Fixed Income

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Fixed Income

Basic concepts

Features
Fixed income securities – promises to pay
a stream of semiannual payments for a
given number of years and then repay the
loan amount at maturity date
 The contract between the borrower and
the lender (the indenture) can be designed
to have any payment stream or pattern
that the parties agree to


Bond indenture
Defines the obligations of and restrictions
on the borrower and forms the basis for all
future transactions between the
bondholder and the issuer
 Contract provision – covenants:


 Negative

– prohibitions on the borrower
 Affirmative – actions that the borrower
promises to perform

Negative covenants
Restrictions on asset sales – the company
can’t sell assets that have been pledged
as collateral
 Negative pledge of collateral – the
company can’t claim that the same assets
back several debt issues in the same time
 Restrictions on additional borrowing – the
company can’t borrow additional money
unless certain financial conditions are met


Affirmative covenants
Maintenance of certain financial ratios and
the timely payment of principal and
interest
 If the values of the agreed ratios are not
maintained, then the bonds could be
considered in technical default


Straight (option free) bond
Treasury bond
 Coupon 6%
 Maturity 5Y
 Notional (face value) USD 1000
 Annual interest paid in two semiannual
installments
 Stream of payments fixed


Coupon rate structures


Zero-coupon bonds:
 Do

not pay periodic interest
 Initially sold at a price below par value
(discount to par value)
 Pay the par value at maturity

Coupon rate structures


Step-up notes
 Have

coupon rates that increase over time at
a specified rate
 Increase may take place one or more times
during the life of the issue

Deferred coupon bonds
Carry coupons, but the initial coupon
payments are deferred for some period
 The coupon payments accrue, at a
compound rate, over the deferral period
and are paid as a lump sum at the end of
that period
 After the initial deferment period has
passed, pay regular coupon interest for
the rest of the life of the issue


Floating rate securities
Bonds for which the coupon interest
payments over the life of the security vary
based on a specified interest rate or index
 They have coupons that are reset
periodically (normally every 3, 6 or 12
months) based on prevailing market
interest rates


Floating rate securities
New coupon rate =
reference rate +/- quoted margin
Reference rate: LIBOR, EURIBOR, ROBOR
Quoted margin may vary over time
according to a schedule that is stated in the
indenture


Inverse floater
Floating rate security with a coupon
formula that actually increases the coupon
rate when a reference interest rate
decreases and vice versa
 Eg
Coupon = 12% - reference rate


Inflation-indexed bonds
Coupon formula based on inflation
 Eg
Coupon = 3% + annual change in the
Consumer Price Index (CPI)


Protection against extreme fluctuations


Placing upper and lower limits on the
coupon rate:
 Upper

limit – cap – puts a maximum on the
interest rate paid by the borrower/issuer
 Lower limit – floor – puts a lower limit on the
periodic coupon interest payments received
by the lender
 Both limits – collar


Eg. floater with a coupon at issuance of
5%, a 7% cap and a 3% floor

Clean and dirty price
When a bond trades between coupon
dates, the seller is entitled to receive any
interest earned from the previous coupon
date through the date of the sale –
accrued interest
 Calculated as a fraction of the coupon
period that has passed times the coupon
 Full (dirty) price = clean price + accrued
interest


Redemption of bonds
Redemption provisions refer to how, when
and under what circumstances the
principal will be repaid
 Nonamotizing – bullet bond or bullet
maturity – at maturity the entire par or face
value is repaid
 Amortizing –make periodic payments of
interest and principal


Redemption of bonds - options







Prepayment options – give the issuer/borrower
the right to accelerate the principal repayment
on a loan
Call provisions – give the issuer the right (but not
the obligation) to retire all or a part of an issue
prior to maturity
Call protection – a period of years after issuance
during which the bonds cannot be called
Call schedule – specify when the bonds can be
called and at what price (declining)

Nonrefundable vs noncallable
Nonrefundable bonds prohibit the call of
an issue using the proceeds from a lower
coupon bond issue
 A bond may be callable but not refundable
 A bond that is noncallable has absolute
protection against a call prior to maturity
 A callable but not refundable bond can be
called for any reason other than refunding


Sinking fund


Provides for the repayment of principal through
a series of payments over the life of the issue,
which can be accomplished via cash or delivery
 Cash

retire the applicable portion of bonds by using a
selection method such as lottery
 Delivery of securities: purchase bonds (at market
price) with a total par value equal to the amount that
is to be retired in that year in the market


Accelerated sinking fund – choice of retiring
more than amount specified in sinking fund

Embedded options
Integral part of the bond contract and are
not a separate security
 Some are exercisable at the option of the
issuer and some at the option of the
purchaser of the bond


Security owner options


Option granted to the security holder and
gives additional value to security

Conversion options – convert bond into a
fixed number of securities
 Put options – right to sell the bond to the
issuer at a special price prior to maturity
 Floors – set a minimum on the coupon
rate for a floating-rate bond


Security issuer options (1)


Exercisable at the option of the issuer of the
fixed income security and gives lower value to
security



Call provision – gives the issuer the right to
redeem the issue prior to maturity
Prepayment option – gives the issuer the right to
prepay the loan balance prior to maturity in
whole or in part without penalty



Security issuer options (2)
Accelerated sinking fund provisions are
embedded options held by the issuer that
allow the issuer to (annually) retire a larger
proportion of the issue than is required by
the sinking fund provision, up to a
specified limit.
 Caps set a maximum on the coupon rate
for a floating rate note


Exercises

Exercise
A 10 year bond pays no interest for three
years, then pays USD 229.25, followed by
payments of USD 35 for seven years and
additional USD 1000 at maturity. This is a:
a) Step-up bond
b) Zero coupon bond
c) Deferred-coupon bond


Exercise
Consider a USD 1 Mio semi-annual pay,
floating rate issue where the rate is reset
on Jan. 1 and Jul. 1 each year. The
reference rate is 6M Libor and the stated
margin is + 1.25%. If 6M Libor is 6.5% on
Jul. 1 what will be the next semi-annual
coupon on this issue?
A) 38,750
B) 65,000
C) 77,500


Exercise
An investor paid a full price of USD
1,059.04 each for 100 bonds. The
purchase was between coupon dates, and
accrued interest was USD 23.54 per bond.
What was the bond clean price?
A. 1000.00
B. 1035.50
C. 1082.58


Exercise
Consider a USD 1 Mio par value, 10Y, 6.5%
coupon bond issued on Jan. 1 2005. The bonds
are callable and there is a sinking fund
provision. The market rate for similar bonds is
currently 5.7%. The main points of the
prospectus are summarized as follows:
Call dates and prices:
 2005 through 2009: 103
 After Jan. 1, 2010: 102


Exercise – additional info
The bonds are non-refundable
 The sinking fund provision requires that
the company redeem USD 0.1 Mio of the
principal amount each year. Bonds called
under the terms of the sinking fund
provision will be redeemed at par
 The credit rating of the bonds is currently
the same as at issuance


Questions
Using only the preceding information, an
analyst should conclude that
A. The bonds do not have call protection
B. The bonds were issued and currently
trade at a premium
C. Given current rates, the bonds will likely
be called and new bonds issued


Questions
Which of the following statements about the
sinking fund provisions for these bonds is most
accurate?
A. An investor would benefit from having his
bonds called under the provision of the sinking
fund
B. An investor would receive a premium if the
bond is redeemed prior to maturity under the
provision of the sinking fund
C. The bonds do not have an accelerated sinking
fund provision


Risks associated
with investing in
bonds

Interest rate risk
Effect of changes in the prevailing market
rate of interest on bond values
 When interest rates rise, bond values fall.
This is the source of interest rate risk
which is approximated by a measure
called duration.


Price – yield relation

Bond’s characteristics vs interest rate risk
Characteristic

Interest Rate Risk

Duration

Maturity up

Interest rate risk up

Duration up

Coupon up

Interest rate risk down

Duration down

Add a call

Interest rate risk down

Duration down

Add a put

Interest rate risk down

Duration down

Example of the coupon effect


Consider the durations of a 5-year and 20year bond with varying coupon rates
(semi-annual coupon payments):

Zero coupon
6% coupon
9% coupon

5 year bond

20 year bond

5
4.39
4.19

20
11.90
10.98

Impact of embedded options




A call feature limits the upside price movement
of a bond when interest rates decline. Hence
the value of a callable bond will be less sensitive
to interest rate changes than an otherwise
identical option-free bond.
A put feature limits the downside price
movement of a bond when interest rates rise.
Hence the value of a putable bond will be less
sensitive to interest rate changes than an
otherwise identical option-free bond

Price - yield callable bond

Callable bond value =
Value of an option-free bond – value of embedded call option

Interest rate risk in a floating rate security






The objective of the resetting mechanism is to bring the
coupon rate in line with the current market yield so the
bond sells at or near its par value. This will make the
price of a floating-rate security much less sensitive to
changes in market yields than a fixed-coupon bond of
equal maturity
Between coupon dates, there is a time lag between any
change in market yield and a change in the coupon rate
(which happens on the next reset date)
The longer the time period between the two dates, t he
greater t he amount of potential bond price fluctuation.
Hence the longer (shorter) the reset period, the greater
(less) the interest rate risk of a floating-rate security at
any reset date

Interest rate risk in a floating rate security
Presence of a cap (maximum coupon
rate) can increase the interest rate risk of
a floating-rate security
 If the reference rate increases enough that
the cap rate is reached, further increases
in market yields will decrease the floater's
price


Duration
Is a measure of the price sensitivity of a
security to changes in yield
 It can be interpreted as an approximation
of the percentage change in the security
price for a 1% change in yield
 Also can be interpreted as the ratio of the
percentage change in price to the change
in yield in percent


Duration - examples
If a bond has a duration of 5 and the yield
increases from 7% to 8%, calculate the
approximate percentage change in the
bond price.
 A bond has a duration of 7.2. If the yield
decreases from 8.3% to 7. 9%, calculate
the approximate percentage change in the
bond price.


Dollar duration
Sometimes the interest rate risk of a bond
or portfolio is expressed as its dollar
duration, which is simply the approximate
price change in dollars in response to a
change in yield of 100 basis points (1%).
 Another measure is Basis Point Value –
BPV which is the approximate price
change in dollars in response to a change
in yield of 1 basis point.


Duration examples
If a bond's yield rises from 7% to 8% and
its price falls 5%, calculate the duration.
 If a bond's yield decreases by 0.1% and its
price increases by 1.5%, calculate its
duration.
 A bond is currently trading at $1,034.50,
has a yield of 7.38%, and has a duration of
8.5. If the yield rises to 7.77%, calculate
the new price of the bond.


Yield curve risk
Arises from the possibility of changes in
the shape of the yield curve (which shows
the relation between bond yields and
maturity).
 While duration is a useful measure of
interest rate risk for equal changes in yield
at every maturity (parallel changes in the
yield curve), changes in the shape of the
yield curve mean that yields change by
different amounts for bonds with different
maturities.


Yield curve shifts

Duration for a bond portfolio





Computed as a weighted average based on
individual bond durations and the proportions of
the total portfolio value invested in each bond
Is an approximation of the price sensitivity of a
portfolio to parallel shifts of the yield curve
For a non-parallel shift in the yield curve, the
yields on different bonds in a portfolio can
change by different amounts, and duration alone
cannot capture the effect of a yield change on
the value of the portfolio.

Key rate durations


To estimate the impact of non-parallel
shifts, bond portfolio managers calculate
key rate durations, which measure the
sensitivity of the portfolio's value for
changes in yields for specific maturities (or
portions of the yield curve)

Call risk
When interest rates fall, a callable bond
investor's principal may be returned and
must be reinvested at the new lower rates.
 When interest rates are more volatile,
callable bonds have relatively more call
risk because of an increased probability of
yields falling to a level where the bonds
will be called.


Prepayment risk
Prepayments are principal repayments in
excess of those required on amortizing
loans
 If rates fall, causing prepayments to
increase, an investor must reinvest these
prepayments at the new lower rate
 As with call risk, an increase in interest
rate volatility increases prepayment risk


Reinvestment risk





When market rates fall, the cash flows (both
interest and principal) from fixed-income
securities must be reinvested at lower rates,
reducing the returns an investor will earn.
Reinvestment risk is related to call risk and
prepayment risk.
Coupon bonds are also subject to reinvestment
risk, because the coupon interest payments
must be reinvested as they are received

Reinvestment risk


A security has more reinvestment risk
under the following conditions:
 The

coupon is higher so that interest cash
flows are higher
 It has a call feature
 It is an amortizing security
 It contains a prepayment option

Credit risk
Is the risk that the creditworthiness of a
fixed-income security's issuer will
deteriorate, increasing the required return
and decreasing the security's value
 It is reflected by the credit rating of the
issuance


Rating


A bond's rating is used to indicate its
(relative) probability of default, which is the
probability of its issuer not making timely
interest and principal payments as
promised in the bond indenture

Rating agencies
Rate specific debt issues
 The ratings are issued fo indicate the
relative probability that all promised
payments on the debt will be made over
the life of the security and, therefore, must
be forward looking.
 Ratings on long-term bonds will consider
factors that may come into play over at
least one full economic cycle.


Firm specific factors considered in rating










Past repayment history
Quality of management, ability to adapt to changing
conditions
The industry outlook and firm strategy
Overall debt level of the firm
Operating cash flow, ability to service debt
Other sources of liquidity (cash, salable assets)
Competitive position, regulatory environment, and union
contracts/history
Financial management and controls.
Susceptibility to event risk and political risk

Bond Ratings
Moody's
Aaa
Aa1
Aa2
Aa3
A1
A2
A3
Baa1
Baa2
Baa3
Ba1
Ba2
Ba3
B1
B2
B3
Caa1
Caa2
Caa3
Ca

S&P
AAA
AA+
AA
AAA+
A
ABBB+
BBB
BBBBB+
BB
BBB+
B
BCCC+
CCC
CCC-

Bond Ratings by Agency
Fitch
DBRS
DCR
AAA
AAA
AAA
AA+
AA+
AA+
AA
AA
AA
AAAAAAA+
A+
A+
A
A
A
AAABBB+
BBB+
BBB+
BBB
BBB
BBB
BBBBBBBBBBB+
BB+
BB+
BB
BB
BB
BBBBBBB+
B+
B+
B
B
B
BBBCCC
CCC+
CCC
CCC
CCC-

C
DDD
D
DD
DD
D
D
DP
Source: http://www.bondsonline.com/asp/research/bondratings.asp

Definitions
Prime. Maximum Safety
High Grade High Quality

Upper Medium Grade

Lower Medium Grade

Non Investment Grade
Speculative
Highly Speculative

Substantial Risk
In Poor Standing
Extremely Speculative
May be in Default
Default

Bond Ratings



There is virtually no risk of default within 1 year, and very little over
longer periods, if investing in investment grade securities.
Once go below investment grade, however, the risk of default rises
dramatically.

Bond Ratings
Default Rate by S&P Bond Rating
(15 Years)
60.00%

Default Rate

50.00%
40.00%
30.00%
20.00%
10.00%
0.00%
AAA  

AA   

A    

BBB  

Default Rate 0.52%

1.31%

2.32%

6.64%

BB   

B    

CCC  

19.52% 35.76% 54.38%

S&P Bond Rating

Bond Ratings
Who Rates Bonds?
Each company's share of the total global revenue in 2001 for credit rating agencies
Other
6%
Fitch
14%

Moody's
38%

Standard &
Poor's
42%

Source: Wall Street Journal, 6 January 2003, p. C1 and M oody's

Transition matrix (S&P)
percent

End of year rating

Initial
rating

AAA

AA

A

AAA

89.37

6.04

0.44

AA

0.57

87.76

A

0.05

BBB

BBB

CCC

D

No
rating

BB

B

0.14

0.05

0.00

0.00

0.00

3.97

7.30

0.59

0.06

0.11

0.02

0.01

3.58

2.01

87.62

5.37

0.45

0.18

0.04

0.05

4.22

0.03

0.21

4.15

84.44

4.39

0.89

0.26

0.37

5.26

BB

0.03

0.08

0.40

5.50

76.44

7.14

1.11

1.38

7.92

B

0.00

0.07

0.26

0.36

4.74

74.12

4.37

6.20

9.87

CCC

0.09

0.00

0.28

0.56

1.39

8.80

49.72

27.87

11.30

Source: Standard & Poor’s (Special Report: Ratings Performance 2002, 2003)

Transition matrix (Moody’s)
percent*
End of year rating
Initial
Rating
Aaa

Aa

A

Baa

Ba

B

Caa-C

Aaa

Aa

A

Baa

Ba

B

Caa-C

Faliment

Rating
retras

86.34

8.21

0.19

0.00

0.00

0.00

0.00

0.00

5.26

87.69

6.13

0.42

0.00

0.08

0.00

0.00

0.00

5.68

0.76

86.71

9.13

0.10

0.00

0.00

0.00

0.00

3.30

0.72

85.21

8.75

0.45

0.12

0.02

0.00

0.00

4.74

0.00

5.05

84.80

3.63

0.10

0.02

0.00

0.02

6.39

0.08

2.32

87.15

5.34

0.64

0.24

0.03

0.02

4.18

0.74

0.25

4.82

78.83

2.86

1.16

0.04

0.00

11.31

0.07

0.30

5.55

83.01

4.54

0.99

0.08

0.18

5.28

0.00

0.00

0.64

10.52

71.40

9.29

0.68

0.25

7.22

0.03

0.04

0.65

5.18

73.90

8.57

0.47

1.45

9.71

0.00

0.00

0.33

1.03

9.40

65.52

8.28

3.29

12.17

0.01

0.06

0.23

0.64

5.06

73.94

3.84

7.18

9.04

0.00

0.00

0.00

0.00

0.00

22.41

48.58

14.53

14.47

0.00

0.00

0.00

1.18

1.66

5.18

59.51

21.75

10.72

First line European companies, second line US companies
Source: Moody’s 2002 (Default and Recovery Rates of European Corporate bond Issuers, 1985-2001)

Credit risk events


ISDA (1999):
 Bankruptcy
 Rating

downgrade
 Merger/acquisition
 Restructuring
 Accelerating of obligation
 Bankruptcy of a related entity
 Default on coupon/interest
 Debt repudiation

Downgrade risk
The risk that a credit rating agency will
lower a bond's rating
 The resulting increase in the yield required
by investors will lead to a decrease in the
price of the bond
 A rating increase – upgrade - will have the
opposite effect, decreasing the required
yield and increasing the price


Credit spread
The difference between the yield on a
Treasury security, which is assumed to be
default risk-free, and the yield on a similar
maturity bond with a lower rating
 Yield on a risky bond =
Yield in a default free bond + credit spread


Credit spread risk
Refers to the fact that the default risk
premium required in the market for a given
rating can increase, even while the yield
on Treasury securities of similar maturity
remains unchanged
 An increase in this credit spread increases
the required yield and decreases the price
of a bond


Corporate (credit) bond spreads
over Treasuries




Bond yields (spreads
over equivalent
Treasuries) increase
as credit ratings
decline
Spreads widen as
maturity increases

Corporate (Industrials) Spreads over Treasuries (in basis points)
Rating
1 yr
2 yr
3 yr
5 yr
7 yr
10 yr
Aaa/AAA
35
40
45
55
69
81
Aa1/AA+
40
45
55
65
79
91
Aa2/AA
45
55
60
70
85
101
Aa3/AA50
60
65
80
95
111
A1/A+
60
70
85
100
116
132
A2/A
70
80
100
115
136
155
A3/A80
95
110
125
152
170
Baa1/BBB+
100
115
130
145
167
190
Baa2/BBB
120
135
150
165
183
200
Baa3/BBB140
150
160
175
195
215
Ba1/BB+
275
300
325
350
375
425
Ba2/BB
300
325
350
400
450
525
Ba3/BB350
400
425
475
525
575
B1/B+
450
475
500
575
650
700
B2/B
525
575
625
700
750
825
B3/B600
650
750
850
975
1075
Caa/CCC
850
900
1050
1150
1250
1400
Source: http://www.bondsonline.com/asp/corp/spreadind.html on 20 Nov 2001

30 yr
92
102
112
123
148
171
193
208
228
248
475
600
750
825
975
1200
1600

Credit Default Swap - CDS





Captures the credit risk of an issuer
Bilateral agreement in which periodical fixed
payments are made to protection seller in exchange
of a single payment the protection buyer will make in
case if a credit event (specified in the CDS contract
occurs).
Flows – payoff:


Periodical payments (premium leg): basis points
applied to the notional of the CDS agreement
 Single payment if credit event occurs (protection leg):
par value of the bond × [100 – bond’s market price
after the credit event occurred]

CDS – evolution
USD bill.
45000
40000
35000
30000
25000
20000
15000
10000
5000
0
2001 S1

2002 S1

2003 S1

2004 S1

2005 S2

Source: International Swaps and Derivatives Association, Bank of International Settlements

2006 S2

CDS – utilization
Credit risk hedging (credit risk transfer)
 Taking exposure on credit risk
 Structured products (credit linked note)
 Informational content


CDS - Valuation




Market value – determined by supply and demand of
such instruments. Cannot be available for some
issuers.
Theoretical value – based on:






Probability of default of the issuer (implied by the credit rating)
Recovery rate (1 – Loss Given Default)
Coupon/interest rate of the bond
Maturity
Market interest rates

CDS - Romania

Source: Bloomberg

CDS term structure Romania

Source: Bloomberg

CDS informational content

Source: Bloomberg

Debt crisis 2010

Source: Bloomberg

Liquidity risk
Risk that the sale of a fixed-income
security must be made at a price less than
fair market value because of a lack of
liquidity for a particular issue
 Since investors prefer more liquidity to
less, a decrease in a security's liquidity will
decrease its price, as the required yield
will be higher


Bid-ask spread
The difference between the price that
dealers are willing to pay for a security
(the bid) and the price at which dealers are
willing to sell a security (the ask)
 If trading activity in a particular security
declines, the bid-ask spread will widen
(increase), and the issue is considered to
be less liquid


Exchange-rate risk


Arises from the uncertainty about the
value of foreign currency cash flows for an
investor in terms of his home-country
currency

Inflation risk
Unexpected inflation risk or purchasingpower risk
 Uncertainty about the amount of goods
and services that a security's cash flows
will purchase


Volatility risk
Is present for fixed-income securities that
have embedded options, such as call
options, prepayment options, or put
options.
 Changes in interest rate volatility affect the
value of these options and, thus, affect the
values of securities with embedded
options


Volatility risk
Value of a callable bond =
Value of an option-free bond – value of a call
 Value of a putable bond =
Value of an option-free bond + value of a put




Volatility risk for callable bonds is the risk
that volatility will increase, and volatility risk
for putable bonds is the risk that volatility
will decrease

Event risk


The risks outside the risks of financial
markets, such as the risks posed by
natural disasters, regulatory changes and
corporate restructurings

Sovereign risk
The credit risk of a sovereign bond issued
by a country other than the investor's
home country
 Law under which the bond is issued


Exercises

Exercise
A bond with a 7.3% yield has a duration of
5.4 and is trading at $985. If the yield
decreases to 7.1%, the new bond price is
closest to:
A. $974.40
B. $995.60
C. $1, 091.40


Exercise
The current price of a bond is 102.50. If
interest rates change by 0.5%, the value of
the bond price changes by 2.50. What is
the duration of the bond?
A. 2.44.
B. 2.50.
C. 4.88.


Question
Which of the following bonds has the
greatest interest rate risk?
A. 5% 1 0-year callable bond
B. 5% 1 0-year putable bond
C. 5% 1 0-year option-free bond


Question
A floating-rate security will have the
greatest duration:
A. the day before the reset date.
B. the day after the reset date.
C. Never - floating-rate securities have a
duration of zero


Exercise


A straight 5% bond has two years remaining to

maturity and is priced at $981.67. A callable
bond that is the same in every respect as the
straight bond, except for the call feature, is
priced at $917.60. With the yield curve flat at
6%, what is the value of the embedded call
option?
A.
$45.80
B.
$64.07
C.
$101.00

Question
Which of the following statements about
the risks of bond investing is most
accurate?
A. A bond rated AAA has no credit risk
B. A bond with call protection has volatility
risk
C. A U.S. Treasury bond has no
reinvestment risk


Bond sectors and
instruments

Sovereign bonds








Bonds issued by a country’s central government
Largest market – sovereign debt of the US
Government which consists of US Treasury and
considered to be essentially free of default risk
Sovereign debt of other countries is considered
to have varying degrees of credit risk
Can be issued on own domestic market, another
country’s foreign bond market or in the
Eurobond market
Issued in own currency but also in other
currencies

U.S. Treasury securities
Bills – matures in one year or less, issued
at a discount
 Notes – matures between 2-10 years,
issued as a coupon security
 Bonds –maturities longer than 10 years
 Treasury inflation protection securities
(TIPS) – principal is indexed to CPI with
real rate being fixed


T-bills
Maturities of less than 1Y (29, 91 and 182
days)
 Do not make explicit interest payments,
paying only the face value at the maturity
date
 Issued at discount


Bonds quotation
Treasury bond and note prices in the
secondary market are quoted in percent
and 32nds of 1% of face value.
 A quote of 102-5 (sometimes 102:5) is
102% plus 5/32% of par, which for a
$100,000 face value T-bond, translates to
a price of:


Bond quotations
Bonds can be quoted also in yield in
format BID – ASK
 Example: 5.00 – 4.50


TIPS







Maturities: 5, 10 and 20Y
Make semi-annual coupon payments at a rate
fixed at issuance
The par value begins at USD 1000 and is
adjusted semi-annually for changes in CPI
The fixed coupon rate is paid semiannually as a
percentage of the inflation adjusted par value
Any increase in the par value taxed as income

TIPS - example






For example, consider a $100,000 par value
TIPS with a 3% coupon rate, set at issuance. Six
months later annual rate of inflation measured
by CPI is 4%. The par value will be increased by
one half of 4% and will be 1.02 x 100,000 =
$102,000.
The first semi-annual coupon will be one half of
3% coupon rate times the inflation adjusted par
value: 1.5% x 102,000 = 1,530
Any percentage change in the CPI over the next
6M period will used to adjust the par value from
102,000

Stripped Treasury Securities


Several major brokerages have created an
investment vehicle from Treasury
securities. They purchase these
securities, deposit them in a bank custody
account and then separate out each
coupon payment and principal. Then a
receipt is issued to investors representing
an ownership in the account. In essence,
the security is stripped.

STRIPS
STRIPS – U.S. Treasury program issues these
direct obligations of the U.S. government, ending
trademark and generic receipts
Treasury strips - zero-coupons or stripped
Treasury securities:
• Treasury coupon strips – created from the future
coupon
• Treasury principal strips - created from the
principal payment at maturity

Agency bonds





Debt securities issued by various agencies and
organizations of the US Government as
Federally related institutions as Government
National Mortgage Association (Ginnie Mae)
which are owned by US Government
Government sponsored enterprises, as Federal
National Mortgage Association (Fannie Mae),
Federal Home Loan Bank Corporation (Freddie
Mac), which are created by the US Congress,
but privately owned. They issue debentures –
securities not backed by collateral (unsecured)

Mortgage-backed securities - MBS
Backed (secured) by pools of mortgage
loans, which not only provide collateral but
also the cash flows to service the debt.
 Security where the collateral for the issued
security is a pool of mortgages.
 The Government National Mortgage
Association (GNMA), the Federal National
Mortgage Association (FNMA), and the
Federal Home Loan Mortgage Corporation
(FHLMC) all issue mortgage-backed
securities.


Securitisation
Process of combining many similar debt
obligations as the collateral for issuing
securities
 Primary reason for mortgage securitization
is to increase the debt's attractiveness to
investors and to decrease investor
required rates of return, increasing the
availability of funds for home mortgages


Cash flows from mortgages
Periodic interest,
 Scheduled repayments of principal
 Principal repayments in excess of
scheduled principal payments


Prepayment risk
Because the borrower can accelerate
principal repayment, the owner of a
mortgage has prepayment risk.
 Prepayment risk is similar to call risk
except that prepayments may be part of or
all of the outstanding principal amount.
 This, in turn, subjects the mortgage holder
to reinvestment risk, as principal may be
repaid when yields for reinvestment are
low


Securitization types
Mortgage pass-through security
 Collateralized mortgage obligations
(CMOs)
 Stripped mortgage-backed securities


Mortgage passthrough security
Passes the payments made on a pool of
mortgages through proportionally to each
security holder
 A holder of a mortgage passthrough
security that owns a 1% portion of the
issue will receive a 1 % share of all the
monthly cash flows from all the mortgages,
after a percentage fee for administration is
deducted


Mortgage passthrough security




Each monthly payment consists of interest,
scheduled principal payments, and prepayments
of principal in excess of the scheduled amount –
therefore prepayment risk
Since prepayments tend to accelerate when
interest rates fall, due to the refinancing and
early payoff of existing mortgage loans, security
holders can expect to receive greater principal
payments when mortgage rates have decreased
since the mortgages in the pool were issued.

CMOs
Created from mortgage passthrough
certificates and referred to as derivative
mortgage-backed securities
 A CMO issue has different tranches, each
of which has a different type of claim to the
cash flows from the pool of mortgages


Sequential CMO - example






Tranche I (the short-term segment of the issue) receives
net interest on outstanding principal and all of the
principal payments from the mortgage pool until it is
completely paid off.
Tranche II ( the intermediate-term) receives its share of
net interest and starts receiving all of t he principal
payments after Tranche I has been completely paid off.
Prior to that, it only receives interest payments.
Tranche III (the long-term) receives monthly net interest
and starts receiving all principal repayments after
Tranches I and II have been completely paid off. Prior of
that, it only receives interest payments.

Stripped mortgage-backed securities





Are either the principal or interest portions of a
mortgage passthrough security
The holder of a principal-only strip will gain from
prepayments because the face value of the
security is received sooner rather than later.
The holder of an interest-only strip will receive
less total payments when prepayment rates are
higher since interest is only paid on the
outstanding principal amount, which is
decreased by prepayments.

Municipal bonds
Debt securities issued by state and local
governments in the United States are
known as municipal bonds (or munis for
short)
 Municipal bonds are often referred to as
tax-exempt or fax-free bonds, since the
coupon interest is exempt from federal
income taxes.
 While interest income may be tax free,
realized capital gains are not.


Secured debt





Backed by the pledge of assets/collateral, which
can take the following forms:
Personal property (e.g., machinery, vehicles,
patents)
Real property (e.g., land and buildings)
Financial assets (e.g., stocks, bonds, notes).
These assets are marked to market from time to
time to monitor their liquidation values.
Covenants may require a pledge of more assets
if values are insufficient. Bonds backed by
financial assets are called collateral trust bonds.

Unsecured debt
Is not backed by any pledge of specific
collateral
 Unsecured bonds are referred to as
debentures
 They represent a general claim on any
assets of the issuer that have not been
pledged to secure other debt


Credit enhancements
The guarantees of others that the
corporate debt obligation will be paid in a
timely manner:
 Third-party guarantees that the debt
obligations will be met. Often, parent
companies guarantee the loans of their
affiliates and subsidiaries.
 Letters of credit are issued by banks and
guarantee that the bank will advance the
funds for service the corporation's debt.


Medium term notes - MTNs
Once registered, such securities can be
"placed on the shelf" and sold in the
market over time and at the discretion of
the issuer.
 MTNs are sold over time, with each sale
satisfying some minimum dollar amount
set by the issuer, typically $1 million and
up.


MTNs
Are issued in various maturities
 Can have fixed or floating-rate coupons
 Can be denominated i n any currency
 Can have special features, such as calls,
caps, floors, and non-interest rate indexed
coupons
 The notes issued can be combined with
derivative instruments to create the special
features that an investor requires


Structured notes
A debt security created when the issuer
combines a typical bond or note with a
derivative
 Example: an issuer could create a
structured note where the periodic coupon
payments were based on the performance
of an equity security or an equity index by
combining a debt instrument with an equity
swap


Types of structured MTNs








Step-up notes - Coupon rate increases over time on a preset
schedule
Inverse floaters - Coupon rate increases when the reference rate
decreases and decreases when the reference rate increases
Deleveraged floaters- Coupon rate equals a fraction of the reference
rate plus a constant margin
Dual-indexed floaters - Coupon rate is based on the difference
between two reference rates.
Range notes - Coupon rate equals the reference rate if the
reference rate falls within a specified range, or zero if the reference
rate falls outside that range.
Index amortizing notes - Coupon rate is fixed but some principal is
repaid before maturity, with the amount of principal prepaid based
on the level of the reference rate.

Commercial paper
Short-term, unsecured debt instrument
used by corporations to borrow money at
rates lower than bank rates
 Is typically issued as a pure discount
security and makes a single payment
equal to the face value at maturity


Certificates of deposit - CDs
Are issued by banks and sold to their
customers
 Negotiable CDs, permit the owner to sell
the CD in the secondary market at any
time
 Negotiable CDs have maturities ranging
from days up to 5Y. The interest rate paid
on them is called the London Interbank
Offering Rate because they are primarily
issued by banks' London branches.


Asset–backed securities - ABSs




Securitization of credit card debt, auto loans,
bank loans, and corporate receivables
The assets are transferred to a special purpose
entity – SPV for bankruptcy protection
External credit enhancements to increase the
rating
 Corporate

guarantees, which may be provided by the
corporation creating the ABS or its parent
 Letters of credit, which may be obtained from a bank
for a fee

Collateralized debt obligation (CDO)






Is a debt instrument where the collateral for the promise
to pay is an underlying pool of other debt obligations and
even other CDOs
Underlying debt obligations can be business loans,
mortgages, debt of developing countries, corporate
bonds of various ratings, asset-backed securities, or
even problem/ non-performing loan
Tranches of the CDO are created based on the seniority
of the claims to the cash flows of the underlying assets,
and these are given separate credit ratings depending
on the seniority of the claim, as well as the
creditworthiness of the underlying pool of debt securities.

Exercises

Exercise
A Treasury security is quoted at 97-17 and
has a par value of $ 100,000. Which of the
following is its quoted dollar price?
A. $97,170.00.
B. $97,531.25.
C. $100,000.00


Exercise






An investor holds $100,000 (par value) worth of
Treasury Inflation Protected Securities (TIPS)
that carry a 2.5% semiannual pay coupon. If t he
annual inflation rate is 3%, what is the inflationadjusted principal value of the bond after six
months?
A. $101,500.
B. $102,500.
C. $103,000.

Question
A Treasury note (T-note) principal strip
has six months remaining to maturity. How
is its price likely to compare to a 6-month
Treasury bill (T-bill) that has just been
issued? The T-note price should be:
A. lower
B. higher
C. the same


Yield curves

Yield curves
A plot of yields by years to maturity
 Shapes of yield curves:




Normal or upward sloping
 Inverted or downward sloping.
 Flat
 Humped

Yield curve shapes

RON yield curve 2008

Source: Bloomberg

RON yield curve 2012 - 2013

Source: Bloomberg

EUR and RON yield curves

Source: Bloomberg

Term structure theories
Pure expectations theory
 Liquidity preference theory
 Market segmentation theory


Pure expectation theory






Yield for a particular maturity is an average (not
a simple average) of the short-term rates that
are expected in the future
If short-term rates are expected to rise in the
future, interest rate yields on longer maturities
will be higher than those on shorter maturities,
and the yield curve will be upward sloping
If short-term rates are expected to fall over time,
longer maturity bonds will be offered at lower
yields

Liquidity preference theory





In addition to expectations about future shortterm rates, investors require a risk premium for
holding longer term bonds
This is consistent with the fact that interest rate
risk is greater for longer maturity bonds
The size of the liquidity premium will depend on
how much additional compensation investors
require to induce them to take on the greater risk
of longer maturity bonds or, alternatively, how
strong their preference for the greater liquidity of
shorter term debt is

Pure expectations vs. liquidity preference

Market segmentation theory
Is based on the idea that investors and
borrowers have preferences tor different
maturity ranges.
 Under this theory, the supply of bonds
(desire to borrow) and the demand for
bonds (desire to lend) determine
equilibrium yields for the various maturity
ranges


Market segmentation

Types of curves
Zero (spot)
 Yield to maturity (YTM)
 Par
 Forward


Zero Rates
A zero rate (or spot rate), for maturity T is the
rate of interest earned on an investment that
provides a payoff only at time T

Example
Maturity
(years)
0.5

Zero Rate
(% cont comp)
5.0

1.0

5.8

1.5

6.4

2.0

6.8

Bond Pricing




To calculate the cash price of a bond we
discount each cash flow at the appropriate zero
rate
In our example, the theoretical price of a twoyear bond providing a 6% coupon semiannually
is
−0.05× 0.5
−0.058 ×1.0
−0.064 ×1.5

3e

+ 3e

+ 3e

+ 103e − 0.068× 2.0 = 98.39

Bond Yield






The bond yield is the discount rate that
makes the present value of the cash flows
on the bond equal to the market price of
the bond
Suppose that the market price of the bond
in our example equals its theoretical price
of 98.39
The bond yield (continuously
compounded) is given by solving

3e − y × 0.5 + 3e − y ×1.0 + 3e − y ×1.5 + 103e − y × 2.0 = 98.39

to get y=0.0676 or 6.76%.

Par Yield




The par yield for a certain maturity is the
coupon rate that causes the bond price
to equal its face value.
In our example we solve
c −0.05×0.5 c −0.058×1.0 c −0.064×1.5
e
+ e
+ e
2
2
2
c  −0.068×2.0

= 100
+ 100 + e
2

to get c=6.87 (with s.a. compoundin g)

Sample Data
Bond

Time to

Annual

Bond Cash

Principal

Maturity

Coupon

Price

(dollars)

(years)

(dollars)

(dollars)

100

0.25

0

97.5

100

0.50

0

94.9

100

1.00

0

90.0

100

1.50

8

96.0

100

2.00

12

101.6

The Bootstrap Method





An amount 2.5 can be earned on 97.5 during 3
months.
The 3-month rate is 4 times 2.5/97.5 or 10.256%
with quarterly compounding
This is 10.127% with continuous compounding
Similarly the 6 month and 1 year rates are
10.469% and 10.536% with continuous
compounding

The Bootstrap Method continued


To calculate the 1.5 year rate we solve

4e −0.10469×0.5 + 4e −0.10536×1.0 + 104e − R×1.5 = 96
to get R = 0.10681 or 10.681%


Similarly the two-year rate is 10.808%

Zero Curve Calculated from the Data
12

Zero
Rate (%)
11

10.681
10.469
10

10.808

10.536

10.127

Maturity (yrs)
9
0

0.5

1

1.5

2

2.5

Forward Rates
The forward rate is the future zero rate
implied by today’s term structure of
interest rates

Calculation of Forward Rates

Year (n )

n-year

Forward Rate

zero rate

for n th Year

(% per annum)

(% per annum)

1

3.0

2

4.0

5.0

3

4.6

5.8

4

5.0

6.2

5

5.3

6.5

Formula for Forward Rates




Suppose that the zero rates for time periods T1
and T2 are R1 and R2 with both rates continuously
compounded.
The forward rate for the period between times T1
and T2 is

R2 T2 − R1 T1
T2 − T1

Embedded options vs. yield
Investors will require a higher yield on a
callable bond, compared to the same bond
without the call feature
 The inclusion of a put provision or a
conversion option with a bond will have
the opposite effect


Question
Under the pure expectations theory, an
inverted yield curve is interpreted as
evidence that:
a. demand for long-term bonds is falling
b. short-term rates are expected to fall in
the future
c. investors have very little demand for
liquidity


Question







With respect to the term structure of interest
rates, the market segmentation theory holds
that:
a. An increase in demand for long-term
borrowings could lead to an inverted yield curve
b. Expectations about the future of short-term
interest rates are the major determinants of the
shape of the yield curve
c. the yield curve reflects the maturity demands
of financial institutions and investors

Bond valuation

Bond valuation
The intrinsic value of a bond, like stocks, is
the present value of its future cash flows.
 Bonds, however, have much more
predictable cash flows and a finite life.
 The cash flows promised by a bond are:


A

series of (usually) constant interest
payments
 The return of the face value of the bond at
maturity

Bond valuation


The value of a bond is determined by four variables:


The Coupon Rate – This is the promised annual rate of interest. It is
normally fixed at issuance for the life of the bond. To determine the
annual interest payment, multiply the coupon rate by the face value of
the bond. Interest is normally paid semiannually, and the semiannual
payment is one-half the annual total payment.
 The Face Value – This is nominally the amount of the loan to the
issuer. It is to be paid back at maturity.
 Term to Maturity – This is the remaining life of the bond, and is
determined by today’s date and the maturity date. Do not confuse this
with the “original” maturity which was the life of the bond at issuance.
 Yield to Maturity – This is the rate of return that will be earned on the
bond if it is purchased at the current market price, held to maturity, and
if all of the remaining coupons are reinvested at this same rate. This is
the IRR of the bond.

Premium, par and discount

Price vs. yield

Yield to maturity
A summary measure and is essentially an
internal rate of return based on a bond's
cash flows and its market price
 Assumes that all cash flows are reinvested
at the YTM


Reinvestment risk


A coupon bond's reinvestment risk will
increase with:
 Higher

coupons - because there's more cash
flow to reinvest
 Longer maturities - because more of the total
value of the investment is in the coupon cash
flows (and interest on coupon cash flows)

Arbitrage-free valuation
Discount each cash flow using a discount
rate that is specific to the maturity of each
cash flow.
 These discount rates are the spot rates
and can be thought of as the required
rates of return on zero-coupon bonds
maturing at various times in the future
 If the market value of the bond different
than the arbitrage-free valuation –
arbitrage opportunity


Arbitrage opportunities
If the bond is selling for more than the sum
of the values of the pieces (individual cash
flows), one could buy the pieces, package
them to make a bond, and then sell the
bond package to earn an arbitrage profit
 If the bond is selling for less than the sum:
buy the bond and sell the pieces


Arbitrage example
Consider a 6% Treasury note with 1.5
years to maturity.
 Spot rates (expressed as yields to
maturity) are: 6 months = 5%, 1 year =
6%, and 1.5 y ears = 7%.
 If the note is selling for $992, compute the
arbitrage profit, and explain how a dealer
would perform the arbitrage


Yield to call
Used to calculate the yield on callable
bonds that are selling at a premium to par
 For bonds trading at a premium to par,
the yield to call may be less than the yield
to maturity
 This can be the case when the call price is
below the current market price


Yield to call - calculation






Similar as the calculation of yield to maturity,
except that the call price is substituted for the
par value in FV and the number of semiannual
periods until the call date is substituted for
periods to maturity, N
When a bond has a period of call protection, we
calculate the yield to first call over the period
until the bond may first be called, and use the
first call price in the calculation as FV
In a similar manner, we can calculate the yield to
any subsequent call date using the appropriate
call price

Yield to call - example
Consider a 20-year, 10% semiannual-pay
bond with a full price of 112 that can be
called in five years at 102 and called at par
in seven years.
 Calculate the YTM, YTC, and yield to first
par call


Yield to call - example
YTM: N = 40; PV = -112; PMT = 5; FV =
100
 Yield to first call: N = 10; PV = -112; PMT
= 5; FV = 102
 Yield to first par call: N = 14; PV = -112;
PMT = 5; FV = 100


Yield to worst


Is the worst yield outcome of any that are
possible given the call provisions of the
bond

Yield to put
Is used if a bond has a put feature and is
selling at a discount
 The yield to put will likely be higher than
the yield to maturity.
 The yield to put calculation is just like the
yield to maturity with the number of
semiannual periods until the put date as N,
and the put price as FV


Yield to put - example
Consider a 3-year, 6%, $1,000
semiannual-pay bond
 The bond is selling for a full price of
$925.40
 The first put opportunity is at par in two
years.
 Calculate the YTM and the YTP


Yield to put - example
YTM: N = 6; PV = -925; PMT = 30; FV =
1000
 Yield to first call: N = 4; PV = -925; PMT =
30; FV = 1000


Spread measures
Nominal spread
 Zero-volatility spread: Z-spread
 Option adjusted spread: OAS


Nominal spread


It is an issue's YTM minus the YTM of a
Treasury security of similar maturity

Zero volatility spread


It is the equal amount that we must add to
each rate on the Treasury spot yield curve
in order to make the present value of the
risky bond's cash flows equal to its market
price

Option adjusted spread
The measure is used when a bond has
embedded options
 The option-adjusted spread takes the
option yield component out of the Zspread measure
 The option-adjusted spread is the spread
to the Treasury spot rate curve that the
bond would have if it were option-free
 Z-spread - OAS = option cost in percent


OAS


For embedded short calls (e.g., callable bonds): option
cost > 0 (you receive compensation for writing the option
to the issuer) - OAS < Z-spread. In other words, you
require more yield on the callable bond than for the
option-free bond.



For embedded puts (e.g., putable bonds), option cost < 0
(i.e., you must pay for the option) - OAS > Z-spread. In
other words, you require less yield on the putable bond
than for an option-free bond

Interest rate risk

Price vs. yield
option free, 8%, 20Y

Callable bond - negative convexity

Putable bond

Duration






Is the slope of the price-yield curve at the bond's
current YTM. Mathematically, the slope of the
price-yield curve is the first derivative of the
price-yield curve with respect to yield.
Is a weighted average of the time (in years) until
each cash flow will be received. The weights are
the proportions of the total bond value that each
cash flow represents.
Is the approximate percentage change in price
for a 1% change in yield. This interpretation,
price sensitivity in response to a change in yield,
is the preferred, and most intuitive, interpretation
of duration.

Duration measures
Effective duration
 Macaulay duration
 Modified duration


Effective duration


The ratio of the percentage change in
price to change in yield

Effective duration

Effective duration - example
Consider a 20-year, semiannual-pay bond
with an 8% coupon that is currently priced
at $908.00 to yield 9%.
 If the yield declines by 50 basis points (to
8.5%), the price will increase to $952.30,
and if the yield increases by 50 basis
points (to 9.5%), the price will decline to
$866.80.
 Based on these price and yield changes,
calculate the effective duration of this bond


Macaulay duration


Macaulay duration is an estimate of a
bond's interest rate sensitivity based on
the time, in years, until promised cash
flows will arrive

Macaulay duration - examples
A 5-year zero-coupon bond has only one
cash flow five years from today, its
Macaulay duration i s five. The change in
value in response to a 1% change in yield
for a 5-year zero-coupon bond is
approximately 5%.
 A 5-year coupon bond has some cash
flows that arrive earlier than five years
from today (the coupons), so its Macaulay
duration is less than five.


Macaulay duration


Weighted average term to maturity




Measure of average maturity of the bond’s promised cash flows

Duration formula:
T

where:

Dm = ∑ (t × wt )
t =1

PV (CFt )
CFt /(1 + y ) t
wt =
=
P
PV ( Bond )
q



t is measured in years

∑w
t =1

t

=1

Macaulay duration
T

Dm = ∑ t × wt
t =1

 PV(Ct ) 
= ∑t × 
 PV(Bond) 
t =1
T

 CN 
 C2 
 C1 
+
...
+
N
+
2
1
1
 (1+ y)N 
 (1+ y)2 
 (1 + y) 
=
C1
CN
C2
+
...
+
+
2
1
(1+ y)N
(1+ y) (1 + y)

Modified duration
Is derived from Macaulay duration and
offers an improvement over Macaulay
duration in that it takes the current YTM
into account
 For option-free bonds, effective duration
(based on small changes in YTM) and
modified duration will be very similar


Modified duration (D*m)
Dm
D =
1+ y
*
m

 Direct

measure of price sensitivity to interest
rate changes
 Can be used to estimate percentage price
volatility of a bond
∆P
*
= − Dm × ∆y
P

Derivation of modified duration
N

Ct
t
t =1 (1+ y)

P =∑

∂P
−1 N 
Ct 
=
t⋅

∂y 1 + y t =1  (1 + y) t 
∂P − Dm
*
=
⋅ P = − Dm ⋅ P
∂y 1 + y
1 ∂P
= − Dm*
P ∂y



So D*m measures the sensitivity of the %
change in bond price to changes in yield
Dm* =

Dm
1+ y

Example


Consider a 3-year 10% coupon bond selling at $107.87 to yield
7%. Coupon payments are made annually.
10
= 9.35
(1.07)
10
= 8.73
PV (CF2 ) =
2
(1.07)
110
= 89.79
PV (CF3 ) =
3
(1.07)
Price of bond = 9.35 + 8.73 + 89.79 = 107.87
PV (CF1 ) =

9.35  
8 .73  
89.79 

Duration ( Dm ) = 1*

 +  3*
 + 2*
 107.87   107.87   107.87 
= 2.7458

Example


Modified duration of this bond:

Dm* =



2.7458
= 2.5661
1.07

If yields increase to 7.10%, how does the bond price change?
The percentage price change of this bond is given by:

∆P
*
× 100 = −Dm × ∆y × 100
P
= –2.5661 × .0010 × 100
= –.2566

Example


What is the predicted change in dollar
terms?
.2566
×P
100
.2566
× $107.87
=−
100
= −$.2768

∆P = −

New predicted price: $107.87 – .2768 = $107.5932
Actual dollar price (using PV equation): $107.5966

Effective vs. modified duration
Modified duration is calculated without any
adjustment to a bond's cash flows for
embedded options.
 Effective duration is appropriate for bonds
with embedded options because the inputs
(prices) were calculated under the
assumption that the cash flows could vary
at different yields because of the
embedded options in the securities


Duration of a portfolio


Duration of a portfolio is the weighted
average of the durations of the individual
securities in the portfolio

Convexity
Is a measure of the curvature of the priceyield curve
 The more curved the price-yield relation is,
the greater the convexity


Price vs. yield - 8%, 20Y

Using duration and convexity


By combining effective duration and
convexity, we can obtain a more accurate
estimate of the percentage change in price
of a bond, especially for relatively large
changes in yield

Example - convexity
Consider an 8% Treasury bond with a
current price of $908 and a YTM of 9%.
 Calculate the percentage change in price
of both a 1% increase and a 1% decrease
in YTM based on a duration of 9.42 and a
convexity of 6 8.33.


Example - convexity


The duration effect, is 9.42 x 0.01 = 0.0942 = 9.42%.
The convexity effect is 68.33 x 0.012 x 100 = 0.00683 x
100 = 0.683%.



The total effect for a decrease in yield of 1% (from 9% to
8%) is 9.42% + 0.683% = + 10.103%, and the estimate
of the new price of the bond is 1.10103 x 908 = 999.74.



The total effect for an increase in yield of 1% (from 9% to
10 %) is -9.42% + 0.683% = -8.737%, and the estimate
of the bond price is (1 - 0.08737)(908) = $828.67.

Convexity - determination


Second derivative of price with respect to yield divided by bond
price
N

 CFt
∂ 2P
1
2
=
⋅(t
+
t)

 (1+ y)t
∂2 y (1 + y)2 ∑
t =1

1 ∂2 P
Convexity =
P ∂2 y

Predicted percentage price change


Recall approximation using only duration:
∆P
*
× 100 = −Dm × ∆y × 100
P



The predicted percentage price change
accounting for convexity is:
∆P
1
2
*
× 100 = (− Dm × ∆y × 100 ) +  × Convexity × (∆y) × 100
2
P

Example with convexity


Consider a 20-year 9% coupon bond
selling at $134.6722 to yield 6%.
Coupon payments are made
semiannually.



Dm= 10.98

10.98
D =
= 10.66
1 + (0.06 / 2)
*
m



The convexity of the bond is 164.106.

Example with convexity


If yields increase instantaneously from 6% to 8%,
the percentage price change of this bond is given
by:


First approximation (Duration):

–10.66 × .02 × 100 = –21.32


Second approximation (Convexity)

0.5 × 164.106 × (.02)2 × 100 = +3.28
Total predicted % price change: –21.32 + 3.28
(Actual price change = –18.40%.)

= –18.04%

Example with convexity



What if yields fall by 2%?
If yields decrease instantaneously from 6% to 4%,
the percentage price change of this bond is given by:


First approximation (Duration):

–10.66 × –.02 × 100 = 21.32


Second approximation (Convexity)

0.5 × 164.106 × (–.02)2 × 100 = +3.28
Total predicted price change: 21.32 + 3.28 = 24.60%
Note that predicted change is NOT SYMMETRIC.

Effective vs. modified convexity






Effective convexity takes into account changes
in cash flows due to embedded options, while
modified convexity does not
The difference between modified convexity and
effective convexity mirrors the difference
between modified duration and effective duration
Effective convexity is the appropriate measure to
use for bonds with embedded options, since it is
based on bond values that incorporate the effect
of embedded options on the bond's cash flows

Price value of a basis point
Is the dollar change in the price/value of a
bond or a portfolio when the yield changes
by one basis point, or 0.01%
 PVBP can be calculated directly for a bond
by changing the YTM by one basis point
and computing the change in value
 As a practical matter, duration can be
used to calculate the price value of a basis
point
 PVBP = duration x 0.0001 x bond value


PVBP - example








A bond has a market value of $100,000 a and a duration
of9.42. What is t he price value of a basis point?
Using t he duration formula, the percentage change in
the bond's price for a change in yield of 0.01% is 0.01%
x 9.42 = 0.0942%.
We can calculate 0.0942% of the original $100,000
portfolio value as 0.000942 x 100,000 = $94.20.
If the bond's yield increases (decreases) by one basis
point, the portfolio value will fall (rise) by $94.20. $94.20
is the (duration-based) price value of a basis point for
this bond.
We can ignore the convexity adjustment here because it
is of very small magnitude

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