Bond Valuation
FINANCE 350
Global Financial Management
Professor Alon Brav
Fuqua School of Business
Duke University
1
Bond Valuation: An Overview
• Bond Markets
– What are they? How big? How important?
• Valuation of bonds
– Zero-coupon bond
– Coupon bonds
• Interest rate sensitivity
• The term structure of interest rates
2
Definition of a Bond
A bond is a security that obligates the issuer to make specified
interest and principal payments to the holder on specified dates.
•
– Coupon rate
– Face value (or par)
– Maturity (or term)
Bonds are also called fixed income securities.
Bonds differ in several respects:
•
•
–
–
–
–
–
Repayment type
Issuer
Maturity
Security
Priority in case of default
3
Types of Bonds: Repayment
•
Pure Discount or Zero-Coupon Bonds
– Pay no coupons prior to maturity.
– Pay the bond’s face value at maturity.
•
Coupon Bonds
– Pay a stated coupon at periodic intervals prior to maturity.
– Pay the bond’s face value at maturity.
•
Floating-Rate Bonds
– Pay a variable coupon, reset periodically to a reference rate.
– Pay the bond’s face value at maturity.
•
Perpetual Bonds (Consols)
– No maturity date.
– 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.
4
Types of Bonds: Issuers
Bonds
Government Bonds
Mortgage-Backed Securities
Municipal Bonds
Corporate Bonds
Asset-Back Securities
Issuer
US Treasury, Government Agencies
Government agencies (GNMA etc)
State and local government
Corporations
Corporations
5
U.S. Government Bonds
•
Treasury Bills
– No coupons (zero coupon security)
– Face value paid at maturity
– Maturities up to one year
•
Treasury Notes
– Coupons paid semiannually
– Face value paid at maturity
– Maturities from 2-10 years
6
U.S. Government Bonds
•
Treasury Bonds
–
–
–
–
•
Coupons paid semiannually
Face value paid at maturity
Maturities over 10 years
The 30-year bond is called the long bond.
Treasury Strips
– Zero-coupon bond
– Created by “stripping” the coupons and principal from Treasury
bonds and notes.
•
•
•
•
No default risk. Considered to be risk free.
Exempt from state and local taxes.
Sold regularly through a network of primary dealers.
Traded regularly in the over-the-counter market.
7
Mortgage and Municipal Bonds
•
Agencies Bonds: Mortgage-Backed Bonds
– Bonds issued by U.S. Government agencies that are backed by a
pool of home mortgages.
– Self-amortizing bonds. (mostly monthly payments)
– Maturities up to 30 years.
– Prepayment risk.
•
Municipal Bonds
– Maturities from one month to 40 years.
– Exempt from federal, state, and local taxes.
– Generally two types:
• Revenue bonds
• General Obligation bonds
– Riskier than U.S. Government bonds.
8
Corporate Bonds
•
Bonds issued by corporations
–
–
–
–
–
Bond indentures or covenants.
Seniority: Secured bonds; Debentures.
Fixed-rate versus floating-rate bonds.
Investment-grade vs. below investment-grade bonds.
Additional features:
• call provisions
• convertible bonds
• puttable bonds
9
Seniority of Corporate Bonds
•
•
In case of default, different classes of bonds have different
claim priority on the assets of a corporation.
Secured Bonds (Asset-Backed)
– Secured by real property.
– Ownership of the property reverts to the bondholders upon default.
•
Debentures
– Same priority as general creditors.
– Have priority over stockholders, but subordinate to secured debt.
10
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
Baa
BBB
Medium quality. Currently adequate, but potentially unreliable.
High-Medium quality. Strong attributes, but potentially vulnerable.
Ba
BB
Some speculative element. Long-run prospects questionable.
B
B
Caa
CCC
Ca
CC
High speculative quality. May be in default.
C
C
Lowest rated. Poor prospects of repayment.
D
-
In default.
Able to pay currently, but at risk of default in the future.
Poor quality. Clear danger of default.
11
The US Bond Market:
Amount ($bil.). Source: U.S. Federal Reserve (Table L.4, September/2006)
Debt Instrument
2006 Q2
Treasury securities
4,759.6
Municipal securities
2,305.7
Corporate and foreign bonds
8,705.3
Consumer Credit
2,327.4
Mortgages
12,757.7
Corporate equities
18,684.5
Mutual fund shares
6,406.4
12
A Few Bond Markets Statistics
U.S. Treasuries, May 20th 2007.
U.S. Treasuries
Bills
3-Month
6-Month
MATURITY
DATE
08/16/2007
11/15/2007
DISCOUNT/YIELD
4.72 / 4.84
4.78 / 4.98
DISCOUNT/YIELD
CHANGE
0.01 / .010
0.01 / .015
TIME
13:41
13:41
Notes/Bonds
COUPON
2-Year
3-Year
5-Year
10-Year
30-Year
4.500
4.500
4.500
4.500
4.750
MATURITY
DATE
04/30/2009
05/15/2010
04/30/2012
05/15/2017
02/15/2037
CURRENT
PRICE/YIELD
99-121⁄4 / 4.84
99-081⁄2 / 4.77
98-281⁄2 / 4.75
97-15 / 4.82
96-17+ / 4.97
PRICE/YIELD
CHANGE
-0-02 / .035
-0-031⁄ 2 / .040
-0-06 / .043
-0-091⁄ 2 / .038
-0-17 / .035
TIME
14:08
14:06
14:07
14:07
14:07
13
Term Structure, May 20th, 2007
14
Bond Valuation: Zero Coupon Bonds
B
F
R
m
i
T
N
Market price of the Bond
Face value
Annual percentage rate
compounding period (typical: semiannual)
Effective periodic interest rate; i=R/m
Maturity (in years)
Number of compounding periods; N = T*m
• Two cash flows to purchaser of bond:
– B0 at time 0
– F at time T
• What is the price of a bond?
Use present value formula:
B0 =
F
(1 + i )N
15
Valuing Zero Coupon Bonds
• What is the current market price of a U.S. Treasury strip that
matures in exactly 5 years and has a face value of $1,000.
The APR is R=7.5% (annual compounding)?
1,000
= $696.56
1.075 5
• What is the APR on a U.S. Treasury strip that pays $1,000
in exactly 7 years and is currently selling for $591.11
(annual compounding)?
591.11 =
1,000
1,000
R=7
1 = 7.8%
7
591.11
(1 + R )
We also call R the yield to maturity.
16
Bond Prices and Interest Rates
The case of zero coupon bonds
• Consider the following 1, 2 and 10- year zero-coupon bonds, all with face
value of F=1000.
– APR of R=10%, compounded annually.
We obtain the following table for increases and decreases of the interest
rate by 1%:
Interest Rate
9.0%
10.0%
11.0%
Bond 1
1-Year
$917.43
$909.09
$900.90
Bond 2
2-Year
$841.68
$826.45
$811.62
Bond 3
10-Year
$422.41
$385.54
$352.18
• Bond prices move up if interest rates drop, decrease if interest rates rise
17
Bond Prices and Interest Rates
• Bond prices are
inversely related to
interest rates
• Longer term bonds
are more sensitive
to interest rate
changes than short
term bonds
• The lower the IR,
the more sensitive
the price.
18
Measuring Interest Rate Sensitivity
Zero Coupon Bonds
• We would like to measure the interest rate sensitivity of a bond
or a portfolio of bonds.
– How much do bond prices change if interest rates change by a small amount
– Why is this important?
• Use “Dollar value of a one basis point decrease” (DV01):
– Basis point (bp): 1/100 of one percentage point =0.01%=0.0001
– Calculate DV01:
• Method 1: Difference of moving one basis point down:
DV01= B(R-0.01)-B(R).
• Method 2: Difference of moving 1/2bp down minus 1/2pb up:
DV01=B(R-0.005%) -B(R+0.005%).
• Method 3: Use calculus: DV 01 = dB 1
dR 10,000
19
Computing DV01: An Example
• Reconsider the 1, 2 and 10- year bonds discussed before:
Interest Rate
Bond 1
1-Year
$909.1736
$909.1322
$909.0909
$909.0496
$0.082652
$0.082645
$0.082645
9.990%
9.995%
10.000%
10.005%
Method 1
Method 2
Method 3
dB
1
$1,000
Bond 2
2-Year
$826.5966
$826.5214
$826.4463
$826.3712
$0.150283
$0.150263
$0.150263
1
Bond 3
10-Year
$385.8940
$385.7186
$385.5433
$385.3681
$0.350669
$0.350494
$0.350494
1
• Method 3: dR 10,000 = T 1.10T +1 10,000 = T * $0.10 * 1.10T +1
20
DV01: A Graphical Approach
• DV01 estimates the slope of the line on the Price-Interest rate curve.
higher slope implies greater sensitivity
21
Prices of Coupon Bonds:
Example 1: Amortization Bond
• Consider Amortization Bond
–
–
–
–
T=2
m=2
C=$2,000 c = C/m = $2,000/2 = $1,000
R=10% i = R/m = 10%/2 = 5%
• How can we value this security?
– Brute force discounting
– Similar to another security we already know how to value?
– Replication
22
Prices of Coupon Bonds:
Example 1: Amortization Bond
• Compare with a portfolio of zero coupon bonds:
Period\Price
1
2
3
4
Coupon Bond
$3,545.95
$1,000.00
$1,000.00
$1,000.00
$1,000.00
Zero 1
$952.38
$1,000.00
$0.00
$0.00
$0.00
Zero 2
$907.03
$0.00
$1,000.00
$0.00
$0.00
Zero 3
$863.84
$0.00
$0.00
$1,000.00
$0.00
Zero 4
$822.70
$0.00
$0.00
$0.00
$1,000.00
23
A First Look at Arbitrage
• Reconsider amortization bond; suppose bond trades at $3,500
– Can make risk less profit
• Buy low: buy amortization bond
• Sell high: Sell portfolio of zero coupon bonds
Period\Price
1
2
3
4
Coupon Bond
Zero 1
Zero 2
Zero 3
Zero 4
($3,500.00)
$952.38
$907.03
$863.84
$822.70
$1,000.00 ($1,000.00)
$0.00
$0.00
$0.00
$1,000.00
$0.00 ($1,000.00)
$0.00
$0.00
$1,000.00
$0.00
$0.00 ($1,000.00)
$0.00
$1,000.00
$0.00
$0.00
$0.00 ($1,000.00)
Total
$45.95
$0.00
$0.00
$0.00
$0.00
• riskless profit of $45.95
• no riskless profit if price is correct
24
Valuation of Coupon Bonds:
Example 2: Straight Bonds
• 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
18
24 ...
120
45
45
45
45
1045
B=
Months
45 1 1000
1 +
= $937.69
0.05 1.05 20 1.05 20
25
Valuing Coupon Bonds
The General Formula
• What is the market price of a U.S. Treasury 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?
– Semiannual coupon is: c = C/m
– Effective periodic interest rate is: i = R/m
– number of periods N = mT
0
1
2
3
4
c
c
c
c
...
…
c
N
c+F
c
1 F + B = [ Annuity ] + [ Zero] = 1
i (1+ i) N (1+ i) N 26
The Concept of a “Yield to Maturity”
• So far we have valued bonds by using a given interest
rate, then discounted all payments to the bond.
• Prices are usually given from trade prices
– need to infer interest rate that has been used
Definition: The yield to maturity is that interest rate
that equates the present discounted value of all future
payments to bondholders to the market price:
• Algebraic:
1
c
1 yield / m (1 + yield / m )N
B=
F
+
(1 + yield / m )N
27
Yield to Maturity
A Graphical Interpretation
$2,500.00
$2,000.00
$1,500.00
$1,000.00
24%
22%
20%
18%
16%
14%
12%
8%
10%
6%
4%
0%
$0.00
2%
$500.00
•Consider a U.S. Treasury bond that has a coupon rate of 10%, a face value of
$1,000 and matures exactly 10 years from now.
– Market price of $1,500, implies a yield of 3.91% (semi-annual
compounding); for B=$1,000 we obviously find R=10%.
28
Bond Yields and Prices
The case of coupon bonds
• Coupon bonds can be represented as portfolios of zero-coupon bonds
– Implication for price sensitivity
• Suppose you purchase the 9% U.S. Treasury bond described earlier
and immediately thereafter interest rates fall:
– APR on the bond is now 8%, compounded semiannually.
– What is the bond’s new market price?
• Suppose the interest rate rises, so that the new interest rate is 12%
compounded semiannually.
– What is the market price now?
• Suppose the interest equals the coupon rate of 9%. What do you
observe?
• What are the pricing implications of these scenarios?
29
Valuing Coupon Bonds (cont.)
•
New Semiannual interest rate = 8%/2 = 4%
c 1 B = 1 i 1 + i N
+ F
(1 + i )N
•
What is the price of the bond if the APR is 8% compounded
semiannually?
1 1 1,000
B=
1 * 45 +
= $1067.95
0.04 1.04 20 1.04 20
•
Similarly:
If R=12%: B=$827.95
If R= 9%: B=$1,000.00
30
Relationship Between Bond Prices
and Interest Rates
•
Bond prices are inversely related to interest rates (or
yields).
•
A bond sells at par only if its interest rate equals the
coupon rate
A bond sells at a premium if its coupon rate is above the
interest rate.
•
•
A bond sells at a discount if its coupon rate is below the
interest rate.
31
Interest Rate Sensitivity of Coupon Bonds
• Consider two bonds with 10% annual coupons with maturities of
5 years and 10 years.
• The APR is 8%
• What are the responses to a .01% (1bp) interest rate change?
Yield
7.995%
8.000%
8.005%
DV01
5-Year Bond
$1,080.06
$1,079.85
$1,079.64
$ Change
$0.21019
($0.21013)
$0.42032
% Change 10-Year Bond
0.0195% $1,134.57
$1,134.20
-0.0195% $1,133.84
$ Change
$0.36585
% Change
0.0323%
($0.36569)
$0.73154
-0.0322%
• Does the sensitivity of a coupon bond always increase with the
term to maturity?
32
Bond Prices and Interest Rates
$2,500.00
5-Year Bond
10-Year Bond
$2,000.00
$1,500.00
$1,000.00
$500.00
24%
22%
20%
18%
16%
14%
12%
8%
10%
6%
4%
2%
0%
$0.00
Longer term bonds are more sensitive to changes in interest rates
than shorter term bonds (holding constant the bond cashflows).
33
Bond Yields and Prices
• Consider the following two bonds:
– Both have a maturity of 5 years
– Both have yield of 8%
– First has 6% coupon, other has 10% coupon, compounded annually.
• Then, what are the price sensitivities of these bonds,
measured by DV01 as for zero coupon bonds?
Yield
6%-Bond
$ Change
7.995%
$920.33
$0.1891
8.000%
$920.15
8.005%
$919.96
% change
10%-Bond
$ Change
$1,080.06
$0.2102
$1,079.85
($0.1891)
$1,079.64
0.0411%
DV01
$0.3782
% change
($0.2101)
0.0389%
$0.4203
• Why do we get different answers for two bonds with the same yield and
same maturity?
34
Maturity and Price Risk
• Zero coupon bonds have well-defined relationship between
maturity and interest rate sensitivity:
– DV01 is direct function of maturity t.
• Coupon bonds can have different volatilities for the same
maturity
– DV01 now depends on maturity and coupon rate.
– Get different results for 6% coupon and 10% coupon bonds
with same maturity.
• Need concept of “average maturity” of coupon bond:
– Duration
35
Duration
• The logical way to measure sensitivity of the bond price to
changes in interest rates is to take the derivative of the
price B with respect to effective rate i (see slide 22):
B
N
n 1
= n c (1 + i ) + N F (1 + i ) N 1 i
n =1
• We adjust the derivative by dividing by minus the bond
price and the number of periods per year m, and multiply
by one plus the effective rate.
• The measure obtained is often called Macaulay Duration.
36
Duration (cont.)
• If we also replace n/m with Tn -- which will be the time (in
years) until the nth cash flow, the formula is:
Duration = (1 + i ) B 1 N
n
N
= Tn c (1 + i ) + TN F (1 + i )
m B i B n =1
• Duration is a weighted average term to maturity where the cash
flows are in terms of their present value. We can rewrite the
above equation in a simpler format:
Duration = T 1
PV (c )
PV (c )
PV (c )
N + T PV (F)
1 +T 2 +L+ T N
2
N
B
B
B
B
37
Duration (cont.)
• The duration of a bond is less than its time to maturity (except
for zero coupon bonds).
• The duration of the bond decreases the greater the coupon
rate. This is because more weight (present value weight) is
being given to the coupon payments.
• As market interest rate increases, the duration of the bond
decreases. This is a direct result of discounting. Discounting
at a higher rate means lower weight on payments in the far
future. Hence, the weighting of the cash flows will be more
heavily placed on the early cash flows -- decreasing the
duration.
• Modified Duration = Duration / (1+yield)
38
Spot Rates
• A spot rate is a rate agreed upon today, for a loan that is to be
made today
– r1=5% indicates that the current rate for a one-year loan is 5%.
– r2=6% indicates that the current rate for a two-year loan is 6%.
– Etc.
• The term structure of interest rates is the series of spot rates
r1, r2, r3 ,…
– We can build using STRIPS or coupon bond yields.
– Explanations of the term structure.
The Term Structure of Interest Rates
An Example
Yield
6.00
5.75
5.00
1
2
3
Maturity
40
Term Structure, July 1st 2005
41
Term Structure, September 12th 2006
42
Term Structure, May 20th, 2007
43
44
45
Summary
• Bonds can be valued by discounting their future cash flows
• Bond prices change inversely with yield
• Price response of bond to interest rates depends on term to
maturity.
– Works well for zero-coupon bond, but not for coupon
bonds
• Measure interest rate sensitivity using ‘DV01’ and duration.
• The term structure implies terms for future borrowing:
– Forward rates
– Compare with expected future spot rates (Appendix).
46