Yield curve

This article is about yield curves as used in finance. For the term's use in physics, see yield curve (physics).

In finance and economics, the yield curve or the term structure of interest rates is the relationship between the cost of borrowing money and the amount of time the money is being borrowed for.

The yield of a financial instrument is the amount of money to be made per year by investing in that instrument. For instance, a bank account that promises an interest rate of 4% per year has a 4% yield. In general the amount per year that can be made 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. Generalizing, investing for a period of time t gives a yield Y(t). This function Y is called the yield curve. The nomenclature "curve" is used rather than "yield function" because when plotted on a graph, the function is a curve.

Y is often an increasing function, but that need not always be the case - yield curves are used by fixed income analysts, who analyse 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 periods of time, the other periods are calculated by interpolation (see Construction of the full yield curve from market data below).

Yield curves carry an implicit forecast of future short-term interest rates: for example if the annual yield on a 10-year bond is 5%, and on an 11-year bond is 5.5%, then the implicit yield in year 11 is

<math>\frac{1.055^{11}}{1.05^{10}} - 1 = 10.6%<math>

Theory

There are three main economic theories attempting to explain how yield varies with term (borrowing) period. Two of the theories are extreme positions, while the third is the combination of the former two. It attempts to take the middle ground.

Market expectations (pure expectations) theory

This theory is also called the expectation hypothesis. In this theory, financial instruments of different durations are considered perfect substitutes. The market expectations theory states that a certificate of deposit for 2 years will have the same yield as a CD for 1 year followed by another CD for 1 year.

<math>(1 + i_{lt})^n=(1 + i_{st}^{year1})(1 + i_{st}^{year2}) \cdots (1 + i_{st}^{yearn})<math>

This theory suggests that the yield on a long-term instrument is equal to the geometric mean of the yield on a series of short-term instruments. This theory perfectly explains the stylized fact that yield tend to move together. However, it fails to explain the other stylized facts regarding a normal yield curve.

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 explain the stylized fact that short-term yield is usually lower than long-term yield. This theory explains the stylized fact about a normal yield curve. However, because the supply and demand of the two markets are still independent, this theory fails to explain the stylized fact that yields of different terms move together.

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.

Liquidity preference theory

This theory is also called preferred habitat hypothesis. This theory attempts to find the middle ground in former two theories. It is also the most accepted theory of the three.

This theory introduces an element called the liquidity premium stating that debtors must pay an incentive to lenders in order to obtain funds for a longer duration. This explains the stylized fact that long-term yield is often higher than short-term yield. In this theory, instruments of different terms are imperfect substitutes, which mean the yield rates are related but not identical.

Historical development of yield curve theory

In August 1971, U.S. President Richard Nixon announced US unilateral withdrawal from the Bretton Woods fixed-exchange rates system, initiating the era of floating exchange rates.

Floating exchanges made life more complicated for bond traders, including importantly those at Salomon Brothers in New York. By the middle of that decade, 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, and accordingly by short-term interest rates. A trader who has held a 5 year bond for 4 years and three quarters will compare herself to a trader then in the market for 3 month bonds, in terms of their respective exploitations of the time value of money.

Both academic research and practical implementation of yield-curve models have made tremendous strides in sophistication since those beginnings, including the development of the economic theories described above.

Practice

The typical shape of the yield curve

Yield curves are usually upward sloping and accelerating; the longer the maturity, the higher the yield. The usual explanation is that longer maturities entail greater risks for the investor (i.e. the lender) and so require higher yields. With longer maturities, more catastrophic events might occur that may impact the investment, hence the need for a risk premium. This explanation depends on the distant future being more uncertain than the near future, and risk of future adverse events (such as default and higher short-term interest rates) being higher than the chance of future positive events (such as lower short-term interest rates). This effect is also referred to as the liquidity spread.

The opposite situation – short term interest rates higher than longer term rates – does occur. For instance, at November 2004, the yield curve for UK Government bonds (i.e. the bonds which the UK Government issues to borrow money - see gilt) was partially inverted. The yield for the 10 year bond stood at 4.68% but only 4.45% on the thirty year bond. Strongly inverted yield curves have historically preceded economic depressions.

Yield curves move on a daily basis; representing the market's reaction to news. A further "stylized fact" observed is that yield curves tend to move in parallel. That is, an increase in the cost of borrowing money for one year is frequently accompanied by a similar shift at points further along the curve.

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 primary 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 30year point. By contrast the GBP curve ranges from 4-5% along its curve.

Even when currency is taken into account, different individuals, companies, institutions and governments can borrow money at different rates. This represents the relative perceived stability or riskiness of the entity. Countries perceived as stable (such as those in North America, Australia, western and central Europe, Scandinavia and east Asia) can borrow most cheaply. The yield curves corresponding to the bonds issued by these governments are the government yield curve. Next banks with the highest credit rating (e.g. Standard and Poors AAA borrow money from each other at the LIBOR rates. These curves are typically a little higher (~0.2%) 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 market curve. The construction of the market curve is described in a later section.

After the LIBOR curve come company-specific curves. They are constructed from corporate bonds issued by finance-seeking companies. Because companies are typically more likely to go bust (and thus be unable to pay the coupons and principal on the bond) than banks and governments, the yields are typically higher. Company yield curves are often quoted in terms of a "spread" over the relevant market yield curve. For instance the five-year yield curve point for Vodafone might be quoted as LIBOR + 0.75%, where 0.75% (often written as 75bps or 75 basis points) is the spread.

Construction of the full yield curve from market data

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

<math>Y(t) = \left(\frac{1}{P(t)} -1 \right)^{\frac{1}{t}}<math>

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" today's LIBOR rates), which determine the "short end" of the curve i.e. for t ≤ y, 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 gives provides with 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). In actual fact 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

<math> A*P = F + \epsilon<math>

where <math>\epsilon<math> 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 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.

Practioners and researchers have suggested many ways of solving the A*P = F equation. It transpires that the most natural method - that of minimizing <math>\epsilon<math> 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 or the Nelson-Siegel family of curves.

Local regression using kernels

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

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• Template:Book reference. See in particular the section Theories of the term structure (section 4.7 in the fourth edition).

External link

• British Banker's Association page with historic yield curve data in various currencies (http://www.bba.org.uk/bba/jsp/polopoly.jsp?d=141&a=627)de:Zinskurve

nl:Yieldcurve