Editing Yield curve (section)

==Construction of the full yield curve from market data==
{{See also|Bootstrapping (finance)|Fixed-income attribution#Modeling the yield curve|Multi-curve framework}}
{| class="toccolours" border="1" cellpadding="4" cellspacing="0" align="right" style="margin: 0em 1em 0em 1em;"
|+ '''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
|-
|colspan="3"|
A list of standard instruments used to build a money market yield curve.
|-
|colspan="3"|
The data is for lending in [[US dollar]], taken from October 6, 1997
|}

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

:<math>Y(t) = P(t)^{-1/t} -1. </math>

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]] use prices of "cash" from today's LIBOR rates, which determine the "short end" of the curve i.e. for ''t''&nbsp;≤&nbsp;3m, [[interest rate future]]s which determine the midsection of the curve (3m&nbsp;≤&nbsp;''t''&nbsp;≤&nbsp;15m) and [[interest rate swap]]s which determine the "long end" (1y&nbsp;≤&nbsp;''t''&nbsp;≤&nbsp;60y).

The example given in the table at the right is known as a [[LIBOR]] 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 of similar maturity is usually positive, meaning that private borrowing is at a premium above government borrowing. This spread is a measure of the difference in the risk tolerances of the lenders to the two types of borrowing. For the U. S. market, a common benchmark for such a spread is given by the so-called [[TED spread]].

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

: <math> AP = F + \varepsilon \, </math>

where <math>\varepsilon</math> is as small a vector as possible (where the size of a vector might be measured by taking its [[norm (mathematics)|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]] 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 <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 [[Spline (mathematics)|splines]], the [[Nelson-Siegel]] family, the [[Fixed income attribution#Modeling the yield curve|Svensson family]], the exponential polynomial<ref>{{Cite web|url=https://www.researchgate.net/publication/323689711|title=The exponential polynomial family|last=Moulin|first=Serge|date=2018|website=Research gate.net}}</ref> 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 [[Kernel (statistics)|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 (finance)|bootstrapping]] from one to the next. At the long end, a regression technique with a cost function that values smoothness might be used.