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 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 overall rate of return available
on 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 increases (that is, as
one moves to the right, the curve flattens out). 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. 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). A
risk
premium is needed by the market, since at longer durations
there is more uncertainty and a greater chance of catastrophic
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, 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 position (short-term interest rates higher than
long-term) can also 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 was only 4.45% for 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 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 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).
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
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 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 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 May 2009, the gap between yields on two-year Treasury notes and
10-year notes widened to 2.75 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
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 risk2) Reinvestment
rate risk
Liquidity preference theory
The Liquidity Preference Theory, also known as the Liquidity
Premium Theory, is an offshoot of the Pure Expectations Theory. The
Liquidity Preference 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 preference theory:
(1 + i_{lt})^n=rp_{n}+((1 + i_{st}^{\mathrm{year\,1}})(1 +
i_{st}^{\mathrm{year\,2}}) \cdots (1 +
i_{st}^{\mathrm{year\,}n}))
Where rp_{n} is the risk premium associated with an {n} year
bond.
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 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).
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 is another guise of the Market
Segmentation 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.
Historical development of yield curve theory
On 15 August 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 importantly those at Salomon
Brothers in New
York
. 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.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
krona and many other currencies including the ecu). This innovation
was a major contribution towards the issuance of long dated zero
coupon bonds and the creation of long dated mortgages.
Construction of the full yield curve from market data
Typical inputs to the money market
curve
| 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 + \epsilon
where \epsilon 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 \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 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
- 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.
See also
Notes
References
- See in particular the section Theories of the term
structure (section 4.7 in the fourth edition).
- 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 J.-B. Wets, Stephen W. Bianchi, "Term and Volatility
Structures" in
- Rise in Rates Jolts Markets --- Fed's Effort to Revive Economy
Is Complicated by Fresh Jump in Borrowing Costs author = Liz
Rappaport. Wall Street Journal. May 28, 2009. p. A.1
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