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Fina2802: Investments and Portfolio Analysis
Spring, 2008
Dragon Tang
Lecture 11
Bond Prices/Yields and Yield Curve
February 25/26, 2008
Readings: Chapter 14
Practice Problem Sets: 4,5,7,8,9,10,13,35,36
FIN 2802, Spring 08 - Tang
Chapter 15: Yield Curve
1
Bond Prices and Yields
Objectives:
1. Analyze the relationship between bond prices and and
bond yields.
2. Calculate how bond prices will change over time for a
given interest-rate projection.
3. Identify the determinants of bond safety and rating.
4. Analyze how callable, convertible, and sinking fund
provisions will affect a bond's equilibrium yield to
maturity.
5. Define the yield curve and study its properties
FIN 2802, Spring 08 - Tang
Chapter 15: Yield Curve
2
Bond Characteristics
• Face or par value
• Coupon rate
– Zero coupon bond
• Compounding and payments
– Accrued Interest
• Indenture
FIN 2802, Spring 08 - Tang
Chapter 15: Yield Curve
3
Provisions of Bonds
•
•
•
•
•
FIN 2802, Spring 08 - Tang
Secured or unsecured
Call provision
Convertible provision
Put provision (putable bonds)
Floating rate bonds
Chapter 15: Yield Curve
4
Bond Ratings
Moody’s
S&P
Aaa(1,2,3)
Aa
A
Baa
AAA
AA
A
BBB
Ba
B
Caa
Ca
C
D
BB
B
CCC
CC
C
D
FIN 2802, Spring 08 - Tang
Investment grade
Speculative grade
(junk bonds)
Chapter 15: Yield Curve
5
Present Value
Value today of an amount to be received
in the future
FVn
PVn 
n
(1  r )
FVn = payment received at time n
n = number of periods
r = interest rate per period
PVn = Present value
FIN 2802, Spring 08 - Tang
Chapter 15: Yield Curve
6
Present Value. Example
Example: What is the present value of $1 to be received (or paid)
in 10 years if the annual interest rate is 5%?
FIN 2802, Spring 08 - Tang
Chapter 15: Yield Curve
7
Bond Pricing
Bond value =Present value of coupons +
present value of par value
Bond
T
Coupon Par value


t
T
value t 1 1  r 
1  r 
 Coupon  Annuityfactor (r, T)  Par Value  PV factor (r, T)
FIN 2802, Spring 08 - Tang
Chapter 15: Yield Curve
8
Bond Pricing
Example: Find the value (price) of a 30-year semiannual $1,000
par bond with coupon rate of 8% when similar bonds are also
paying 8%. Using formula or PRICE function in Excel.
FIN 2802, Spring 08 - Tang
Chapter 15: Yield Curve
9
Bond Pricing
Example: Find the value (price) of a 30-year semiannual $1,000
par bond with coupon rate of 8% when similar bonds are also
paying 10% (I.e., 5% semi-annually).
FIN 2802, Spring 08 - Tang
Chapter 15: Yield Curve
10
Bond Pricing
Market interest rates rise bond prices fall
FIN 2802, Spring 08 - Tang
Chapter 15: Yield Curve
11
Bond Pricing
Example: Find the value (price) of a 30-year semiannual $1,000
par bond with coupon rate of 8% when similar bonds are also
paying 6% (I.e., 3% semi-annually).
FIN 2802, Spring 08 - Tang
Chapter 15: Yield Curve
12
Bond Pricing
Market interest rates fall bond prices rise
FIN 2802, Spring 08 - Tang
Chapter 15: Yield Curve
13
Bond Pricing and Market Interest Rates
Market
Maturity Coupon Interest
Value
Rate
Rate
Bond
Price
$ 1,000
8%
8%
$1,000.00
$ 1,000
8%
10%
$ 810.71
$ 1,000
8%
6%
$1,276.75
FIN 2802, Spring 08 - Tang
Chapter 15: Yield Curve
14
Bond Pricing: Convexity
Convexity - An increase in the interest rate results in a price
decline that is smaller than the price gain that results from a
decrease of equal magnitude in the interest rate.
Bond
Price
Flatter
Market Interest Rate
FIN 2802, Spring 08 - Tang
Chapter 15: Yield Curve
15
Bond Pricing: Maturity and Coupon
The longer the time to maturity/lower coupon
rate of a bond, the greater the sensitivity of
price fluctuations to the interest rate.
Maturity
1yr
10yrs
20yrs
30yrs
FIN 2802, Spring 08 - Tang
4%
$1,038
1,327
1,547
1,695
Price of a 8% Coupon Bond when Mkt
Interest Rate is
6%
8%
10%
12%
$1,019 $1,000
$981
$963
1,148
1,000
875
770
1,231
1,000
828
699
1,276
1,000
810
676
Chapter 15: Yield Curve
16
Yield to Maturity
Yield to maturity is a measure of the average
rate of return that will be earned if the bond is
held to maturity.
YTM assumes reinvestment of the coupons at
the yield to maturity.
YTM is the solution (IRR) of :
T
coupont
par
Bond Price  

t
T
(
1

YTM
)
(
1

YTM
)
t 1
FIN 2802, Spring 08 - Tang
Chapter 15: Yield Curve
17
Yield to Maturity
Example: Find the YTM of a semiannual bond with a price =
$1,276.76, 30 years left to maturity, and a coupon rate of 8%.
(Use trial and error method. In Excel: use Solver or YIELD
function).
FIN 2802, Spring 08 - Tang
Chapter 15: Yield Curve
18
Corporate Bonds
Promised versus Expected Yield
• The promised or stated yield is the maximum
possible return if held to maturity.
• The expected yield weights the promised yield by
the probability of future default.
• The default premium is the differential in
promised yield between a corporate bond and a
government bond.
FIN 2802, Spring 08 - Tang
Chapter 15: Yield Curve
19
Holding Period Return
(B t 1 - B t )  Coupon t
HPR 
Bt
Where
Bt = Bond Price at time t
Bt+1= Bond Price at time t+1
FIN 2802, Spring 08 - Tang
Chapter 15: Yield Curve
20
Bond Prices over Time
Coupon bonds
•Premium bonds
move towards par
as they approach
maturity
•Discount bonds
move towards par
as they approach
maturity
FIN 2802, Spring 08 - Tang
Chapter 15: Yield Curve
21
Spot Rates and the Term Structure
• Spot rate: rate used to discount a cash flow
at a point in time – YTM on a zero-coupon
bond maturing at that time
• Term structure of interest rates, represents
the variation of spot rates over terms to
maturity
• Yield curve often refers to the variation of
YTM over terms to maturity
FIN 2802, Spring 08 - Tang
Chapter 15: Yield Curve
22
Treasury Yield Curves
FIN 2802, Spring 08 - Tang
Chapter 15: Yield Curve
23
Theories of Term Structure of Interest Rates
• If future interest rate is uncertain, we need a
theory of the term structure of interest rates to
explain changes in bond prices and in turn in
interest rates
• Three major theories of yield curve
– Expectation Hypothesis
– Liquidity Preference Hypothesis
– Segmented Market Hypothesis
– Combination (Synthesis)
FIN 2802, Spring 08 - Tang
Chapter 15: Yield Curve
24
The Expectation Hypothesis
Example: Suppose that 2-year bonds offer yields to
maturity of 6%, and 3-year bonds have yields of 7%.
What is the expected one-period rate for the third
year?
FIN 2802, Spring 08 - Tang
Chapter 15: Yield Curve
25
Forward Rates
• From the current term structure of interest rates, it is
possible to glean the market expectation of future interest
rates. These expected future spot rates are called forward
rates .
• With forward rates, we can estimate bond prices at a future
date.
• Forward rates are used most often in pricing fixed income
derivatives, and in particular, swap contracts.
FIN 2802, Spring 08 - Tang
Chapter 15: Yield Curve
26
Figure 15.2 Two 2-Year Investment Programs
FIN 2802, Spring 08 - Tang
Chapter 15: Yield Curve
27
Calculating Forward Rates from the Term Structure
• Consider two alternative investments:
– Buy a 3-year zero-coupon bond
– Buy a 2-year zero-coupon bond and then after 2 years
buy another 1-year zero-coupon bond
• Investors will be indifferent among these two
alternatives if they have the same expected return; i.e., if
we end up with $100 either way, the current initial
investment should be the same.
FIN 2802, Spring 08 - Tang
Chapter 15: Yield Curve
28
Calculating Forward Rates from Term Structure (cont.)
• Alternative one:
100
– The cost is
(1  y3 ) 3
• Alternative two
– The cost is
100
(1  y2 ) 2 (1  f 3 )
• Costs in two alternatives should equal, this implies
(1  y2 ) (1  f 3 )  (1  y3 )
2
3
(1  y3 )
1  f3 
(1  y2 ) 2
3
FIN 2802, Spring 08 - Tang
Chapter 15: Yield Curve
29
An Example
• 2-year spot rate: 8.995%
• 3-year spot rate: 9.660%
• 1-year forward rate 2 years from now is
(1  y3 )
f3 
 1  11%
2
(1  y2 )
3
FIN 2802, Spring 08 - Tang
Chapter 15: Yield Curve
30
Figure 15.3 Short Rates versus Spot Rates
FIN 2802, Spring 08 - Tang
Chapter 15: Yield Curve
31
FIN 2802, Spring 08 - Tang
Chapter 15: Yield Curve
32
FIN 2802, Spring 08 - Tang
Chapter 15: Yield Curve
33
Forward rate
• Frequently, you will see forward rate expressed as f(t,t).
This indicates a yield on a forward loan agreement struck
today. The loan period is from future time t until t+t.
• Generally,
t t


1

y
t t
1  f (t ,t ) t 
, for annual compoundin g
t
1  yt 
2 t t 

1  yt t / 2
2t
1  f (t ,t ) / 2 
, for semi - annual compoundin g
2t
1  yt / 2
FIN 2802, Spring 08 - Tang
Chapter 15: Yield Curve
34
Future Bond Prices and Forward Rates
• Recall that when we need to calculate the expected bond
price sometime in the future, we need to know the
expected spot rates at that time
Pf 
FIN 2802, Spring 08 - Tang
2 n  2 hp

t
1
Pp
Ct 2

t
2 hp
(1  f (t ,t ) 2)
(1  f (t ,2hp) 2)
Chapter 15: Yield Curve
35
Expectation Hypothesis
• Current forward rate equals the market consensus
expectation of future short term interest rate.
– If the current term structure of interest rates is upward
(downward) sloping, then the prediction of future spot
rates is that they will be increasing (decreasing).
– Example: 6 Mo. spot rate = 6.03, 1 year spot rate =
6.23 then 6-mo. forward rate in 6 mo. is 6.43.
• Current long-term rate is a geometric mean of current and
future short-term interest rates expected to prevail over the
maturity.
• This implies that an investor is indifferent to holding either
short term or long term bonds.
FIN 2802, Spring 08 - Tang
Chapter 15: Yield Curve
36
Liquidity Preference Hypothesis
• Suppose an investor, who prefers liquidity, prefers short
term bonds.
– There are two bonds available, a one-year zero with yield 4.5%,
and a two-year zero 5.0%. As she prefers the one-year bond, after
one year, she reinvests in another one-year zero at the thenprevailing yield. The expected return from holding two
consecutive one-year bonds is 1  4.5%1  E[r ],
and the expected return from holding the two-year zero is
1  5%2  (1  4.5%)(1 
f (1)).
– Then to entice her to hold the two-year bond, it has to be that
f (1)  E[r ]
i.e., there is a positive liquidity premium for
the long bond.
– This theory assumes that short-horizon investors dominate in the
market? (What if long-horizon investors dominate?)
FIN 2802, Spring 08 - Tang
Chapter 15: Yield Curve
37
Segmented Market Hypothesis
• Individual and institutional investors have their preferred
maturities as they hedge against their liabilities – Preferred
Habitat Theory
• Long-term bonds and short-term bonds are traded in
essentially distinct or segmented markets
• No longer a popular view, as market participants (and
arbitrageurs) invest in all segments to search for more
profitable opportunities.
FIN 2802, Spring 08 - Tang
Chapter 15: Yield Curve
38
Yield Curve. Combination of Theories
Example: Suppose the liquidity preference theory predicted
a 4% rate for 1-year securities and a 6% rate for 2-year
securities. However, investors expected the interest rate in
year 2 fall by 2%.
Liquidity Preference Yield Curve
Yield
6%
4%
Combined Yield Curve
1
FIN 2802, Spring 08 - Tang
2
Maturity
Chapter 15: Yield Curve
39
“Modern” Term Structure Theories
• Using mathematical models to describe the movement
of interest rates directly.
• Assuming that one or several factors driving the
interest rate movement as well as the evolution of the
term structure
• Can be used to price interest rate derivatives, such as
options on bonds, swaps, etc.
FIN 2802, Spring 08 - Tang
Chapter 15: Yield Curve
40
Summary and Chapter Problems
• Basic features of bonds
• Bond yields, in particular, yield-to-maturity
• Default risk and credit spreads
•
•
•
•
Spot rates and the term structure
Forward rates
Theories of the term structure
Next class: managing bond portfolios
FIN 2802, Spring 08 - Tang
Chapter 15: Yield Curve
41
Summary
•Inverse relationship between bond prices and
bond yields
•Premium and discount bonds
•Corporate bonds and default risk
•Term structure of interest rates
Expectations theory
Liquidity preference theory
•Next Class: Managing Bond Portfolios
FIN 2802, Spring 08 - Tang
Chapter 15: Yield Curve
42