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. For example a company may have to pay interest of 6% per year if it wished to borrow money from a bank for 10 years, but only 5% per year if borrowed for five years. In this case the yield curve, usually represented by a graph of time against interest rate would have points at (5 years,5%) and (10 years,6%). The graph would have the yield or interest rate on the ordinate and term to maturity on the abscissa.

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.

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

1.055¹¹/1.05¹⁰ - 1 = 10.6%

The typical shape of a yield curve

Yield curves are usually upward sloping and accelerating; the longer the maturity, the high 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 are higher than longer term rates; do occur. For instance as at November 2004, the yield curve for UK Government bonds (i.e. the bonds which the UK Government issues to borrow money - see gilt) is partially inverted. The yield for the 10 year bond is 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.

Economies theories describing the yield curve and its movements

There are three main theories attempting to explain how yield varies with term. 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.

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 advanced 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.

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.

Construction of the market yield curve

A typical 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

The significant difficulty in defining a yield curve therefore is to determine the discount function P(t). There are three types of financial instrument used to determine specific points on the curve - "cash" (today's LIBOR rates), which determine the "short end" of the curve i.e. for t<=1y, 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 between these points interpolation is used to determine P(t) for other t.