Bootstrapping (interest)

Bootstrapping is a process with which risk-free interest rates can be derived from yields on coupon bonds that are traded on efficient markets. The yield curve can then be displayed with the interest rates derived . Using the bootstrapping process, it is possible to eliminate the so-called reinvestment risk. Bootstrapping is not possible with zero coupon bonds as they are not traded in efficient markets.

With bootstrapping, the risk-free interest rate for the smallest period, i.e. starting at the short end of the interest rate structure, is derived and then gradually expanded to include the next longer period.

Procedure

The procedure can be described as follows:

The 1-year risk-free interest rate is obtained from the yield on a 1-year coupon bond, the 2-year risk-free interest rate is obtained from a 2-year coupon bond in conjunction with the 1-year coupon bond, etc.

formula

${\displaystyle r_{n}={\sqrt[{n}]{\tfrac {N+k}{\left(K-{\tfrac {k}{(1+r_{1})^{1}}}-{\tfrac {k}{(1+r_{2})^{2}}}-...-{\tfrac {k}{(1+r_{n-1})^{n-1}}}\right)}}}-1}$

With:

• K = price of a bond (corresponds to the present value of a bond)

• k = coupon of a bond

• N = face value of a bond

• R = yield on a bond

• r = risk-free interest rate

Assumption: r ₁ = R ₁

example

Given the price of a bond (100 GE) as well as the 1-year yield on a bond (1%), the 2-year yield on a bond (3%) and the 3-year yield on a bond (5%). Initially, only the first two years are considered and thus the following form is formed:

${\displaystyle 100={\tfrac {3}{(1+0,01)^{1}}}+{\tfrac {103}{(1+r_{2})^{2}}}}$. A simple reshaping gives . You get the risk-free interest rate of around 3.03%. ${\displaystyle r_{2}={\sqrt {\tfrac {103}{100-{\tfrac {3}{1,01}}}}}-1}$ ${\displaystyle r_{2}}$

To now determine the risk-free interest rate , based on the risk-free interest rate , add the yield on the 3-year bond to the formula: ${\displaystyle r_{3}}$ ${\displaystyle r_{2}}$

${\displaystyle 100={\tfrac {5}{(1+0,01)^{1}}}+{\tfrac {5}{(1+0,0303)^{2}}}+{\tfrac {105}{(1+r_{3})^{3}}}}$. By reshaping one obtains . You get the risk-free interest rate of about 5.14%. ${\displaystyle r_{3}={\sqrt[{3}]{\tfrac {105}{\left(100-{\tfrac {5}{1,01}}-{\tfrac {5}{1,0303^{2}}}\right)}}}-1}$ ${\displaystyle r_{3}}$

literature

• William F. Sharpe, Gordon J. Alexander, Jeffery V. Bailey: Investments. Prentice Hall International, 1998. ISBN 0-13-011507-X
• Klaus Spremann, Pascal Gantenbein: Interest, bonds, loans . Oldenbourg, 2007, p. 122, ISBN 3486583735 Google Books