Portfolio Management- Chapter 12

8/3/2019 Portfolio Management- Chapter 12

1/101

Chapter 12

Bond Prices and the

Importance of Duration

Prof. Rushen Chahal 1

Prof. Rushen Chahal


2/101

We cannot gamble with anything

so sacred as money.

- William McKinley



3/101

Outline

Introduction

Review of bond principles

Bond pricing and returns Bond risk



4/101

Introduction

The investment characteristics of bonds rangecompletely across the risk/return spectrum

As part of a portfolio, bonds provide bothstability and income

Capital appreciation is not usually a motive for

acquiring bonds



5/101

Review of Bond Principles

Identification of bonds

Classification of bonds

Terms of repayment Bond cash flows

Convertible bonds

Registration



6/101

Identification of Bonds

A bond is identified by:

The issuer

The coupon

The maturity

For example, five IBM eights of 10 means

$5,000 par IBM bonds with an 8% coupon rateand maturing in 2010



7/101

Classification of Bonds

Introduction

Issuer

Security Term



8/101

Introduction

The bond indenture describes the details of a

bond issue:

D

escription of th

e loan Terms of repayment

Collateral

Protective covenants

Default provisions



9/101

Issuer

Bonds can be classified by the nature of the

organizations initially selling them:

Corporation

Federal, state, and local governments

Government agencies

Foreign corporations or governments



10/101

Security

Definition

Unsecured debt

Secured debt



11/101

Definition

The securityof a bond refers to what backs

the bond (what collateral reduces the risk of

the loan)



12/101

Unsecured Debt

Governments:

Full faith and credit issues (general obligation

issues) is government debt without specific assets

pledged against it

E.g., U.S.Treasury bills, notes, and bonds



13/101

Unsecured Debt (contd)

Corporations:

Debentures are signature loans backed by the

good name of the company

Subordinated debentures are paid off after

original debentures



14/101

Secured Debt

Municipalities issue:

Revenue bonds

Interest and principal are repaid from revenue

generated by the project financed by the bond

Assessment bonds

Benefit a specific group of people, who pay anassessment to help pay principal and interest



15/101

Secured Debt (contd)

Corporations issue:

Mortgages

Well-known securities that use land and buildings as

collateral

Collateral trust bonds

Backed by other securities

Equipment trust certificates Backed by physical assets



16/101

Term

The term is the original life of the debt

security

Short-term securities have a term of one year or

less

Intermediate-term securities have terms ranging

from one year to ten years

Long-term securities have terms longer than tenyears



17/101

Terms ofRepayment

Interest only

Sinking fund

Balloon

Income bonds



18/101

Interest Only

Periodic payments are entirely interest

The principal amount of the loan is repaid atmaturity



19/101

Sinking Fund

A sinking fundrequires the establishment of a

cash reserve for the ultimate repayment of

the bond principal

The borrower can:

Set aside a potion of the principal amount of the debt

each year

Call a certain number of bonds each year



20/101

Balloon

Balloon loans partially amortize the debt with

each payment but repay the bulk of the

principal at the end of the life of the debt

Most balloon loans are not marketable



21/101

Income Bonds

Income bonds pay interest only if the firm

earns it

For example, an income bond may be issued

to finance an income-producing project



22/101

Bond Cash Flows

Annuities

Zero coupon bonds

Variable rate bonds

Consols



23/101

Annuities

An annuitypromises a fixed amount on a

regular periodic schedule for a finite length of

time

Most bonds are annuities plus an ultimate

repayment of principal



24/101

Zero Coupon Bonds

A zero coupon bondhas a specific maturity

date when it returns the bond principal

A zero coupon bond pays no periodic income

The only cash inflow is the par value at maturity



25/101

Variable Rate Bonds

Variable rate bonds allow the rate to fluctuate

in accordance with a market index

For example, U.S. Series EE savings bonds



26/101

Consols

Consols pay a level rate of interest

perpetually:

The bond never matures

The income stream lasts forever

Consols are not very prevalent in the U.S.



27/101

Convertible Bonds

Definition

Security-backed bonds

Commodity-backed bonds



28/101

Definition

A convertible bondgives the bondholder the

right to exchange them for another security or

for some physical asset

Once conversion occurs, the holder cannot

elect to reconvert and regain the original debt

security



29/101

Security-Backed Bonds

Security-backed convertible bonds are

convertible into other securities

Typically common stock of the company that

issued the bonds

Occasionally preferred stock of the issuing firm,

common stock of another firm, or shares in asubsidiary company



30/101

Commodity-Backed Bonds

Commodity-backed bonds are convertible into

a tangible asset

For example, silver or gold



31/101

Registration

Bearer bonds

Registered bonds

Book entry bonds



32/101

Bearer Bonds

Bearer bonds:

Do not have the name of the bondholder printed

on them

Belong to whoever legally holds them

Are also called coupon bonds

The bond contains coupons that must be clipped

Are no longer issued in the U.S.



33/101

Registered Bonds

Registered bonds show the bondholders

name

Registered bondholders receive interest

checks in the mail from the issuer



34/101

Book Entry Bonds

The U.S.Treasury and some corporation issue

bonds in book entry form only

Holders do not take actual delivery of the bond

Potential holders can:

Open an account through the Treasury DirectSystem at

a Federal Reserve Bank Purchase a bond through a broker



35/101

Bond Pricing and Returns

Introduction

Valuation equations

Yield to maturity Realized compound yield

Current yield

Term structure of interest rates Spot rates



36/101

Bond Pricing and Returns (contd)

The conversion feature

The matter of accrued interest



37/101

Introduction

The current price of a bond is the marketsestimation of what the expected cash flowsare worth in todays dollars

There is a relationship between:

The current bond price

The bonds promised future cash flows

The riskiness of the cash flows



38/101

Valuation equations

Annuities

Zero coupon bonds

Variable rate bonds Consols



39/101

Annuities

For a semiannual bond:


? A

2

0

1

0

1 ( / 2)

where term of the bond in years

cash flow at timeannual yield to maturity

current price of the bond

N

tt

t

t

CPR

N

C t

R

P

!!

!

!!

!


40/101

Annuities (contd)

Separating interest and principal components:


? A ? A

2

0 21 1 ( / 2) 1 ( / 2)

where coupon payment

N

t Nt

C Par P

R R

C

!

!

!


41/101

Annuities (contd)

Example

A bond currently sells for $870, pays $70 per year (Paid

semiannually), and has a par value of $1,000. The bond has aterm to maturity of ten years.

What is the yield to maturity?



42/101

Annuities (contd)

Example (contd)

Solution:Using a financial calculator and the following input provides thesolution:

N = 20

PV = $870

PMT = $35

FV = $1,000

CPTI = 4.50

This bonds yield to maturity is 4.50% x 2 = 9.00%.



43/101

Zero Coupon Bonds

For a zero-coupon bond (annual and

semiannual compounding):


0

0 2

(1 )

(1 / 2)

t

t

ParP

R

ParP

R

!

!


44/101

Zero Coupon Bonds (contd)

Example

A zero coupon bond has a par value of $1,000 and currently

sells for $400. The term to maturity is twenty years.

What is the yield to maturity (assume semiannual

compounding)?



45/101

Zero Coupon Bonds (contd)

Example (contd)

Solution:


0 2

40

(1 / 2)

$1,000$400

(1 / 2)

4.63%

t

ParP

R

R

R

!

!

!


46/101

Variable Rate Bonds

The valuation equation must allow for variable

cash flows

You cannot determine the precise present

value of the cash flows because they are

unknown:


2

0

1 (1 )

where interest rate at time

N t

tt t

t

CPI

I t

!

!

!


47/101

Consols

Consols are perpetuities:


0CPR

!


48/101

Consols (contd)

Example

Aconsol is selling for $900 and pays $60 annually in perpetuity.

What is this consols rate of return?



49/101

Consols (contd)

Example (contd)

Solution:


0

$606.67%

$900

CP

R

R

!

! !


50/101

Yield to Maturity

Yield to maturitycaptures the total return

from an investment

Includes income

Includes capital gains/losses

The yield to maturity is equivalent to the

internal rate of return in corporate finance



51/101

Realized Compound Yield

The effective annual yield is useful to compare

bonds to investments generating income on a

different time schedule


? AEffective annual rate 1 ( / ) 1

where yield to maturitynumber of payment periods per year

x

R x

Rx

!

!

!


52/101

Realized Compound

Yield (contd)Example

A bond has a yield to maturity of 9.00% and pays interest

semiannually.

What is this bonds effective annual rate?



53/101

Realized Compound

Yield (contd)Example (contd)

Solution:


? A

? A2

Effective annual rate 1 ( / ) 1

1 (.009 / 2) 1

9.20%

x

R x!

!

!


54/101

Current Yield

The current yield:

Measures only the return associated with the

interest payments

Does not include the anticipated capital gain or

loss resulting from the difference between par

value and th

e purch

ase price



55/101

Current Yield (contd)

For a discount bond, the yield to maturity is

greater than the current yield

For a premium bond, the yield to maturity is

less than the current yield



56/101

Current

Yield (contd)Example

A bond pays annual interest of $70 and has a current price of

$870.

What is this bonds current yield?



57/101

Current

Yield (contd)Example (contd)

Solution:

Current yield = $70/$870 = 8.17%



58/101

Term Structure of

Interest Rates

Yield curve

Theories of interest rate structure



59/101

Yield Curve

The yield curve:

Is a graphical representation of the term structureof interest rates

Relates years until maturity to the yield tomaturity

Is typically upward sloping and gets flatter forlonger terms to maturity



60/101

Information Used to

Build AYield Curve



61/101

Theories of

Interest Rate Structure

Expectations theory

Liquidity preference theory

Inflation premium t

heory



62/101

Expectations Theory

According to the expectations theoryof

interest rates, investment opportunities with

different time horizons should yield the same

return:


2

2 1 1 2

1 2

(1 ) (1 )(1 )

where the forward rate from time 1 to time 2

R R f

f

!

!


63/101

Expectations Theory (contd)

Example

An investor can purchase a two-year CD at a rate of 5 percent.

Alternatively, the investor can purchase two consecutive one-year CDs. The current rate on a one-year CD is 4.75 percent.

According to the expectations theory, what is the expected

one-year CD rate one year from now?



64/101

Expectations Theory (contd)

Example (contd)

Solution:


2

2 1 1 2

2

1 2

2

1 2

1 2

(1 ) (1 )(1 )

(1.05) (1.045)(1 )

(1.05)(1 )(1.045)

5.50%

R R f

f

f

f

!

!

!

!


65/101

Liquidity Preference Theory

Proponents of the liquidity preference theorybelieve that, in general:

Investors prefer to invest short term rather than

long term Borrowers must entice lenders to lengthen their

investment horizon by paying a premium for long-term money (the liquidity premium)

Under this theory, forward rates are higherthan the expected interest rate in a year



66/101

Inflation Premium Theory

The inflation premium theorystates that risk

comes from the uncertainty associated with

future inflation rates

Investors who commit funds for long periods

are bearing more purchasing power risk than

short-term investors

More inflation risk means longer-term investmentwill carry a higher yield



67/101

Spot Rates

Spot rates:

Are the yields to maturity of a zero coupon

security

Are used by the market to value bonds

The yield to maturity is calculated only after learning

the bond price

The yield to maturity is an average of the various spot

rates over a securitys life



68/101

Spot Rates (contd)


Spot Rate Curve

Yield to Maturity

Time Untilthe Cash Flow

InterestRate


69/101

Spot Rates (contd)

Example

A six-month T-bill currently has a yield of 3.00%. A one-year T-

note with a 4.20% coupon sells for 102.

Use bootstrapping to find the spot rate six months from now.



70/101

Spot Rates (contd)

Example (contd)

Solution: Use the T-bill rate as the spot rate for the first six

months in the valuation equation for the T-note:


2

2

2

2

2

2

2

21.00 1, 0211,020

(1 .03 / 2) (1 / 2)

1,021999.31(1 / 2)

(1 / 2) 1.022

2.16%

r

r

r

r

!

!

!

!


71/101

The Conversion Feature

Convertible bonds give their owners the right toexchange the bonds for a pre-specified amount orshares of stock

Th

econversion ratio

measures th

e number of sh

aresthe bondholder receives when the bond is converted

The par value divided by the conversion ratio is theconversion price

The current stock price multiplied by the conversion ratio

is the conversion value


Th C i


72/101

The Conversion

Feature (contd) The market price of a bond can never be less than its

conversion value

The difference between the bond price and the

conversion value is thepremium over conversionvalue

Reflects the potential for future increases in the common

stock price

Mandatory convertibles convert automatically intocommon stock after three or four years



73/101

The Matter ofAccrued Interest

Bondholders earn interest each calendar day

they hold a bond

Firms mail interest payment checks only twice

a year

Accrued interestrefers to interest that has

accumulated since the last interest payment

date but whichhas not yet been paid


Th M tt f


74/101

The Matter of

Accrued Interest (contd)

At the end of a payment period, the issuer

sends one check for the entire interest to the

current bondholder

The bond buyer pays the accrued interest to the

seller

The bond sells receives accrued interest from thebond buyer


Th M tt f


75/101

The Matter of

Accrued Interest (contd)Example

A bond with an 8% coupon rate pays interest on June 1 and

December 1.

The bond

currently sells for $920.

What is the total purchase price, including accrued interest,

that the buyer of the bond must pay if he purchases the bond

on August 10?


Th M tt f


76/101

The Matter of

Accrued Interest (contd)Example (contd)

Solution: The accrued interest for 71 days is:

$80/365 x 71 = $15.56

Therefore, the total purchase price is:

$920 + $15.56 = $935.56



77/101

Bond Risk

Price risks

Convenience risks

Malkiels interest rate th

eories Duration as a measure of interest rate risk



78/101

Price Risks

Interest rate risk

Default risk



79/101

Interest Rate Risk

Interest rate riskis the chance of loss because

of changing interest rates

The relationship between bond prices and

interest rates is inverse

If market interest rates rise, the market price of

bonds will fall



80/101

Default Risk

Default riskmeasures the likelihood that a

firm will be unable to pay the principal and

interest on a bond

Standard & Poors Corporation and Moodys

Investor Service are two leading advisory

services monitoring default risk



81/101

Default Risk (contd)

Investment grade bonds are bonds rated BBB

or above

Junk bonds are rated below BBB

The lower the grade of a bond, the higher its

yield to maturity



82/101

Convenience Risks

Definition

Call risk

Reinvestment rate risk Marketability risk



83/101

Definition

Convenience riskrefers to added demands on

management time because of:

Bond calls

The need to reinvest coupon payments

The difficulty in trading a bond at a reasonableprice because of low marketability



84/101

Call Risk

If a company calls its bonds, it retires its debtearly

Call riskrefers to the inconvenience ofbondholders associated with a companyretiring a bond early

Bonds are usually called when interest rates arelow



85/101

Call Risk (contd)

Many bond issues have:

Call protection

A period of time after the issuance of a bond when the

issuer cannot call it

A call premium if the issuer calls the bond

Typically begins with an amount equal to one years

interest and then gradually declining to zero as thebond approaches maturity



86/101

Reinvestment Rate Risk

Reinvestment rate riskrefers to the

uncertainty surrounding the rate at which

coupon proceeds can be invested

The higher the coupon rate on a bond, the

higher its reinvestment rate risk



87/101

Marketability Risk

Marketability riskrefers to the difficulty of

trading a bond:

Most bonds do not trade in an active secondary

market

The majority of bond buyers hold bonds until

maturity

Low marketability bonds usually carry a widerbid-ask spread


Malkiels


88/101

Malkiel s

Interest Rate Theorems

Definition

Theorem 1

Theorem 2

Theorem 3

Theorem 4

Theorem 5



89/101

Definition

Malkiels interest rate theorems provide

information about how bond prices change as

interest rates change

Any good portfolio manager knows Malkiels

theorems



90/101

Theorem 1

Bond prices move inversely with yields:

If interest rates rise, the price of an existing bond

declines

If interest rates decline, the price of an existing

bond increases



91/101

Theorem 2

Bonds with longer maturities will fluctuate

more if interest rates change

Long-term bonds have more interest rate risk



92/101


93/101

Theorem 4

When comparing two bonds, the relative

importance ofTheorem 2 diminishes as the

maturities of the two bonds increase

A given time difference in maturities is more

important with shorter-term bonds



94/101

Theorem 5

Capital gains from an interest rate decline

exceed the capital loss from an equivalent

interest rate increase


Duration as A Measure of Interest


95/101

Duration as A Measure ofInterest

Rate Risk

The concept of duration

Calculating duration



96/101

The Concept ofDuration

For a noncallable security:

Duration is the weighted average number of years

necessary to recover the initial cost of the bond

Where the weights reflect the time value of

money


The Concept of


97/101

The Concept of

Duration (contd)

Duration is a direct measure of interest rate

risk:

The higher the duration, the higher the interest

rate risk



98/101

Calculating Duration

The traditional duration calculation:


1 (1 )

where duration

cash flow at time

yield to maturity

current price of the bond

years until bond maturity

time at which a cash flow is received

Nt

tt

o

t

o

Ct

RD P

D

C t

R

P

N

t

!

v

!

!

!

!

!

!

!


99/101

Calculating Duration (contd)

The closed-end formula for duration:


1

2

(1 ) (1 ) ( )

(1 ) (1 )

where par value of the bondnumber of periods until maturity

yield to maturity of the bond per period

N

N N

o

R R R N F N

C R R RD

P

F

N

R

v v

- !

!!

!


100/101


Example

Consider a bond that pays $100 annual interest and has a

remaining life of 15 years. The bond currently sells for $985 and

has a yield to maturity of 10.20%.

What is this bonds duration?


( )


101/101


Example (contd)

Solution: Using the closed-form formula for duration:

1

2

31

2 30 30

(1 ) (1 ) ( )

(1 ) (1 )

(1.052) (1.052) (0.052 30) 1,000 30500.052 (1.052) (1.052)

985

15.69 years

N

N N

o

R R R N F NC

R R RD

P

v v - !

v v - !

!

Portfolio Management- Chapter 12

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