portfolio management- chapter 12
TRANSCRIPT
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Chapter 12
Bond Prices and the
Importance of Duration
Prof. Rushen Chahal 1
Prof. Rushen Chahal
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We cannot gamble with anything
so sacred as money.
- William McKinley
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Outline
Introduction
Review of bond principles
Bond pricing and returns Bond risk
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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
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Review of Bond Principles
Identification of bonds
Classification of bonds
Terms of repayment Bond cash flows
Convertible bonds
Registration
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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
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Classification of Bonds
Introduction
Issuer
Security Term
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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
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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
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Security
Definition
Unsecured debt
Secured debt
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Definition
The securityof a bond refers to what backs
the bond (what collateral reduces the risk of
the loan)
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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
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Unsecured Debt (contd)
Corporations:
Debentures are signature loans backed by the
good name of the company
Subordinated debentures are paid off after
original debentures
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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
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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
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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
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Terms ofRepayment
Interest only
Sinking fund
Balloon
Income bonds
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Interest Only
Periodic payments are entirely interest
The principal amount of the loan is repaid atmaturity
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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
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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
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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
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Bond Cash Flows
Annuities
Zero coupon bonds
Variable rate bonds
Consols
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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
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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
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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
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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.
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Convertible Bonds
Definition
Security-backed bonds
Commodity-backed bonds
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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
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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
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Commodity-Backed Bonds
Commodity-backed bonds are convertible into
a tangible asset
For example, silver or gold
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Registration
Bearer bonds
Registered bonds
Book entry bonds
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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.
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Registered Bonds
Registered bonds show the bondholders
name
Registered bondholders receive interest
checks in the mail from the issuer
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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
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Bond Pricing and Returns
Introduction
Valuation equations
Yield to maturity Realized compound yield
Current yield
Term structure of interest rates Spot rates
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Bond Pricing and Returns (contd)
The conversion feature
The matter of accrued interest
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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
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Valuation equations
Annuities
Zero coupon bonds
Variable rate bonds Consols
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Annuities
For a semiannual bond:
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? 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
!!
!
!!
!
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Annuities (contd)
Separating interest and principal components:
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? A ? A
2
0 21 1 ( / 2) 1 ( / 2)
where coupon payment
N
t Nt
C Par P
R R
C
!
!
!
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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?
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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%.
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Zero Coupon Bonds
For a zero-coupon bond (annual and
semiannual compounding):
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0
0 2
(1 )
(1 / 2)
t
t
ParP
R
ParP
R
!
!
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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)?
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Zero Coupon Bonds (contd)
Example (contd)
Solution:
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0 2
40
(1 / 2)
$1,000$400
(1 / 2)
4.63%
t
ParP
R
R
R
!
!
!
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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:
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2
0
1 (1 )
where interest rate at time
N t
tt t
t
CPI
I t
!
!
!
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Consols
Consols are perpetuities:
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0CPR
!
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Consols (contd)
Example
Aconsol is selling for $900 and pays $60 annually in perpetuity.
What is this consols rate of return?
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Consols (contd)
Example (contd)
Solution:
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0
$606.67%
$900
CP
R
R
!
! !
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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
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Realized Compound Yield
The effective annual yield is useful to compare
bonds to investments generating income on a
different time schedule
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? AEffective annual rate 1 ( / ) 1
where yield to maturitynumber of payment periods per year
x
R x
Rx
!
!
!
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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?
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Realized Compound
Yield (contd)Example (contd)
Solution:
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? A
? A2
Effective annual rate 1 ( / ) 1
1 (.009 / 2) 1
9.20%
x
R x!
!
!
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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
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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
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Current
Yield (contd)Example
A bond pays annual interest of $70 and has a current price of
$870.
What is this bonds current yield?
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Current
Yield (contd)Example (contd)
Solution:
Current yield = $70/$870 = 8.17%
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Term Structure of
Interest Rates
Yield curve
Theories of interest rate structure
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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
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Information Used to
Build AYield Curve
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Theories of
Interest Rate Structure
Expectations theory
Liquidity preference theory
Inflation premium t
heory
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Expectations Theory
According to the expectations theoryof
interest rates, investment opportunities with
different time horizons should yield the same
return:
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2
2 1 1 2
1 2
(1 ) (1 )(1 )
where the forward rate from time 1 to time 2
R R f
f
!
!
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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?
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Expectations Theory (contd)
Example (contd)
Solution:
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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
!
!
!
!
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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
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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
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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
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Spot Rates (contd)
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Spot Rate Curve
Yield to Maturity
Time Untilthe Cash Flow
InterestRate
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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.
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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:
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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
!
!
!
!
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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
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Th C i
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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
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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
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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
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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?
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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
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Bond Risk
Price risks
Convenience risks
Malkiels interest rate th
eories Duration as a measure of interest rate risk
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Price Risks
Interest rate risk
Default risk
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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
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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
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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
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Convenience Risks
Definition
Call risk
Reinvestment rate risk Marketability risk
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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
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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
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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
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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
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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
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Malkiels
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Malkiel s
Interest Rate Theorems
Definition
Theorem 1
Theorem 2
Theorem 3
Theorem 4
Theorem 5
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Definition
Malkiels interest rate theorems provide
information about how bond prices change as
interest rates change
Any good portfolio manager knows Malkiels
theorems
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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
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Theorem 2
Bonds with longer maturities will fluctuate
more if interest rates change
Long-term bonds have more interest rate risk
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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
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Theorem 5
Capital gains from an interest rate decline
exceed the capital loss from an equivalent
interest rate increase
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Duration as A Measure of Interest
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Duration as A Measure ofInterest
Rate Risk
The concept of duration
Calculating duration
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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
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The Concept of
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The Concept of
Duration (contd)
Duration is a direct measure of interest rate
risk:
The higher the duration, the higher the interest
rate risk
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Calculating Duration
The traditional duration calculation:
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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
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v
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8/3/2019 Portfolio Management- Chapter 12
99/101
Calculating Duration (contd)
The closed-end formula for duration:
Prof. Rushen Chahal 99
1
2
(1 ) (1 ) ( )
(1 ) (1 )
where par value of the bondnumber of periods until maturity
yield to maturity of the bond per period
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8/3/2019 Portfolio Management- Chapter 12
100/101
Calculating Duration (contd)
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?
Prof. Rushen Chahal 100
( )
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8/3/2019 Portfolio Management- Chapter 12
101/101
Calculating Duration (contd)
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
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