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Topic 4: Bond Valuation
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Learning Outcomes
discounted cash flows (DCF) approach
bond terminology zero-coupon bonds and coupon bonds
interest rate risk
credit risk and bond ratings (extra)
yield curve
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Valuation Model
in making financial decisions, we have to
estimate the fair value of a financialinstrument/firm in many occasions
mergers and acquisitions, company research
report, expert opinion in court, corporaterestructuring
we can use a valuation model to do so
one of the most commonly used valuation
models is the discounted cash flow (DCF)
approach
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Discounted Cash Flow Approach
steps
estimate the future cash flows generated bythe financial instrument
estimate the required rate of return (or known
as the discount rate) based on time value ofmoney and risk of the financial instrument(use modern financial theories)
calculate the present value of the future cashflows and it is the fair value of the financialinstrument
application: bond valuation and stock valuation
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Example: DCF Model
$24.87.beto
estimatedisinstrumentfinancialtheofvaluefairThe
87.24$ %)121(
32$
%)121(
5.1$
%121
1$PV
32
=
+
+
+
+
+
=
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Bond
when a company wishes to borrow money frominvestors on a long-term basis, it does so by
issuing debt securities that are generally knownas bonds
contractual obligation of issuer (borrower) to paya specific amount of money to investors (calledbondholders) as creditors at some time in thefuture
regular interest/coupon payments
principal repayment at maturity
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Bond
0
…
N321 N-1
bond
price
CPN CPNCPNCPNCPN
CPN = coupon payment
FV = face value
FV
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Bond
total return on bond
1.2.
(sold before maturity)
(held to maturity)
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Bond Certificate
bond certificate: a document states the terms
and conditions of a bond as well as the amounts
and dates of the payments to be made
mostly scripless trading nowadays
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Bond Certificate
Bond trading is
scripless withelectronic book entry
in Hong Kong!
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Bond Indenture/Agreement
interest payment dates: the specified dates formaking interest payments
face/par/nominal value or principal amount:notional amount of bond to compute interestpayments and is due at bond’s maturity
coupon/nominal rate: specified quotedinterest rate (APR) to determine periodicinterest payments at the time of issue
coupon (payment): the promised interestpayment of a bond, paid periodically until thematurity date of a bond
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Bond Indenture/Agreement
annual coupon = face value * coupon rate
periodic coupon = annual coupon/numberof payments per year, e.g. annual coupon/2for semi-annual coupon payments
description of property used as security/collateral secured bond (with collateral) vs. unsecured
bond (without collateral)
seniority of claims upon company’sliquidation (which is senior?)
other things being equal, which offers a
higher market interest rate?
less risky(lower market interest rate)
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Bond Indenture/Agreement
protective covenant that limits certain actions the
issuing company may otherwise wish to take
during the term of the bond (positive or negative)
credit rating: show credit worthiness of issuer
embedded option: an option-like feature includedin a bond to give a specified right to either the
bond issuer or the bondholder, e.g. a callable
bond allows the bond issuer to redeem the bondat a specified redemption price before maturity
and it will affect the price and yield of the bond
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Case Study on Bond Terms
source: Fantasia
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Case Study on Bond Terms
source: Fantasia
embedded option
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Case Study on Bond Terms
source: Fantasia
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Case Study on Bond Terms
source: Fantasia
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Example: Bond Basics
Consider a 3-year 5% $50,000 par bond at a
prevailing market interest rate of 4%. The coupon
are made on a semi-annual basis.
coupon rate: 5%
par/face/nominal value: $50,000 annual coupon = $50,000*5% = $2,500
semi-annual coupon = $2,500/2 = $1,250 term to maturity or tenor: 3 years
yield (to maturity): 4% (to be discussed later)
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Types of Bonds
government bond: bond issued by government
short-term: Treasury bill long-term: Treasury note and bond
corporate bond: bond issued by a company
coupon bond: receive both regular coupons and
face value at maturity
zero-coupon (pure discount) bond: receive facevalue of bond at maturity only; coupon rate = 0%
(what to earn?)
perpetual bond: bond without a maturity date
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Zero-Coupon Bond
source: HKMA
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Bond Valuation Model
application of discounted cash flow approach
cash flows to bondholders periodic coupon payments CPN
par value at maturity FV
discount rate = market interest rate y (not the
same as coupon rate – what is the difference
between them?) term to maturity = N years
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Bond Valuation Model
N
N
1tNt
y)(1FV pricebond
bondcoupon -zero
)y1(
FV
)y1(
CPN pricebond
bondcoupon
+
=
+
+
+
=∑=
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Example: Coupon Bond
Consider a 5%, 3-year, $50,000 par bond with
annual coupon payments. If the market interest
rate on the bond is 4%, what is its value now?
premium.aattradesbondthe
value,facethethanhigherispricebondtheWhen
.55.387,51$
%)41(
500,52$
%)41(
500,2$
4%)(1
$2,500 pricebond 32
=
++
++
+=
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Example: Zero-Coupon Bond
Consider a 2-year, $50,000 par zero-coupon bond.
If the market interest rate on the bond is 3%,
what is its value now?
discount.aattradetosaidisbondthe
value,facethethanlessispricebondtheWhen
80.129,47$%)31(
000,50$ pricebond 2 =
+=
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Yield to Maturity
yield (to maturity) is a measure of the average
rate of return on a bond under two conditions
1.
2.
calculated average rate of return that makes the
present value of future cash flows generated by abond equal to its bond price
1. reinvest the coupons in the bond2. hold the bond to maturity
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Yield to Maturity
N
N
1tNt
YTM)(1
FV
pricebond
bondcoupon -zero
) YTM1(FV
YTM)(1CPN pricebond
bondcoupon
+=
+
+
+
=∑=
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Yield to Maturity
for a zero-coupon bond with term-to-maturity of N
years, the yield to maturity must be equal to N-
year market interest rate
N
N
y)(1
FV pricebond
) YTM1(
FV
pricebond
+
=
+=
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Example: Coupon Bond
A bond has a price of $1,030.42. It has a par
value of $1,000, an annual coupon of $30, and a
maturity of two years. What is its yield to
maturity?
%38.4 YTM
) YTM1(
060,1$
YTM)(1
$60 $1,030.42
2
=
+
+
+
=
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Example: Zero-Coupon Bond
Consider a 3-year, $50,000 par zero-coupon bond.
The current bond price is $48,000. What is its
yield to maturity?
%37.1TM
) YTM1(000,50$ $48,000 3
=
+
=
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Risk-Free Interest Rates
spot (risk-free) interest rates: default-free, zero-
coupon yields of risk-free bond
yield curve: a graph to show the bond yields (YTM)
(a measure of the market interest rates) as afunction of the bonds’ term to maturity (TTM)
(usually upward-sloping)
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Risk-Free Interest Rates
zero-coupon yield curve: a plot of the yield of risk-
free zero-coupon bonds (STRIPS – Separate
Trading of Registered Interest and Principal of
Securities) as a function of the bond’s maturity
date
can also plot coupon-paying yield curve by
using on-the-run coupon bonds
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Risk-Free Interest Rates
yield to maturity
term to maturity
zero-coupon yield curve
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Market Practice
many bond professionals use the yield to
maturity to represent the market interest rate,
e.g. they use the yield to determine the bond
price (instead of the other way around)
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Example: Coupon Bond
Consider a 2%, 2-year, $1,000 par bond with
semi-annual coupons. If the yield to maturity is
3%, what is the bond price?
$980.731.5%)(1
010,1$
%)5.11(10$
%)5.11(10$
%5.11$10$ pricebond
10$2
2%*$1,000 couponannualsemi
%5.12
3% YTMannual -semi
4
32
=+
+
+
+
+
+
+
=
==−
==
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Example: Zero-Coupon Bond
Consider a 5-year, $1,000 par zero-coupon bond.
If the yield to maturity 5%, what is the bond price?
$783.535%)(1
000,1$ pricebond
5 =
+
=
Valuing a Coupon Bond with Zero-
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Valuing a Coupon Bond with Zero
Coupon Prices and Yields consider a 5%, 2-year, $1,000 par bond with
annual coupon payments
0 1 2
$50 $1,050
year
-bond price
equivalent to a1-year $50 par
zero-coupon
bond
equivalent to a2-year $1,050
par zero-
coupon bond
STRIPS*
Valuing a Coupon Bond with Zero-
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Valuing a Coupon Bond with Zero
Coupon Prices and Yields
in the bond market
1-year zero-coupon bond trades at 96.62 (% of
face value) at a yield of 3.5% (YTM1)
2-year zero-coupon bond trades at 92.45 at a
yield of 4% (YTM2) assume all bonds have the same risk
Valuing a Coupon Bond with Zero-
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Valuing a Coupon Bond with Zero
Coupon Prices and Yields
%99.3TM
) YTM1(
050,1$
)TM Y (1
$5009.019,1$
error)roundingtoduee(differenc
09.019,1$%)41(
050,1$3.5%)(1
$50 pricebond
04.019,1$92.45%*$1,050%62.96*50$pricebond
2
2
=
+
+
+
=
=
+
+
+
=
=+=
Valuing a Coupon Bond with Zero-
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Valuing a Coupon Bond with Zero
Coupon Prices and Yields
if the bond price deviates from $1,019.04, itgives rise to an arbitrage opportunity
for example, the bond price is $1,000
buy the bond and sell (issue) $50 par 1-yearzero-coupon bond at 96.62 and sell (issue)$1,050 par 2-year zero-coupon bond at 92.45
P&L = $50*96.62%+$1,050*92.45%-$1,000= $19.04 (arbitrage profit)
You receive $50 from the bond in year 1 andpay off 1-year zero-coupon bond and $1,050 inyear 2 to pay off 2-year zero-coupon bond
Valuing a Coupon Bond with Zero-
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g p
Coupon Prices and Yields
yield to maturity is a weighted average of thezero-coupon yields (remember that they are the
same as the market interest rates) if zero-coupon yield curve is upward-sloping,
yield-to-maturity decreases with coupon rate
(why?) if zero-coupon yield curve is downward-sloping,
yield to maturity increases with coupon rate
(why?)
with a flat zero-coupon yield curve, same yieldto maturity for all coupon rates
Wh B d P i Ch g ?
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Why Bond Prices Change?
most issuers choose to set a coupon rate close to
the market interest rate so that the bonds will
initially issued at par (not for zero-coupon bonds
which must be issued at a discount)
relationship between bond price and market
interest rate/yield (why?)
market interest rate usually increases with risk
and term to maturity of a bond
Wh B d Pri Ch g ?
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Why Bond Prices Change?
subsequently, market interest rate changes andbond price is not necessary equal to the face
valuewhen coupon rate is higher than market
interest rate/yield, bond trades at a and
bond price is than face value (why?)when coupon rate is equal to market interest
rate/yield, bond trades at and bond price
is face valuewhen coupon rate is lower than market
interest rate/yield, bond trades at a and
bond price is than face value
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Example: Premium Par and Discount
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Example: Premium, Par and Discount
Consider a 5%, 2-year, $1,000 par bond with
annual coupon payments. What is the bond price
if the yield to maturity is (a) 6%; (b) 5% and (c)
4%?
(premium)$1,018.86
4%)(1
$1,050
%)4(1
50$ pricebond(c)
(par)$1,0005%)(1
$1,050
%)5(1
50$
pricebond(b)
(discount)$981.676%)(1
$1,050%)6(1 50$ pricebond(a)
2
2
2
=
+
+
+
=
=+++=
=
+
+
+
=
Time and Bond Prices
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Time and Bond Prices
as the next coupon from a bond grows nearer, the
bond price increases to reflect the increasing
present value of the cash flow
it will peak right before the coupon is made and
will drop when coupon is made (buyer cannotreceive the coupon any more)
convergence property: bond price moves
gradually to the face value and it is equal to theface value on the maturity date when the last
coupon is made (why?)
Time and Bond Prices
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Time and Bond Prices
yield = 5%
Example: Time and Bond Prices
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Example: Time and Bond Prices
Consider a 3-year, $1,000 par zero-coupon bond.
If the yield of 5% is constant over time, what is
the bond price in year 0, year 1, year 2 and year 3right after the coupon is made.
$1,0003yearinpricebond
$952.385%)(1
$1,000 2yearinpricebond
$907.03
5%)(1
$1,000 1yearinpricebond
$863.84%)5(1
$1,000 0yearinpricebond
2
3
=
=
+
=
=
+
=
=+
=
Interest Rate Risk and Bond Prices
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Interest Rate Risk and Bond Prices
interest rate risk: the risk that arises for
bondholders from fluctuating interest rates
the longer the term to maturity, the higher the
interest rate risk (why?)
the lower coupon rate, the higher the interestrate risk (why?)
Example: Interest Rate Risk and Term to
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Maturity
consider two bonds (5-year and 10-year) with 5%
annual coupons and a par value of $100
$20
$40
$60
$80
$100
$120
$140
$160
0% 5% 10% 15% 20%
bond price
yield
5% 5-year bond
5% 10-year bond
Example: Interest Rate Risk and Term to
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Maturity
5% 10-year bond 5% 5-year bond
0% 150.00 125.00
1% 137.89 119.41
2% 126.95 114.143% 117.06 109.16
4% 108.11 104.45
5% 100.00 100.00
6% 92.64 95.79
7% 85.95 91.80
8% 79.87 88.02
9% 74.33 84.44
10% 69.28 81.05
11% 64.66 77.82
12% 60.45 74.77
13% 56.59 71.8614% 53.05 69.10
15% 49.81 66.48
16% 46.83 63.98
17% 44.10 61.61
18% 41.58 59.35
19% 39.25 57.19
20% 37.11 55.14
bond priceyield
change in bond price for 5%10-year bond = ($92.64-
$100)/$100 = -7.36%
change in bond price for 5%5-year bond = ($95.79-
$100)/$100 = -4.21%
Example: Interest Rate Risk and Coupon
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Rate
consider two 5-year bonds with 1% and 10%
annual coupons and a par value of $100
$20
$40
$60
$80
$100
$120
$140
$160
0% 5% 10% 15% 20%
bond price
yield
10% 5-year bond
1% 15-year bond
Example: Interest Rate Risk and Coupon
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Rate
10% 5-year bond 1% 5-year bond
0% 150.00 105.00
1% 148.53 100.00
2% 147.13 95.293% 145.80 90.84
4% 144.52 86.64
5% 143.29 82.68
6% 142.12 78.94
7% 141.00 75.40
8% 139.93 72.059% 138.90 68.88
10% 137.91 65.88
11% 136.96 63.04
12% 136.05 60.35
13% 135.17 57.79
14% 134.33 55.37
15% 133.52 53.07
16% 132.74 50.89
17% 131.99 48.81
18% 131.27 46.84
19% 130.58 44.9620% 129.91 43.18
bond priceyield
change in bond price for 10%5-year bond = ($142.12-
$143.29)/$143.29 = -0.82%
change in bond price for 1%5-year bond = ($78.94-
$82.68)/$82.68 = -4.53%
Bond Prices and Accrued Interest
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Bond Prices and Accrued Interest
dirty/invoice/settlement price: a bond’s actualcash price
accrued interest: amount of the next couponpayment that has already accrued, i.e. accrued
interest = coupon * (days since last couponpayment up to that day/days in current couponperiod)
calculated based on day count convention, e.g.Actual/360, Actual/365, Actual/Actual,30/360
Bond Prices and Accrued Interest
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Bond Prices and Accrued Interest
clean price: a bond’s cash price less an
adjustment for accrued interest, i.e. clean price =
dirty price – accrued interest, and dealers usuallyquote this price (as a percentage of face value)
from the perspective of an investor, weobserve the clean price and calculate the
accrued interest; hence, dirty price = clean
price + accrued interest
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Bond Prices and Accrued Interest
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source: HKEx
face value =
$10,000
clean price =
$10,000*106.3%= $10,630
dirty price =
$10,630 +$105.86 =
$10,735.86
Example: Bond Prices and Accrued
I t t
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Interest
The dirty price of a 5%, 5-year $1,000 par bondwith semi-annual coupon payments is $985. The
number of days from the last coupon payment upto the settlement date of the bond trade is 45days. The number of days between the last
coupon payment to the next coupon payment is182. Calculate the accrued interest and the cleanprice.
accrued interest = ($1,000*5%/2)*45/182 =$6.18
clean price = $985 - $6.18 = $978.82
Credit Risk and Bond Rating
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g
corporate bonds are risky because there is a
chance that the issuer may not be able to settle
the debt obligations
credit risk: the risk of default by the issuer of any
risky bond that the bond’s cash flows are not
known with certainty
investors pay for a bond with higher credit
risk yield to maturity is for a bond with higher
credit risk
lower
Credit Risk and Bond Rating
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g
yield to maturity (based on promised cash
flows) is than the expected return (based
on expected cash flows) because of the chanceof default (why?)
bond/credit rating: an assessment to show the
possibility of default by the bond issuer
different yield curves for bonds with different
bond ratings credit rating agencies: Standard and Poor’s,
Moody’s and Fitch
higher
for yield to maturity--> we expect no credit risk
Credit Risk and Bond Rating
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Moody's S&P FitchAaa AAA AAA
Aa1 AA+ AA+
Aa2 AA AA
Aa3 AA- AA-A1 A+ A+
A2 A A
A3 A- A-
Baa1 BBB+ BBB+
Baa2 BBB BBB
Baa3 BBB- BBB-
Ba1 BB+ BB+
Ba2 BB BB
Ba3 BB- BB-
B1 B+ B+
B2 B BB3 B- B-
Caa1 CCC+ CCC+
Caa2 CCC+ CCC+
Caa3 CCC- CCC-
Ca CC CC
C C CD D
investmentgrade bonds
non-investment
grade, speculative,
junk or high yield
bonds
if you buy these, you are subjected a low
credit risk
Challenging Questions
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1. Give two examples of a positive covenant in abond indenture.
A. B.
2. Give two examples of a negative covenant in abond indenture.
A.
B.3. Explain the difference between the coupon rate,
the yield to maturity and the market interest
rate.
put up collateral to back up the bond
make regular interest payments and principal repayment at maturity
cannot issue a new bond with a claim priority than the old one
cannot announce a dividend higher than a specified level before principal repayment
upward sloping yield to maturity(calculated number)
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4. Why is the yield curve upward-sloping more
often than not?
5. What is the implication of an upward-sloping
yield curve?
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6. Other things being equal, which of the following
bonds should have the highest yield?
A. a term to maturity of 10 years and a credit
rating of AA
B. a term to maturity of 10 years and a creditrating of BB
C. a term to maturity of 5 years and a credit
rating of AA
D. a term to maturity of 5 years and a credit
rating of BB
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7. A very important concept in finance is the risk-
free rate, i.e. the rate of return of a risk-free
financial instrument. Which of the following islikely to be used as a proxy for the risk-free rate?
Explain why.
A. Hong Kong Interbank Offered Rate, HIBOR
B. total return on stock of Tracker Fund (an
index fund replicating Hang Seng Index) C. yield on bond issued by HKSAR Government
D. yield on bond issued by AAA-rated company
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8. Other things being equal, which of the following
bonds should have the highest interest rate risk?
A. a term to maturity of 10 years and a
coupon rate of 10%
B. a term to maturity of 10 years and acoupon rate of 5%
C. a term to maturity of 5 years and a coupon
rate of 10%
D. a term to maturity of 5 years and a coupon
rate of 5%
face value will constitute a higher rate
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9. Other things being equal, if the interest rate isexpected to fall and you have to buy a bond, is it
better to buy a bond with a shorter maturity orone with a longer maturity? Explain.
10.Is the price calculated from the bond valuation
model the clean price or the dirty price? Explain.11.If a bond’s yield to maturity does not change,
how does its price change between coupon
payments?12.Explain why the expected return of a corporate
bond does not equal its yield to maturity.
bond price
bond price rise, coupon payments
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13.The market interest rate increases when
the credit standing of the bond issuer is ;
the term to maturity of the bond is ; and
the demand for the bond is in the
market.14.Explain what is meant by credit risk and interest
rate risk.
15.If short-term interest rates are lower than long-
term rates, why might a borrower still choose to
finance with long-term debt?
lower
longer
lower
bond price lower bond price and market interest rate (inverse)
demand and supply for short period of time
short term is more fluctuate
1. stable source of funds
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16.The price-yield relationship of a bond is a convex
curve (as shown). The impact on the size in the
bond price change will not be symmetrical foran increase and a decrease of 1% change in
interest rate. If the bond price effect from a 1%
decrease in the interest rate is usually than
that from a 1% increase in the interest rate. This
is known as positive convexity.
higher
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bond price
yieldi
P positiveconvexity
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17.Nowadays, a bond issuer tends to include anoption-like feature to a bond. For example, it
may include a redemption provision in the bondindenture to allow it to redeem the bonds at aspecified redemption price before maturity of
the bond. The bond is called a callable bond. Ifan investor buys a callable bond, he effectivelybuys a straight bond (without any option-likefeatures) and buys an embedded option. Otherthings being equal, as compared to a straightbond, the callable bond tends to have aprice and a yield. Explain.
sell to the issuer
higher
lower
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18.Most bonds are traded over-the-counter through
a dealer market where large financial
institutions (especially banks) act as the dealers.In the dealer quotation of the bond price, which
price is likely to be higher, the bid price or the
ask price? Which yield is likely to be higher, thebid yield or the ask yield?
dealer: for profit
thus ask price must be higherthe bid yieldif ask price is higherask yield is lower
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