SS15 Fixed-Income: Basic Concepts
SS15 Fixed-Income: Analysis of Risk
SS 15
• R52 FI Securities: Defining Elements
• R53 FI Markets: Issuance, Trading, and Funding
• R54 Introduction to FI Valuation
SS 16
• R55 Understanding FI Risk and Return
• R56 Fundamentals of Credit Analysis
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Framework for Fixed Income
Basic Concepts
Valuation
Return
Securities
Markets
Yield Measures
Yield Spreads
Interest Rate Risk
Risk
Credit Risk
Liquidity Risk
Slide 2
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Contents for Sept. 14
 Overview of a Fixed-Income Security
 Bonds with Contingency Provisions
• Callable Bonds
• Putable Bonds
• Convertible Bonds
 Fixed-Income Valuation
• Pricing Bond with a Market Discount Rate
• Pricing Bond with Spot Rates
• Relationships Between Bond Price and Bond Characteristics
• Flat Price, Accrued Interest, and Full Price
 Sources of Return
 Interest Rate Risk
Slide 3
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Overview of a Fixed-Income Security
 Fixed-Income Security
 An instrument that allows governments,
companies, and other types of issuers to
borrow money from investors.
 A financial obligation of an entity that promises
to pay a specified sum of money at a specified
future dates.
 FI Security Issuers - Borrowers
 FI Security Holders/Investors – Lenders
Slide 4
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Overview of a Fixed-Income Security
 Basic Features of a Bond
 Issuer
 Maturity
 Par value
 Coupon rate and frequency
 Currency denomination
Slide 5
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Callable Bonds
 Callable bond = option-free bond – call option
 Options for issuers
 The right to redeem all or part of the bond
before the specified maturity date.
 Protect issuers against a decline in interest
rate.
• Market interest rates fall.
• Issuer’s credit quality improves.
 Obligations for investors
 Higher level of reinvestment risk → Higher
yield and lower price
Slide 6
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Call
Callable Bond
Putable Bonds
 Putable bond = option-free bond + put option
 Options for bondholders
 The right to sell back bond to the issuer
before maturity.
 Increasing int. rate → decreasing bond price
→ benefit bondholders because of the predetermined price
 Increasing int. rate → higher reinvestment
income
 Obligations for issuers
 Higher price and lower yield
Slide 7
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Putable Bond
Put
Convertible Bond
 Convertible bond
 = straight bond + embedded equity call option
 Option for bondholders to convert bond into a
specified number of the issuer’s shares.
 Advantages for bondholders
 Downside protection & equity upside potential
 Advantages for issuers
 Reduced interest expense
 Eliminating debt once converted
 Especially attractive for newer companies
 Disadvantages for existing shareholders
 Dilution
of interests
Slide 8
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Convertible Bond
 Conversion price
 The price at which the convertible bond can
be converted into shares.
 Conversion ratio
 The number of common shares that each bond
can be converted into.
 = par value÷conversion price
Slide 9
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Convertible Bond
 Conversion value
 = current share price×conversion ratio
 Conversion premium
 = convertible bond price - conversion value
 Conversion parity
 Parity: conversion value = convertible bond’s
price
 Below parity: conversion value < bond price
 Above parity: Vconversion> Pbond
 value/parity value
Slide 10
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Convertible Bond
Example: Convertible Bond
A convertible bond with the par value of $1,000 is trading at $900
now. It could be converted into common shares at the price of $50
per share. Now the market price of the common share is $60 per
share.





Conversion price = $50
Conversion ratio = $1000÷$50 = 20
Conversion value = $60×20 = $1,200
Conversion premium = $900 - $1,200 = -$300
Conversion parity: above parity
Slide 11
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Pricing Bond with a Market Discount Rate
 Cash flows
 Coupon payments & Principal repayment
 Discount Rate
 Mkt. discount rate/Required yield/Required
N
Rate of Return
PMT
FV
PV  

i
(1  r ) N
 Valuation
i 1 (1  r )
 PV = present value/price of the bond
 PMT = coupon payment per period
 FV = future value paid at maturity/par value
 r = mkt. discount rate/required rate of return
 N = number of periods to maturity
Slide 12
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Pricing Bond with a Market Discount Rate
Example 1: Annual-Payment Bond Valuation
The coupon rate on a bond is 4% and the payment is made once a year. If the
time-to-maturity is five years and the market discount rate is 6%, calculate the
price of the bond per 100 of par value.

•
•

•
•
Cash flows
Coupons
Principal
Discount
Discount periods
Discount rate
Cash Flows
Discounting Factor
4
(1+6%)
4
(1+6%)2
4
(1+6%)3
4
(1+6%)4
104
(1+6%)5
2ND FV → N=5 I/Y=6% PMT=4 FV=100 → CPT → PV=91.5753
Slide 13
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Pricing Bond with a Market Discount Rate
 Coupon rate
 Indicate interests given by issuers.
 Mkt. discount rate
 Reflect interest required by investors to pay full
par value for the bond.
 Relationship
 Coupon rate = mkt. discount rate →
Price = Par Value
 Coupon rate < mkt. discount rate →
Price < Par Value (discount)
 Coupon rate > mkt. discount rate →
Price CFA
> Par
Value (premium)
Slide 14
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Pricing Bond with a Market Discount Rate
Example 2: Annual Compounding Zero-Coupon Bond Valuation
A zero coupon bond matures in 15 years. At a market discount rate of 4.5% per
year and assuming annual compounding. Calculate the price of the bond per 100 of
par value.
Example 3: Semi-Annual-Payment Bond Valuation
The coupon rate on a bond is 8% and the payments are made semiannually on
15 June and 15 December. If there are three years to maturity and the market
discount rate is 3%, calculate the bond price.
Example 4: Semi-Annual Compounding Zero-Coupon Bond Valuation
A semi-annual pay zero coupon bond matures in 15 years. The market discount
rate of 4.5% per year. Calculate the price of the bond per 100 of par value.
Slide 15
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Pricing Bonds with Spot Rates
 Spot Rates
 YTM on zero-coupon bonds
 Bootstrap Method
 Value a bond as a package of cash flows.
 View each cash flow as a zero-coupon bond
 Discount each cash flow at spot rate.
 No-arbitrage value
 Sum of PVs of zero-coupon bonds.
PV 
Slide 16
PMT
PMT
PMT  FV


...

(1  Z1 )1 (1  Z 2 )2
(1  Z N ) N
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Pricing Bonds with Spot Rates
Example: Bond Valuation with Spot Rates
Calculate the price for a four-year 3% annual coupon payment bond with the par
value of $100, using spot rates. And calculate the YTM for the bond.
Maturity
Spot Rates
1 year
0.39%
1
3
1.0039
3
3
1.0250
2 years
1.40%
2
3
1.0140
4
103
1.0360
3 years
2.50%
4 years
3.60%
Period CFs
Disc. Factor
Period CFs Disc. Factor
Price Calculation
3
3
3
103



 98.104
1
2
3
4
(1.0039) (1.0140) (1.0250) (1.0360)
YTM Calculation
3
3
3
103



(1  r )1 (1  r ) 2 (1  r )3 (1  r ) 4
r  3.516%
98.104 
Slide 17
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Relationships Between Bond Price and Bond Characteristics
 Four Relationships about the change in the bond
price given the market discount rate change are
 Inverse Effect
 Convexity Effect
 Coupon Effect
 Maturity Effect
 The change in the bond price over time for a
given market discount rate
 Constant-yield price trajectory
Slide 18
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Relationships Between Bond Price and Bond Characteristics
Slide 19
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Relationships Between Bond Price and Bond Characteristics
 Inverse Effect
 Mkt. discount rate ↑bond price↓
N
PV  
i 1
PMT
FV

(1  r )i (1  r ) N
 Convexity Effect
 The percentage price change is greater when
the market discount rate goes down than
when it goes up.
 Price appreciation > price decline for a same
percentage of price change
Slide 20
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Relationships Between Bond Price and Bond Characteristics
Price
Option-Free Bond
Par
Coupon Rate
Slide 21
Market
Yield
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Relationships Between Bond Price and Bond Characteristics
 Coupon Effect
 Coupon rate↓price volatility↑Lower-coupon
bonds have more price volatility than highercoupon bonds, other things being equal.
 Maturity Effect
 Generally, longer-term bonds have more price
volatility than shorter-term bonds, other
things being equal.
 Exceptions occur for low-coupon (but not
zero-coupon), long-term bonds trading at a
discount.
Slide 22
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Relationships Between Bond Price and Bond Characteristics
 Constant-Yield Price Trajectory/Pull to Par
 Bond price approaches par value as time
passes.
 At maturity date bond value/price = par value
• bond selling at a premium → price declines
• bond selling at a discount → price increases
• bond selling at par value → unchanged price
Premium
Par
Discount
Slide 23
Premium Bond
Par Bond
Maturity
Discount Bond
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Relationships Between Bond Price and Bond Characteristics
Example: Bond Price and Bond Characteristics
An investor is considering the following six annual coupon payment government
bonds. Which bond will go up in price the most on a percentage basis if all yields
go down from 5% to 4.9% ? Which bond will go down in price the least on a
percentage basis if all yields go up from 5% to 5.1% ?
Slide 24
Bond
Coupon Rate
Maturity
YTM
A
0%
2 years
5%
B
5%
2 years
5%
C
8%
2 years
5%
D
0%
4 years
5%
E
5%
4 years
5%
F
8%
4 years
5%
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Flat Price, Accrued Interest, and Full Price
 Accrued interest
 The interest earned by the seller but received
by the buyer between the last coupon payment
date and the settlement date.

t
AI  ×coupon payment per period
T
• t=days between last coupon payment and the
settlement date
• T=number of days in the coupon period
Slide 25
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Flat Price, Accrued Interest, and Full Price
 Full price and Flat Price
Accrued
Interest
Flat
Price
Full
Price
T
P
t
C
C+P
PFull
 Accrued interest increases every day.
 Flat prices are quoted to not misrepresent the
daily increase in the full price as a result of
interest accruals.
Slide 26
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Flat Price, Accrued Interest, and Full Price
 Full price and Flat Price
PV
Full
PMT
PMT
PMT  FV


 ... 
1t /T
2 t / T
(1  r )
(1  r )
(1  r ) N t /T
PV Full  PV ×(1  r )t /T
PV Flat  PV Full  Accrued Interst
Slide 27
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Flat Price, Accrued Interest, and Full Price
Example: Full Price, Accrued Interest, and Flat Price
Bond G, described in the exhibit below, is sold for settlement on 16 June 2014.
Annual Coupon
Coupon Pmt Frequency
Interest Payment Dates
Maturity Date
Day Count Convention
Annual YTM
5%
Semiannual
10 April and 10 Oct.
10 Oct. 2016
30/360
4%
Calculate Full Price, Accrued Interst, and Clean Price on the settlement date of 16
June 2014.
Slide 28
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Flat Price, Accrued Interest, and Full Price
T
P
t
C
C+P
PFull
N=5 I/Y=2% PMT=2.5 FV=100 → PV=102.36
T=180 t=66
PVFull=PV×(1+r)66/180=102.36×(1+2%)66/180=103.10
Accrued Interest=PMT×(t/T)=2.5×(66/180)=0.92
PVFlat=PVFull – Accrued Interest=103.10 – 0.92=102.18
Slide 29
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Sources of Return
 Receipt of promised coupon and principal
payments on the scheduled dates.
 Reinvestment of coupon payments
 Amortization of discount or premium as the
bond’s carrying value “pulls to par”
 Potential capital gains or losses on the sale of
the bond prior to maturity
Slide 30
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Sources of Return
 Constant-Yield Price Trajectory:the relationship between bond
price and time based on the YTM when the bond is purchased
Carrying Value
Amortization of Discount
Slide 31
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Sources of Return
 Carrying Value
 The price implied by any point on the constant-yield price
trajectory.
 Carrying value = purchase price + amortization of discount
 Carrying value = purchase price - amortization of premium
 Amortization of Discount/Premium
 The change in price between two points on the trajectory.
 Capital Gain/Loss
 Return associated with a change in the value of a bond as indicated
by a change in the YTM
 Capital gain = sale price – carrying value
 CapitalCFAloss
= carrying valueCFA– 学习网站：lms.finance365.com
sale price
Slide 32
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Return Calculation
 Total Return
 Coupon payments
 Coupon reinvestment income
 Redemption of principal on maturity date / sale
price prior to maturity
 Horizon Yield
 Realized Rate of Return
 Internal rate of return (IRR) between the total
return and the purchase price
 Annualized holding-period rate of return
Slide 33
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Return Calculation
Example:
An investor purchases a nine-year, 7% annual coupon payment bond at a
price equal to par value. After the bond is purchased and before the first
coupon is received, interest rates increase to 8%. The investor sells the
bond after five years. Assume that interest rates remain unchanged at 8%
over the five-year holding period. Please calculate:
a) Coupon and coupon reinvestment income.
b) Capital gain/loss per 100 of par value from the sale of the bond at the
end of five-year holding period.
c) Total return for five-year holding period.
d) Horizon yield for five-year holding period.
Slide 34
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Return Calculation
a) coupon=7 reinvestment rate=8% no. of coupons=5
7×(1+8%)4+7×(1+8%)3+ 7×(1+8%)2 …+7=41.07
b)Issuing at par→carrying value=100
N=4 I/Y=8% PMT=7 FV=100→market price=96.69
capital loss=96.69 – 100=-3.31
c)Total return = coupon & coupon reinvestment income
+ sale price
= 41.07+96.69=137.76
d)Purchase Price=Total Return/(1+Horizon Yield)N
100 = 137.76/(1+r)5 r=6.62%
Slide 35
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Return Calculation
Example:
An investor purchases a nine-year, 7% annual coupon payment bond at a
price equal to par value. After the bond is purchased and before the first
coupon is received, interest rates increase to 8%. The investor sells the
bond after 8 years. Assume that interest rates remain unchanged at 8%
over the 8-year holding period. Please calculate:
a) Coupon and coupon reinvestment income.
b) Capital gain/loss per 100 of par value from the sale of the bond at the
end of five-year holding period.
c) Total return for 8-year holding period.
d) Horizon yield for 8-year holding period.
Slide 36
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Return Calculation
a) coupon=7 reinvestment rate=8% no. of coupons=8
7×(1+8%)7+7×(1+8%)6+ 7×(1+8%)5 …+7=74.4564
b)Issuing at par→carrying value=100
N=1 I/Y=8% PMT=7 FV=100→market price=99.07
capital loss=99.07 – 100=-0.93
c)Total return = coupon & coupon reinvestment income
+ sale price
= 74.4564+99.07=173.53
d)Purchase Price=Total Return/(1+Horizon Yield)N
100 = 173.53/(1+r)8 r=7.13%
Slide 37
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Interest Rate Risk
Interest Rate Risk
Coupon Reinvestment Risk
Market Price Risk
offset each other
When int. rate↑reinvestment↑& capital loss
When int. rate↓reinvestment ↓& capital gain
coupon reinvestment risk dominates
when the investor has a long-term
horizon relative to the time-tomaturity (e.g. a buy-and-hold investor)
market price risk dominates when
the investor has a short-term
horizon relative to the time-tomaturity
Investment Horizon is important for the analysis of interest rate risk.
Slide 38
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Interest Rate Risk
 Interest Rate Risk
 The risk that interest rate will change (from
the initial market discount rate), which affects
the reinvestment of coupon payments and the
market price if the bond is sold prior to
maturity.
Slide 39
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