In
finance, bootstrapping is a method for constructing a (
zero-coupon) fixed-income
yield curve
In finance, the yield curve is a graph which depicts how the Yield to maturity, yields on debt instruments - such as bonds - vary as a function of their years remaining to Maturity (finance), maturity. Typically, the graph's horizontal or ...
from the prices of a set of coupon-bearing products, e.g.
bonds and
swaps.
A ''bootstrapped curve'', correspondingly, is one where the prices of the instruments used as an ''input'' to the curve, will be an exact ''output'', when these same instruments are
valued using this curve.
Here, the term structure of spot returns is recovered from the bond yields by solving for them recursively, by
forward substitution: this iterative process is called the ''bootstrap method''.
The usefulness of bootstrapping is that using only a few carefully selected zero-coupon products, it becomes possible to derive par
swap rates (forward and spot) for ''all'' maturities given the solved curve.
Methodology
As stated above, the selection of the input securities is important, given that there is a general lack of data points in a
yield curve
In finance, the yield curve is a graph which depicts how the Yield to maturity, yields on debt instruments - such as bonds - vary as a function of their years remaining to Maturity (finance), maturity. Typically, the graph's horizontal or ...
(there are only a fixed number of products in the market). More importantly, because the input securities have varying coupon frequencies, the selection of the input securities is critical. It makes sense to construct a curve of zero-coupon instruments from which one can price any yield, whether forward or spot, without the need of more external informatio
Note that certain assumptions (e.g. the
interpolation
In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points.
In engineering and science, one often has ...
method) will always be required.
General methodology
The general methodology is as follows: (1) Define the set of yielding products - these will generally be coupon-bearing bonds; (2) Derive discount factors for the corresponding terms - these are the internal rates of return of the bonds; (3) 'Bootstrap' the zero-coupon curve, successively
calibrating
In measurement technology and metrology, calibration is the comparison of measurement values delivered by a device under test with those of a calibration standard of known accuracy. Such a standard could be another measurement device of known a ...
this curve such that it returns the prices of the inputs. A generically stated
algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
for the third step is as follows; for more detail see .
For each input instrument, proceeding through these in terms of increasing maturity:
*
solve analytically for the zero-rate where this is possible (see side-bar example)
*if not,
iteratively solve (initially using an approximation) such that the price of the instrument in question is exactly made output when calculated using the curve (note that the rate corresponding to this instrument's maturity is solved; rates between this date and the previously solved instrument's maturity are interpolated)
*once solved, save these rates, and proceed to the next instrument.
When solved as described here, the curve will be
arbitrage free in the sense that it is exactly consistent with the selected prices; see and . Note that some analysts will instead construct the curve such that it results in a
best-fit "through" the input prices, as opposed to an exact match, using a method such as
Nelson-Siegel.
Regardless of approach, however, there is a requirement that the curve be arbitrage-free in a second sense: that all
forward rates The forward price (or sometimes forward rate) is the agreed upon price of an asset in a forward contract. Using the rational pricing assumption, for a forward contract on an underlying asset that is tradeable, the forward price can be expressed in t ...
are positive. More sophisticated methods for the curve construction — whether targeting an exact- or a best-fit — will additionally target
curve "smoothness" as an outpu
http://www.math.ku.dk/~rolf/HaganWest.pdf] and the choice of
interpolation, interpolation method here, for rates not directly specified, will then be important.
Forward substitution
A more detailed description of the forward substitution is as follows. For each stage of the iterative process, we are interested in deriving the n-year
zero-coupon bond
A zero coupon bond (also discount bond or deep discount bond) is a bond in which the face value is repaid at the time of maturity. Unlike regular bonds, it does not make periodic interest payments or have so-called coupons, hence the term ze ...
yield, also known as the
internal rate of return
Internal rate of return (IRR) is a method of calculating an investment’s rate of return. The term ''internal'' refers to the fact that the calculation excludes external factors, such as the risk-free rate, inflation, the cost of capital, or ...
of the zero-coupon bond. As there are no intermediate payments on this bond, (all the interest and principal is realized at the end of n years) it is sometimes called the n-year spot rate. To derive this rate we observe that the theoretical price of a bond can be calculated as the present value of the cash flows to be received in the future. In the case of swap rates, we want the par bond rate (Swaps are priced at par when created) and therefore we require that the present value of the future cash flows and principal be equal to 100%.
:
therefore
:
(this formula is precisely
forward substitution)
:where
:*
is the coupon rate of the n-year bond
:*
is the length, or
day count fraction
A day is the time period of a full rotation of the Earth with respect to the Sun. On average, this is 24 hours, 1440 minutes, or 86,400 seconds. In everyday life, the word "day" often refers to a solar day, which is the length between two solar ...
, of the period