The construction of a zero-coupon yield curve by the method of bootstrapping

The diversity of yield curves

In financial markets, there is, at any given time, not just one, but a multitude of yield curves.
One can broadly distinguish two types of curves:

• Observed curves or market curves that are built directly from quotations on the markets (e.g. swap curves, government bond yield curves)
• Implicit curves, which are derived from market quotes, but are obtained via transformation (e.g. zero-coupon yield curves, par yield curves)

In order to have a consistent set of data, yield curves are always constructed using the yield rates of a set of homogeneous instruments.
For bond yield curves, for example, this means in particular that one always uses instruments from the same issuer or, if it is a sector curve, from issuers which belong to the same sector.

But building a yield curve from “classic” coupon bonds would create a curve which suffers from a number of inconsistencies.
Thus, for example, two bonds with the same maturity but a very different duration, will not have the same yield. Also, two identical coupons belonging to two bonds with different maturities will not be discounted at the same yield, whereas they generate the exact same cash flow.

The bootstrapping method

To overcome these problems, one constructs a zero-coupon yield curve from the prices of these traded instruments. As a reminder, the zero-coupon rate is the yield of an instrument that does not generate any cash flows between its date of issuance and its date of maturity.
The technique used to achieve this is called bootstrapping, a term which describes a self-contained process that is supposed to proceed without external input.
This method is based on the assumption that the theoretical price of a bond is equal to the sum of the cash flows discounted at the zero-coupon rate of each flow.

To illustrate this, let's take the example of a bond with a remaining lifetime of five years and an annual coupon of 3.5 %. Using a series of zero-coupon rates, the determination of which we will see later, we can determine the theoretical price of the bond :

Based on this concept, we can build a zero-coupon curve starting from a set of bonds with different maturities. For the construction of our zero-coupon curve, we will take for example the following list of securities as a starting point :

The first securities (A, B and C) actually are already zero coupons, since they generate no cash flows before maturity. This is generally the case for money market securities with a maturity of less than, or equal to, one year which remunerate their holders not through a coupon but through the difference between the issue price, which is below par, and the price at which they will be redeemed.

For these securities we can calculate the zero-coupon rates directly from their market price:

Security A (6 months) :

\[ r_{6m} = \left( \frac{100}{99.05} - 1\right ) \cdot \frac{12}{6} = 1.9182\% \]

Security B (9 months) :

\[ r_{9m} = \left( \frac{100}{98.45} - 1\right ) \cdot \frac{12}{9} = 2.0992\% \]

Security C (1 year) :

\[ r_{1y} = \left( \frac{100}{97.85} - 1\right ) \cdot \frac{12}{12} = 2.1972\% \]

The preceding three yields have been calculated using this formula

We can then successively carry out the deduction of zero-coupon rates for the 2 and 3 year maturities . Already knowing the rate for one year maturity ( 2.1972 %) , we can deduce the rate two years as described hereafter.

Keeping in mind that a bond can be considered a set of zero-coupon instruments, its (theoretical) price is, therefore, equivalent to the sum of present values ​​of the zero-coupons.
To calculate the zero-coupon rate for the 2-year maturity, we will strip security D into two zero-coupons : the first with a nominal amount of 3.5 ( the 1st year coupon) and a maturity of one year, and the second with a nominal amount of 103.5 (2nd year coupon plus redemption of the bond's nominal) and a maturity of two years.

Discounting the coupon of the first year of security D (3.50 %) using the zero-coupon rate from security C above (2.1972%), we will obtain the following present value:

\[ PV(cpn_{1y}) = \frac{3.500}{(1+2.1972\%)} = 3.42475 \]

By subtracting the result of this calculation from the price of security D, we now can determine the present value of the second “zero-coupon instrument”:

\[ PV(cpn_{2y}) = 101.40 - 3.42475 = 97.97525 \]

And we now have all the information required in order to calculate the zero-coupon rate as illustrated in the table below:

In order to obtain the 2-year zero-coupon rate, we just have to calculate, by iteration, the rate at which an amount of 97.97525 would have to be placed in order to receive 103.50 after two years:

\[ 97.97525 \cdot (1+i)^{2} = 103.50 \]

The rate i obtained is 2.7808%.

Having obtained the zero-coupon rates for the maturities 1 year ( 2.1972 %) and 2 years ( 2.7808 %), we will now use the same approach for determining the 3-year zero-coupon rate.

The 3-year zero-coupon rate we are looking for is, therefore, the one that verifies the following equation:

\[ 94.4995 \cdot (1+i)^{3} = 104.00 \]

The rate obtained, again by iteration, is 3.2447%.

Our resulting zero-coupon yield curve thus presents itself as follows: