CHAPTER 14, 15 and 16
Ch 14 Bond Prices and Yields
Ch 15 The Term Structure of Interest Rates
Ch 16 Managing the Bond Portfolios
Comparative Performance of Stocks and Bonds
(1997-2008)
3
Bond Prices and Yields
CHAPTER 14
4
Bond Characteristics
• Bonds are debt – “IOU” securities. Issuers
are borrowers and holders are creditors.
– The indenture is the contract between the
issuer and the bondholder.
– The indenture gives the coupon rate, maturity
date, and par value.
5
Bond Characteristics
• Face or par value is typically $1000; this is the
principal repaid at maturity.
• The coupon rate determines the interest
payment.
– Interest is usually paid semiannually.
– The coupon rate can be zero.
– Interest payments are called “coupon
payments”.
6
U.S. Treasury Bonds
• Bonds and notes may be purchased
directly from the Treasury.
• Maturities vary.
– Note maturity is 1-10 years
– Bond maturity is 10-30 years
• Denomination can be as small as $100,
but $1,000 is more common.
• Bid price of 100:08 means 100 8/32 or
$1002.50
7
8
Corporate Bonds
• Callable bonds can be repurchased before
the maturity date.
• Convertible bonds can be exchanged for
shares of the firm’s common stock.
• Puttable bonds give the bondholder the
option to retire or extend the bond.
• Floating rate bonds have an adjustable
coupon rate
9
List of Corporate Bonds
10
Callable Bonds
• Thus far, we have calculated bond prices
assuming that the actual bond maturity is the
original stated maturity.
• However, most bonds are callable bonds.
• A callable bond gives the issuer the option to buy
back the bond at a specified call price anytime
after an initial call protection period.
• Therefore, for callable bonds, YTM may not be
useful.
11
YTM and YTC
• Suppose a 20-year bond has a coupon of 8 percent, a
price of 98.851, and is callable in 10years. The call
prices 105. What are its yield to maturity and yield to
call?

$80 
1
$988.51 
1

YTM 
1  YTM

2


• YTM = 8.12 percent

$80 
1
$988.51 
1

YTC 
1  YTC

2

• YTC = 8.50 percent


$1,000
 
220 
1  YTM

2


$1,050


210 
1  YTC

2



220
210
Inflation-Indexed Treasury
Securities
• Recently, the U.S. Treasury has issued securities called Treasury
Inflation Protected Securities (TIPS) that guarantee a fixed rate of
return in excess of realized inflation rates .
• That is, they are indexed to inflation in order to protect investors
from the negative effects of inflation.
• TIPS can be purchased directly from the government through the
TreasuryDirect system in $100 increments with a minimum
investment of $100 and are available with 5-, 10-, and 20-year
maturities.
• You can hold a TIPS until it matures or sell it in the secondary
market before it matures.
12
13
Example: TIPS
•
Their par value rises with inflation and falls with deflation, as measured
by the CPI.
– TIPS pay interest twice a year, at a fixed coupon rate on their
current principal
– But, principal are adjusted semiannually according to the most
recent inflation rate.
– So, like the principal, interest payments rise with inflation and fall
with deflation.
– When a TIP matures, you are paid the adjusted principal or original
principal, whichever is greater.
•
Suppose an inflation-indexed note is issued with a coupon rate of 3.5%
and an initial principal of $1,000.
– Six months later, the note will pay a coupon of $1,000 × (3.5%/2) =
$17.50.
– Assuming 2 percent inflation over the six months since issuance,
the note’s principal is then increased to $1,000 × 102% = $1,020.
– Six months later, the note pays $1,020 × (3.5%/2) = $17.85
– Its principal is again adjusted to compensate for recent inflation.
Principal and Interest Payments for a
Treasury Inflation Protected Security
14
15
Bond Pricing
PB = Price of the bond
Ct = interest or coupon payments
T = number of periods to maturity
r = semi-annual discount rate or the semi-annual
yield to maturity
ParValue
C
PB  

T
t
(1 r )
t 1 (1 r )
T
16
Bond Pricing
Price of a 30 year, 8% coupon bond
with semi-annual coupon payments.
Market rate of interest is 10%.
60
$40
$1000
Price  

t
60
1.05
t 1 1.05
Price  $810.71
17
Bond Prices and Yields
• Prices and yields (required rates of return) have
an inverse relationship
• The bond price curve is convex.
– For a given absolute change in a bond’s YTM, the
magnitude of the price increase caused by a decrease in
yield is greater than the price decrease caused by an
increase in yield.
• The longer the maturity, the more sensitive the
bond’s price to changes in market interest rates.
• The lower the coupon rate, the more sensitive
the bond’s price to changes in market interest
rates.
The Inverse Relationship Between Bond
Prices and Yields
18
Bond Prices at
Different Interest Rates
19
20
Yield to Maturity
• Interest rate that makes the present value
of the bond’s payments equal to its price is
the YTM.
Solve the bond formula for r
ParValue
C
PB  

T
t
(1 r )
t 1 (1 r )
T
21
Yield to Maturity Example
Suppose an 8% coupon paid semiannually,
30 year bond is selling for $1276.76. What is
its average rate of return?
$40 1000
$1276.76  

60
t
(1 r )
t 1 (1 r )
60
r = 3% per half year
Bond equivalent yield = 6%
EAR = (1.03)2 - 1 = 6.09%
22
YTM vs. Current Yield
YTM
Current Yield
• The YTM is the bond’s
internal rate of return.
• YTM is the interest rate
that makes the present
value of a bond’s
payments equal to its
price.
• YTM assumes that all
bond coupons can be
reinvested at the YTM
rate.
• The current yield is the
bond’s annual coupon
payment divided by the
bond price.
• For bonds selling at a
premium, coupon rate >
current yield>YTM.
• For discount bonds,
relationships are
reversed.
23
Yield to Call
• If interest rates fall, price of straight bond
can rise considerably.
• The price of the callable bond is flat over a
range of low interest rates because the
risk of repurchase or call is high.
• When interest rates are high, the risk of
call is negligible and the values of the
straight and the callable bond converge.
Bond Prices:
Callable and Straight Debt
24
Realized (Horizon) Yield vs.
Promised YTM
• YTM is a promised yield, which may not equal the realized yield.
• Two assumptions with YTM
– YTM assumes the investor’s holding bond until maturity.
– YTM assumes that coupons are reinvested at YTM.
• That is, YTM will equal the realized return only if all coupons are
reinvested at YTM and investors hold the bond until maturity..
• Two questions on the Promised YTM.
– What if the investor does not hold the bond until maturity?
– What if the investor is unable to reinvest coupons at the promised YTM?
• Horizon Yield or Holding Period Return (HPR) measure the
expected rate of return of a bond that you expect to sell prior to its
maturity with varying reinvestment rates.
25
26
YTM vs. HPR
YTM
• YTM is the average
return if the bond is held
to maturity.
• YTM depends on coupon
rate, maturity, and par
value.
• All of these are readily
observable.
HPR
• HPR is the rate of
return over a particular
investment period.
• HPR depends on the
bond’s price at the end
of the holding period,
an unknown future
value.
• HPR can only be
forecasted.
27
Growth of Invested Funds
28
Example
• Suppose we bought a par bond with 14%, 20-year
bond.
– This means that YTM is 14%.
• Assume the holding period is 2 years.
• Assumes that the market interest rate is expected
to decline to 10 percent.
• Expected price at the end of year 2
– N = 36, I = 10 (2P), PMT = 70, FV = 1000
– Price at the end of year 2 = $1,330.94
• Horizon Yield
– N = 4, PV = -1000, PMT = 70, FV = 1330.94,
– I = 27.50%
Prices over Time of 30-Year Maturity,
6.5% Coupon Bonds
29
The Price of a 30-Year Zero-Coupon
Bond over Time
30
31
Default Risk and Bond Pricing
• Rating companies:
– Moody’s Investor Service, Standard & Poor’s,
Fitch
• Rating Categories
– Highest rating is AAA or Aaa
– Investment grade bonds are rated BBB or Baa
and above
– Speculative grade/junk bonds have ratings
below BBB or Baa.
32
Factors Used by Rating Companies
•
•
•
•
•
Coverage ratios
Leverage ratios
Liquidity ratios
Profitability ratios
Cash flow to debt
Financial Ratios and Default Risk by
Rating Class, Long-Term Debt
33
34
Protection Against Default
• Sinking funds – a way to call bonds early
• Subordination of future debt– restrict
additional borrowing
• Dividend restrictions– force firm to retain
assets rather than paying them out to
shareholders
• Collateral – a particular asset bondholders
receive if the firm defaults
35
Default Risk and Yield
• The risk structure of interest rates refers to
the pattern of default premiums.
• There is a difference between the yield based
on expected cash flows and yield based on
promised cash flows.
• The difference between the expected YTM
and the promised YTM is the default risk
premium.
36
Yield Spreads
37
Credit Default Swaps
• A credit default swap (CDS) acts like an insurance
policy on the default risk of a corporate bond or loan.
• CDS buyer pays annual premiums.
• CDS issuer agrees to buy the bond in a default or pay
the difference between par and market values to the
CDS buyer.
• Institutional bondholders, e.g. banks, used CDS to
enhance creditworthiness of their loan portfolios, to
manufacture AAA debt.
• CDS can also be used to speculate that bond prices
will fall.
• This means there can be more CDS outstanding than
there are bonds to insure!
38
Prices of Credit Default Swaps
Credit Risk and Collateralized Debt
Obligations (CDOs)
• Major mechanism to reallocate credit risk in
the fixed-income markets
– Structured Investment Vehicle (SIV) often
used to create the CDO
– Loans are pooled together and split into
tranches with different levels of default risk.
– Mortgage-backed CDOs were an
investment disaster in 2007
39
40
Collateralized Debt Obligations
41
The Term Structure of Interest Rates
CHAPTER 15
42
Overview of Term Structure
• The yield curve is a graph that displays the
relationship between yield and maturity.
• Information on expected future short term
rates can be implied from the yield curve.
43
Money Market
Interest Rates
44
The Treasury Yield Curve
• Because of its sheer size, the leading world
market for debt securities is the market to
U.S. Treasury securities.
• The Treasury yield curve is a plot of Treasury
yields against maturities.
• It is fundamental to bond market analysis,
because it represents the interest rates for
default-free lending across the maturity
spectrum.
45
Example: The Treasury Yield Curve
46
Treasury Yield Curves
47
The Term Structure of Interest Rates
• The term structure of interest rates is
the relationship between time to maturity
and the interest rates for default-free, pure
discount instruments.
• The term structure is sometimes called the
“zero-coupon yield curve” to distinguish
it from the Treasury yield curve, which is
based on coupon bonds.
48
Two Types of Yield Curves
On-the-run Yield
Pure Yield Curve
Curve
• The pure yield curve
• The on-the-run yield
uses stripped or zero
curve uses recently
coupon Treasuries.
issued coupon bonds
selling at or near par.
• The pure yield curve
may differ significantly • The financial press
from the on-the-run
typically publishes onyield curve.
the-run yield curves.
49
Bond Pricing
• Yields on different maturity bonds are not all
equal.
• We need to consider each bond cash flow
as a stand-alone zero-coupon bond.
• Bond stripping and bond reconstitution
offer opportunities for arbitrage.
• The value of the bond should be the sum
of the values of its parts.
Prices and Yields to Maturities on ZeroCoupon Bonds ($1,000 Face Value)
50
51
Valuing Coupon Bonds
• Value a 3 year, 10% coupon bond using
discount rates from Table 15.1:
$100 $100 $1100
Price 


2
1.05 1.06
1.07 3
• Price = $1082.17 and YTM = 6.88%
• 6.88% is less than the 3-year rate of 7%.
52
Yield Curve Under Certainty
• Suppose you want to invest for 2 years.
– Buy and hold a 2-year zero
- or
– Rollover a series of 1-year bonds
• Equilibrium requires that both strategies
provide the same return.
53
Two 2-Year Investment Programs
54
Yield Curve Under Certainty
• Buy and hold vs. rollover:
(1  y2 )  (1  r1 ) x(1  r2 )
2
1  y2   (1  r1 ) x(1  r2 ) 
1
2
• Next year’s 1-year rate (r2) is just enough
to make rolling over a series of 1-year
bonds equal to investing in the 2-year
bond.
55
Spot Rates vs. Short Rates
• Spot rate – the rate that prevails today for
a given maturity
• Short rate – the rate for a given maturity
(e.g. one year) at different points in time.
• A spot rate is the geometric average of its
component short rates.
56
Short Rates and Yield Curve Slope
• When next year’s
short rate, r2 , is
greater than this
year’s short rate, r1,
the yield curve slopes
up.
– May indicate rates are
expected to rise.
• When next year’s
short rate, r2 , is less
than this year’s short
rate, r1, the yield
curve slopes down.
– May indicate rates are
expected to fall.
57
Short Rates versus Spot Rates
58
Forward Rates from Observed Rates
(1  yn )
(1  f n ) 
n 1
(1  yn 1 )
n
fn = one-year forward rate for period n
yn = yield for a security with a maturity of n
n 1
(1  yn )  (1  yn1 ) (1  f n )
n
59
Forward Rates
• The forward interest rate is a forecast of a
future short rate.
• Rate for 4-year maturity = 8%, rate for 3year maturity = 7%.

1  y4 
1 f4 
3
1  y3 
4
f 4  11.06%
4
1.08

 1.1106
3
1.07
60
Interest Rate Uncertainty
• Suppose that today’s rate is 5% and the
expected short rate for the following year
is E(r2) = 6%. The value of a 2-year zero
is:
$1000
1.051.06
 $898.47
• The value of a 1-year zero is:
$1000
 $952.38
1.05
61
Interest Rate Uncertainty
• The investor wants to invest for 1 year.
– Buy the 2-year bond today and plan to
sell it at the end of the first year for
$1000/1.06 =$943.40.
– Or,
– Buy the 1-year bond today and hold to
maturity.
62
Interest Rate Uncertainty
• What if next year’s interest rate is more (or
less) than 6%?
– The actual return on the 2-year bond is
uncertain!
63
Interest Rate Uncertainty
• Investors require a risk premium to hold a
longer-term bond.
• This liquidity premium compensates shortterm investors for the uncertainty about
future prices.
64
Theories of Term Structure
• Expectations
• Liquidity Preference
– Upward bias over expectations
65
Expectations Theory
• Observed long-term rate is a function of today’s
short-term rate and expected future short-term rates.
• The term structure of interest rates reflects financial
market beliefs about future interest rates.
• Long-term and short-term securities are perfect
substitutes.
• Forward rates that are calculated from the yield on
long-term securities are market consensus expected
future short-term rates.
• fn = E(rn) and liquidity premiums are zero.
66
Liquidity Premium Theory
• Long-term bonds are more risky, and therefore, long-term interest
rates contain a maturity premium necessary to induce lenders
into making longer term loans.
– Long-term bonds are more risky; therefore, fn generally exceeds E(rn)
– The excess of fn over E(rn) is the liquidity premium.
• Investors will demand a premium for the risk associated with
long-term bonds.
• Forward rates contain a liquidity premium and are not equal to
expected future short-term rates.
• The yield curve has an upward bias built into the long-term rates
because of the liquidity premium.
67
Yield Curves
68
Interpreting the Term Structure
• The yield curve reflects expectations of future
interest rates.
• The forecasts of future rates are clouded by
other factors, such as liquidity premiums.
• An upward sloping curve could indicate:
– Rates are expected to rise
– And/or
– Investors require large liquidity premiums to
hold long term bonds.
69
Interpreting the Term Structure
• The yield curve is a good predictor of the
business cycle.
– Long term rates tend to rise in anticipation
of economic expansion.
– Inverted yield curve may indicate that
interest rates are expected to fall and
signal a recession.
Term Spread: Yields on 10-year vs. 90day Treasury Securities
70
71
Dynamic Yield Curve
• http://stockcharts.com/freecharts/yieldcurv
e.html
Inverted Yield Curve = Prognosis for
Recession?
72
Inverted Yield Curve = Prognosis for
Recession?
73
74
Managing the Bond Portfolios
CHAPTER 16
75
Bond Pricing Relationships
1. Bond prices and yields are inversely related.
2. An increase in a bond’s yield to maturity
results in a smaller price change than a
decrease of equal magnitude.
3. Long-term bonds tend to be more price
sensitive than short-term bonds.
76
Bond Pricing Relationships
4. As maturity increases, price sensitivity
increases at a decreasing rate.
5. Interest rate risk is inversely related to the
bond’s coupon rate.
6. Price sensitivity is inversely related to the
yield to maturity at which the bond is selling.
77
Change in Bond Price as a Function of
Change in Yield to Maturity
78
Prices of 8% Coupon Bond (Coupons Paid
Semiannually)
79
Prices of Zero-Coupon Bond
(Semiannually Compounding)
80
Duration
• Bondholders know that the price of their bonds change when interest
rates change. But,
– How big is this change?
– How is this change in price estimated?
• Since price volatility of a bond varies inversely with its coupon and
directly with its term to maturity, it is necessary to determine the best
combination of these two variables to achieve our objective.
• A composite measure considering both coupon and maturity would
be beneficial.
• Macaulay Duration, or Duration, is the name of concept that helps
bondholders measure the sensitivity of a bond price to changes in
bond yields. That is:

Two bonds with the same duration, but not necessarily the same
maturity, will have approximately the same price sensitivity to a
(small) change in bond yields.
81
Macaulay’s Duration or Duration
• Macaulay’s Duration values are stated in years, and are often
described as a bond’s effective maturity.
• The weighted average of the times until each payment is
received, with the weights proportional to the present value of
the payment
– For a zero-coupon bond, duration = maturity.
– For a coupon bond, duration = a weighted average of individual
maturities of all the bond’s separate cash flows
• Duration is shorter than maturity for all bonds except zero
coupon bonds.
• Duration is a measure of interest rate sensitivity or elasticity
of a liability or asset.
82
Macaulay Duration
T
DMAC
CFt
(t )

t
T
T
PV
(
CF
)
(1

y
)
t
 t 1T
 t 
  t  weightt
CFt
price
t 1
t 1

t
t 1 (1  y )
Where
DMAC = duration
t = number of periods in the future
CFt = cash flow to be delivered in t periods
T= time-to-maturity
y = yield to maturity
PV = present value
83
Example: Annual Bond
84
Example: Semi-annual Bond
Duration of 2-year, 8% bond:
Face value = $1,000, YTM = 12%
t
years CFt
PV(CFt)
1
0.5
40
37.736
Weight
(W)
0.041
2
1.0
40
35.600
0.038
0.038
3
1.5
40
33.585
0.036
0.054
4
2.0
1,040 823.777 0.885
1.770
P = 930.698 1.000
W × years
0.020
D=1.883
(years)
85
Durations of Coupon and Zero
coupon Bonds
86
87
Duration/Price Relationship
• Price change is proportional to duration and not to maturity
 (1  y ) 
 y 
P
 D 
 D


P
1

y
1

y




Pct. Change in Bond Price  Duration 
Change in y
1  y 
2

• Some analysts prefer to use a variation of Macaulay’s
Duration, D*, known as Modified Duration.
D* = DMAC / (1+y/2)
Modified Duration 
Macaulay Duration
y

1



 2
88
Modified Duration
• The relationship between percentage changes in bond prices and
changes in bond yields is approximately:
P
  D * y
P
Pct. Change in Bond Price  -Modified Duration  Change in YTM
or
Pct. Change in Bond Pric
 -Modified Duration
Change in YTM
• Duration is a measure of interest rate sensitivity or elasticity of a
liability or asset.
89
Example 1
• Example: Suppose a bond has a Macaulay Duration of 11 years,
and a current yield to maturity of 8%.
• If the yield to maturity increases to 8.50%, what is the resulting
percentage change in the price of the bond?


0.085  0.08 
Pct. Change in Bond Price  - 11
1 0.08 2
 -5.29%.
• The bond’s price will decline by approximately 5.29% in response to
50 basis point increase in yields.
90
Example 2
• Consider two bonds that have the same durations
of 1.8852 years.
– One is a 2-year, 8% coupon bond with YTM=10%.
– The other bond is a zero coupon bond with maturity of
1.8852 years.
• Duration of both bonds is 1.8852 x 2 = 3.7704
semiannual periods.
– Alternatively, keep DMAC = 1.8852 years
• Modified D = 3.7704/1.05 = 3.5909 semiannual
periods.
– Alternatively, DMOD = 1.8852 / 1.05=1.7954 years
91
Example 2, cont’d
• Suppose the semiannual interest rate increases by
0.01% (This means 0.02% annual rate). Bond
prices fall by:
P
  D y
*
P
= -3.5909 x 0.01% = -0.03591%
Alternatively, -1.7954 x 0.02% = -0.03591%
• Both bonds with equal duration have the same
interest rate sensitivity.
92
Example 2, cont’d
Coupon Bond
Zero
• The coupon bond,
which initially sells at
$964.540, falls to
$964.1942 when its
yield increases to
5.01%
• percentage decline of
0.0359%.
• The zero-coupon
bond initially sells for
$1,000/1.05 3.7704 =
$831.9704.
• At the higher yield, it
sells for
$1,000/1.053.7704 =
$831.6717. This price
also falls by 0.0359%.
93
Example 3
• Suppose DMAC = 8 year, YTM = 0.10. Assume that you
expect the bond’s YTM to decline by 75 basis point .
• DMOD = 8/(1+0.10/2) = 7.62
• Percent change in bond price = - DMOD * Change in YTM
• %∆P = -7.62*(-.0075) = 0.0572 or 5.72%
• The bond price should increase approximately 5.72
percent.
94
Example 4
• The 6-year Eurobond with an 8% coupon and 8% yield,
had a duration of DMAC = 4.99 years. If yields rose 1
basis point, then:
• Since C= YTM, it’s a par bond priced at $1,000.
• DMOD = 4.99 / (1+ 8%) = 4.6204, assuming that it is an
annual bond.
• dP/P = -(4.6204) (.0001) = -.000462 or -0.0462%
• To calculate the dollar change in value, rewrite the
equation,
• dP = -DMOD * P *dR = (-4.6204)($1,000)(.0001)= -$0.462
• The bond price falls to $999.538 after a one basis point
increase in yields.
95
Examples 5
96
Calculating Macaulay’s Duration
• In general, for a bond paying constant semiannual
coupons, the formula for Macaulay’s Duration is:
Duration 
1  YTM  MC  YTM 
2
2
2M
YTM

YTM
YTM  C 1 
 1
2


1  YTM


• In the formula, C is the annual coupon rate, M is
the bond maturity (in years), and YTM is the yield
to maturity, assuming semiannual coupons.
Calculating Macaulay’s Duration
for Par Bonds
• If a bond is selling for par value, the
duration formula can be simplified to:

1  YTM 
1
2


Par Value Bond Duration 
1
2M 
YTM 
YTM
1

2 


97
Example
98
99
Calculating Duration Using Excel
• We can use the DURATION and MDURATION
functions in Excel to calculate Macaulay Duration and
Modified Duration.
• The Excel functions use arguments like we saw
before:
=DURATION(“Today”,“Maturity”,Coupon Rate,YTM,2,3)
• You can verify that a 5-year bond, with a 9% coupon
and a 7% YTM has a Duration of 4.17 and a Modified
Duration of 4.03.
Calculating Macaulay and Modified
Duration
100
101
Rules for Duration
Rule 1 The duration of a zero-coupon bond
equals its time to maturity
Rule 2 Holding maturity constant, a bond’s
duration is higher when the coupon rate
is lower
Rule 3 Holding the coupon rate constant,
a bond’s duration generally increases
with its time to maturity
102
Rules for Duration
Rule 4 Holding other factors constant,
the duration of a coupon bond is higher
when the bond’s yield to maturity is
lower
Rules 5 The duration of a level perpetuity
is equal to: (1+y) / y
103
Bond Duration versus
Bond Maturity
104
Bond Durations (Yield to Maturity = 8%
APR; Semiannual Coupons)