Finance 100 Prof. Michael R. Roberts docx

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Copyright © Michael R. Roberts
Bonds
Finance 100
Prof. Michael R. Roberts
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Copyright © Michael R. Roberts
Topic Overview
z Introduction to bonds and bond markets
z Zero coupon bonds
»Valuation
» Yield-to-Maturity & Yield Curve
» Spot Rates
» Interest rate sensitivity – DVO1
z Coupon bonds
»Valuation
»Arbitrage
» Bond Prices Over Time
» Yield Curve Revisited
» Interest rate sensitivity – Duration & Immunization
z Forward Rates
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Copyright © Michael R. Roberts
What is a Bond and What are its Features?
z A bond is a security that obligates the issuer to make interest and principal
payments to the holder on specified dates.
» Maturity (or term)

» Face value (or par): Notional amount used to compute interest payments
» Coupon rate: Determines the amount of each coupon payment, expressed as an
APR
z Bonds differ in several respects:
» Repayment type
» Issuer
» Maturity
» Security
» Priority in case of default
Coupon Rate Face Value

Number of Coupon Payments per Year
Coupon
×
=
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Copyright © Michael R. Roberts
Repayment Schemes
z Bonds with a balloon (or bullet) payment
» Pure discount or zero-coupon bonds
– Pay no coupons prior to maturity.
» Coupon bonds
– Pay a stated coupon at periodic intervals prior to maturity.
» Floating-rate bonds
– Pay a variable coupon, reset periodically to a reference rate.
z
Bonds without a balloon payment
» Perpetual bonds
– Pay a stated coupon at periodic intervals.
» Annuity or self-amortizing bonds

– Pay a regular fixed amount each payment period.
– Principal repaid over time rather than at maturity.
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Copyright © Michael R. Roberts
Who Issues Bonds?
z US Government (Treasuries)
» T-bills: 4,13,16-week maturity, zero coupon bonds
» T-notes: 2,3,5,10 year, semi-annual coupon bonds
» T-bonds: 20 & 30-year, semi-annual coupon bonds
» TIPS: 5,10,20-year, semi-annual coupon bond, principal π-adjusted
» Strips: Wide-ranging maturity, zero-coupon bond, IB-structured
z Foreign Governments
z Municipalities
» Maturities from one month to 40 years, semiannual coupons
» Exempt from federal taxes (sometimes state and local as well).
» Generally two types: Revenue bonds vs General Obligation bonds
» Riskier than government bonds (e.g., Orange County)
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Copyright © Michael R. Roberts
Who Issues Bonds? (Cont.)
z Agencies:
» E.g. Government National Mortgage Association (Ginnie Mae),
Student Loan Marketing Association (Sallie Mae)
» Most issues are mortgage-backed, pass-through securities.
» Typically 30-year, monthly paying annuities mirroring underlying
securities
»Prepayment risk.
z Corporations

» 4 types: notes, debentures, mortgage, asset-backed
» ~30 year maturity, semi-annual coupon set to price at par
» Additional features/provisions:
– Callable: right to retire all bonds on (or after) call date, for call price
– convertible bonds
– putable bonds
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Bond Ratings
Moody’s S&P Quality of Issue
Aaa AAA Highest quality. Very small risk of default.

Aa AA High quality. Small risk of default.

A A High-Medium quality. Strong attributes, but potentially
vulnerable.
Baa BBB Medium quality. Currently adequate, but potentially
unreliable.
Ba BB Some speculative element. Long-run prospects
questionable.
B B Able to pay currently, but at risk of default in the future.
Caa CCC Poor quality. Clear danger of default.

Ca CC High speculative quality. May be in default.

C C Lowest rated. Poor prospects of repayment.

D - In default.

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Copyright © Michael R. Roberts
The US Bond Market – Flows
Amount ($bil.). Source: Flow of Funds Data 2005-2007
132.3104.494.5Consumer Credit
1417.5
53.6
195
307.3
2005
1397.1
213.4
177.3
183.7
2006
1053.2
314.1
214.6
237.5
2007
Mortgages
Corporate
Municipal
U.S. Gov.
Debt
Instrument
Dollar volume of bonds traded daily is 10 times that of equity markets!

Outstanding investment-grade dollar denominated debt is about $8.3 trillion (e.g.,
treasuries, agencies, corporate and MBSs
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Copyright © Michael R. Roberts
Zero Coupon Bonds
(a.k.a. Pure Discount Bonds)
z Notation Reminder:
» V
n
= B
n
= Market price of the bond in period n
» F = Face value
» R= Annual percentage rate
» m = compounding periods (annual Æ m = 1, semiannual Æ m = 2,…)
» i = Effective periodic interest rate; i=R/m
» T= Maturity (in years)
» N = Number of compounding periods; N = T*m
» r = discount rate
z Two cash flows to buyer of a zero coupon bond (a.k.a. “zero”):
»-V
0
at time 0
» F at time T
z What is the price of a bond?
() ()
00 00
or

11
TN
FF
VB VB
ri
⎛⎞
== ==
⎜⎟
⎜⎟
++
⎝⎠
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Copyright © Michael R. Roberts
Zero Coupon Bond
Examples
z Value a 5 year, U.S. Treasury strip with face value of $1,000.
The APR is 7.5% with quarterly compounding?
» Approach 1: Using R (APR) and i (effective periodic rate)
» Approach 2: Using r (EAR)
» Approach 3: Using r (periodic discount rate)
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Copyright © Michael R. Roberts
Yield to Maturity
z The Yield to Maturity (YTM) is the one discount rate that
sets the present value of the promised bond payments equal to

the current market price of the bond
» Doesn’t this sound vaguely familiar…
z Example: Zero-Coupon Bond
» But this is just the IRR since
()
1/
0
0
1
1
T
T
FF
Vr YTMy
V
r
⎛⎞
=⇒=−==
⎜⎟
+
⎝⎠
?
()
1/
0
0
01
1
T
T

FF
VIRRYTMy
V
IRR
⎛⎞
=− ⇒ = −= =
⎜⎟
+
⎝⎠
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Yields for Different Maturities
z Note: bonds of different maturities have different YTMs
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Spot Rates, Term Structure, Yield Curve
z A spot rate is the interest rate on a T-year loan that is to be made today
» r
1
=5% indicates that the current rate for a one-year loan today is 5%.
» r
2
=6% indicates that the current rate for a two-year loan today is 6%.
»Etc.
» Spot rate = YTM on default-free zero bonds.
z The term structure of interest rates is the series of spot rates r
1
,r

2
,r
3
,…
relating interest rates to investment term
z The yield curve is just a plot of the term structure: interest rates against
investment term (or maturity)
» Zero-Coupon Yield Curve: built from zero-coupon bond yields (STRIPS)
» Coupon Yield Curve: built from coupon bond yields (Treasuries)
» Corporate Yield Curve: built from corporate bond yields of similar risk (i.e.,
credit rating)
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Term Structure of Risk-Free U.S. Interest
Rates, January 2004, 2005, and 2006
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Using the Yield Curve
z We should discount each cash flow by its appropriate discount
rate, governed by the timing of the cash flow
z Example: What is the present value of $100, 10 years from
today (Use the term structure from January 2004)
z Generally speaking, we must use the appropriate discount rate
for each cash flow:
12
2
1
12

1 (1 ) (1 ) (1 )
=
=+ ++ =
++ + +
∑
"
N
NN
Nn
n
Nn
CC
CC
PV
rr r r
?
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Copyright © Michael R. Roberts
A Cautionary Note
z All of our valuation formulas (e.g., perpetuity, annuity)
assume a flat term structure.
» I.e., there is only one discount rate for cash flows received at any point
in time
z Recall:
» Growing Annuity:
» Growing Perpetuity:
– “r” is implicitly assumed to be the same every period…
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1

( ) (1 )
N
g
PV C
rg r
⎛⎞
⎛⎞
+
⎜⎟
=× −
⎜⎟
⎜⎟
−+
⎝⎠
⎝⎠

( )
C
PV
rg
=
−
9
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Copyright © Michael R. Roberts
Interest Rate Sensitivity
Zero Coupon Bonds
z Why do zero-coupon bond prices change? Interest rates
change!

z The price of a zero-coupon bond maturing in one year from
today with face value $100 and an APR of 10% is:
z Example: Now imagine that immediately after you buy the
bond, the interest rate increase to 15%. What is the price of the
bond now
()
0
1
N
F
V
i
=
+
()
0
1
100
$90.91
10.10
V ==
+
?
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Copyright © Michael R. Roberts
Characterizing the Price Rate Sensitivity
of Zero Coupon Bonds
z Consider the following 1, 2 and 10-year zero-coupon bonds, all with
» F=$1,000
» APR of R=10%, compounded annually.

$0
$200
$400
$600
$800
$1,000
$1,200
0.0% 5.0% 10.0% 15.0% 20.0% 25.0%
1-Year
2-Year
10-Year
Note 4 things:
1. Bond prices are
inversely related
to IR
2. Fix the interest
rate: Longer term
bonds are less
expensive
3. Longer term
bonds are more
sensitive to IR
changes than
short term bonds
4. The lower the IR,
the more sensitive
the price.
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Copyright © Michael R. Roberts
Quantifying the Interest Rate Sensitivity
of Zero Coupon Bonds – DV01
z What’s the natural thing to do? Compute the derivative
» If we change the interest rate by a little (e.g., 0.0001 or 1 basis point) than
multiplying this number by the derivative should tell me how much the price
will change, all else equal (i.e., DV01 = Dollar Value of 1 Basis Point)
z Alternatively, we can just compute the prices at two different interest rates
and look at the difference: B
0
(i) – B
0
(i+0.0001)
()
()
()
()()
()
0
1
0
2
2
0
2
1
1 0 (Negative slope in )
1 1 0 (Convex function of )
N
N

N
F
V
i
V
FN i i
i
V
FN N i i
i
−+
−+
=
+
∂
⇒=− + <
∂
∂
⇒= ++ >
∂
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Copyright © Michael R. Roberts
Valuing Coupon Bonds
Amortization Bonds
z Consider an amortization bond maturing in two years with
semiannual payments of $1,000. Assume that the APR is 10%
with semiannual compounding
z How can we value this security?
1. Brute force discounting
2. Recognize the stream of cash flows as an annuity

()
()()()
0
234
1000 1000 1000 1000
$3545.95
10.10/2
1 0.10 / 2 1 0.10 / 2 1 0.10 / 2
V =+ + + =
+
+++
()
4
0
1000
1 (1 0.10 / 2) $3,545.95
0.10 / 2
V
−
=−+ =
()()()()
0
0.5 1 1.5 2
1000 1000 1000 1000
$3545.95
1 0.1025 1 0.1025 1 0.1025 1 0.1025
V =+++=
++++
(i):
EAR (r):

or
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Replication
z Can we construct the same cash flows as our amortization
bond using other securities?
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A First Look at Arbitrage
z What if the bond is selling for $3,500 in the market?
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Copyright © Michael R. Roberts
Valuation of Straight Coupon Bond
Example
z What is the market price of a U.S. Treasury bond that has a
coupon rate of 9%, a face value of $1,000 and matures
exactly 10 years from today if the interest rate is 10%
compounded semiannually?
0 6 12 108 120Months
Cash Flows 45 45 45 1045
Timeline:
Present Value = Current Price =
?
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Copyright © Michael R. Roberts
Valuation of Straight Coupon Bond

General Formula
z What is the market price of a bond that has an annual coupon
C, face value F and matures exactly T years from today if the
required rate of return is R, with m-periodic compounding?
» Coupon payment is: c = C/m
» Effective periodic interest rate is: i = R/m
» number of periods N = Tm
» Note the assumption of a flat term structure…
[]
[
]
()
⎥
⎦
⎤
⎢
⎣
⎡
+
+
⎥
⎦
⎤
⎢
⎣
⎡
+−
⋅=
+
=

−
N
N
i
F
i
i
c
ZeroAnnuityV
1
)1(1
0
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Copyright © Michael R. Roberts
Relationship Between Coupon Bond Prices
and Interest Rates
z Bond prices are inversely related to interest rates (or yields).
z A bond sells at par only if its interest rate equals the coupon
rate.
» Most bonds set the coupon rate at origination to sell at par
z A bond sells at a premium if its coupon rate is above the
interest rate.
z A bond sells at a discount if its coupon rate is below the
interest rate.
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Copyright © Michael R. Roberts
The Effect of Time on Bond Prices
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Copyright © Michael R. Roberts
YTM and Bond
Price Fluctuations
Over Time
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Copyright © Michael R. Roberts
Yield to Maturity
Coupon Bonds
z Recall: The Yield to Maturity is the one discount rate that sets the
present value of the promised bond payments equal to the current market
price of the bond
z Prices are usually given from trade prices
» need to infer interest rate that has been used
» This is not the annualized yield, which equals yield* = ( 1 + yield / m)
m
-1
z Typically must solve using a computer
» E.g., IRR function in excel or your calculator since:
()()
NN
myield
F
myield
myield
c
B
/1/1
1

1
/
+
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
−=
()()
NN
myield
F
myield
myield
c
B
/1/1
1
1
/
+
+
⎟
⎟

⎠
⎞
⎜
⎜
⎝
⎛
+
−=
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Copyright © Michael R. Roberts
The Yield Curve Revisited
z Treasury Coupon-Paying Yield Curve
» Often referred to as “the yield curve”
» Same idea as the zero-coupon yield curve except we use the
yields from coupon paying bonds, as opposed to zero-
coupon bonds.
– Treasury notes and bonds are semi-annual coupon paying bonds
» We often use On-the-Run Bonds to estimate the yields
– On-the-Run Bonds are the most recently issued bonds
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Copyright © Michael R. Roberts
Interest Rate Sensitivity
Duration
z The Duration of a security is the percent sensitivity of the
price to a small parallel shift in the level of interest rates.
» A small uniform change dy across maturities might by 1 basis point.
» Duration gives the proportionate decline in value associated with a rise
in yield

» Negative sign is to cancel negative first derivative
z Alternatively, given a duration DB of a security with price B, a
uniform change in the level of interest rates brings about a
change in value of
1
B
dB
Duration D
Bdy
==−
B
dB D dy B
=
−××
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Copyright © Michael R. Roberts
Duration of a Coupon Bond
z The mathematical expression for Duration is:
which we can rearrange
()
1
1
1
111
1/ (1/)
N
n
N

n
n
dB
nc ym NF ym
Bdy Bm
−−
−−
=
⎡⎤
−= ⋅⋅+ +⋅⋅+
⎢⎥
⎣⎦
∑
()
()
()
N
() ()
1
1
1
1
Time in Years
"Weight" on
until n payment
n payment
1/
(1 / )
1/
1/

th
th
n
N
N
n
n
N
n
n
cym
nNFym
Dym
mB mB
PV c PV F
nN
ym
mBmB
−
−
−
=
−
=
⎡
⎤
⋅+
⋅+
=+ ⋅ + ⋅
⎢

⎥
⎢
⎥
⎣
⎦
⎡⎤
⎢⎥
⎢⎥
=+ ⋅ + ⋅
⎢⎥
⎢⎥
⎢⎥
⎣⎦
∑
∑
 
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Copyright © Michael R. Roberts
Duration of a Coupon Bond
Example
z Compute the duration of a two-year, semi-annual, 10%
coupon, par bond, with face value of $100.
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More on Duration
z Duration is a linear operator: D(B
1
+ B

2
) = D(B
1
) + D(B
2
)
» The duration of a portfolio of securities is the value-weighted sum of
the individual security durations
» DVO1 is also a linear operator
z Duration is a local measure
» Based on slope of price-yield relation at a specific point
» Based on a bond of fixed maturity but maturity declines over time
z Duration of a zero is
()
1
1/
N
Dym
m
−
=+
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Copyright © Michael R. Roberts
Duration Matching
Example
z Bank of Philadelphia balance sheet (Figures in $billions, D=duration
assuming flat spot rate curve)
z Duration of liabilities =
z The problem:
» Increases in interest rates will decrease value of liabilities by more than assets

because of duration mismatch.
Liabilities & Shareholders EquityAssets
$25 Total Liabilities (D = ?)25Total Assets (D = 1)
$5 Shareholder Equity
$10 2-Year Notes (D = 1.77)
$10 Commercial Paper (D = 0.48)
?
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Copyright © Michael R. Roberts
Duration Matching
Example (Cont.)
z What is the change in assets value when interest rates change
uniformly
z What is the change in liability value when interest rates
change uniformly
z We want our assets and liabilities to experience similar value
changes when interest rates change, so set these two
expressions to be equal and solve for D
L
(D
A
=1.0):
?
?
?
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Copyright © Michael R. Roberts
Duration Matching

Example (Cont.)
z What fraction of the bank’s liabilities should be in CP and
Notes in order to get a liability duration of 1.25
z How much money should the bank hold in CP and Notes in
order to get a liability duration of 1.25
z How should the bank alter their liabilities to achieve this
structure
?
?
?
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Copyright © Michael R. Roberts
Forward Rates
z A forward rate is a rate agreed upon today, for a loan that is
to be made in the future. (Not necessarily equal to the future
spot rate!)
»f
2,1
=7% indicates that we could contract today to borrow money at 7%
for one year, starting two years from today.
z Example: Consider the following term structure
r
1
=5.00%, r
2
=5.75%, r
3
=6.00%

» Consider two investment strategies:
1. Invest $100 for three years Æ how much do we have?
2. Invest $100 for two years, and invest the proceeds at the one-year forward
rate, two periods hence Æ how much do we have?
» When are these two payoffs equal? (i.e. what is the implied forward
rate?)
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Copyright © Michael R. Roberts
Forward Rates
z Strategy #1: Invest $100 for three years Æ how much do we
have
z Strategy #2: Invest $100 for two years and then reinvest the
proceeds for another year at the one year forward rate, two
periods hence Æ how much do we have
z When are these two payoffs equal? (i.e. what is the implied
forward rate?)
?
?
?
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Copyright © Michael R. Roberts
Arbitraging Forward Rates
Example
z What if the prevailing forward rate in the market is 7%, as
opposed to what calculated in the previous slide?
z Step 1: Is there a mispricing and, if so, what is mispriced
z Step 2: Is the forward loan cheap or expensive
z Step 3: Given your answer to Step 2, what is the first step in

taking advantage of the mispricing
?
?
?
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Copyright © Michael R. Roberts
Arbitraging Forward Rates
Example
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Copyright © Michael R. Roberts
General Forward Rate Relation
z Forward rates are entirely determined by spot rates (and vice
versa) by no arbitrage considerations.
z General Forward Rate Relation: (1+r
n+t
)
n+t
=(1+r
n
)
n
(1+f
n,t
)
t
z Think of this picture for intuition:
Time 0 1 2
(1+r

2
)
2
(1+r
3
)
3
(1+f
2,1
)
(1+f
2,1
)(1+f
1,1
)(1+r
1
)
(1+r
1
)
(1+f
1,2
)
2
3
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Copyright © Michael R. Roberts
Summary
z Bonds can be valued by discounting their future cash flows
z Bond prices change inversely with yield

z Price response of bond to interest rates depends on term to
maturity.
» Works well for zero-coupon bond, but not for coupon bonds
z Measure interest rate sensitivity using duration.
z The term structure implies terms for future borrowing:
» Forward rates
» Compare with expected future spot rates