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
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
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]
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
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
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.
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
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 ) =
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:
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