Yield Curve - Construction of The Full Yield Curve From Market Data

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 October 6, 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

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 use 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 midsection of the curve (3m ≤ t ≤ 15m) and interest rate swaps which determine the "long end" (1y ≤ t ≤ 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, usually positive, meaning private borrowing is at a premium above government borrowing, of similar maturity is a measure of risk tolerance of the lenders. For the U. S. market, a common benchmark for such a spread is given by the so call 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

where 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 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.

Read more about this topic:  Yield Curve

Famous quotes containing the words construction of the, construction of, market, data, curve, construction, full and/or yield:

    There is, I think, no point in the philosophy of progressive education which is sounder than its emphasis upon the importance of the participation of the learner in the formation of the purposes which direct his activities in the learning process, just as there is no defect in traditional education greater than its failure to secure the active cooperation of the pupil in construction of the purposes involved in his studying.
    John Dewey (1859–1952)

    There is, I think, no point in the philosophy of progressive education which is sounder than its emphasis upon the importance of the participation of the learner in the formation of the purposes which direct his activities in the learning process, just as there is no defect in traditional education greater than its failure to secure the active cooperation of the pupil in construction of the purposes involved in his studying.
    John Dewey (1859–1952)

    It is a sign of our times, conspicuous to the coarsest observer, that many intelligent and religious persons withdraw themselves from the common labors and competitions of the market and the caucus, and betake themselves to a certain solitary and critical way of living, from which no solid fruit has yet appeared to justify their separation.
    Ralph Waldo Emerson (1803–1882)

    Mental health data from the 1950’s on middle-aged women showed them to be a particularly distressed group, vulnerable to depression and feelings of uselessness. This isn’t surprising. If society tells you that your main role is to be attractive to men and you are getting crow’s feet, and to be a mother to children and yours are leaving home, no wonder you are distressed.
    Grace Baruch (20th century)

    And out again I curve and flow
    To join the brimming river,
    For men may come and men may go,
    But I go on forever.
    Alfred Tennyson (1809–1892)

    There’s no art
    To find the mind’s construction in the face.
    William Shakespeare (1564–1616)

    There was somewhat military in his nature, not to be subdued, always manly and able, but rarely tender, as if he did not feel himself except in opposition. He wanted a fallacy to expose, a blunder to pillory, I may say required a little sense of victory, a roll of the drum, to call his powers into full exercise.
    Ralph Waldo Emerson (1803–1882)

    I would not deny you; but, by this good day, I yield upon great persuasion; and partly to save your life, for I was told you were in a consumption.
    William Shakespeare (1564–1616)