Division of Applied Mathematics
School of Education, Culture and Communication
MMA 708 Analytical Finance II
Lecturer: Jan Röman
Bootstrapping with bond prices
Written by: Dmytro Sheludchenko
Arad Tahmidi
Daria Novoderezhkina
Abstract
The purpose of this assignment was to create a bootstrap application in Excel with help of
linear interpolation/extrapolation, given Swedish Government Bonds data. Zero coupon
rates, forward rates, as well as discount factors had to be calculated, and their term
structures plotted accordingly.
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Table of Contents
1.Introduction……………………………………………………………………………………………………………………….4
2.Bootstrappingof bonds.…………………………..…………………………………………………………………………4
3.Spot rates……………….……………………………………………………………………………………………………….…5
4.Forward rates……………….……………………………………………………………………………………………………5
5.Discount factors…………………………………………………………………………………………………………………6
6.Interpolation………………………………………………………………………………………………………………………6
Conclusion…………………………………………………………………………………………………………………….….…..7
References…………………………………………………………………………………………………………………………….8
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1.Introduction
Bonds can be considered as quite simple fixed-income instruments. The concept is as
follows: a bonds issuer is borrowing money from an investor through the bonds sale, and
compensates by making interest payments and repaying the principal amount on the bonds
maturity. A substantial amount of bonds can be traded in a secondary market, just like
stocks and other securities. The main components that must be taken into consideration
when pricing a bond are:
 Principal (notional amount of a loan), or it is also known as par value or face value.
Bonds principal, as it has been already mentioned above, is paid at the end of
bonds life.
 Maturity – expiration of the instrument
 Interest payments, or coupons. These cash flows are received periodically by
owners of majority of bonds.
 Call provisions and other special features
 Credit quality of the issuer
Theoretical price of the bond can be found by discounting all the cash flows, that are to be
received by bond holder, to their present values, and then summing them all up. Often bond
traders use the same discount rate for all the coupon payments, which is not really accurate.
For each cash flow underlying a bond, there should be a different zero rate considered.
Understanding of how to correctly approach bond pricing is vital for traders and investors
nowadays.
2.Bootstrapping of bonds
Bootstrap method can be summarized as an algorithm for calculating the zero-coupon yield
curve from the market data.
Term structure of interest rate:
Yield curve – is simply a plot of bonds yield to maturity against time. It can be derived from
zero-coupon bonds. Shape of the yield curve can be either concave or convex. There exist a
number of theories that try to explain why it is shaped in this manner. In order to observe
the dynamic behavior of interest rates, we have to use a term structure model. What is
really significant, is that term structure models imply the possibility of derivation of an entire
term structure from the stochastic behavior of just few variables. The reason why it is so
important to calculate the whole term structure lies in our desire to maintain internal
consistency between model prices.
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Yield and Yield to Maturity:
As has already been mentioned above, we are interested in calculating zero-coupon rates
using Bootstrap method. If the yield curve is already given to us, then it is possible to
implement bootstrapping technique to find the spot rates and plot their term structure. This
implies “stripping” bonds with purpose to construct artificial zero-coupon bonds of the
coupons and the principal. STRIPS is an abbreviation for “Separate Trading for Registered
Interest and Principal of Securities”. To strip the bond means to separate coupons and the
principal.
Bootstrapping:
Linear bootstrapping can be used to calculate and build a yield curve, or as it can also be
referred to: plotted spot and forward rates’ term structures. In order to do that, we first
have to derive bonds prices from the yield. This can be done with the following formula(for
the bond that pays dividends):
𝑛
𝑁
𝐶
𝑃=
+ ∑
𝑇
(1 + 𝑦𝑡𝑚)
(1 + 𝑦𝑡𝑚)𝑡𝑖
𝑖=1
Where C is the coupon, N – face value and ytm – yield to maturity.
3.Spot rates
We start with the first spot rate. If the first bond doesn’t bear any coupon, then it is a zerocoupon bond and its spot rate can be found directly. For the other bonds, we follow the
same algorithm: we make the price of the bond equal to the discounted values of the
coupons and the principal, by simply substituting known to us values in the equation above.
As long as the bonds’ prices have already been calculated, the only unknown value that we
are looking for in each equation will be the next spot rate. We repeat the same procedure
until all the spot rates have been found out and then plot them against dates on which
coupon payments are due.
4.Forward rates
The relationship between spot and forward rates can be expressed as follows:
𝑓𝑜𝑟𝑤𝑎𝑟𝑑
𝑟𝑡2 −𝑡1
(1 +
= (
(1 +
1
𝑠𝑝𝑜𝑡 𝑡2 𝑡2 −𝑡1
𝑟𝑡2 )
𝑡1 )
𝑟𝑡𝑠𝑝𝑜𝑡
)
1
−1
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5.Discount factors
Discount factors are calculated as:
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(1+𝑠𝑝𝑜𝑡 𝑟𝑎𝑡𝑒)𝑡𝑠𝑝𝑜𝑡
6.Interpolation
During our calculation of the spot rates we have used linear interpolation in order to find all
values that we were interested in. Interpolation between two known points can be
summarised as below:
𝑦 = 𝑦0 + (𝑥 − 𝑥0 )
𝑦1 − 𝑦0
𝑥1 − 𝑥0
Where the two known points are given by the coordinates (𝑥0 , 𝑦0 ) 𝑎𝑛𝑑 (𝑥1 , 𝑦1 ).
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Conclusion
In this assignment we calculated zero-coupon rates using Swedish bonds market data,
obtained forward rates from them and plotted both term structures. Graph of the derived
discount factors dynamics had also been constructed.
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References
1). John C. Hull (2003) Options, Futures and Other Derivatives, Fifth Edition, Prentice Hall,
Upper Saddle River
2). Lecture Notes: Jän Roman
3). http://nasdaqomxnordic.com/bonds/sweden
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