NKEMKA, CHRISTOPHER CHUKWUKA PG/M.Sc/03/34508 THE TERM STRUCTURE OF INTEREST RATES AND BOND VALUATION MODELLING IN A PERIOD OF ECONOMIC DISTORTION Mathematics BEING A PROJECT SUBMITTED TO THE DEPARTMENT OF MATHEMATICS IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE AWARD OF MASTER OF SCIENCE (M.SC) DEGREE IN MATHEMATICS. Webmaster 2010 UNIVERSITY OF NIGERIA
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NKEMKA, CHRISTOPHER CHUKWUKA
PG/M.Sc/03/34508
THE TERM STRUCTURE OF INTEREST RATES AND
BOND VALUATION MODELLING IN A PERIOD OF
ECONOMIC DISTORTION
Mathematics
BEING A PROJECT SUBMITTED TO THE DEPARTMENT OF
MATHEMATICS IN PARTIAL FULFILLMENT OF THE REQUIREMENT
FOR THE AWARD OF MASTER OF SCIENCE (M.SC) DEGREE IN
MATHEMATICS.
Webmaster
2010
UNIVERSITY OF NIGERIA
2
THE TERM STRUCTURE OF INTEREST
RATES AND BOND VALUATION
MODELLING IN A PERIOD OF
ECONOMIC DISTORTION
BY
NKEMKA, CHRISTOPHER CHUKWUKA
PG/M.Sc/03/34508
DEPARTMENT OF MATHEMATICS
UNIVERSITY OF NIGERIA
NSUKKA.
SEPTEMBER, 2010.
i
THE TERM STRUCTURE OF INTEREST
RATES AND BOND VALUATION
MODELLING IN A PERIOD OF
ECONOMIC DISTORTION
BY
NKEMKA, CHRISTOPHER CHUKWUKA
PG/M.Sc/03/34508
Being a project submitted to the Department of
Mathematics in partial fulfillment of the
requirement for the award of Master of Science
(M.Sc) degree in Mathematics.
SEPTEMBER, 2010.
ii
CERTIFICATION
This is to certify that this work was carried out by Nkemka,
Christopher Chukwuka with registration number PG/M.Sc/03/34508 of the
Department of Mathematics, Faculty of Physical Sciences, University of
Nigeria, Nsukka.
___________________ _____________
Dr. G.C.E. Mbah Date
(Supervisor)
____________________ _____________
Prof. F. I, Njoku Date
(Head of Department)
____________________ _____________
External Examiner Date
iii
DEDICATION
To my wife Ezinwanne and my daughter Ifunanyachukwu.
iv
ACKNOWLEDGEMENT
My greatest thanks and praises go to the Almighty God for his
promises are ever sure. Without Him, I would have done nothing tangible.
I am profoundly grateful to my mentor, motivator and supervisor Dr.
G.C.E. Mbah for his timely encouragement and suggestions towards the
development of this work and beyond. I gratefully acknowledge every one
who in one way or the other motivated, helped and encouraged me to this
far.
I am immensely grateful to Prof. J.C.Amazigo who is a great
father to me, he has never reserved his love to students. Prof, you are
blessed. I have the pleasure to acknowledge my lecturers: Dr. E.C Obi, Dr.
G.C.E Mbah, Prof. F.I. Njoku, Prof. M.O. Oyesanya, Prof. A.N. Eke, Prof.
M.O. Osilike and all the staff of the Department of Mathematics, University
of Nigeria, Nsukka.
Nevertheless, I appreciate the special efforts of special people like my
wife Mrs. Nkemka Ezinwanne, who has been there for me, my father Elder
Nkemka Ezekiel, my brother, Mr. Nkemka Emmanuel, my sisters and my
wonderful daughter Ifunanyachukwu Chimdimma. You are great.
I reserve special kudos to Miss Ngozi Ezema for her relentless effort
in typesetting this work.
Above all, to God be the glory.
v
ABSTRACT
This work presents the term structure of interest rate and bond valuation
modeling in a period of economic distortion. In real life, we do not expect
interest rate to be constant. Government policies affect the interest rate of
debt instrument.
By the theory of economic fluctuations, there will be economic shocks that
distort the lending rates. With these shocks, investors tend to limit potential
losses. With the equation that determines the market price of the bond at
time t, the market price at which the stream of continuous cash flows would
trade (if arbitrage is avoided) is formulated. Thus the sensitivity of market
price due to interest rate, duration and convexity of the market price due to
interest rates are formulated and solved.
vi
CONTENTS
Title Page i
Certification ii
Dedication iii
Acknowledgement iv
Abstract v
Table of Contents vi
CHAPTER ONE:
Introduction 1
1.1 Aims and Objectives 3
1.2 Scope of the Study 3
1.3 Limitation of the study 4
CHAPTER TWO:
2.0 Literature Review 6
CHAPTER THREE:
3.0 Debt Instruments 16
3.1 Bond Yield 24
3.1.1 Bond Duration 29
3.2 Convertible Bonds 32
vii
3.2.1 Determinants of the Market Prices of a Convertible Bond 37
3.3 Factors Influencing Bond Prices and Interest Rates. 39
3.4 The Term Structure of Interest Rates. 41
3.4.1 The Spot Rates 43
3.4.2 The Forward Rates 45
3.4.3 The Swap Curves/Rates 47
CHAPTER FOUR:
4.1 The Model 49
4.2 Method of Solution 50
4.3 The Analysis of the Model 56
CHAPTER FIVE:
5.1 Discussion of Results 59
5.2 Summary 61
5.3 Conclusion 62
References
Appendix/Glossary
1
CHAPTER ONE
1.0 INTRODUCTION
When the government uses its powers to influence total spending
either by directly changing its purchases of goods and services or indirectly
by altering the disposable incomes of persons through changes in the level of
taxation or transfer outlays, we have fiscal policy. The effects of fiscal
policies of the state and local government are usually harsh. The reason is
just that such sub-national governments cannot conduct systematic fiscal
policy because they cannot run unlimited deficits. They must try to make
ends meet or will lose their credit ratings and they are therefore bound by a
budgetary constraint that does not apply to the federal government.
Government expenditure and tax polices has three major macroeconomic
effects; the expenditure impact, the financial, and the supply impact. If the
government embarks on a high way construction program, the spending
increase directly raises economic activity. If government finances the
resulting deficit by selling bonds to the private sector of the economy, the
wealth of the private sector will increase, and this financial impact will have
subsequent spending effects.
A simple macroeconomic model of income determination shows that
national income Y is given as
2
Y = C + I + G
Where C = consumption, I = Investment and
G = government expenditure
Also C = Co + b Y; 0 < b < 1
Real national income (Y) equals Realized national expenditure (E)
Y = Yd + T
Yd = disposable income
T = All taxes minus government transfer
payments.
E = C + Ir + G
C = consumption; Ir = Realized net investment.
G = Government purchase of goods and services.
Hence Y = E = C + I + G.
Investment is the most volatile of the major component of aggregate
expenditure and for the purpose of analysis, we can split it into three
components – Business fixed investment, residential construction and net
change in business inventories. Considering investment, a holder of wealth
must remember that different assets yield different return and have different
level of risk attached to them. The wealth holder must decide whether and in
what proportion to hold long term bonds, short term bonds, equities (stocks)
3
and other types of assets. These are financial assets which are traded on the
factor market (financial market to be precise).
1.1 AIMS AND OBJECTIVES
The study has the following aims
(a) Exploring the concept and mathematical technology of bond
pricing.
(b) Analyzing the potential return for investments with different
maturities.
(c) Analyzing bond valuation and bond option valuation.
With the general objectives of looking at
(a) Bond valuation formulas in continuous time and (b) the term
structure of interest rate in continuous time. The ultimate objective is to
formulate models for bond valuation in continuous time when short term
rates are variables.
1.2 SCOPE OF THE STUDY
The study is not concerned with detailed debt instrument theories and
laws. The study is concerned with debt instrument which are bonds and bank
loans, where it focuses at yield on bonds of the same credit quality at
4
different maturities; valuation of bonds, bond option under the term structure
environment. Principles governing the valuation of debts instruments, the
interest rate that makes the present value of the cash flow and the market
value of the instrument shall be discussed using an iterative technique like
Newton-Raphson. The par value which will be extracted as a ratio of price
of the dept instrument P to the maturity value M, the yield measures which
are: the yield to maturity on country’s bench mark government bonds, the
spot rates, the forward rates, and the swap rate will be explained and as they
are used in construction of the yield curve. The study will adopt arbitrage
free models.
Classical economic theories like expectation theories (which include
pure expectation liquidity theory and preferred habitat theory) and the
market segmentation theories shall be used to x-ray investors’ behavior.
The bond valuation models and structure in bond, continuous time
will be formulated and method of solution explained. Interpretation and
analysis of the model will follow immediately as they may help investors.
1.3 LIMITATION OF THE STUDY
All glory comes from daring to start. Initial stress of how best to
model this study of interest subjected the researcher to a pause, think and act
5
sections. To hold the bull by the horns, the financial economics and
econophysics terminologies were reviewed for smooth flow of application.
The stochastic description of interest rate is challenging from the point
of view of both mathematics and economic theory. The mathematical
difficulties stem from the fact that one should consider not just one interest
rate but the entire term structure of interest rates, which is a difficult
problem of infinite dimensionality.
Financial and time constraints on the part of the researcher added to
the difficulties. For any research work to be successful, the fund and time
should be readily available.
6
CHAPTER TWO
LITERATURE REVIEW Fixed income securities can be studied looking through the
frameworks like debt instrument classification the market interest rates and
bond portfolio strategy. Such securities are always issued by entities like
corporations, the government, political sub-division and non-profit
organization. Accordingly, the securities issued by the above entities differ
in liquidity, risk, coupon payment, size, maturity, taxability and other
features depending on the state of the world at the very time of issuance. A
bond is a debt instrument used as an evidence of indebtedness specifying the
right of the holder and the duties of the issuer. Bonds are traded on the floor
of organized exchanges or over the counter.
The market value of this debt instrument can be evaluated as the
discounted value of the cash flows expected from the bond. However, the
value of bond is dependent on its life and on assumed reinvestment rate. The
coupon may or may not be the same value over the life (Sakis. J. Khoury and
Torrence D Parsons, 1981) Two models were formulated one for coupon
with the same value over life and the other varying values. The resulting
equation involves geometric series. Considering the bond yield to maturity,
the approximation method developed by the two researchers above was
found less reliable, the higher the interest rates. The bond duration is always
7
equal to or less than the term to maturity (Fredrick Macaulay, 1956).
Comparing a pure discount bond and coupon bond, in pure discount bond,
the investor prefers to allow the duration equal to its term of structure but for
coupon bond, once there is a realization of a portion of the expected wealth,
then maturity is reached. For a given change in market yield, bond prices
vary proportionally with duration, that is, the duration of a bond or a
portfolio of bonds contains information about risk (M. Hopewell and G.
Kaufman 1973). The concept of bond duration is more powerful than that of
bond maturity. This concept of bond duration is very useful in bond portfolio
management. In 1970, Burton G. Malkiel developed the theorems of bond
price. These five theorems are most helpful in explicating bond price
movements. James Walter and Augustine V. Que in 1973 attempted to
improve on the conventional models by using the Monte Carlo simulation to
forecast rates of return on convertible bonds conditional upon the simulated
behavior of the underlying stock. Their conclusion was that behavioural
input derived for the simulation model attested to the powerful influence of
the relationship between conversion values and straight bond values upon
convertible bond premiums and to the asymmetry of premiums, depending
on whether conversion values or straight bond values dominated (Walter and
Que, 1973).
8
Interestingly, many factors influence the bond prices. Amongst six of
these factors is the quality of the bond, that is, the probability of default on
interest and/or principal payment. This could be seen in various rating given
by rating services. Default risk is non-diversifiable risk of a portfolio and is
simply the weighted average of the default risk of the securities making up
the portfolio. Interest rates are also influenced by some factors. Interest
represents the price paid for exchanging future naira for current naira.
The invisible hands of demand and supply have a high influence on
the interest rates. Put differently the determination of price of credit is
largely a function of supply and demand. The total reserves for the banking
system equal the sum of the required reserves and the excess reserves.
(Polakoff et al 1970). A negative excess reserve that is becoming more
negative over times is a strong indication of tight credit condition because it
indicates that the ability of the banking system to expand credit is being
stretched further and further towards its limits. However, interest rates
reported in the financial papers are nominal ex-ante of interest. Their real-
rate component must be positive; otherwise the lender would willingly be
transferring wealth to the borrowers. Although inflation rates may impose
adverse influence on the rate of interest, an expansionary monetary policy
may well result in higher rather than lower rates of interest if it increases
9
inflationary expectations (Rachel Balbach 1977). This determines how large
the range in which ex-ante yields on long term bonds relative to short term
bonds vary if term pressure are to account for a significant fraction of the
variance of the holding period yields on long term bonds (James Pesando,
2008).
The use of treasury method for counting yield curves was criticized
from the academic communities. They derived testable equations which
were used to fit the curve using econometric techniques instead of the
treasury’s freehand approach (Martin E. Ecols and Jan Walter Elliott. 1976).
The arising equation was tested using various measures of goodness of fit
and other measures dealing with auto regression and multicolinearity. It was
tested against a model proposed by Cohen et al (1966) The difficulty in the
estimation of the term structure results from two factors, (1) Most
observable securities are not pure discount bonds (2) the maturities of
observable securities are scattered throughout the future and not necessarily
at points in time for which term structure estimates are needed. (Deborah H.
Miler, 1979). The first factor mentioned by H. Miller corroborates with the
findings of E. Ecols and W. Elliott. It deals with the differential impact
bonds or a set of bonds have on yields to maturity depending on whether the
bond carry coupons or not and on the size of the coupon payment. Pure
10
discount (zero coupon) bonds have only one cash flow equal to the face
value of the bond while a coupon bond has one or many cash flows prior to
the payment of face value.
A fundamental based crisis arises when some state variables such as
foreign exchange reserve, reaches a critical level and triggers an
abandonment of the fixed rate. A self-fulfilling crisis is triggered by an
autonomous change in the belief of speculators (Alan Suther land 2006). If
interest rates are expected to change over the life of a bond, then the size and
timing of coupon payments can greatly affect the return from a bond. The
more of a bond’s return which comes from its coupon payments rather than
from a payment of principal at maturity, the more important reinvestment
rates become in determining a bond’s return. Thus yield to maturity has the
potential of causing greater distortion as a spot rate estimator when coupon
is large (Deborah H. Miller 1979). Short term loans have cost advantage
over but incur higher refinancing and interest risk than longer term loans.
Only firms with greater financial flexibility and financial strength can use
proportionately more short term loans. Financially, strong firms take
advantage of lower interest rate of short term debt. There are proportionately
more short term-loans when the term premium is high. (Jun Sang-Gyang et
al, 2003).
11
The expectation theory of the term structure is a demand based theory.
The expectations of investors about the future course of interest rates
determine their demand for certain maturities. The theory asserts that no
supplier of securities not even the federal government is large enough to
exert influence over the structure of interest rates. From the assumptions, the
expectation theory argues that the long term rate is geometric average of
short rates. Investors should be indifferent between investing in long-term
securities or in a series of short term securities. This indifference is solely
dependent on the absence of transactions cost and of a preference function
for certain maturities. Any changes in the supply of bonds of a given
maturity would not affect the term structure unless it some how affected
expectations. (Richard W Lanz and Robert H. Rasche 1978).
The model proposed by David Meiselman (1962) in testing the
validity of the expectation hypothesis is the “error – learning” model. As
investors observed that the actual rate of interest is different from the
forward rate, which they have anticipated, they could revise their forecast of
the next one period rate by a fraction of the previous error. He found out that
the results were in support of the hypothesis confirming the validity of the
expectation theory. J.A. Grant (1967) tried to duplicate Meiselman’s result
with British data and found that the error – learning model is not supportable
12
using unsmoothed data. (As opposed to Durand’s smoothed Data) (used by
Meiselman. A. Buse (1967) using smoothed yield curves for British and US
government securities, found the constant term to be positive and significant,
which contradicts the error-learning model. A. M Santomero,(1975) using
Euro-dollar spot rate observation-which bypasses the problems of yield to
maturity and the differential in coupon rate – found substantial support for
the error-learning model. Thus expectation alone does not determine the
shape of the yield curve. Furthermore, and more fundamentally, the
argument that the yield curve allows for the discovery of expected rates does
not lead to the conclusion that expectations are the sole determinants of the
shape of the yield curve. Moreover, observable yield curves are “market”
yield curves that do not necessarily coincide with those of individual
investors.
From a macro economic perspective, the short term interest rate is a
policy instrument under the direct control of the central bank. From a
financial perspective, long rates are adjusted average of expected future
short rates. (Francis Die-boid, 2005). Borrowers have a propensity to
borrow long to lock in the interest rate costs and ensure the availability of
funds. Lenders are more interested in lending short term. They require
compensation for assuming short term maturities because of the probability
13
that the bond will be called before their maturity or that interest rates will
rise in the interim and investors will be unable to take advantage of higher
rates as their capital is locked in the size of premium, its value over time, its
sign and even its very existence are subject to much controversy (John R.
Hicks 1986). In equilibrium, the expectations hypothesis asserts that the
liquidity premium is equal to zero, while the liquidity preference theory
asserts that the value is positive. It is the size, sign and behaviour of the term
premium that distinguishes one theory from another (D.H. Miller 1979).
Some may argue that the liquidity theory can only explain rising yield
curves. The shape of the yield curves may well be determined by factors that
offset the rising liquidity premium. If the expected-rate portion of the
forward rate is expected to fall by a value larger than the rise in the liquidity
premium, the forward rate would be falling and consequently also the yield
curve. (A. Reuben Kessel, R. H. Scott and J. Gray 1973).
The demand for loans comes from government (Federal, state and
local government), business and consumers, while the supply of funds
basically comes from three sources: savings; changes in the money supply
those which impact bank reserves in particular; and changes in the money
balances held for speculative purpose (hoarding, dishoarding). Empirical
evidence does not support the extreme cases of the segmentation hypothesis.
14
Our various findings each support a single conclusion that the demands for
various maturities of debt are not infinitely elastic at going rates, and
therefore the changes in the relative supplies of different maturities can alter
the term structure” (E. Kane and B. Malkiel 1987). The so called term
structure models are driven by the assumption that arbitrage opportunities
are absent. The intuitive concept of absence of arbitrage can be linked
directly to the existence of a pricing kernel and a risk neutral probability
measure. (Konstaantijn Maes, 2003).
The financial market to a large extent determines the forecasting
technique. To forecast or not forecast therefore depends on the market. If the
financial markets were strongly efficient, no amount of forecasting skill
would produce yields higher than those obtained by buying a bond and
holding it to maturity. Successive trades based on forecasts of interest rates
would not out perform the naive buy-and-hold strategy in an efficient
market. The best predictor therefore of the future course of interest rates is
the yield curve which represents all the investors’ expectation that is, the
collective wisdom of the market. Investors do not have homogeneous
expectations and bonds of differing maturities are not perfect substitutes for
each other (J. C. Cox et al 1981).
15
Changes in the volatility of the real interest rate at which small
emerging economics borrow have a qualitatively important effect on real
variables like output, consumption, investment, and hours worked. From the
findings, an increase in real interest rate volatility triggers a fall in output,
consumption, investment and hours worked and notable change in the
current account of the economy. (Jesus Ferncendez – Villaverde, 2009).
Interest rate forecasting is an inexact science or art. Bond portfolio managers
faced with the uncertainty of interest rates have devised a method of
immunizing their bond portfolios from interest rate changes. The method of
eradicating the interest rate risk is realized only under certain circumstance.
The assumption underlying bond immunization principle; Constant coupon
value, fixed time horizon, yield to maturity is constant and reinvestment rate
equals yield to maturity. If interest rate changes after the coupon payment is
made, the realized yield would be lower or higher than yield to maturity
depending on the relationship between the duration of the bond and the
holding period (H. Guilford Babcock 2000). The realized yield is a weighted
average of the yield to maturity and the interest rate equals the yield to
maturity if the duration of the bond were equal to the time horizon of the
investor (Richard McEnally, 1980).
16
CHAPTER THREE
3.0 DEBT INSTRUMENT
Debt securities are fixed obligations that evidence a debt, usually
repayable on a specific future date or dates and which carry a specific date
or dates and which carry a specific rate or rates of interest payable
periodically. They may be non-interest bearing also. A bond is evidence of
indebtedness specifying the rights of the holder and the duties of the issuer.
Bonds and bank loans are examples of debt instruments. Debt instrument are
financial assets which posses the following properties that determine or
influence their attractiveness to different classes of investors. The properties
are moneyness; divisibility and denomination, reversibility, term to maturity;
liquidity, convertibility; currency, cash flow and return predictability; and
tax status.
The market value of bond is the discounted value of the cash flows
expected from the bond. Specifically the market price of a bond at time zero
is a function of the value of the coupon, the face value of the bond and the
assumed reinvestment rate.
If P0 = Market price of the bond at time zero
Ct = Value of the coupon = coupon rate x 1000
V = face value of the bond or principal value or par value
17
or maturity value.
r = assumed reinvestment rate = cost of debt. = discount rate.
Then
n
n
n
tt
t
t
n
n
n
n
n
r
V
r
CP
r
V
r
C
r
C
r
C
r
CP
)1()1(
.)1()1(
...)1()1()1(
1
0
3
3
3
2
2
2
1
1
10
From equation (3.1) and (3.2), it is obvious that the value of the bond
is dependent on its life n reinvestment rate (or discount rate) rt. It is assumed
that they can be reinvested at rt at any time t. This is the nature of
compounded interest. Equation (3.2) assumes that coupon payments are
made annually. If coupon payments were paid m times a year, then equation