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Topic 6: Relationship Between Bond Prices and YieldsBUS 442 Investment Theory and Portfolio Management

The goal of the previous topic is to help you become more familiar with the different features of a bond the

different types of bonds and the different ratings a bond can receive! "nce you have a better understanding of the

basic anatomy of a bond it is time to learn more about the relationship between the price of a bond and its yield #or

mar$et interest rate%!

1. Determining the Intrinsic Value of a Bond

&hen you invest in a bond you are e'pected to receive a steady stream of fi'ed interest payments from the bond!

&hat do you do with the interest payments( )ou are e'pected to reinvest them in the mar$et at the mar$et interest

rate!

*eep in mind that we are dealing with the nominal mar$et interest rate which is made up of three parts+ #a% a real

ris$,free rate of return #b% a compensation for e'pected inflation and #c% a compensation for bond,specific

characteristics #such as ratings call features li-uidity etc!%! This means that there will be a different mar$et interest

rate for each bond that reflects the ris$s associated with that bond #because of its characteristics%! To $eep thingssimple for now we will assume that there is a single nominal mar$et interest rate that applies to all bonds!

To determine the intrinsic value of a bond we simply apply the concept of time value of money to determine the

present values of all the cash flows #i!e! interest payments and principal% generated by the bond throughout its

lifetime! In other words the intrinsic value of a bond is simply the present value of all its interest payments plus the

present value of its principal which is e'pressed by the formula below+

( ) ( ) ( )

T

T

t

t

t

TT

T

r

Par

r

C

r

Par

r

C

r

C

r

C

r

CP

%.#%.#

%.#.!!!!!

...

.

/

/

2

2.0

++

+=

++

+++

++

++

+=

=

whereP0is the intrinsic value of a bond Ciis the coupon #or interest% payment from period i and ris the nominal

mar$et interest rate!

Refer to Inclass !"ample 1

#. $essons $earned from Valuing a Bond

In the previous e'ample we $now that the characteristics of a bond #e!g! coupon rate term to maturity and paymentschedule% will have an impact on the intrinsic value #i!e! fair mar$et price% of the bond! &hat are some of the lessons

we learned from valuing a bond( &e will loo$ at five of these lessons through a number of e'amples!

Refer to Inclass !"ample #

$esson 1:There is an inverse relationship between the market interest rate and the price of a

bond, i.e. if the market interest rate goes up (down), then the price of a bond will go down (up).

&e can easily plot the data we have collected from 1'ample . and illustrate what we have learned in esson . in the

following graph!

Investment Theory ,., Topic 3 andout

8/13/2019 Relationship Between Bond Prices and Yields

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Price

Mar$et

interest rate

$esson #: The price of a bond is affected by the relationship between its coupon rate and the

market interest rate.

Coupon rate !arket interest Price of bond Par "old at premium

Coupon rate # market interest Price of bond # Par "old at par

Coupon rate $ market interest Price of bond $ Par "old at discount

&e have learned esson 2 by observing a pattern that emerges from our calculations in 1'ample 2! 5an you provide

an intuitive e'planation as to why this 6pattern7 is observed(

Refer to Inclass !"ample %

$esson %: The lower the coupon rate of a bond (with the same term to maturity), the more

sensitive it is to interest changes.

Refer to Inclass !"ample &

Price

Mar$et

interest rate

Investment Theory ,2, Topic 3 andout

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$esson &:%onger&term bonds are more sensitive to interest rate changes than shorter&term bonds

(with the same coupon rate). 's a result, longer&term bonds are considered to be riskier than

shorter&term bonds.

Investment Theory ,/, Topic 3 andout

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So far the e'amples we loo$ed at focused on studying the impacts of interest rate changes on the fair mar$et price

#i!e! intrinsic value% of a bond! &e would now turn our attention to understanding how the price of a bond changes

as it gets closer to its maturity date #in a scenario where the mar$et interest rate remained the same%!

Refer to Inclass !"ample '

Price

Time

$esson ':f the market interest rate remains unchanged, the intrinsic values of ero couponbonds, discount bonds, and premium bonds all converge to their face values.

%. The (an) Yields of a Bond

)ou have probably heard the term yieldused in a financial conte't before! The term yieldis generally used to

represent the return of an investment! owever do you $now that there are different types of yield associated with a

bond each representing a different way to measure the return of the bond( In this section we will loo$ at #a% normalyield #b% current yield #c% yield to maturity and #d% yield to call!

#a% 8ormal yield

The normal yield of a bond is simply the -uoted coupon rate of the bond! 9or e'ample a bond with an :;

coupon rate has a normal yield of :;! *eep in mind that the normal yield of a bond is fi'ed #in general%

and does not ta$e into consideration the current mar$et condition which means that you should nor rely on

the normal yield to help you determine the return of a bond!

#b% 5urrent yield

The current yield of a bond does ta$e the current mar$et condition into consideration by incorporating the

current mar$et price into its calculation! < bond=s current yield is determined as follows+

pricebondcurrent

paymentcoupon

8/13/2019 Relationship Between Bond Prices and Yields

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T

T

t

t

t

*T!

Par

*T!

CP

%.#%.#. ++

+=

=

Basically the yield to maturity is simply the discount rate that ma$es the present values of a bond=s couponpayments and principal the same as its current mar$et price! )ou can solve for the )TM using a financial

calculator or estimate it with the following formula+

[ ]%priceMar$et2#>value/

.

maturityuntil)ears

priceMar$etvalue9ace>paymentcooupon

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Investment Theory ,3, Topic 3 andout

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&. Term *tructure of Interest Rates and the Yield +ur,e

To ma$e li$e simple it was assumed earlier that there is a single mar$et interest rate #or yield to maturity% for all the

e'isting bonds! &e $now this not true because the yield to maturity of a bond is affected by the bond=s

characteristics such as its term to maturity coupon rate call provisions li-uidity default ris$ and ta' status!

In this section we will focus solely on how the yield of a bond is affected by its term to maturity! The relationshipbetween the yield to maturity of a bond and its term to maturity is $nown as the terms structure of interest rates and

it is represented graphically by the yield curve! )ou can loo$ up the yield curve daily in the 5redit section of the

all "treet -ournal! It is important to $now that the yield curve assumes all the bonds have the same ris$ li-uidity

and ta' status! The yield curve can ta$e on any of the following four shapes+

.! ormal yield curve+ The short,term yield is lower than thelong,term yield i!e! it is cheaper to borrow short,term than

it is to borrow long,term!

Time to maturity

)TM

2! nverted yield curve+ The short,term yield is higher than the

long,term yield i!e! it is more e'pensive to borrow short,term than it is to borrow long,term!

Time to maturity

)TM

/! +lat yield curve+ The short,term yield is the same as thelong,term yield i!e! the short,term cost of borrowing is the

same as the long,term cost of borrowing!

Time to maturity

)TM

4! /umped yield curve+ The intermediate yield is higher than

both the short,term and long,term yields!

Time to maturity

)TM

There are three ma@or theories that attempt to e'plain the shape of a yield curve+ #a% pure e'pectation theory #b%

li-uidity preference theory and #c% mar$et segmentation theory! Since you will be learning more each of these three

theories in your financial institution course #BUS 444% and money and ban$ing course #15" 4.0% this will simply

be a brief overview of the three theories!

Investment Theory ,A, Topic 3 andout

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#a% Pure e'pectation theory

This theory claims that the term structure of the interest rate is based on the current e'pectations of future

short,term interest rates! In other words long,term interest rates are simply the #geometric% mean of the

short,term interest rate in the same time period! This relationship is represented by the following formula+

[ ] .%.#!!!%.#%.#%.#.

...2... ++++= nnn rrr00