Chapter 6
Bonds
Chapter Outline
1. Why Issue Debts?
2. Bond Basics
3. Interest Rate
a)
Yield Curve
4. Bond Valuation
a) Zero-Coupon Bonds
b) Coupon Bonds
5. Why Bond Price Changes
a) Premium, Discount, Par
b) Capital gain / loss yield, Current yield, Yield to Maturity
c) Interest Rate Sensitivity
6. Bond Credit Risk
a) Credit Spread and Credit Rating
2
WHY ISSUE DEBT?
3
Investing in Bonds
Advantage of Debt over Equity
• Interest expense is tax-deductible but
dividend is not.
• Avoid earning/ownership dilution
• Avoid a high flotation cost for issuing
stock.
– Flotation cost = Underwriting fee, Fee to
investment banker
5
Example of Tax Saving with Debt
• Income Statement
Revenue
−COGS
Profit Margin
− Op. Cost
Firm w/o Debt
EBIT
$5
−Int. Exp
0
EBT
5
−Tax (30%)
1.5
Net Income
3.5
Firm w/ Debt
$5
1
4
1.2
2.8
• 30 cents savings for every dollar of interest
expense
6
Disadvantage of Debt
• A high level of debt increases credit /
default risk.
– Interest payments can be a burden, while
dividend payments can be skipped or reduced.
• A high level of debt may make it difficult to
obtain additional funding.
• Debt covenants can be a burden.
7
BOND BASICS
8
Types of Bonds
• Domestically,
– Treasury bill, note, or bond: Issued by federal
government, Called “risk-free” securities, about $4
trillion market
– Municipal bond: “munis”
– Corporate bond: about $5 trillion market
• Internationally,
– Euro bond (Dollar-denominated bonds sold in
Germany by GM)
– Foreign bond: “Yankee” bond (dollar-denominated
bond sold in U.S. by non-U.S. issuer), “Samurai”
bond (Yen bonds sold in Japan by a non-Japanese
borrower), etc
9
Types of Bonds
• Callable bond: The seller has an option to buy
back their bonds from bond investors.
• Convertible bond: The seller grant bondholders
the right to exchange each bond for a designated
number of common stock shares of the issuing
firm.
• Zero-coupon bonds: “zeros” or “deep discount”
bonds
• Floating-rate bonds: The coupon payments are
adjustable.
• Inflation-indexed bonds: Protecting against
inflation, Fairly new.
10
AMD Bonds
11
Bond Pricing: Cash Flow
AMD
(Issuer, Seller, or
Borrower)
Coupons at
t=1,2, …. T
Investor
(Buyer, Lender)
Face Value at T
Price?
Main Question: At how much would a buyer be willing to pay
for this bond?
12
Elements of Bond Pricing
1. Par value (par): Face amount. paid at maturity. Assume
$1,000.
2. Coupon interest rate: Stated interest rate on the bond
certificate. Multiply by par value to get dollars of interest
to be paid. Generally fixed.
3. Maturity: Years until bond must be repaid. Declines over
time.
4. Yield-to-Maturity (YTM): The current market interest rate
that is used to discount the future coupon payments and
face amount. Or, the required rate of return to be earned
from other bonds with same level of risk.
13
INTEREST RATES REVIEW
14
U.S. Interest Rates and Inflation Rates,
1955–2009
15
The Determinants of Interest Rates
• The Yield Curve and Discount Rates
– Term Structure
• The relationship between the investment term and
the interest rate
– Yield Curve
• A plot of bond yields as a function of the bonds’
maturity date
– Risk-Free Interest Rate
• The interest rate at which money can be borrowed or
lent without risk over a given period.
16
Term Structure of Risk-Free U.S.
Interest Rates, November 2006,
2007, and 2008
17
The Determinants of Interest Rates
• The Yield Curve and Discount Rates
– Present Value of a Cash Flow Stream Using a
Term Structure of Discount Rates
C1
C2
PV =


2
1 + r1 (1 + r2 )

CN
(1 + rN )
N
(Eq. 5.7)
18
Using the Term Structure to
Compute Present Values
• Compute the present value of a risk-free five-year annuity
of $2,500 per year, given the following yield curve for July
2009.
Term
Years
1
2
3
4
5
0
1
$2,500
PV 
Date
July-09
0.54%
1.05%
1.57%
2.05%
2.51%
2
$2,500
3
$2,500
4
$2,500
5
$2,500
$2,500 $2,500 $2,500 $2,500 $2,500




 $13,227
2
3
4
5
1.0054 1.0105 1.0157 1.0205 1.0251
19
Using the Term Structure to
Compute Present Values
• The yield curve tells us the market interest rate per
year for each different maturity.
• In order to correctly calculate the PV of cash flows
from five different maturities, we need to use the five
different interest rates corresponding to those
maturities.
• Note that we cannot use the annuity formula here
because the discount rates differ for each cash flow.
• However, in a typical finance class like this one, we
assume that interest rates will remain the same, and
this assumption allows us to use one interest rates for
all different maturities. Then we can use the annuity
formula.
20
The Determinants of Interest Rates
• Interest Rate Determination
– Federal Funds Rate
• The overnight loan rate charged by banks with excess reserves at a
Federal Reserve bank to banks that need additional funds to meet
reserve requirements
• The Federal Reserve determines very short-term interest rates
through its influence on the federal funds rate
– If interest rates are expected to rise, long-term interest rates
will tend to be higher than short-term rates to attract investors
– If interest rates are expected to fall, long-term rates will tend
to be lower than short-term rates to attract borrowers.
21
Yield Curve Shapes

Steep
– Long-term rates are much higher than short-term rates

Inverted
– Long-term rates lower than short-term rates

Check the dynamic yield curve at
• http://stockcharts.com/freecharts/yieldcurve.html
22
Short-Term versus Long-Term U.S.
Interest Rates and Recessions
23
BOND VALUATION
24
Financial Asset Valuation
0
1
2
r
...
Value
PV =
n
CF1
CF1
1+ r 
1
+
CF2
CF2
1+ r 
2
+ ... +
CFn
CFn
1+ r 
n
.
The value of any financial asset (e.g., a bond, a stock, a loan,
etc) is simply the present value of the cash flows the asset is
expected to produce.
25
6.2 Zero-Coupon Bonds
• Zero-coupon bonds
– Only two cash flows
• The bond’s market price at the time of purchase
• The bond’s face value at maturity (n)
– Treasury bills are zero-coupon U.S. government bonds
with maturity of up to one year.
• We often refer to this as the risk-free interest rate for
that period (n).
• Because a default-free zero-coupon bond that matures on
date n provides a risk-free return over that period, the Law
of One Price guarantees that the risk-free interest rate
equals the yield to maturity on such a bond.
26
6.2 Zero-Coupon Bonds
• A one-year, risk-free, zero-coupon bond
with a $100,000 face value has an initial
price of $96,618.36.
– If you purchased this bond and held it to
maturity, you would have the following cash
flows:
27
6.2 Zero-Coupon Bonds
• Yield to Maturity of a Zero-Coupon Bond
– The discount rate that sets the present value of
the promised bond payments equal to the
current market price of the bond
– Yield to Maturity of an n-Year Zero-Coupon
Bond:
1/ n
 Face Value 
1  YTM n  

 Price 
(Eq. 6.2)
28
Example 6.1
Yields for Different Maturities
Problem:
• Suppose the following zero-coupon bonds are
trading at the prices shown below per $100 face
value. Determine the corresponding yield to
maturity for each bond.
Maturity
1 year
2 years
3 years
4 years
Price
$96.62
$92.45
$87.63
$83.06
29
Example 6.1
Yields for Different Maturities
Solution:
• We can use Eq. 6.2 to solve for the YTM of the bonds.
• Solving for the YTM of a zero-coupon bond is the same
process we used to solve for the compounding rate of return,
given the present and future values, by using your financial
calculator.
• Indeed, the YTM is the rate of return of buying the bond.
YTM1  (100 / 96.62)1/1  1  3.50%
YTM 2  (100 / 92.45)1/ 2  1  4.00%
YTM 3  (100 / 87.63)1/ 3  1  4.50%
YTM 4  (100 / 83.06)1/ 4  1  4.75%
30
6.3 Coupon Bonds
• Coupon bonds
– Pay face value at maturity & Also make regular
coupon interest payments
– Two types of U.S. Treasury coupon securities:
• Treasury notes: original maturities from one to ten
years
• Treasury bonds: original maturities of more than ten
years
31
Coupon Bond
32
Example 6.3
The Cash Flows of a Coupon Bond or Note
• Assume that it is May 15, 2010 and the U.S. Treasury has
just issued securities with May 2015 maturity, $1000 par
value and a 2.2% coupon rate with semiannual coupons.
Since the original maturity is only 5 years, these would be
called “notes” as opposed to “bonds”. The first coupon
payment will be paid on November 15, 2010. What cash
flows will you receive if you hold this note until maturity?
33
6.3 Coupon Bonds
• Yield to Maturity of a Coupon Bond:
– Cash flows shown in the timeline below:
– The coupon payments are an annuity, so the yield to
maturity is the interest rate y that solves the following
equation:
Annuity Factor using the YTM (y )
P
CPN 
1
1
1 
y
(1  y ) N



Present Value of all of the periodic coupon payments

FV
(1  y ) N
Present Value of the
Face Value repayment
using the YTM (y )
(Eq. 6.3)
34
Example 6.4 Computing the Yield to
Maturity of a Coupon Bond
Problem:
• Consider the five-year, $1000 bond with a 2.2% coupon rate
and semiannual coupons described in Example 6.3.
• If this bond is currently trading for a price of $963.11, what
is the bond’s yield to maturity?
Solution:
• Because the bond has ten remaining coupon payments, we
compute its yield y by solving Eq.(6.3) for this bond:

1
1
1000
963.11  11  1 

10 
10

y
(1  y)  (1  y)
35
Example 6.4 Computing the Yield to
Maturity of a Coupon Bond
(cont’d):
• We can solve it by trial-and-error, financial calculator, or a
spreadsheet. To use a financial calculator, we enter the price
we pay as a negative number for the PV (it is a cash
outflow), the coupon payments as the PMT, and the bond’s
par value as its FV. Finally, we enter the number of coupon
payments remaining (10) as N.
Given:
Solve for:
10
-963.11
11
1,000
1.50
Excel Formula: =RATE(NPER,PMT,PV,FV)=
RATE(10,11,-963.11,1000)
36
Example 6.4 Computing the Yield to
Maturity of a Coupon Bond
(cont’d):
• Therefore, y = 1.50%.
• Because the bond pays coupons semiannually, this yield is for a
six-month period.
• We convert it to an APR by multiplying by the number of coupon
payments per year.
• Thus the bond has a yield to maturity equal to a 3.0% APR
with semiannual compounding.
• As the equation shows, the yield to maturity is the discount rate
that equates the present value of the bond’s cash flows with its
price.
• Note that the YTM is higher than the coupon rate and the price
is lower than the par value. We will discuss why in the next
section.
37
Example 6.5 Computing a Bond Price
from Its Yield to Maturity
Problem:
• Consider again the five-year, $1000 bond with a
2.2% coupon rate and semiannual coupons in
Example 6.4. Suppose interest rates drop and the
bond’s yield to maturity decreases to 2%
(expressed as an APR with semiannual
compounding). What price is the bond trading for
now? And what is the effective annual yield on this
bond?
38
Example 6.5 Computing a Bond Price
from Its Yield to Maturity
Execute:
• Using Eq. 6.3 and the 6-month yield of 1.0%, the bond price
must be
1 
1  1000
P  11 

 $1009.47
1 
10 
10
0.01  1.01  1.01
Given:
Solve for:
10
1.0
25
1,000
-1,009.47
Excel Formula: = PV(RATE,NPER,PMT,FV)=PV(.01,10,11,1000)
• The effective annual yield corresponding to 1.0% every six
months is (1+.01)2-1=0.0201, or 2.01%
39
Example 6.5 Computing a Bond Price
from Its Yield to Maturity
Evaluate:
• The bond’s price has risen to $1009.47, lowering
the return from investing in it from 1.5% to 1.0%
per 6-month period. Interest rates have dropped,
so the lower return brings the bond’s yield into
line with the lower competitive rates being offered
for similar risk and maturity elsewhere in the
market.
40
6.3 Coupon Bonds
• Coupon Bond Price Quotes
– Prices and yields are often used
interchangeably.
– Bond traders usually quote yields rather than
prices.
– One advantage is that the yield is independent
of the face value of the bond.
– When prices are quoted in the bond market,
they are conventionally quoted per $100 face
value.
41
WHY BOND PRICE
CHANGES
42
6.4 Why Bond Prices Change
• Zero-coupon bonds always trade for a
discount.
• Coupon bonds may trade at a discount or
at a premium
• Most issuers of coupon bonds choose a
coupon rate so that the bonds will initially
trade at, or very close to, par.
• After the issue date, the market price of a
bond changes over time.
43
6.4 Why Bond Prices Change
• Interest Rate Changes and Bond Prices
– If a bond sells at par the only return investors
will earn is from the coupons that the bond
pays.
– Therefore, the bond’s coupon rate will exactly
equal its yield to maturity.
– As interest rates in the economy fluctuate, the
yields that investors demand will also change.
44
6.4 Why Bond Prices Change
• Interest Rate Risk and Bond Prices
– Effect of time on bond prices is predictable, but
unpredictable changes in rates also affect
prices.
– Bonds with different characteristics will respond
differently to changes in interest rates
• Investors view long-term bonds to be riskier than
short-term bonds.
• Investors view low coupon bonds to be riskier than
high coupon bonds.
45
Figure 6.3 A Bond’s Price vs. Its Yield
to Maturity
46
Table 6.3 Bond Prices Immediately
After a Coupon Payment
47
Definitions
Annual
coupon
pmt
Current yield =
Current price
Change
in
price
Capital gains yield =
Beginning price
Exp total
Exp
Exp cap
= YTM =
+
return
Curr yld
gains yld
48
Find current yield and capital
gains yield for a 8%, 10-year
bond when the bond sells for
$827.97 and YTM = 10.91%.
$80
Current yield =$827.97
= 0.0966 = 9.66%.
49
YTM = Current yield + Capital gains yield.
Cap gains yield = YTM - Current yield
= 10.91% - 9.66%
= 1.25%.
Could also find values in Years 0 and 1,
get difference, and divide by value in
Year 1. Same answer.
50
Premium and Discount Bonds
All 10-year
Bonds
Premium
C = 15%
YTM = 10.91%
Discount
C = 8%
YTM = 10.91%
Current
Yield
12.08%
9.66%
Capital Gain ─1.17%
or Loss
Yield
1.25%
Example 6.6 Determining the
Discount or Premium of a Coupon
Bond
Problem:
• Consider three 30-year bonds with annual coupon
payments.
• One bond has a 10% coupon rate, one has a 5%
coupon rate, and one has a 3% coupon rate.
• If the yield to maturity of each bond is 5%, what
is the price of each bond per $100 face value?
• Which bond trades at a premium, which trades at
a discount, and which trades at par?
52
Example 6.6 Determining the
Discount or Premium of a Coupon
Bond
Evaluate:
• The prices reveal that when the coupon rate of the bond is higher
than its yield to maturity, it trades at a premium.
• When its coupon rate equals its yield to maturity, it trades at par.
• When its coupon rate is lower than its yield to maturity, it trades at
a discount.
P(10% coupon)  10 
1 
1 
100
1


 $176.86 (trades at a premium)

30 
30
0.05  1.05  1.05
P(5% coupon)  5 
1 
1  100
1


 $100.00 (trades at par)

30 
30
0.05  1.05  1.05
P(3% coupon)  3 
1 
1  100
1


 $69.26

30 
30
0.05  1.05  1.05
(trades at a discount)
53
Figure 6.4 The Effect of Time on Bond
Prices
54
6.4 Why Bond Prices Change
• Bond Prices in Practice
– Bond prices are subject to the effects of both
passage of time and changes in interest rates.
– Prices converge to face value due to the time
effect, but move up and down because of
changes in yields.
55
Figure 6.5
Yield to
Maturity and
Bond Price
Fluctuations
over Time
56
Changes in Bond Price over Time:
Reality
57
INTEREST RATE
SENSITIVITY OF BOND
PRICES
58
Long-Term versus Short-Term Loans
• You work for a bank that has just made two
loans. In one, you loaned $909.09 today in return
for $1,000 in one year. In the other, you loaned
$909.09 today in return for $15,863.08 in 30
years. The difference between the loan amount
and repayment amount is based on an interest
rate of 10% per year.
59
Long-Term versus Short-Term Loans
• Imagine that immediately after you make the loans,
news about economic growth is announced that
increases inflation expectations so that the market
interest rate for loans like these jumps to 11%. Loans
make up a major part of a bank’s assets, so you are
naturally concerned about the value of these loans.
What is the effect of the interest rate change on the
value to the bank of the promised repayment of these
loans?
• Note that each of these loans has only one repayment
cash flow at the end of the loan. They differ only by the
time to repayment.
• The effect on the value of the future repayment to the
bank today is just the PV of the loan repayment,
calculated at the new market interest rate.
60
Long-Term versus Short-Term Loans
For the one-year loan:
For the 30-year loan:
$1,000
PV 
 $900.90
1
1.11
PV 
$15,863.08
$692.94
30
1.11
• The value of the one-year loan decreased by $909.09 $900.90 = $8.19, or 0.9%, but the value of the 30-year
loan decreased by $909.09 - $692.94 = $216.15, or almost
24%!
• The small change in market interest rates, compounded
over a longer period, resulted in a much larger change in
the present value of the loan repayment.
• You can see why investors and banks view longer-term
loans as being riskier than short-term loans!
61
Example 6.8 The Interest Rate
Sensitivity of Bonds
Problem:
• Consider a 10-year coupon bond and a 30-year coupon bond, both
with 10% annual coupons.
• By what percentage will the price of each bond change if its yield to
maturity increases from 5% to 6%?
• The price of the 10-year bond changes by (129.44 - 138.61) /
138.61 = -6.6% if its yield to maturity increases from 5% to 6%.
• For the 30-year bond, the price change is (155.06 - 176.86) /
176.86 = -12.3%.
62
Example 6.8 The Interest Rate
Sensitivity of Bonds
Evaluate:
• The 30-year bond is twice
as sensitive to a change in
the yield than is the 10year bond.
• In fact, if we graph the
price and yields of the two
bonds, we can see that the
line for the 30-year bond,
shown in blue, is steeper
throughout than the green
line for the 10-year bond,
reflecting its heightened
sensitivity to interest rate
changes.
63
Example 6.9 Coupons and Interest
Rate Sensitivity
Problem:
• Consider two bonds, each pays semi-annual
coupons and 5 years left until maturity.
• One has a coupon rate of 5% and the other has a
coupon rate of 10%, but both currently have a
yield to maturity of 8%.
• How much will the price of each bond change if its
yield to maturity decreases from 8% to 7%?
64
Example 6.9 Coupons and Interest
Rate Sensitivity
Execute:
•
The 5% coupon bond’s price changed from $87.83 to $91.68, or
4.4%, but the 10% coupon bond’s price changed from $108.11 to
$112.47, or 4.0%.
Given:
Solve for:
10
4
2.50
100
-87.83
Excel Formula: =PV(RATE,NPER,PMT,FV)=PV(.04,10,2.5,100)
65
Example 6.9 Coupons and Interest
Rate Sensitivity
Evaluate:
• The bond with the smaller coupon payments is more
sensitive to changes in interest rates.
• Because its coupons are smaller relative to its par
value, a larger fraction of its cash flows are received
later.
• Later cash flows are affected more greatly by changes
in interest rates, so compared to the 10% coupon
bond, the effect of the interest change is greater for
the cash flows of the 5% bond.
66
CREDIT RISK
67
6.5 Corporate Bonds
• Credit Risk
– U.S. Treasury securities are widely regarded to
be risk-free.
– Credit risk is the risk of default, so that the
bond’s cash flows are not known with certainty
• Corporations with higher default risk will
need to pay higher coupons to attract
buyers to their bonds.
68
6.5 Corporate Bonds
• Corporate Bond Yields
– Yield to maturity of a defaultable bond is not equal
to the expected return of investing in the bond
– The promised cash flows used to determine the yield
to maturity are always higher than (or equal to, if
not defaulted) the expected cash flows investors
may receive.
– As a result, the yield to maturity will always be
higher than the expected return of investing in the
bond.
– Therefore, a higher yield to maturity does not
necessarily imply that a bond’s expected return is
higher.
69
6.5 Corporate Bonds
• Bond Ratings
– Several companies rate the creditworthiness of
bonds
• Two best-known are Standard & Poor’s and Moody’s
– These ratings help investors assess creditworthiness
– Investment-grade bonds
– Speculative bonds
• junk bonds
• high-yield bonds
– The rating depends on
• the risk of bankruptcy
• bondholders’ claim to assets in the event of bankruptcy.
70
Table 6.6 Bond Ratings and the Number of U.S.
Public Firms with those Ratings at the End of 2009
71
6.5 Corporate Bonds
• Corporate Yield Curves
– We can plot a yield curve for corporate bonds just as we can for
Treasuries.
– The credit spread is the difference between the yields of
corporate bonds and Treasuries.
Corporate
Yield Curves
for Various
Ratings,
March 2010
72
YTM is a function of many factors
YTM = (r* + DRP) + IP + Others.
• YTM
•
•
•
•
= Required rate of return on a
debt security.
r*
= Real risk-free rate. T-bond
rate if no inflation; 1% to 4%.
DRP
= Default risk premium.
IP
= Inflation premium.
Others = Liquidity premium and/or
Maturity risk premium.
73
Example 6.10 Credit Spreads and
Bond Prices
Problem:
• Your firm has a credit rating of A.
• You notice that the credit spread for 10-year
maturity debt is 90 basis points (0.90%).
• Your firm’s ten-year debt has a coupon rate of
5%.
• You see that new 10-year Treasury notes are
being issued at par with a coupon rate of 4.5%.
• What should the price of your outstanding 10-year
bonds be?
74
Example 6.10 Credit Spreads and
Bond Prices
Solution:
Plan:
• If the credit spread is 90 basis points, then the yield to
maturity (YTM) on your debt should be the YTM on similar
treasuries plus 0.9%.
• The fact that new 10-year treasuries are being issued at par
with coupons of 4.5% means that with a coupon rate of
4.5%, these notes are selling for $100 per $100 face value.
• Thus their YTM is 4.5% and your debt’s YTM should be 4.5%
+ 0.9% = 5.4%.
1 
1 
100
2.50 

 $96.94
1 
20 
20
0.027  1.027  1.027
75
Bond Risk Premium over Time
Source: Federal Reserve.
76
Table 6.5 Interest Rates on Five-Year
Bonds for Various Borrowers, July 2013
77
Risk Premiums of Junk Bonds versus
Other Corporate Bonds over Time
Source: Federal Reserve.
78
Example 6.10 Credit Spreads and
Bond Prices
Evaluate:
• Your bonds offer a higher coupon (5% vs. 4.5%) than
treasuries of the same maturity, but sell for a lower
price ($96.94 vs. $100).
• The reason is the credit spread.
• Your firm’s higher probability of default leads investors
to demand a higher YTM on your debt.
• To provide a higher YTM, the purchase price for the
debt must be lower.
• If your debt paid 5.4% coupons, it would sell at $100,
the same as the treasuries.
• But to get that price, you would have to offer coupons
that are 90 basis points higher than those on the
treasuries—exactly enough to offset the credit spread.
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Credit Rating Example
• Andrew Industries is contemplating issuing a 30-year bond
with a coupon rate of 7% (annual coupon) and a face value
of $1,000.
– Andrew believes it can get a rating of A from S&P.
– However, due to recent financial difficulties at the company, S&P is
warning that it may downgrade Andrew’s bonds to BBB.
– Yields on A-rated, long-term bonds are currently 6.70%, and yields on
BBB-rated bonds are 7.20%.
• What is the price of the bod if Andrew Industries maintains
the A rating for the bond issue?
• What will the price of the bond be if it is downgraded?
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Chapter Quiz
1.
What types of cash flows does a bond buyer receive?
2.
How are the periodic coupon payments on a bond
determined?
3.
Why would you want to know the yield to maturity of a
bond?
4.
What is the relationship between a bond’s price and its
yield to maturity?
5.
What cash flows does a company pay to investors holding
its coupon bonds?
6.
What do we need in order to value a coupon bond?
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Chapter Quiz
7.
Why do interest rates and bond prices move in
opposite directions?
8.
If a bond’s yield to maturity does not change,
how does its cash price change between coupon
payments?
9.
What is a junk bond?
10. How will the yield to maturity of a bond vary
with the bond’s risk of default?
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