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

(3.2) becomes.

mn

mr

mn

tt

mr

t

mnt

VmCP

1)1(

/

1

0 -----------------------------------------------(3.2.1)

If the coupons are equal in value, we have

mn

mr

mn

tt

mr mnt

VP

11

1

1

0 -------------------------------------------------- (3.2.2)

If the coupons are equal and the assumed reinvestment rate fixed, we have

mn

mr

mn

tt

mr t

V

m

CP

11

1

1

0 ---------------------------------------- (3.2.3)

Using 3.2.2

----------------- (3.1)

------------------------------------- (3.2)

18

mn

mn

mn

mn

mn

mr

mn

tt

mr

m

r

V

m

r

m

r

m

r

m

rm

C

V

m

CP

mnt

11

1...

1

1

1

1

1

1

11

1

3

3

2

21

1

0

By Bootstrapping technique one observes that o< r1< r2 < … < rn.

mn

mn

mn

mn

mn

mn

mn

mnrm

Vm

rm

cmn

rm

Vm

m

rm

Vn

m

r

V

m

r

mn

m

CP

111

0

11

Also mn

mn

mn

mn

mn

mn

mn

mn

mn

mnrm

Vm

rm

ncm

m

r

V

m

r

mn

m

CP

11

0

mn

mn

mn

omn

mn

mn

mn

mn

mn

rm

Vm

rm

cmnP

rm

Vm

rm

ncm

1

which shows that P0 is bounded for any given r.

mn

mr

mn

tt

mr

t

o

VmCP

1)1(

/

1

------------------------------------------------------ (3.2)

If the coupons are equal in value, the subscript t is dropped and we have

mn

mn

tto

m

r

V

m

r

mCP

11

/

1

--------------------------------------------------- (3.3)

Recall mn

mn

tt

m

r

m

r

m

r

m

r

m

r

1

1...

1

1

1

1

1

1

1

132

1

19

Using the sum of a geometric sequence we have

m

r

m

r

m

r

m

r

mn

mn

mn

tt

1

11

1

11

1

1

1

1

1

=

mnm

r

m

r

m

r

m

r

1

11

1

1

1

=

mnm

r

m

r

m

r

m

r1

11

1

1

1

Therefore

mnm

r

mn

tt

m

r

m

r1

11

1

1

1

1

=> mn

mn

tt

m

r

m

rm

r

1

11

1

1

1

20

=>

mn

ttmn

m

rm

r

m

r 11

11

1

1 ------------------------------------- (3.4)

Combining (3.3) and (3.4) by substitution we have

t

mn

t

mn

tt

mc

o

m

rm

rV

m

rP

1

11

111

mn

tt

mn

tt

mc

m

rm

VrV

m

r 111

1

1

m

Vr

m

C

m

rV

m

rm

Vr

m

rV

mn

tt

mn

tt

mn

tt

mc

1

11

1

1

1

1

1

Let A =

mn

tt

m

r11

1 then an equivalent to equation (3.3) is

m

Vr

m

CAVPo …………………………………………….. (3.5)

Where A = present value of an annuity of N1 received every year for

n years.

21

If the bond has an infinite life, (3.3) gives.

mnt

t

m

r

V

m

r

m

C

P

11

1

0

as n , mn

m

r

V

1

0

Leaving

1

0

1t

t

m

r

m

C

P

=

...

1

1

1

1

1

12

1

m

r

m

rm

C

m

rm

C

tt

m

r

m

r

m

CP

1

11

1

1

0 Sum to infinity

m

rm

r

m

rm

C

m

rm

rm

r

m

C1

1

1

1

1

1

22

= r

C

r

m

m

C

=> r

CPo ……………..……………………………………. (3.6)

This means rate of return on actual investment is the ratio of the

coupon to present value. (If the bond has an infinite life).

In perfect markets, all risk less instruments have the same short term

return which must coincide with the risk less short term rate for that period.

To enforce this condition, we expect simultaneous purchase and sales of

instruments in different markets to profit from the price differences. The 1-

period rate of return from say, an instrument with maturity n, and a cash

flow denoted by (a1, a2, …an) consists of the cash payment a1, plus capital

gain, or the difference between the next period price and the current price of

the security, expressed as a percentage of initial value. Let the price j period

(j<n) from the present of an instrument maturing n periods later be nPj; the

capital gain for the current period is n-1P1 - nPo and for the short term rate r1,

we have

r1 = a1 + (n-1P1 – nPo) . . . …………………………………..(3.7)

nPo

solving for the current price, nPo;

nPor1 = a1 + n-1P1 - nPo

23

=> nPo + nPor1 = a1 + n-1P1

nPo (1+ r1) = a1 + n-1P1

nPo = 1

111

1 r

Pa n

…………………………………………………….(3.8)

If nPo is greater than the right hand side, then the 1-period return of

the debt instrument, given by equation (3.7) would be smaller than the return

r1 obtainable by investing in the 1-period debt instrument. As a result, no one

would want to hold it, causing its price to drop. Similarly, if nPo is smaller

than the right hand side of equation (3.8), this yield for the debt instrument

would be larger than r1, and everyone would want to hold it.

Observe that n-1P1 must satisfy an equation like equation (3.8)

or 2

22111

1 r

PaP n

n

. ………………………………………………… (3.9)

Substituting this equation (3.9) into equation (3.8) we get

We get )1)(1()1( 21

222

1

1

rr

Pa

r

aP n

on

. ………………………...…. (3.10)

Recursively respecting the same substitution up to maturity of the debt

instrument gives

21

2

1

1

111 rr

a

r

aPon

- . . .

n

n

rrr

a

1...11 21

……….….. (3.11)

24

This shows that the debt instrument must equal to the sum of the

present value of the payment that the debtor is required to make until

maturity.

3.1 BOND YIELDS.

The three types of returns to be considered when investing in a debt

instrument are

(1) The Coupon Rate: This rate represents the rate of return on the

face value of the bond.

(2) The Current Yield: This rate represents the rate of return on

the actual investment (current price) of the bond. CY = C/Po.

(3) Yield to Maturity: This rate represents the rate on the face

value of the bond adjusted for the amortization of the premium

(paid) or the discount (saved) on the bond at the time of

purchase. Put differently, it is the interest rate that makes the

present value of the cash flow equal to the market value (price)

of the instrument. In equation (3.11) If y = r1 = r2 = r3 =. . . =

rn, where y is the yield to maturity or the interest rate that

satisfies the outcome of equation (3.11) which is

nn

ony

a

y

a

y

a

y

aP

1...

1113

3

2

2

1

1 ……………………. (3.12)

25

To find y is not often easy for n>2 or for large n. Hence we must find

y by trial and error or by using iterative technique like Newton-Raphson. For

bonds the cash flow (a1, a2, a3, . . ., an) can be written as (C, C, C, . . ., C+M)

where C is the coupon payment and M, the maturity value. Hence equation

(3.12) can be rewritten as

ny

MC

y

C

y

C

y

CP

1...

111321

…………………….. (3.13)

nny

My

Cy

Cy

Cy

CP

1

1

1

1...

1

1

)1(

1

1

1321

Dividing both sides of the equation by M

nnyM

M

yM

C

yM

C

yM

C

yM

C

M

P

1

1

1

1...

1

1

1

1

1

1321

=> nt

n

t yyM

C

M

P

1

1

1

1

1

……………………………(3.14)

The first part of the right hand side of equation (3.14) is a geometric

series. Therefore we can readily rewrite it as

n

n

y

y

yy

M

C

M

p

1

1

1

11

1

11

)1(

1

26

= nn

yy

y

y

y

M

C

1

1

1

11

1

1

1

= n

n

yy

yM

C

1

1)1(1

1

=>

nn

yy

y

M

C

M

P

1

111 ...............................................(3.15)

Solving for y in equation (3.15) we get the yield to maturity for an n-

period bond. P/M in equation (3.15) is called the par –value relation usually

expressed in percentage. When it is equal to one, the bond sells “at par”,

when it is larger than one, it sells at a “premium”; and if it is less than one, it

sells at “discount” C/M is the coupon rate expressed as a ratio.

Interest rate (and maturity) customarily are quoted per year (e.g. 5%

p.a, 7% p.a etc).Thus in equation (3.15) the coupon rate is c per year and

paid once a year. If for instance bonds pay interest s times a year, each

coupon payment there fore amounts to c/s which must be discounted s times

a year at 1/s of the annual yield or y/s.

ns

sy

sy

sm

sy

SM

C

M

P

1

111 …………………………...…..(3.16)

27

Consider a bond whose coupon rate is such that the corresponding

value of p/m is one. That is to say, the bond sells at Par. Then equation

(3.15) turns to n

n

yy

y

M

C

1

1)1(11 …………………………. (3.17)

Equation (3.17) can always be solved for all values of n є R or

R* (i.e. Real or Extended real).

For y. the solution is C/m, to see this

n

n

yy

y

M

C

1

11

11

1

1

11)1(

y

y

M

Cy

n

n

y

y

M

Cy

n

n 111)1(

dividing both side by 11 n

y

=>

yM

C 11

=> M

Cy …………………………………………………... (3.18)

MC is the coupon rate. In general, if a bond sells at par, then its yield

to maturity is the same as its coupon rate. For example, if a 5.6% 10years

bond sells at par; its yield to maturity is 5.6%. Thus, the coupon rate of an n-

28

period bond selling at par may be labeled the n-period par yield. Multiplying

both sides of equation (3.13) by (1+y)n we obtain.

P(1+y)n = C (1+y)

n-1 + C (1+y)

n-2 + …+ C+M …………….. (3.19)

Given that the yield to maturity y takes into account the coupon

income and any capital gain or loss that the investor will realize by holding

the bond to maturity. The measure has its shortcomings, however for

terminal value to be P(1+y)n each of the coupon payments must be

reinvested until maturity at an interest rate equal to the yield to maturity. If

the coupon payment is semiannual, then each semiannual payment must be

reinvested at the yield y.

Equation (3.19) show that investors will realize the yield to maturity

that is calculated at the time of purchase if (1) all the coupon payments can

be reinvested at the yield to maturity, and (2) the bond is held to maturity.

The first assumption presents condition for reinvestment risk. With respect

to the first assumption, the risk that an inventor faces is that future interest

rates at which the coupon can be reinvested will be less than the yield to

maturity at the time the bound is purchased. If the bond is not held to

maturity, it may have to be sold for less than its purchase price, resulting in a

return that is less than the yield to maturity. This risk that a bond will be sold

at a loss is referred to as interest rate risk. Bonds that do not make coupon

29

payments are called zero-coupon bonds. The advantage of these bonds is

that they do not expose the investor to reinvestment risk.

3.1.1 BOND DURATION

Duration recognizes the fact that two bonds having the same maturity

but different levels and patterns of coupon payments cannot be considered

equivalent. The duration concept is best understood if one compares the

bond for a period under pure discount (no coupon) with coupon bond of the

same maturity. The coupon payments allow for the realization of a portion of

the expected wealth before the bond’s maturity date. Hence its duration is

less than its term to maturity. The pure discount bond would always have

duration equal to its term to maturity. Hence duration is always equal to or

less than the term to maturity.

Duration, d, simply accounts for the difference in the cash flow stream

between bonds of equivalent maturities.

dr

m

r

V

m

rd

dr

rPdd

mn

mn

t

t

mCmn

ti

o

11

)(

1

…………………………..……………. (3.20)

30

Practitioners are really interested in the percentage change of a bond

price with respect to small parallel changes in interest rates. The percentage

change is the price change divided by the bond price.

)(

11

)(

)(

0

1

0

0

rP

m

r

V

m

rdr

d

rPdr

rdP

d

mn

mn

t

t

mCmn

t

)(

11

...

111

3

3

2

2

1

1

rP

m

r

V

m

r

m

r

m

r

m

rdr

d

o

mn

mn

mn

mn

mC

mC

mC

mC

)(

1

1.

1

1.

...

1

1.3

1

1.2

1

1.1

0

113

3

3

2

2

1

rP

m

r

mVmn

m

r

mm

cmn

m

r

m

m

r

m

m

r

mmn

mn

mn

mn

mC

mC

mC

)(

1

.

1

111

2

rP

m

r

nV

m

rt

o

mn

mn

t

t

mCmn

t

……………………………… (3.20.1)

If the bond does not have coupons, d would equal

31

.111

.1

m

r

n

m

r

V

m

r

nV

mnmn

mmn

mn

mn

ndras

m

r

nd mn

mn

,0

.1

)21.3.......(................................................................................

.1

0 mn

mn

m

r

VP

dr

m

r

vn

m

r

drm

Vmn

Pmn

mn

mn

mn .1

.

1

1.

110

orasandndset

r

m

r

n

P

P

r

m

r

nVP

mn

mn

mn

1

1

0

0

10

)( rdP

P

o

o

………………..…………………… ………..(3.22)

32

Therefore, for a given change in yields to maturity, say an increase of

1 percent, a bond with duration equal to 4 would experience a higher change

in its price (-4 percent) than a bond with a duration equal to 3 (-3 percent)

other thing being equal.

i.e. 4100

4

100

14

o

o

P

P percent when duration = 4 with %1r .

3100

3

100

13

o

o

P

P percent with d = 3; %1r

3.2 CONVERTIBLE BONDS

Convertible bonds are debt instruments that can be converted into

equity securities at option of the holder during a specific period of time.

They are usually debenture bonds with no collateral pledged by the issuing

corporation. A convertible bond represents a combination of a straight bond

(non-convertible) and a warrant, a long term option to purchase stock from

the issuing corporation under specified terms.

To understand convertible bonds knowledge of the following is

required.

1. Investment Value (I) – The price at which a convertible bond

would have to sell in order to provide a yield equivalent to that

of a non convertible bond of equal maturity and risk. If the

33

bond were to sell for this price, the value of the conversion

privilege would be zero. Investment value represents a support

level, a cushion in the event of excessive decline in the price of

the common stock; assuming no accompanying changes in the

bond risk.

2. Conversion Ratio (R): The number of shares to which a bond

can be converted. This number is stated in the indenture

agreement.

3. Conversion Price (CP): The reciprocal of the conversion ratio

multiplied by face value (V). It is equal to R

1000

4. Conversion Value (CV): The market value of the bond if

conversion takes place. It equals the conversion ratio multiplied

by the market value of the common stock. CV = R.Ps . Where

Ps = market price per common stock.

5. Premium over conversion value (PC): The percentage

difference between the conversion value and the market prices

of the convertible bond P.

PC = (PB - CV)/CV.

34

6. Premium Over Investment Value: The percentage difference

between the investment value and the market price of the

convertible bond.

PI = (P-I)/I

PI measures the worth of the conversion privilege and currently the

proportion of the market value of the convertible bond subject to risk

resulting from the fluctuation in the price of the common stock.

7. Price of Latent Warrant: Represents the value of the

conversion privilege per warrant that is per share to which the

bond can be converted.

W = price per warrant = (P - I)/R

(or number of latent warrants)

A conservative investor would, therefore choose a convertible bond

with a higher PC and a low PI because the behaviour of the bond under these

conditions is less dependent on the behaviour of the stock. An aggressive

investor would choose a convertible bond with low PC and a higher P1

because its price more closely follows that of the common stock.

The following observations are worth making

a. The percentage increase in the market price of the bond lags

behind that of the stock.

35

b. The lower the premium over conversion value, the lower the

yield.

c. The investment value represents a cushion of considerable

importance in a bear market. When the market price of the bond

is equal to its investment value, the convertible bond behaves

exactly like a straight bond and its price is then determined by

market rates of interest, supply and demand and the financial

position of the issuing company.

Price

Figure 1: (Relationship among the various values of a invertible bond).

200

400

600

800

1000 878

CV

=

I = P

B

=

0 1 2 3 4 5 6 7 8 9 10

time

V(face value = N1000)

b s

c

d

1BP cv1

36

The line from I through S to CV1 represents the minimum market

price of the bond. The bond cannot sell below its conversion value.

Otherwise inventors would buy bonds, convert them immediately, and sell

the stock in the market. The conversion value CV to CV1 is sloped as such

because of the assumed constant geometric growth rate in price of the

underlying stock.

Pt = Po (1+g)t This obviously excludes the 50% decline in the price of

the stock. The curve PB to

1BP represents the bond market value curve over

a portion of the bonds life. This value is higher than the conversion value

curve, initially because of the protection convertible bonds provide (in a bear

market) through their investment value. The difference between PB and CV

represents the value of the “safety net” that convertible bond provide. This

protection diminishes in significance as the prices of the stock rises in value.

At t = 5, in this particular case, the market value of the bond is equal

to its conversion value (premium over conversion value is equal to zero) and

the bond is equivalent to holding 20 shares of underlying security. The

investment value of the bond is shown above to converge linearly to the face

value of the bond as the maturity date of the bond approaches.

37

3.2.1 DETERMINANTS OF THE MARKET PRICE OF A

CONVERTIBLE BOND.

The problem in pricing convertible bonds results primarily from the

asymmetric effects on the convertible bond price when different conditions

hold. This asymmetry can best be understood through a re examination of

figure 1. The market price of the bond equal or exceeds the investment value

because of the equity aspect of the convertible bond. That is because of the

price appreciation potential the convertible bond offers. This is obverted at t

= O, where PB > I > CV. The difference P

B – I represents the value of the

conversion privilege. Beyond point S, the market price PB of the convertible

bond would exceed the conversion value. At t = 3, difference PB – CV

equals cd. And represent the value of the safety net provided by the

investment value. The safety value is non existent if the convertible bond is

replaced by an equivalent number of common stocks; it becomes

meaningless when the conversion premium equal zero. A stock position

could theoretically fall to zero, but an equivalent convertible bond would fall

to the investment value. It is important to note however, that the investment

value is not constant. it is affected by changes in market yields and in the

riskiness of the firm. A falling stock price is general reflective of a

deteriorating position within the firm. This increases the risk of default and

38

bankruptcy. As this risk increases, the investment value falls. The safety net

is not as strong as it may appear to be.

improvements have been made on the conventional model using

Monte Carlo Simulation to forecast rates of return on convertible bonds

conditional upon “ the simulated behaviour of the underlying stock” The

conclusion was that “Behavioral input derived for the simulation model

attested to the powerful influence of the relationship between conversion

values and straight bond values upon convertible bond premium and t o the

asymmetry of premium, depending on whether conversion values or straight

bond values dominated” If the range of expectation about the course of

future stock prices is restricted to certain dimension and if discrete time

intervals are used, the calculation of market value will be considerably

simpler.

An investor expecting the conversion value of the bond always to

exceed its investment value and the stock price to grow at a constant rate g,

would calculate the market price of bond as

)24.3.........(............................................................1

.1

11n

nc

ns

on

tt

ct

B

r

NgP

r

CP

where n – length of holding period

s

oP - Market price of stock at time of purchase

39

c - value of coupon = coupon rate x N1000

rtc - cost of capital of the firm = wd rd + we re

wd, we – The weights of long-term debt and equity in the

capital structure respectively

rd, re - cost of debt and equity, respectively

N - Number of shares to which the bond can be

converted.

3.3 FACTORS INFLUENCING BOND PRICES AND INTEREST

RATES.

Different bonds will sell at different prices since all investors do not

have homogeneous needs and all bonds do not have homogeneous

characteristics.

The factors that affect the prices of bonds are

(i) The frequency of the coupon payment and the number of days

over which interest is accrued and the size of the coupon

payment on the bond.

(ii) The quality of the bond that is the probability of default on

interest and/or principal payments. The quality is reflected in

the rating given the bond by the various rating services.

40

(iii) The liquidity of the bond

(iv) The yield on the bond

(v) The peculiar features of the bond

(vi) The investors’ personal preference for a certain type of bond,

issuer, or maturity.

Interest rates are influenced by demand and supply of money. The

supply of money influences interest rates if the government has a hand in the

following

(a) By changing the discount rates it charges its member banks.

(b) Charging the reserve requirements on various deposits held by

member bank or by subjecting to or exempting some types of

deposits from reserve requirements

(c) By buying and selling securities in the open market – open

market operation - with intent of influencing the level of excess

reserves of depository institution and

(d) By moral suasions.

The Federal Reserve System where policies have considerable (if not

exclusive) impact on the interest rates can influence the demand for credit or

money in several indirect or direct ways.

The indirect ways include:

41

i. Through the actual imposition or the threat of credit control.

ii. Through the pursuit of other policies that affect the expectation

of borrowers in the market place.

An unduly expansionary monitory policy and accommodating

monetary policy to the treasury operation - can stroke inflationary

expectation, which in turn, can influence the level as well as the direction of

interest rates because of the high inflation components in the nominal rate of

interest. The demand for funds comes from primarily three sources:

government, business and consumers.

3.4 THE TERM STRUCTURE OF INTEREST RATES.

The yield curves summarize the terms structure of interest rates. The

term structure and the yield curve are not exactly the same. They become

equivalent if and only if the debt instruments are pure discount bonds (bills).

The yield curve shows the term structure as the relationship between a

point in time and time to maturity to yield to maturity on fixed income

securities within a given risk class. The term structure is not the yield curve

for coupon bearing bonds. The yield on a coupon bond maturing in two

years, for instances, is some weighted average of the term structure. The

42

graphical depiction of the term structure of interest rates is called the yield

curve.

To constrict the yield curve, there exist four different measures that

can be used. They are:

1. The yield to maturity on a country’s bench mark government

bond;

2. The spot rate;

3. The forward rates; and

4. The swap rates.

Using the treasuring yield curve to determine the appropriate yield at

which to discount the cash flow of a bond showcases the limitations of using

the yield to value a bond. The yield on a treasury security with the same

maturity as the bond plus an appropriate risk premium or spread explains the

interest rate. consider two hypothetical 5-year treasury bonds, A and B. the

difference between these two treasury bonds is the coupon rate, which is

20% for A and 10% for B. Assuredly, the cash flow for these two bonds per

N100 of par-value for the 10 six month period to maturity would be

43

Period Cash flow A Cash Flow B

1-9 N10 N5

10 N110 N105

It is therefore appropriate to use a particular interest rate to discount

cash flow such that is favourable within the time the cash flow will be

received.

To avoid arbitrage opportunities, look at the conditions and packages

of cash flows on bond A and B as packages of zero coupon instruments.

Here interest earned is the difference between maturity price and the price

paid. The value of each coupon bond is equal to the total value of its

component zero –coupon instrument.

3.4.1 THE SPOT RATES

The process of extracting the theoretical spot rates from the Treasury

yield curved is called boot strapping. The spot rate curve is the graphical

depiction of the relationship between the spot rates at its maturity. The yield

on a zero – coupon bond is called the spot rate. The theoretical spot rate

curve is a curve derived from theoretical consideration as applied to the

44

yields of actual treasury securities. Given the data for the prices, annualized

yield (yield to maturity) and maturity, one can bootstrap the spot rates using

the values. If we need semiannual rates, then the first six months and one

year of annual par yield to maturity will coincide with the spot rate. The

subsequent spot rate can be calculated using the discount function. The

discount function is related to the spot rate.

Let Dn represent the discount function and zn, the spot rates. Then,

from equ. (3.2.2), where Zn =m

rt ,

nn

nz

D

1

1 ……………..…………………………….……….(3.25)

The spot rates arise as a solution of prices function. The price of bond

is simply the sum of the products of the cash flows expected from the bond

at time t and the discount function for time t. which means that for a bond

with a maturity n and a cash flow of C for periods 1, 2, 3,…, n -1 and

maturity value of M, the price P is given by

)(1

1

MCDCDP n

n

t

t

………………………….……...……….(3.26).

45

3.4.2 THE FORWARD RATES

Forward rates can be extracted or generated from the treasury yield

curve by using arbitrage arguments as in the case of spot rates above. If two

increments A and B have the same cash flows and the same risk, they should

have the same value.

Consider an investor who has a 1 year investment horizon and is faced

with the following two alternatives.

Alternative 1: Buy a I –year Treasury security

Alternative 2: Buy a 6 month treasury security and when it matures in

6 months, buy another 6 month treasury security.

The investor will be indifferent towards the two alternatives if they

produce the same return over the 1-year investment horizon. The investor

knows the spot rate on the 6-month treasury security and 1-year Treasury

security. However, he does not know what yield will be available on a 6-

month treasury security that will be purchased six months form now. That is,

he does not know the 6-month forward rate six months from now. Given the

spot rates from the 6-month Treasury security and 1-year Treasury security,

the forward rate on a 6-month treasury is the rate that equalizes the naira

return between the two alternatives.

46

If NH represents the face amount of the 6-month treasury security, z1,

the one half bond equivalent yield (BEY) as y extracted from equation (3.16)

earlier – of the theoretical 6 – month spot rate and z2 represents one –half the

BEY of the theoretical 1-year spot rate, then the investor will be indifference

towards the two alternatives if

H(1+z1) (1+f) = H(1+z2)2……..………………………………..(3.27)

where f is the 6 – month forward rate six months from now. Solving

for f, we have

11

1

1

2

2

z

zf …………………………….. (3.28).

The BEY for the 6 month forward rate six month from now is gotten

by doubling f. To generate the 1 – period forward rates, let fn devote the 1-

period forward rate contract that will begin at time n. then fo is simply the

current 1 – period spot rate. The set of forward rates

f = {1fo, 1f1, 1f2,…, 1fn-1} are in one to one correspondence to the set

of spot rates. S = {z1, z2, z3, . . ., zn} according to the annualized rates on a

bond equivalent basis. F is called the short term forward rate curve. The

relationship between the n-period spot rates, the current 6-month spot rate,

and the 6-month forward rates is

47

11...1111

1121 n

nn fffzz …………………………... (3.29)

i.e.

1211 1...1111 n

n

n fffzz

Taking reciprocals we have

121 1...111

1

1

1

nn

n

nfffzz

…………………………. (3.30)

Comparing equations (3.25) and (3.30)

1211 1...111

1

n

nfffz

D ………………………………. (3.31)

But the discount function can be expressed in terms of forward rates

as from equation (3.29) as nz

1 i.e.

n

nz

D1

.

3.4.3 THE SWAP CURVE/RATES.

The swap curve is the London inter bank offered Rate (LIBOR) curve.

The swap curve is derived from the swap rates in the interest rate swap

market. A swap transaction is the simultaneous purchase and sale of spot and

forward exchange or two forward transactions of different maturities.

48

Usually, this is between two parties who agree to exchange cash flow based

on the following considerations.

(a) One of the parties pays a fixed rate and receives a floating rate.

(b) Other party agrees to pay a floating rate and receives a fixed

rate. Here the fixed rate is called the swap rate. The swap curve

shows the credit risk of the banking sector. Thus it can be seen

as an inter bank related curve. Investors and issuers use the

swap market for hedging and arbitrage purposes. The swap

curve x-rays the performance of fixed –income securities and

the pricing of fixed-income securities. This swap curve is more

useful to funded investor than a government yield curve.

49

CHAPTER FOUR

THE MODEL

Assuming interest rates are known at any given time. That is to say for

each time we can calculate the interest rate. Let T denote maturity time at a

point s<T, the time of maturity is t = T-s. Within this very little time dt, the

bond value P(t) changes by an amount dP. Assuming V is the principal to be

repaid at maturity, then the initial condition will be V = P (0). When there is

a parallel shift (i+r)(t) in interest rate due to economic activities with

cumulated capital C(t) which is the cumulative cash flow plus interest earned

due to distortions. In the interval (t, t+dt), the capital increments by the cash

K(t) dt plus interest due to parallel shift A(t) C(t)dt earned on capital C(t) in

period dt is A(t) C(t) + K(t). We have the following:

dt

dC = (i+r)C + K --------------------------------------------- (4.1)

dt

dP = - (i+r)P ------------------------------------------------(4.2)

where for a very long term case

ri

CP

------------------------------------------------(4.3)

let A = (i+r)(t).

and = the increment rate.

50

dt

dP = -A(t)P(t) ----------------------------------------(4.4)

dt

dC = A(t) C (t) + k(t) --------------------------------(4.5)

from (4.3) above

)(

)(

tA

tCP -----------------------------------------------(4.6)

P(t) = Bond value

P0 = Market price of the bond at time zero

C(t) = Value of the coupon = coupon rate x 1000

V = Face value of the bond or maturity value

r = Assumed reinvestment rate

i = the shift or disturbance in interest rate.

4.2 METHOD OF SOLUTION

With equation (2)

dt

dP = - A(t)P(t)

The solution of this equation (ordinary differential equation with

separable variables) is

dttAP

dP)(

dttAInP )(

51

dttA

VeP)( ----------------------------------------------------------- (4.7.1)

from (4.3) )(tA

CP

(4.3) and (4.7.1)

dttA

VetA

C )(

)(

dtAVetAC

)()(

dttAdttA

dttAdttA

VetAVetA

etVAtAtAVedt

dC

)(2)('

.)(')(

))(.)(

)()()(.

.)()(..)(2'

dttA

eVtAtAdt

dC-----------------------------------------(4.8)

Comparing (4.1) and (4.8)

)()()( 2')(tAtAVeKCtA

dttA

.)()()( 2')(

ctAtAtAVeKdttA

CtAtAtAVeKdttA

)()()('1 2)(

CtAVetAVetAdttAdttA

)()()(1 )(2)('

but from 4.7.2

))(

)(dttA

VetAC

CtACtAVetAK odttA

)()()(1 )(

'

52

CtAVetA

t

odttA

)(2)(1 )(

'

---------------------------------------- (4.9)

i.e capital increment,

ACVeAK

t

oAdt

21 '

Consider the stochastic differential equation

dX = b(X,t) dt + B(X, t) dw …. is linear provided the coefficients b

and B have this form.

b(x,t) : = c(t) + D(t) x,

for c :[0,T] IRn,

D : [0,T] Mnxn

, and

B (X,t) : = E(t) + F(t)x

for E :[0,T) Mnxn

F :[0,T] L (Rn, M

nxn)

The space of bounded linear mappings from Rn to M

nxm.

(A linear SDE is called homogeneous if .00 TtforEC It is

called linear in the narrow sense if 0F ) the SDE.

= X = X(0)

dw (t)X} {E+dt X} D(t) + {c(t) = dX

0

has a unique solution, provided E(IXoI2) in finite and let

ktctaXtDXtA t )()(,)()(

53

------------------------------ (4.10)

This equation has an absolutely continuous solution in the domain

t0 consider the matrix differential equation

)(tAdt

d t0 . )(t is

a non singular matrix for each t. The matrix )(t that solves this equation is

called the fundamental solution of the equation (or the integrating factor).

)()( taXtAdt

dX

where A(t) and a(t) are functions of t. one method of solving (t) is to

find a function )(t such that if the equation is multiplied by , the left

side becomes the derivative of the production X

i.e )()( taXtAdt

dX

Then try to impose upon the condition that

)()( Xdt

dXtA

dt

dX

)(

)(

tAdt

d

dt

dX

dt

dXXtA

dt

dX

Using ito’s formula

tdBtdttaXtAdX

dwtFtEdBt

ttt

t

0.,)()()(

)()()(

54

tdBssdssastxt

o

t

ost 0;)()()()()( 11

x0 =

where s

t

odBss )()(1

is the cumulated stochastic term

Translating (4.10) with 0)( tdBt we have

dC = tdttktCtA 0)()()( -------------------------

(4.11)

A(t) = (i+r)(t)

xo = = C (0) = 0 say

Applying ito’s formula, the equation (4.11) admits the following

unique solution.

C(t) = tdsskst

t

o0;)()(1 -------------------

(4.12.1)

where (t), called the fundamental solution solves the equation, but

tAdt

d0,

dtAd

t

odssA

e.)(

Substituting )1.12.4(.)(

infore

t

odssA

55

t

o

duuAdssA

dsesKetC

s

o

t

o)(.)(

)()( ……………….. (4.12.2)

The value at time 0 of the capital C(t) given using (12)

s

o

t

oduuAt

o

dusA

esKetC.)(.)(

)()( ----------------- (4.12.3)

Po = t

odssA

etC.)(

)( -----------------------------------------(4.13)

=> = dseskPt

o

duuA

o

s

o

)(

)( -----------------------(4.14)

but )9.4()(2)(1

0)(

' fromCtAVetAK

t

dssA

dsesCsAVesAP

ss

duuAt duuA

00)(

0

)('

0 )()(2)(

dsesCsAVesA

ss

duuAt duuA

00)(

0

)(2' )()(2)( ……………… (4.15)

This is the market price at which the stream of continuous cash flows

would trade if arbitrage is to be avoided.

The sensitivity of the market price due to interest rate (A (t)) is

A

dsesCsAVesA

A

P

ss

duuAt duuA

00)(

0

)(2'

0

)()(2)(

A

dsesCsAVesA

A

P

ss

duuAt duuA

00)(

0

)(2'

0

)()(2)(

56

dsAeACCeeAVAVeAt AduAduAduAdu

ssss

0

22" 0000 222

dsCeACeVeAAVeAt AduAduAduAdu

ssss

0

22

'2

" 0000 222

dseCACeVAAVAt AduAdu

ss

0

22

'" 00 222 ----------(4.16)

Duration of P

dseCACeVAAVAPPA

P t AduAduss

0

22

'" 00 22211

. …

………………………………………………………………...…….. (4.17)

Equation (4.17) gives the duration for continuously compounding

variable interest rate in continuous time case.

Convexity of P gives

A

dseCACeVAAVA

A

P

t AduAduss

0

22

'"

2

2

00 222

----- (4.18)

4.3 ANALYSIS

The linear stochastic equation above is a type of an arithmetic

Brownian motion with drift written as follows.

dXt = dt + todBt ,

57

xo = , are constants, where is a function of time.

The function (t) is some times called the instantaneous variance and

covariance of the process.

Is called the drift of the process and

, the volatility of the process.

Above we assumed zero volatility with drift = AC + k.

Physically, bond yield and bond price move in opposite direction.

With the distortion in interest rate due time effect trickling down from the

economic policies, reforms, disturbance among others, bond yield A(t)

moves in opposite direction to price p. The cash received over the

infinitesimal change in time is k(t) dt. Its value at time 0 is therefore

K(t)dt e-A(t)

If the short term interest rate is constant.

If short term interest rate is variable

We have duuAs

etK)(

0)(

This shows that the value at time 0 of the entire cash flow stream is

the infinite sum of all the elementary elements, for constant short term rate

we have

58

dseskP Ast

)(0

0

and for general case of variable (but known) short-term interest rates

dseskPduuAt

s

0)(

00 )(

In continuous time where short-term interest rate is constant the bond

valuation formula is

nAAA

nAAA

eVCnCeCedAdP

witheVCCeCeP

)(...2

)(...

2

21

In continuous time where short-term interest rate is variable, the bond

valuation formula is

AdsdsAAds

n

eVCCeCeP

0

2

0

1

0 )(...

with

n

oAdsAdsAds

eVCnCeCedA

dP0

2

01

)(...2

The market price is a sum of parallel shift effect of on capital and face

value of the bond. Showing that the bond value depends on the three

variables interest rate A(t), capital rate C(t) and face value V. Equations

(4.13) and (4.14) show that bond price depends on capital increment and

interest rate inter play.

59

CHAPTER FIVE

5.1 DISCUSSION OF THE RESULTS

This interest rate here with shift A(t) shows that for a very long term

the bond price formula is the relationship between the coupon and the effect

subject to the parallel shift. With i(t) varying with respect to time A(t) also

varies. If interest rates are constant, the yield curve is a straight line and a

change in the interest rates can be thought of as a parallel shift of the yield

curve. The rate of change of the capital with respect to time here according

to the model shows that it is the capital increment on the cash plus interest

factor impact due to parallel shift on capital. This push i(t) could be negative

or positive. From (4.7.1) if the value of the parallel shift due to time

approaches zero (very low), there should be a very high bond price.

A phenomenon that characterized the great depression of the 1930s

both in the United States and elsewhere was the accumulation of large

amounts of money balances that could not possibly be required for

transaction purpose alone. When bonds are expensive, and their yield is low

relative to their cost, they will not be very attractive assets. Conversely, if

there is expectation that interest rates will rise subsequently, bond prices are

expected to fall. Consequently, bond holders would suffer a capital loss.

60

From equation (4.7.1) capital gain or loss expected (g) can be approximated

as

g = P

PPe

The above capital gain or loss expected on the bond is expressed as a

percentage of its current value.

The cumulative capital is given by equation (4.12.2) which is the

solution of equation (4.11) using ito’s formula. Cumulative capital is the

product of interest earned over the interval of time (0,t) and the product of

capital increments by cash and value over (0,t) of the interest earned.

Equation (4.13) Therefore gives the present value at time 0 of capital C(t).

The solution (4.14) shows that in a very long time the present value is the

value at time 0 of the product of the capital increment K(t) and exponential

impact of the interest rate A(t).Consider interest rate going very high for the

period, the present value shrinks to a value infinitesimally small. This

accounts for a sharp chance for capital loss on investors. Equation (4.15)

therefore gives the sensitivity of the bond price P to a change due to the

parallel shift A(t) in the interest rate for constant rates.

Finally equation (16) gives the duration for continuously

compounding variable interest rate in continuous time case which depends

61

on time, parallel shift A(t), coupon value C, market price, P and principal to

be repaid at maturity V.

5.2 SUMMARY

The closed and open economy systems are not void of investment. As

investors invest, they consider several things. To an economist, investment

means the purchase of new factories, houses or equipment. In everyday

language, however, investment usually refers to an individual’s putting away

some funds for the future perhaps in the stock market or bonds. Note that

money balances that exceed required transaction-precautionary holdings

may be exchanged for earning assets. Therefore why hold money and give

up interest earnings?

Interest rates play important role in bond valuation and term structure

modeling since it explains investor’s liquidity preference. When there is

expectation that interest rate will rise, bond prices are expected to fall. Bond

yields are faced with the shocks on interest rates, coupon rates and current

yield. The par value relationship shows up when the quotient of bond price

to maturity value is one. Above one we have premium and below one,

discount price. The yield curve depends on the four measures, yield to

maturity, spot rates forward rates and swap rates. The swap rates curve

62

reflects the credit risks of the banking sector effectively since it is an inter-

bank curve. Since interest rate is practically non-negative, the present values

of bonds are also positive values. The percentage price change the present

value can be expressed in maclaurin power series with interplay of the initial

value, duration, convexity and the residual terms.

5.3 CONCLUSION

The term structure of interest rates is a function of two variables f(t,s)

the yield computed at time t of a zero-coupon risk-free bond with maturity s.

the bond valuation formula especially in the continuous time is a relationship

related to yield and interest rates in a stochastic environment. In

deterministic environment where interest rates are known function of time,

the present value of a continues cash flow k(t) is the integral over the given

time. With the parallel shift in interest rate due to economic reforms,

policies, instability among others, the present value of a bond with maturity

value V shrinks as interest rate gets small. This effect has engulfed global

economic activities since 2009 till date. Thus investors’ marginal

propensities to save are on the decline. The global effect is that interest rate

will tend to rise. In the dramatic move investors will be encouraged to invest

in bonds. Obviously, a claim that cannot be collected until far into the future

63

has very little present value. What is the use of having a claim for N1million

if you have to wait 100 years to collect it? Investors engage in bonds which

represents a series of claims collectible at different times in the future.

Today’s value of the bond must be the present value of the sum of all the

discounted future returns plus the discounted valued of the value at maturity.

64

REFERENCES

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University Press Uk.

Alpha C.Chiang. 1984. Fundamental methods of Mathematical Economics.

McGraw-Hill Book Co. Singapore.

Ann de Schepper, 2002. Bonds for present value function with stochastic

interest rates and stochastic volatility, Katholieke University Lenven.

Anthony M. Santomero, June 1975. “The Error- Learning Hypothesis and

the Term Structure of Interest Rates in Eurodollars” Journal of

Finance.

Atkinson, T.R 1967. Trends in Corporate Bond Quality; National Bureau of

Economic Research, New York.

Bierwag, G.O. December 1977. “Immunization, Duration and Term

Structure of interest rates” Journal of Financial and Quantitative

Analysis, pp725-242.

Bikas K. Chakrabarti and Arnab Chatterjee, 2006. Econoplysics and

Sociophysics, Trends and perspectives. Wiley – vcH, Berlin.

Braddock Hickman, W. 1967. Corporate and Quality. National Bureau of

Economic Research New York.

Burton G. Malkiel, 1970. The Term Structure of Interest Rates: Theory,

empirical evidence, and Application McCabb – Sciler, New York.

Buse, A. Feb. 1967. Interest Rates, the Meiselman Model and Random

Numbers” Journal of Political Economy.

David Meiselman, 1962. The Term Structure of Interest Rates, Prentice Hall,

Eaglewood Cliffs,.

Deborah H. Miller, 1979 “Estimating the Yield curve: Alternatives and

Implication, University of Pennsylvania, Philadelphia.

65

Edward J. Kane and Burton G. Malkiel, August 1967. “The Term Structure

of Interest Rates: An Analysis of a Survey of Interest Rate

Expectations” Review of Economics and Statistics.

Francis Dieboid 2005. NBER, working paper for National Bureau of

Economic Review vol. 17.

Franco Modigliani and Richard Sutch, May 1966. “Innovations in Interest

Rate Policy. American Economics Review.

Frederick Macaulay, 1956. Some Theoretical Problems Suggested by the

Movements of Interest Rates, Bond Yields, and stock Prices in the

United States since 1938. National Bureau of Economic Research,

Columbia university Press, New York,.

Grant, J.A.G. Feb. 1967. “Meiselman on the Structure of Interest Rates. A

British Test” Economica.

Guilford Babcock, June 1975 “A modified Measure of Duration” Paper

Presented at the Annual Meeting of the Western Finance Association,

San Francisco.

Hopewell M. and G. Kaufman, September, 1973. “Bond Price Volatility and

Term to Maturity: A Generalized Respecification” American

Economic Review.

James E. Walter and Augustin V Que 28 June 1973. On the Valuation of

Convertible Bonds.” Journal of Finance vol. 28.

James Pesando 2008. On the effects of interest rates. National Bureau of

Economic Research Inc.

Jean. M. Gray, June 1973. “New Evidence on the Term Structure of Interest

Rates 1984-1900” Journal of Finance Vol 28.

Jesus Ferncendez – Villaverde, 2009. NBER working paper for national

Bureau of Economic Research Inc.

John B. Taylor 2004 Principles of Macro Economics. Honghton Mifflin

Company Boston. New York.

66

John C. Cox, J. E Inger Soll, Jr. and S. Ross, May 1981. “The Relation

between Forward Prices and Future Prices.” Center for the Study of

Futures Markets, Columbia University, New York.

John R. Hicks, 1946. Value and Capital, Oxford: Uaredin Press.

Jun, Sang – Gyang and Frank C. Jen, 2003. Review of Quantitative finance

and accounting, vol. 30.

Konataantijn Maes 2003 The term structure analysis. International

Economics paper.

Martin E. Ecols and Jan Walter Elliott, March 1976. A Quantitative Yield

Curve model of Estimating the Term Structure of Interest Rates”.

Journal of Finance and Quantitative Analysis.

Otto H. Peonsgen, 1965, Spring 1966. “The Valuation of Convertible

Bonds” Industrial Management Review, Fall.

Polakoff et al., 1970. Financial Institution and Markets. Honghton Mifflin,

Boston, New York.

Reuben Kassel, A. 1965. “The cyclical Behaviour of the Term Structure of

Interest Rates” Occasional Paper 91, National Bureau of Economic

Research, New York,

Richard W. Lanz and Robert H. Rasche, 1978. “A Comparison of Yield on

Futures Contracts and Implied Forward Rates” Review, Federal

Reserve Bank of St. Louis.

Richard W. McEnally, Spring 1980. “How to Neutralize Reinvestment Rate

Risk. “Journal of Portfolio Management. Vol. 6 No. 3.

Santoni G. J. and Courtenay C. Stone, (March 1981). “Navigating through

the Interest Rate Morass: Some Basic Principles.” Review Vol. 63,

No.3 pp.11-18.

Sarkis J. Khoury 1983 Investment Management Theory and Application.

New York.

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Scott, R.H. August 1963. A “Liquidity” Factor Contributing to those

Downward Sloping Yield Curves. Review to Economics and

Statistics.

Sergio –M. Focardi & Frank. J. Fabozzi. 2004, Financial Modeling and

Investment Management John Wiley & Sons, Inc. Hoboken, New

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Thomas F. Dernburg. 1985 Macro Economics Concept, Theories and

Polices. McGraw Hill Book Company.

William F. Sharpe, Dec. 1973. “Bonds Versus Stocks: Some Lessons from

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68

GLOSSARY

1. Amortization: The payment of a debt by making regular payments.

2. Annuity: A guaranteed series of payments at regular intervals bought

for a lump sum. With a deferred annuity payment begins at a specific

future date; no income tax is paid on the interest earned during the

accumulation period, only when repayment begins;

3. Arbitrage: The simultaneous purchase and sale of mortgages, future

contracts, or mortgage – backed securities in different markets to

profit from price differences.

4. Bear A very popular term indicating the behavior of someone who

anticipates that prices will fall. He sells shares which he does not own,

in the belief that he could buy them at a lower price when the time

comes for delivery of shares.

5. Bear Market: A market in which prices are declining.

6. Bearish and Bullish: When conditions suggest lower prices, a bearish

situation is said to exit. If higher prices appear warranted, the situation

is said to be bullish.

7. Bond (i) An official document promising that a government or

company will pay back money that it has borrowed often with interest.

(ii) The series of claims collectible at different times in the future.

69

8. Bond Immunization: This is the eradication of interest rate risk. A

technique that allows for the transformation of a coupon bond into a

pure discount bond.

9. Bootstrapping: The process of extracting spot rates from the treasury

yield curve.

10. Broker: An agent, working on commission, who handles orders

between buyers and sellers in a market.

11. Brownian Motion: Is a continuous-time stochastic process or the

limit of a discrete random walk. Put differently, it is a continuous limit

of a simple random walk.

12. Bull Market: A market in which prices are rising.

13. Bull: One who expects prices to rise.

14. Call: (i) Investment: To demand payment of an installment of the

price of bonds or stocks that has been subscribed. (ii) Option: the

right to purchase a given number of shares at a stated price fixed date.

15. Capital Markets: Markets for long term funds.

16. Capital: Wealth in the form of money or property; sometimes the

basic sum in an investment enterprise.

17. Closed Economy: An economy that functions in isolation.

70

18. Consumption – Personal purchases of goods and services. i.e

purchases of final goods and service by individuals.

19. Convertible Debentures: Like Bonds, these carry a fixed interest rate

and have a set maturity date. They may be traded in for a given

amount of stock at any time at the option of the investor. The issuer,

however, has the right to call them in to be redeemed either in cash or

for common stock.

20. Coupon – (i) Bonds: the portion of a bond that is redeemable at a

given date for interest payments. (ii) Securities: The interest rate on

the debt security the issuer promise to pay to a holder at maturity,

expressed as an annual percentage of face value.

21. Coupon Rate – This rate represents the rate of return on the face

value of the bond.

22. Date of Maturity: The date on which a debt must be paid usually

applied to those debts evidenced by a written agreement, such as a

note bond and so on.

23. Debenture: A promissory note (IOU), such as bond that is backed by

the general credit of the company.

24. Debt Securities: Fixed obligation that evidence a debt usually

repayable on a specified future date or dates and which carry a

71

specific rate or rates of interest payable periodically. They may be

non-interest bearing also.

25. Depreciation: Reduction in value

26. Discount Rate: An interest rate used to discount a future payment

when computing present (discounted) value.

27. Discount: the price of a stock or bond, less than its face value i.e.

below par value.

28. Discounting: The process of translating a future payment into a value

in the present.

29. Disposable Income – Income left after taxes.

30. Duration: The derivative of a bond’s value with respect to interest

rates divided by the value itself. This accounts for the difference in the

cash flow streams between bonds of equivalent maturities. The

duration coupon bond is less that its term to maturity while the

duration of a pure discount (no coupon) bond is equal to its term to

maturity, therefore duration is always equal to or less than the term to

maturity.

31. Equity Capital: stock holder’s owner investments made in an

organization.

32. Equity Securities: Any stock issue, common or preferred

72

33. Equity: The value of an individuals or corporation that exceeds its

debt.

34. Face Value: The principal that will be paid back when a bond

matures.

35. Fiscal Policy is the use of the budget of the federal government in

order to influence the level of total spending in the economy by means

of changing the amount of the governments’ spending for goods and

services or altering the incomes of the private sector by changing

taxes or government transfer outlays to individuals.

36. Forward Exchange – A forward exchange contract in an agreement

to buy (sell) a certain amount of foreign currency at a specified price

at a specified date in the future.

37. Government Expenditure: Government purchases of goods and

services.

38. Hedge: To avoid committing oneself, or counter balancing

investments to limit potential losses in the event of a charge in price.

39. Inflation: The rise of prices generally over time.

40. Interest: Payments to a borrower from a lender for the use of money.

41. Investment - (i) Purchases of final goods by firms plus purchases of

newly produced residence by home hold. (ii) The money that people

73

or organizations have put into a company, business or bank in order to

get profit or to make a business activity successful.

42. Investors: A persons or an institution who uses his saving or

borrowings to buy securities.

43. Issue: any security of a company, or its distribution.

44. Liquidity: The ease with which holdings can be converted into cash.

Liquidity depends on market conditions

45. Market Position: Term applied to describe the supply and demand

relationship of a given security at a given price.

46. Monetary Policy is conducted by the Federal Reserve System which

has the capacity and the authority to alter the supply of money and

credit in the economy.

47. Money Market: money markets are markets for short-term (less than

one year) debt instruments.

48. Mortgage: The pledge of an asset to a creditor as security until a loan

is repaid; often used in borrowing money from a bank or savings

institution to purchase a home.

49. Open Economy: An economy that is heavily inter-dependent with

other economies.

74

50. Option: The purchased right to buy or sell at a specified price within

a given time, should the holder of the option choose to do so.

51. Par Value: The price of a stock or bond equal to its face value.

52. Portfolio: The total securities held by any investor.

53. Premium: The price of a stock or bond, higher than its face value. ie

above par value.

54. Present Value: The Value in the present of a future payment. A

borrower sacrifices a larger sum of future money for a smaller sum of

present money. The future sum can be brought down to its present

value by of present money.

55. Pure Discount Bond: A pure discount bond trades at a price below

face value and pays face value upon maturity.

56. Rate of Return: The pure rate of return on an investment is the

discounted rate which will just bring the present value of cash flows

down to the actual cost of the investment.

57. Real national income (Y) is the disposable income plus all taxes less

government transfer payments.

58. Reinvestment Rate: This is the discount rate. That is to say as

coupons are clipped and cashed in, it is assumed that they can be

reinvested at a discount rate.

75

59. Securities: Shares or interest in a corporation; properties pledged for

the payment of an obligation.

60. Spot Exchange – A spot exchange contract is a special case of a

forward contract where the time period shrinks to a few days (two

days or less, usually).

61. Stochastic Process: Given a probability space ( , , ) stochastic

process is a parameterized collection of random variables {Xt}, t in [0,

T] that are measurable with respect to . The parameter t is often

interpreted on time.

- A probability measure

- The event – algebra

- The set of all possible outcomes

62. Swap Transaction – A swap is the simultaneous purchase and sale of

spot and forward exchange or two forward transactions of different

maturities.

63. The Current Yield – The rate of return on the actual investment

current price of the bond.

64. Treasury bill is short term obligations of the government. They do

not carry a coupon and are sold at a discount from par value under

76

competition and non-competitive bidding: They generally are not

redeemable before maturity.

65. Treasury Securities are notes and bonds with medium and long-term

obligation respectively. They are coupon bearing instruments issued

with ten to thirty years maturity. They are callable.

66. Yield to Maturity – The rate of return on the face value of the bond

adjusted for the amortization of the premium (paid) or the discount

(saved) on the bond at the time of purchase.

67. Yield: The annual rate of return on a bond if the bond were held to

maturity.

77

INDEX

mn

mr

mn

tt

mr

t

o

VmCP

1)1(

/

1

------------------------------------------------------ (3.2)

If the coupons are equal in value, the subscript t is dropped and we have

mn

mn

tto

m

r

V

m

r

mCP

11

/

1

------------------------------------------- (3.3)

Recall mn

mn

tt

m

r

m

r

m

r

m

r

m

r

1

1...

1

1

1

1

1

1

1

132

1

Using the sum of a geometric sequence we have

m

r

m

r

m

r

m

r

mn

mn

mn

tt

1

11

1

11

1

1

1

1

1

=

mnm

r

m

r

m

r

m

r

1

11

1

1

1

=

mnm

r

m

r

m

r

m

r1

11

1

1

1

Therefore

mnm

r

mn

tt

m

r

m

r1

11

1

1

1

1

78

=> mn

mn

tt

m

r

m

rm

r

1

11

1

1

1

=>

mn

ttmn

m

rm

r

m

r 11

11

1

1 ------------------------------------ (3.4)

Combining (3.3) and (3.4) by substitution we have

t

mn

t

mn

tt

mc

o

m

rm

rV

m

rP

1

11

111

mn

tt

mn

tt

mc

m

rm

VrV

m

r 111

1

1

m

Vr

m

C

m

rV

m

rm

Vr

m

rV

mn

tt

mn

tt

mn

tt

mc

1

11

1

1

1

1

1

Let A =

mn

tt

m

r11

1 then an equivalent to equation (3.3) is

m

Vr

m

CAVPo ……………………………………….. (3.5)

Where A = present value of an annuity of N1 received every year for n years.

If the bond has an infinite life, (3.3) gives.

79

mnt

to

m

r

V

m

r

m

C

P

11

1

as n , mn

m

r

V

1

O

Leaving

1

1t

to

m

r

m

C

P

=

...

1

1

1

1

1

12

1

m

r

m

rm

C

m

rm

C

tt

m

r

m

r

m

CPo

1

11

1

1

Sum to infinity

m

rm

r

m

rm

C

m

rm

rm

r

m

C1

1

1

1

1

1

= r

C

r

m

m

C

=> r

CPo …………….. …………………………………. (3.6)

80

another equation

nt

tn

t

n

tnt

t

r

V

r

C

r

nV

r

Ct

d

11

1

.

1

1

1 . ……………………………..……….. (3.20)

That is

o

n

tnt

t

P

r

nV

r

Ct

d

1 1

.

1

If the bond does not have coupons, d would equal

o

n

P

rVn

1 = n

because Po = nr

V

1 ………...……………………..…… 3.21)

That implies d = n

For a given change in the market yield, bond prices vary proportionately with

duration; that is the duration of a bond or a portfolio of bond contains information about

risk.

Po = nr

v

1

orasandnd

rr

n

P

P

rr

nVP

o

o

no

1

11

)( rdP

P

o

o

…………………………………………… (3.22)

81

3.2 THEOREMS ON BOND PRICES

Theorem 1: Bond prices are inversely related to bond yield. This can be

recovered easily from equation (3.1) and (3.6). The presence of r in the denominator

certifies the inverse relationship. Using equation (3.6),.

)23.3........(..........................................................................................2r

C

r

P

r

cP

o

o

Hence the inverse relationship assumes that the coupon rate is fixed.

Theorem 2: Bond price changes are an increasing function of maturity n for any

given differences between the coupon rate and the yield to maturity. Consider two bonds

A and B. of equivalent risk, Bond A offers 10% semiannual coupon payment of the N50

and matures in five years. Bond B offers a 10% semiannual coupon payment of N50 but

matures in ten years. Both bonds are purchased at par. if the market yield on instruments

of comparable risk increases to 11%, price of A and B becomes N962.31 and N940.25

respectively. Leaving a drop of N37.69 and N59.75 respectively in the N1000. Different

price changes are observed for the same increase in yield to maturity, 11% with all other

characteristics of the bond being the same. Therefore, bond price changes are an

increasing function of maturity.

Theorem 3: The percentage change in the price of the bond increases at a

diminishing rate as n increases. That is to say, the marginal percentage price changes are

decreasing as n increases. For a given maturity structure, the percentage price drop

increases at a decreasing rate for a given rise in yields.

Theorem 4: Given the maturity, the capital gains resulting from a decrease in

yields are always higher them the capital loses resulting from an increase in yields. In all

cases, the capital gain resulting from a percent decline in market yield is greater than the

capital loss (in absolute terms) resulting from a 1 percent rise in yields.

Theorem 5: The higher the coupon rate on a bond, the smaller the percentage

change resulting from a given change in yields. The bond with a higher coupon

experiences a price change that is smaller in percentage term (larger in absolute naira)

82

than the bond with lower coupon. The implication to the investor is quite clear. An

aggressive investor would speculate using low-coupon bonds because their price

fluctuations, in percentage terms, are larger than those of bonds carrying higher coupons.