the market is anticipating a rise in the risk-free 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 long-term investments.

However, interest rates can fall just as they can rise. Another explanation is that longer maturities entail greater risks for the investor (i.e. the lender). Risk premium should be paid, since with longer maturities, more catastrophic events might occur 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, and the risk of future adverse events (such as default and higher short-term interest rates) is higher than the chance of future positive events (such as lower short-term interest rates). 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 — short-term interest rates higher than long-term — also can occur. 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 only 4.45% on the 30 year bond. The market's anticipation of falling interest rates causes such incidents. Negative liquidity premiums can 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 depressions.

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 move on a daily basis, 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).

### 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 it is denominated. The economic situation of the countries and companies using each currency is a primary factor in determining the yield curve. For example the sluggish economic growth of Japan throughout the late 1990s and early 2000s has meant the yen yield curve is very low (rising from virtually zero at the three month point to only 2% at the 30 year point). By contrast, during that time the British pound curve ranged from 4-5% along its 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 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 might be quoted as LIBOR +0.25%, where 0.25% (often written as 25 basis points or 25bps) is the credit spread.

#### Normal yield curve

From the post-Great 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 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.

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.

#### Steep yield curve

Historically, the 20-year Treasury bond 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 relatively 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.

#### 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 back 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

An inverted yield curve occurs when long-term yields fall below short-term yields. Under unusual circumstances, long-term investors will settle for lower yields now if they think the economy will slow or even decline in the future. An inverted curve has indicated a worsening economic situation in the future 5 out of 6 times since 1970. The New York Federal Reserve regards it as a valuable forecasting tool in predicting recessions two to six quarters ahead. 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. This was seen in 1998 during the Long Term Capital Management failure when there was a slight inversion on part of the curve.

## Theory

There are four 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

$(1 + i_{lt})^n=(1 + i_{st}^{year 1})(1 + i_{st}^{year 2}) \cdots (1 + i_{st}^{year n})$

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. These expected rates, along with an assumption that arbitrage opportunities will be minimal, is enough information to construct a complete yield curve. For example, if investors have an expectation of what 1-year interest rates will be next year, the 2-year interest rate can be calculated as the compounding of this year's interest rate by next year's interest rate. More generally, rates on a long-term instrument are equal to the geometric mean of the yield on a series of short-term instruments. This theory perfectly explains 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: Neglects the risks inherent in investing in bonds (because forward rates are not perfect predictors of future rates). 1) Interest rate risk 2) Reinvestment rate risk

### Liquidity preference theory

The Liquidity Preference Theory, an offshoot of the Pure Expectations 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, 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.

### 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 the markets for short-term and long-term instruments is determined independently. Prospective investors would have to decide in advance whether they need short-term or long-term instruments. Due to the fact that 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).

In an empirical study, 2000 Alexandra E. MacKay, Eliezer Z. Prisman, and Yisong S. Tian found segmentation in the market for Canadian government bonds, and attributed it to differential taxation.

For a brief period in the last week of 2005, and again in early 2006, the US Dollar yield curve inverted, with short-term yields actually exceeding long-term yields. Market segmentation theory would attribute this to an investor preference for longer term securities, particularly from pension funds and foreign investors who prefer guaranteed longer term yields.

### Preferred habitat theory

The Preferred Habitat Theory 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 represents a middle ground between the Market Segmentation Theory and the Market Expectations Theory. Moreover, it seems to explain 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.

### Historical development of yield curve theory

On , 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 importantly those at Salomon Brothers in New York. By the middle of the 1970s, due to the prodding of 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, and 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 improvements on 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 the 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 (lira french franc, deutsche mark, danish krona and many other currencies including ecu). This innovation was a major contribution for the issuance of long dated zero coupon bonds and for the creation of long dated mortgages

## Construction of the full yield curve from market data

 Type Settlement date Rate (%) Cash Overnight rate 5.58675 Cash Tomorrow next rate 5.59375 Cash 1m 5.625 Cash 3m 5.71875 Future Dec-97 5.76 Future Mar-98 5.77 Future Jun-98 5.82 Future Sep-98 5.88 Future Dec-98 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 6 October 1997

The usual representation of the yield curve is 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

$Y(t) = \left(\frac{1}{P(t)} \right)^{\frac{1}{t}} -1.$

The significant difficulty in defining a yield curve therefore is to determine the function P(t). P is called the discount factor function.

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 uses prices of "cash" from today's LIBOR rates, which determine the "short end" of the curve i.e. for t ≤ 3m, futures which determine the mid-section of the curve (3m ≤ t ≤ 15m) and interest rate swaps which determine the "long end" (1y ≤ t ≤ 60y).

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 = A*P (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

A * P = F + ε

where ε is as small a vector as possible (where the size of a vector might be measured by taking its norm, for example).

Note that 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 ε 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:

1. Approximation using Lagrange polynomials
2. Fitting using parameterised curves (such as splines, the Nelson-Siegel family, the Svensson family or the Cairns restricted-exponential family of curves). Van Deventer, Imai and Mesler summarize three different techniques for curve fitting that satisfy the maximum smoothness of either forward interest rates, zero coupon bond prices, or zero coupon bond yields
3. Local regression using kernels
4. Linear programming

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.

## References

• Jessica James & Nick Webber (2001). Interest Rate Modelling. John Wiley & Sons. ISBN 0-471-97523-0.
• Riccardo Rebonato (1998). Interest-Rate Option Models. John Wiley & Sons. ISBN 0-471-97958-9.
• Nicholas Dunbar (2000). Inventing Money. John Wiley & Sons. ISBN 0-471-89999-2.
• N. Anderson, F. Breedon, M. Deacon, A. Derry and M. Murphy (1996). Estimating and interpreting the yield curve. John Wiley & Sons. ISBN 0-471-96207-4.
• Andrew J.G. Cairns (2004). Interest Rate Models - An Introduction. Princeton University Press. ISBN 0-691-11894-9.
• John C. Hull (1989). Options, futures and other derivatives. Prentice Hall. ISBN 0-13-015822-4.  See in particular the section Theories of the term structure (section 4.7 in the fourth edition).
• Damiano Brigo, Fabio Mercurio (2001). Interest Rate Models - Theory and Practice. Springer. ISBN 3-540-41772-9.
• Donald R. van Deventer, Kenji Imai, Mark Mesler (2004). Advanced Financial Risk Management, An Integrated Approach to Credit Risk and Interest Rate Risk Management. John Wiley & Sons. ISBN-13: 978-0470821268.
• Ruben D Cohen (2006) "A VaR-Based Model for the Yield Curve [download]" Wilmott Magazine, May Issue.
• Paul F. Cwik (2005) "The Inverted Yield Curve and the Economic Downturn [download]" New Perspectives on Political Economy, Volume 1, Number 1, 2005, pp. 1-37.
• Roger Wets, Stephen Bianchi & Liming Yang (2002), "Serious Zero-Curves" [download]"