Investment Course - 2005
Day Three:
Fixed-Income Analysis and Portfolio Strategies
3-0
The Role of Fixed-Income Securities in the Financial Markets and
Portfolio Management
3-1
U.S. & Chilean Yield Curves: Feb 2004 – Feb 2005
3-2
U.S. Yield Curve and Credit Spreads: February 2005
3-3
Historical Data on U.S. Credit Spreads: Rating-Class Average Yields
Year
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
Treasury Aaa-rated
7.89%
8.73%
8.74
9.63
10.81
11.94
12.87
14.17
12.23
13.79
10.84
12.04
11.99
12.71
10.75
11.37
8.14
9.02
8.64
9.38
8.98
9.71
8.58
9.26
8.74
9.32
8.16
8.77
7.52
8.14
6.45
7.22
7.41
7.97
6.93
7.59
6.80
7.37
6.67
7.27
5.69
6.53
6.14
7.05
6.23
7.04
5.70
6.51
5.43
6.11
4.96
5.66
Aa-rated
8.92%
9.94
12.50
14.75
14.41
12.42
13.31
11.82
9.47
9.68
9.94
9.46
9.56
9.05
8.46
7.40
8.15
7.72
7.55
7.48
6.80
7.36
7.18
6.65
6.27
6.11
A-rated
9.12%
10.20
12.89
15.29
15.43
13.10
13.74
12.28
9.95
9.99
10.24
9.74
9.82
9.30
8.62
7.58
8.28
7.83
7.69
7.54
6.93
7.53
7.46
6.99
6.52
6.38
Baa-rated
9.49%
10.69
13.67
16.04
16.11
13.55
14.19
12.72
10.39
10.58
10.83
10.18
10.36
9.80
8.98
7.93
8.63
8.20
8.05
7.87
7.22
7.88
8.10
7.73
7.11
6.76
3-4
Historical Data on U.S. Credit Spreads: Spreads over Treasury
Year
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
TSY
7.89%
8.74
10.81
12.87
12.23
10.84
11.99
10.75
8.14
8.64
8.98
8.58
8.74
8.16
7.52
6.45
7.41
6.93
6.80
6.67
5.69
6.14
6.23
5.70
5.43
4.96
Aaa - T
84 bp
89
113
130
156
120
72
62
88
74
73
68
58
61
62
77
56
66
57
60
84
91
81
81
68
70
Aa - T
103 bp
120
169
188
218
158
132
107
133
104
96
88
82
89
94
95
74
79
75
81
111
122
95
95
84
115
A-T
123 bp
146
208
242
320
226
175
153
181
135
126
116
108
114
110
113
87
90
89
87
124
139
123
129
109
142
Baa - T
160 bp
195
286
317
388
271
220
197
225
194
185
160
162
164
146
148
122
127
125
120
153
174
187
203
168
180
3-5
Latin American Long-Term Credit Ratings: February 2005
3-6
Par vs. Spot Yield Curves
Generally speaking, bonds are considered capital market instruments in that they have maturities
greater than one year. Another important way in which bonds differ from more basic short-term
money market securities (such as T-bills or Commercial Paper) is that they can provide both
capital gains and periodic interest to the investor. Thus, the bond's yield-to-maturity reflects the
expected annual return to the investment, taking into account both sources of payoff throughout the
remainder of the bond's life.
A yield curve is a graphical representation of the investment possibilities in the bond market at a
particular moment in time. Specifically, the yield curve depicts the yields-to-maturity available to
investors on a series of bonds that are alike in all respects (e.g., default risk, liquidity, tax
considerations) except maturity.
Yield-toM aturity
10%
9%
8%
1 yr
2 yr
3 yr
M aturity
3-7
Par vs. Spot Yield Curves (Cont.)

A par value yield curve summarizes the yields for coupon-bearing
instruments where the coupon rate is equal to the yield-to-maturity.
Assuming that the above example is based a collection of Eurobonds (i.e.,
bonds that pay an annual coupon), the 10% yield for the three-year
instrument can be interpreted as the average annual return that the investor
can expect if he or she:
(i) Holds the bond until maturity,
(ii) Reinvests all intermediate cash flows (i.e., the first two coupons) at the
same 10% rate for the remaining time until maturity.

A spot, or zero coupon, yield curve summarizes the yields for non-couponbearing instruments (i.e., pure discount bonds). These yields can therefore
be interpreted as more of a "pure" return since there is no concern about
having to reinvest intermediate coupon cash flows. For example, if the
above yield curve corresponded to zero coupon securities, the 10%, threeyear yield would represent the average annual price appreciation in the
bond if it were held to maturity.
3-8
U.S. Par and Spot Yield Curves: February 2005
3-9
Par vs. Spot Yield Curves (cont.)
In general, there is no specific shape that the yield curve must take. That is, at any point in time the
yield curve can be upward-sloping, flat, or downward-sloping. There are several classical theories
that purport to explain yield curve shapes:
(i) Unbiased Expectations, which suggests that the presence of intertemporal
arbitrage forces securities of all maturities to reflect the consensus beliefs of
the investing public as to future economic conditions;
(ii) Segmented Markets, which suggests that yields for various maturities are
determined by the supply and demand characteristics of investors in a
specific market, with no regard to what happens in the markets for securities
with different maturities.
(iii) Liquidity Preference, which essentially says that yields are determined by
unbiased expectations but with the addition of a premium to the longer
maturity instruments to reflect the additional default exposure; and
(iv) Preferred Habitat, which is a more general version of the liquidity preference
hypothesis. It assumes that borrowers and lenders alike have a natural
maturity for securities based on institutional characteristics and will venture
outside of that range to attain a higher expected rate of return (or lower cost
of funds) only if the expected gain outweighs the risk. If in each maturity
range the demand for funds equals the supply, then the level of the interest
rate depends on expectations for future rates (so that the expectations theory
holds). If not, the rate will adjust to bring the market into equilibrium such
that no further maturity shifting is desirable.
Empirical evidence provides some evidence to support all four theories, with the unbiased
expectations hypothesis working best for short-term securities.
3 - 10
Implied Spot Yield Curves
The implied zero-coupon (or "strip") yield curve is the sequence of zero-coupon (or pure discount)
interest rates satisfying the no-arbitrage condition that both coupon-stripping and bond
reconstitution exactly break even.
• Coupon-stripping is a bond arbitrage strategy whereby the dealer buys a coupon-bearing bond
and then sells the coupons and principal separately as zero-coupon securities. Profit obtains when
the total sale price of the zeros exceeds the purchase price of the coupon bond.
• Bond reconstitution is a bond arbitrage strategy whereby the dealer buys zero-coupon bonds in
sufficient amounts and with appropriate maturity dates to constitute the exact same cash flows as
on a particular coupon bond. Profit obtains when the sale price of the coupon bonds exceeds the
total purchase price of the zeros.
Example: Assume semiannual coupon payments on the following bonds.
Maturity
0.5
1.0
1.5
2.0
Coupon Rate
3.00%
3.25%
4.00%
4.00%
Price
Yield-to-Maturity
99.926
99.795
100.318
100.000
3.15% (s.a.)
3.46% (s.a.)
3.78% (s.a.)
4.00% (s.a.)
Notice that the 0.5-year security already is a zero-coupon bond since there is only one coupon
remaining.
3 - 11
Implied Spot Yield Curves (cont.)
First, diagram the cash flows on the 0.5-year and 1.0-year bonds:
101.5
101.625
1.625
0.5
0.5
99.926
1.0
99.795
Suppose that the 1.0-year bond is bought for 99.795. The first coupon of 1.625 can be sold as a
0.5-year zero-coupon security priced at 1.600 to yield 3.15% (s.a.) [i.e., (1.625) ÷ (1.01575), where
1.575% is the periodic equivalent of the nominal annual yield of 3.15%]. If coupon-stripping were
to break even, the 1.0-year cash flow of 101.625 must then sell for 98.195, calculated as the
difference between the purchase price of 99.795 and the receipt of 1.600. The implied 1.0-year
zero-coupon rate is therefore 3.463% (s.a.), which is founding by solving 98.195 = (101.625) ÷ (1
+ Z2/2)2 for Z2.
Second, diagram the 1.5-year bond:
102
2
2
0.5
1.0
1.5
100.318
The 1.5-year bond can be bought for 100.318. Following the process established above, the first
two coupons of 2 can be sold as a 0.5-year and 1.0-year zeros priced at 1.969 and 1.932 to yield
3.15% (s.a.) and 3.463% (s.a.), respectively. If coupon-stripping were then to break even, the 1.5year cash flow of 102 must sell for 96.417 [= 100.318 - 1.969 - 1.932]. The 1.5-year implied zerocoupon rate therefore must be 3.788% (s.a.)
3 - 12
Implied Spot Yield Curves (cont.)
Next, diagram the 2.0-year security:
102
2
2
2
0.5
1.0
1.5
2.0
100
Consider a dealer reconstituting this bond and selling it for par value of 100. The first initial
coupon of 2 presumably can be bought for 1.969 to yield 3.15% (s.a.). Then assume that the
second coupon of 2 can be bought for 1.932 to yield 3.463% (s.a.). The third coupon of 2 can be
bought for 1.891 to yield 3.788% (s.a.). The final cash flow of 102 must have been purchased for
94.208 if the strategy were to break even [= 100 - 1.969 - 1.932 - 1.891 = 94.208]. Hence, the 2.0year implied zero-coupon rate must be 4.013% (s.a.).
In general, if the yield curve on coupon bonds (i.e., the par curve) is upward-sloping, the implied
zero curve will lie above it. If the par curve is downward-sloping, the implied zero curve will lie
below.
3 - 13
Implied Spot Yield Curves (cont.)
Yield
(s.a.)
Implied
Zero Curve
3.788%
3.463%
4.013%
4.00%
3.78%
3.46%
3.15%
Par Curve
0.5
1.0
1.5
2.0
Maturity
The difference between the zero curve and the par curve (known as the coupon bias) depends on
the level of rates, the steepness of the yield curve, and the time to maturity.
3 - 14
Implied Forward Rates
Suppose that the yield curve for default free, zero-coupon bonds is as follows:
Maturity
Yield
1
2
3
4
5
7.0%
7.5%
8.0%
8.5%
9.0%
If an investor wishes to make a two year investment, there are several different ways that this can
be achieved. Two of these are:
(i) Purchase a single two year bond,
(ii) Purchase two consecutive one year bonds.
With the first alternative, the investor is certain to have (1 + .075)2 at the end of the two year
holding period.
With the second alternative, the investor is certain to have (1 + .07) at the end of the first one year
holding period.
3 - 15
Implied Forward Rates (cont.)
The question the investor must answer is: What is the rate expected to be on a one year investment
one year from today?
One way to answer this question is to calculate the yield one year from now that will allow the
investor to break even between the two strategies. That is, calculate IFR1x2 such that:
(1 + .075)2 = (1 + .07) x (1 + IFR1x2)
or,
2
(1 + .075)
-1
(1
+
.07)
IFR1x2 =
= 8%
The breakeven yield, IFR1x2, is called the implied forward rate of a one year security available
one year in the future.
If investors are pricing securities consistent with the unbiased expectations view, the implied
forward rate also represents the market's prediction as to where one year interest rates will be one
year forward.
3 - 16
Implied Forward Rates (cont.)
For the yield curve listed above, the series of one year implied forward rates can be calculated as:
Period
IFR1x2
IFR2x3
IFR3x4
IFR4x5
Implied Forward Rate
[(1.075)2÷(1.07)1]-1= 8%
[(1.08)3÷(1.075)2]-1= 9%
[(1.085)4÷(1.08)3]-1=10%
[(1.09)5÷(1.085)4]-1=11%
Notice also that the five-year spot bond rate of 9% is simply the geometric average of the initial
one-year spot rate and the four one-year implied forward rates. That is:
9% =
5
(1.07)(1.08)(1.09)(1.10)(1.11) - 1
In general, letting "A" represent the shorter maturity and "B" the longer one, this process can be
illustrated:
Year: 0
A
B
Actual Rate (B)
Actual Rate (A)
Implied Forward (B-A)
3 - 17
Implied Forward Rates (cont.)
An important, sometimes overlooked, aspect to calculating implied forwards is
that the cash market yields generally must have the same assumed number of
periods in the year. Assume that the current (date 0) zero-coupon yields are
denoted 0 x A and 0 x B. These are the annualized yields for compounding PER
times per year, for A and B years to maturity, where B > A.
With these assumptions, the generalized formula for calculating the implied
forward yield between year A and year B is denoted A x B.
(Yield 0 x A ) 

1 

PER


A Years x PER
(Yield A X B ) 

x 1 

PER


(Yield 0 X B ) 

 1 

PER


(B - A years) x PER
BYears x PER
The terms in parentheses are one plus the periodic yields, that is, the annualized
yields divided by the number of periods in the year. Note that the "PER" in the
exponents cancels out from each side of the equation.
Example: What is the implied forward rate of a three-year bond available two years
forward? Assume PER = 1
(1 +.09)
IFR2x5 =
1
(
)
5-2
5
(1 + .075)
2
-1
= 10%
3 - 18
Uses for Implied Forward Rates

Predictions of Future Spot Rates: This assumes that investors set
yield curves with unbiased expectations, which is seldom true.
Generally, implied forward rates are upward-biased predictions of
future spot rates because of liquidity premiums attached to yields of
longer-term maturity bonds relative to shorter-term instruments

Maturity Choice Decisions: Helps fixed-income investors decide on
appropriate maturity structure for a bond portfolio by quantifying the
reinvestment rate embedded in longer-term securities compared to
shorter-term ones

Pricing Interest Rate Derivatives: Sets the arbitrage boundaries for
the rates attached to actual forward agreements (e.g., bond futures,
interest rate swaps)
3 - 19
U.S. Implied Forward Rates: February 2005
3 - 20
Basics of Bond Valuation

Bonds are simply loans from bondholder to issuer (e.g., firm or
government). Just like loans, bonds require interest payments and
repayment of principal (also called face value or par) at a pre-specified
future date. Interest payments are called coupon payments and bond
principal repayments are usually non-amortizing (i.e., paid all at once at
maturity)

The current market value of a fixed-income bond is the present value of its
future coupon and principal cash flows. In theory, the interest rates used
to discount those future cash flows are the zero-coupon (or pure discount)
rates corresponding to the dates of each cash flow.
 PMT
PMT
PMT 
FV

PV  


.
.
.


1
2
N
N
(1

z
)
(1

z
)
(1

z
)
(1

z
)
1
2
N
N


3 - 21
Basics of Bond Valuation (cont.)

Consider a five-year, 9% (annual coupon payment) Eurobond. The market value
of the bond, 103.99 (% of par value), can be obtained by calculating the present
value of each scheduled cash flow using a sequence of zero-coupon rates
commensurate with the riskiness of the bond.
Period
Cash Flow
Zero-Coupon
Rate
1
9
6.00%
8.49
2
9
7.00
7.86
3
9
7.50
7.24
4
9
7.80
6.66
5
109
8.13
73.74
Value =
Present
Value
103.99
3 - 22
Basics of Bond Valuation (cont.)

The yield to maturity (y) of the bond is the constant interest rate per
period that solves the following equation:
 PMT
PMT
PMT 
FV

PV  


.
.
.

1
2
N 
N
(1

y
)
(1

y
)
(1

y
)
(1

y
)



The yield-to-maturity is the internal rate of return of all cash flows.
It is the rate such that the present values of the cash flows, each
discounted by that same rate, exactly equal the market value of the
bond. The yield to maturity of this bond turns out to be 8.00%.
3 - 23
Basics of Bond Valuation (cont.)
Period
Cash Flow
Yield-toMaturity
1
9
8.00%
8.33
2
9
8.00
7.72
3
9
8.00
7.14
4
9
8.00
6.62
5
109
8.00
74.18
Value =
Present
Value
103.99

The yield to maturity is a statistic about the rate of return on the bond that includes
both the coupon cash flows as well as any inevitable capital gain or loss if the bond is
held to maturity (a gain if the bond is purchased at a discount below par value, a loss if
the bond is purchased at a premium above par value).

Therefore, it contains more information than the current yield, which is simply the
coupon rate divided by the current price, e.g., 9  103.99 = .0865 . The current yield of
8.65% overstates the investor’s rate of return since it neglects the capital loss.
3 - 24
Basics of Bond Valuation (cont.)

Notice that the yield to maturity can be interpreted as a "weighted
average" of the sequence of zero-coupon rates, with most of the
weight placed on the last cash flow since that is when the principal
is redeemed, in that both deliver the same present value:
9
9
9
9
109




 103.99
1.0600
(1.0700) 2
(1.0750) 3
(1.0780) 4
(1.0813) 5


9
9
9
9
109




1.0800
(1.0800) 2
(1.0800) 3
(1.0800) 4
(1.0800) 5
Clearly, the yield to maturity must lie within the range of the zerocoupon rates.
3 - 25
Current Coupon, Premium, and Discount Bonds

A Current Coupon (or Par-Value) Bond is one for which the current market price equals the face value. In that
case, the coupon rate (C/F) will equal the current yield (C/P), which will equal the yield-to-maturity (y).
P=F
<===>
C/F = C/P = y
The bond is priced at par value since its coupon rate is "fair" in that it equals the current market interest rate as
represented by the yield-to-maturity.

A Premium Bond has a current market price that exceeds the face value. In this case, the coupon rate will be
higher than the current yield, which in turn will be higher than the yield-to-maturity.
P>F
<===>
C/F > C/P > y
The bond is priced at a premium above par value since its coupon rate is "high" given current market rates. A
par-value, current coupon bond would have a lower coupon rate, so the premium represents the value of the
"excessive" coupon cash flows. In fact, the amount of the premium is the present value of the annuity
represented by the difference between the coupon rate and the bond's yield, discounted at that yield.

A Discount Bond has a current market price that is less than the face value. The coupon rate will be less than
the current yield, which will be less than the yield-to-maturity.
P<F
<===>
C/F < C/P < y
The bond is priced at a discount below par value since its coupon rate is "low" given current market rates. The
amount of the discount is the present value of the annuity represented by the difference between the yield and
the coupon rate. For example, a zero-coupon bond will usually be at a deep discount to par value.
3 - 26
Current Coupon, Premium, and Discount Bonds (cont.)

Example: Calculate the yield-to-maturity statistic on a seven-year, 6-3/4%
Treasury note priced at 98.125. Assume that a semi-annual coupon
payment has just been made so that exactly 14 periods remain until the
principal is refunded at maturity.

Algebraically, the yield is the solution “y” to the following equation:
98.125 
14

t 1
3.375
y

1



2

t

100
14
y

1



2


Solving for the periodic yield (i.e., y/2) on a financial calculator (such as
the HP 12C) obtains 3.5472 [100 FV, 14 n, 3.375 PMT, -98.125 PV, i ….
3.5472].

The annualized yield-to-maturity would then be reported as y = 7.0944%
(i.e., 3.5472 x 2).
3 - 27
Interpreting Bond Information: Chile Govt 5.50% of January 2013
3 - 28
Interpreting Bond Information: ENDESA 8.35% of August 2013
3 - 29
Valuing the ENDESA 8.350% of 2013 Bond Between Coupon Dates
|
2/1/05
|
|
|
2/18/05
8/1/05
8/1/13
n = 17 actual days (17 on “30/360” basis)
N = 181 actual days (180 on “30/360” basis)
Bond Valuation:
17
On February 1, 2005:
P0 =

t =1
4.175
 .0561315 
1 

2


t
+
100
17
 .0561315 
1 

2


= 118.3007
On February 18, 2005:
Pn + AIn = (118.3007)(1 + 0.0561315/2)(17/180) = 118.6103
where:
AIn = (4.175)(17/180) = 0.3943
so that:
“Clean” or “Flat” Price = 118.6103 - 0.3943 = 118.2160
3 - 30
ENDESA 8.350% Bond of August ’13 (cont.)
3 - 31
Sources of Bond Risk

Primary:



Default: Will the borrower honor its promise to repay?
Interest Rate: How will changing market conditions
affect the value of the bond?
 Price risk component
 Reinvestment risk component
Secondary:



Call: Will the borrower refinance the loan under
conditions that are disadvantageous to investor?
Liquidity: How easily can bond be bought or sold?
Tax: Will changes in the tax code affect bond values?
3 - 32
Bond Yields, Pricing, and Volatility

Theorem #1: Bond prices are inversely related to bond yields.
Implication: When market rates fall, bond prices rise, and vice versa.

Theorem #2: Generally, for a given coupon rate, the longer is the term to maturity, the greater is
the percentage price change for a given shift in yields. (The maturity effect)
Implication: Long-term bonds are riskier than short-term bonds for a given shift in yields, but
also have more potential for gain if rates fall.

Theorem #3: For a given maturity, the lower is the coupon rate, the greater is the percentage
price change for a given shift in yields. (The coupon effect)
Implication: Low-coupon bonds are riskier than high-coupon bonds given the same maturity, but
also have more potential for gain if rates fall.

Theorem #4: For a given coupon rate and maturity, the price increase from a given reduction in
yield will always exceed the price decrease from an equivalent increase in yield. (The convexity
effect)
3 - 33
Bond Yields, Pricing, and Volatility (cont.)
Implication: There are potential gains from structuring a portfolio to be
more convex (for a given yield and market value) since it will outperform a
less convex portfolio in both a falling yield market as well as a rising yield
Price
Convex Price-Yield Curve
Yield
3 - 34
Bond Yields, Pricing, and Volatility: Example


Consider the following bonds:
Bond
Coupon
Maturity
Initial Yield
A
8%
5 yrs
10%
B
5
20
8
C
8
20
8
Initial Prices:
PA =
5 

80
1000


+
= 924.18


t
5
(1 + .10)
t =1 (1 + .10) 
20 

50
1000

 +
PB =  
t
(1 + .08) 20
t =1 (1 + .08) 
PC =
20 

80
1000

 +

t
(1 + .08) 20
t =1 (1 + .08) 
= 705.46
= 1000.00
3 - 35
Bond Yields, Pricing, and Volatility: Example (cont.)

Prices after yields increase by 50 bp:
PA =
PB =
PC

=
5 

 +
t
t =1 (1 + .105) 
 
20 
80

 +
t
t =1 (1 + .085) 
 
20 
(1 + .105) 5
50


t
t =1 (1 + .085) 
 
1000
80
+
1000
(1 + .085) 20
1000
(1 + .085) 20
= 906.43
= 668.78
= 952.68
Percentage price changes:
Bond A: (906.43 - 924.18) / (924.18) = -1.92% (least)
Bond B: (668.78 - 705.46) / (705.46) = -5.20% (most)
Bond C: (952.68 - 1000.00) / (1000.00) = -4.73% (middle)
3 - 36
Bond Yields, Pricing, and Volatility: Example (cont.)

Question: Where would Bond D, which has a coupon rate of 6% and a
maturity of 19 years, fit into this price sensitivity spectrum? (Assume its
initial yield is also 8%.)
Initial:
PD =
19 
60

1000
 +
t
(1 + .08)19
t =1 (1 + .08) 
19 
60
 
= 807.93
After:
PD =

1000
 +
t
(1 + .085)19
t =1 (1 + .085) 
 
= 768.31
So, percentage change:
Bond D: (768.31 - 807.93) / (807.93) = -4.90%
3 - 37
Motivating the Concept of Bond Duration
In general, from the bond pricing theorems, the longer the term-to-maturity and
the lower the coupon rate, the greater the percentage price change for a given
change in yield. This suggests that many bonds can be ranked according to price
volatility simply on the basis of their coupon rates and maturity dates, that is, by
the "name" of the bond.
Identifying Price Risk by Inspection
of the ‘Name’ of the Bond
(i.e. its Coupon Rate and Maturity Date)
Coupon NW
Rate
12%
NE
* * * * * * *
* ** **
* * *
* ***
* * ** * * * *** *
* *
**
* * *
*
*
*
* * *
* * ** * * * **** **
*
****
* *
* * *
* ** ** *
* *
* * *
* * * * * ** *
***
* *
***
* ** * *
* * *
***
** ***
* * SE
SW
* * *
***
* *
5 Years
Maturity
The duration statistic will provide a one-to-one mapping between observable
characteristics of a bond (its coupon rate, maturity, and yield) and its price risk
arising from a volatile yield curve.
3 - 38
Motivating the Concept of Bond Duration (cont.)
The key point is that more than one bond will "map" into the same point on the
duration scale.
Assuming each bond has a yield –to-maturity of 6%
Coupon
Rate
15.71% coupon,
7.0-year bond
11.25% coupon
*
6.5-year bond
7.32% coupon
*
6.0-year bond
3.63% coupon
5.5-year bond
0% coupon
5.0-year bond
*
*
*
Maturity
5
Duration
So, at a first order of approximation each of these five bonds will experience the
same percentage price change if the yield curve were to shift up or down in a parallel
manner, i.e., a uniform (or shape-preserving) shift. The bonds are price-risk
equivalents. That's because they have the same duration statistic despite having
different coupon rates and times to maturity. Also, each of these bonds would
experience about half the percentage price change as would bonds that have a
duration statistic of 10 and twice the change as bonds having a duration of 2.5.
Duration can be used as a relative measure of price risk.
3 - 39
Calculating the Duration Statistic

The duration of a bond is a weighted average of the
payment dates, using the present value of the relative
cash payments as the weights:

  PMT    
  PMT

  
 
 
1 
2
(1

y)





  2 x   (1  y)
DUR  1 x 
 PV0
 

PV0

 




 

 




  PMT  FV   
  

 
  
 
N
(1

y)


   ...  N x 






PV0




 





This statistic is the Macaulay duration, named after
Frederick Macaulay who first developed it, and can be
interpreted as the point in the life of the bond when the
average cash flow is paid.
3 - 40
Calculating the Duration Statistic: Example

or:
Consider a five-year, 12% annual payment bond having a face value of $1,000. Suppose that the bond is
priced at a premium to yield 10% (p.a.). The price of the bond is $1,075.82 and the Macaulay duration is
4.074:
Year
Cash Flow
PV at 10%
PV/Price
Yr x (PV/Pr)
1
120
109.09
0.1014
0.1014
2
120
99.17
0.0922
0.1844
3
120
90.16
0.0838
0.2514
4
120
81.96
0.0762
0.3047
5
1120
695.43
0.6464
3.2321
$1075.82
1.0000
4.074 yrs

 (109.09)   
 99.17   
 90.16  




DUR 0  1 x 

2
x

3
x
 
1,075.82   
1,075.82  
1,075.82

 

 




 81.96   
 695.43  


  4 x 

5
x
 
1,075.82    4.0740
1,075.82

 



3 - 41
Calculating the Duration Statistic: Closed-Form Equation
The general formula for Macaulay duration can be reduced to a closed-form solution by taking the
sums of the geometric series and algebraically rearranging the terms.
+ (n x T) C - Y
1 + Y
1+Y
n
n
F
n D =
Y
C 1 + Y n xT - 1 + Y
n
n
n
F
where n is the number of payments per year, Y/n is the period yield, C/F is the coupon rate per
period, and T is the years to maturity. Note that duration is calculated on the basis of the
underlying periodic cash flows, even though it is typically is annualized when reporting the statistic
by dividing by n.
In the preceding numerical example, Y = .10, n = 1, T = 5, C/F = .12, and Y/n = .10. The bond's
duration can be solved as:
1 + .10 + 5 (.12 - .10)
= 4.0740
D = 1 + .10 5
.10
.12 1 + .10 - 1 + .10
3 - 42
Duration as a Measure of Price Volatility

Basic Price-Yield Elasticity Relationship:
 MV 


MV 

D    (1  y/n) 


(1

y/n)



Convert to “Volatility Prediction” Equation:
 (1  y/n) 
 y 
MV
  - (D) 
 - (D) 
 when n  1
MV
(1

y/n)
(1

y)





Prediction Equation in Modified Form (% price change):
 D 
MV
 -
 (y)  - (Mod D)( y)
MV
(1

y)


3 - 43
Duration as a Measure of Price Volatility (cont.)

Convert to dollar (or cash) sensitivity:
MV ~ -(Mod D)( y)(MV)

Sensitivity to a one bp yield change (i.e., y = 0.0001):
MV ~ -(Mod D)(0.0001)(MV) = Basis Point Value = BPV
3 - 44
Duration as a Measure of Price Volatility (cont.)
Duration is a conservatively biased estimate of actual price changes from the perspective of the
asset-holder. Duration underestimates price increases when rates fall and overestimates price
reductions when rates rise. The reason for the bias is the convexity of the price-yield curve for a
fixed-income instrument that contains no embedded put or call options. The bias is greater, the
greater is the change in yield. For small rate changes, duration can be a very accurate predictor of
the price change.
Price
Underestimate of Price Increase
When the Yield Falls
Overestimate of Price Decrease
When the Yield Rises
Convex Price-Yield
Curve
Yield
3 - 45
Duration and Price Volatility: Example

Consider again the five-year, 12% coupon bond with a yield to
maturity of 10%:

Macaulay D: 4.074

Modified D: 3.704 (= 4.074 / 1.1)


This means that an increase in yields of 100 bp will change the
bond’s price by approximately 3.7% in opposite direction
Basis Point Value: $0.0398 [= (3.704)(.0001)(107.582)]

This means that a one bp change in yields will cause the
bond’s price to move by about 4 cents per $100 of par value
(which would correspond to a 40 cent movement for a bond
with a par value of $1000)
3 - 46
Duration Example: ENDESA 8.350% of 2013
3 - 47
Duration of a Bond Portfolio
The duration of a portfolio can be estimated as a weighted average of the
durations of the securities in the portfolio, whereby the shares of total market
value are the weights. Consider the following portfolio of semiannual
payment bonds.
Face Value Coupon Rate Maturity Yield (s.a.) Market Value
$50 Million
8%
12 yrs
9.62%
$44,306,787
$25 Million 11%
8 yrs
9.38%
$27,243,887
$25 Million
9%
6 yrs
9.10%
$24,886,343
$96,437,017
Duration
7.652
5.633
4.761
The duration of the portfolio can be estimated as:
44,306,787  
27,243,887  
24,886,343 

 7.652 x
   5.633 x
   4.761 x

96,437,017
96,437,017
96,437,017

 
 

= 6.336
3 - 48
LVACL Corporate Bond Index: Description & Performance
3 - 49
Example of Portfolio Duration: LVACL Corporate Bond Index
3 - 50
Example of Portfolio Duration: MBA Investment Fund Endowment Portfolio
3 - 51
Bond Convexity: An Overview
On average, other things being equal, a more convex portfolio will outperform a
less convex portfolio in both bull markets (falling rates, rising prices) and bear
markets (rising rates, falling prices). This suggests that there might be gains
from convexity that can be exploited in asset allocation. In general, this could
mean holding a "barbell" or "laddered" portfolio of diverse maturities rather than
a "bullet" portfolio of concentrated maturities.
Ladder
Barbell
Bullet
For example, consider two portfolios with an equivalent duration--one, for
example, a zero-coupon bond, and the other, a collection of coupon bonds. The
coupon bonds will be more convex.
Price
Two Portfolios That Have the
Same Yield, Same Market Value,
and the Same Duration but Different
Degrees of Convexity
More Convex
Portfolio
Less Convex
Portfolio
Yield
3 - 52
Using Bond Convexity in Estimating Price Volatility
Consider the 8%, semiannual coupon payment, 20-year bond that is priced at
110.678 to yield 7.00% (s.a.). The Macaulay duration is 10.760, the modified
duration 10.396. The change in market value (MV) can be estimated as the
product of the modified duration and the market value (i.e., the money duration)
and the change in the yield to maturity (YLD):
MV  - (Modified Duration x MV) x YLD
But that linear estimator works best with small changes in the yield. For larger
changes, the inherent convexity in the market-value-to-yield relationship leads to
large errors.
The estimate can be much improved by including the convexity statistic:
MV  - (Modified Duration x MV) x YLD
 1/2 x (Convexity x MV) x (YLD) 2
3 - 53
Using Bond Convexity in Estimating Price Volatility (cont.)
The convexity statistic on this 8%, 20-year bond yielding 7.00% is 158.548.
Consider a drop in the yield to 6.00%. Duration alone estimates a change in
market value of + 11.506, whereas the actual change would be +12.437, a
difference of +0.931.
Including the convexity statistic reduces that difference to only +0.054:
MV  [- (10.396 x 110.678) x (-.0100)]  [1/2 x (158.548 x 110.678) x (-.0100) 2 ]
= 11.506 + 0.8777 = 12.383
3 - 54
Convexity Trades: An Example
(Source: R. Dattareya and F. Fabbozi)


Consider the following hypothetical U.S. Treasury bonds:
Bond
Coupon %
Maturity
(yrs)
Invoice
Price
Yield %
Dollar
Duration
Dollar
Convexity
A
8.50
5
100
8.50
4.005
19.8164
B
9.50
20
100
9.50
8.882
124.1702
C
9.25
10
100
9.25
6.434
55.4506
Consider two different bond portfolios:

Bullet Portfolio: 100% of Bond C
 Barbell Portfolio: 50.2% of Bond A, 49.8% of Bond B

Notice the following:

Duration of Barbell: (.502)(4.005)+(.498)(8.882) = 6.434


Same as Bullet Portfolio
Convexity of Barbell: (.502)(19.82)+(.498)(124.17) = 71.7846

Greater than Bullet Portfolio
3 - 55
Relative Performance (Bullet Rtn – Barbell Rtn) Over Six-Month Period
3 - 56
Embedded Bond Options and Negative Convexity
A callable bond allows the issuer to buy the bond back from the investor at a
preagreed price (the call price) on certain dates (call dates). The call price on
corporate bonds is typically a premium above par that declines as maturity
nears. Treasuries, however, when callable are callable at par. Similarly,
homeowners usually can prepay their mortgages without penalty.
The call option will be exercised if the issuer in able to refinance the debt at a
lower cost of funds due to lower market rates in general (a lower benchmark
Treasury) or to an improvement in the issuer’s credit quality (a lower spread
over the benchmark Treasury). Naturally the bondholder suffers when the
call option is exercised because the proceeds would have to be reinvested at a
lower rate or in a less creditworthy firm.
The constraint on upside price appreciation as yields fall is known as
negative convexity of the price-yield relationship:
Price
Convex Price-Yield Relationship
on a Bond with No Embedded Options
Negative Convexity on
a Callable Bond
Yield
3 - 57
Embedded Bond Options and Negative Convexity (cont.)
The presence of call risk to the investor requires an increase in the yield on a
callable bond depending on the design of the option:


The higher the call price, the lower the yield.
The longer the call protection period, the lower the yield.
From the perspective of the investor, buying a callable bond is equivalent to
buying an otherwise comparable noncallable and writing a call option.
+ Callable bond = + Noncallable bond - Call option
whereby "+" indicates a long position and "-" a short position. The higher
yield received on the callable corresponds to the amortized value of the
premium on the embedded option that has been written.
Issuing a callable bond is equivalent to issuing a noncallable and buying a
call option.
- Callable bond = - Noncallable bond + Call option
Similarly, the higher cost of funds on issuing the callable corresponds to the
amortized value of the option (to refinance if yields fall) that has been
purchased.
The yield statistic on a callable bond is complicated by the presence of the option.
One simple remedy is to calculate the yield to each call date, using the call price that
would apply to the redemption amount for that date. The lowest of the yields-to-call
is referred to as the yield-to-worst.
3 - 58
Callable Bond Example: SBC 6.28% of October 2010-04
3 - 59
Callable Bond Example: SBC 6.28% of October 2010-04
3 - 60
Callable Bond Example: ENTEL 7.00% of January 2010-04
3 - 61
Overview of Bond Portfolio Strategies
Generally speaking, bond portfolio management strategies can be broken down into
four groups:
I. Passive portfolio strategies
a. Buy and Hold: This involves selecting securities with the desired
characteristics (e.g., credit quality, coupon, maturity) and then not trading
them again until they mature.
b. Indexing: This is an attempt to design a portfolio of bonds that mimics a
certain index (e.g., Lehman Brothers Index). There are two approaches to
indexing: (1) full replication, in which all of the securities represented in the
index are held in their exact proportions, and (2) stratified sampling, which
divides the index into cells based upon parameters such as coupon, maturity,
country, etc. and concentrates on indexing within certain cells.
3 - 62
Overview of Bond Portfolio Strategies (cont.)
3 - 63
Overview of Bond Portfolio Strategies (cont.)
II. Active management strategies
The active approach to bond portfolio management involves altering the portfolio
so that the characteristics of the securities held provide the best chance of taking
advantage of the manager’s “view” of anticipated conditions in the macro economy or
with the specific security. Generally, the following types of views are employed in
practice:
a. Interest Rate Anticipation: This involves trading in and out of existing
securities based on uncertain forecasts of future interest rate conditions; for
this reason it is considered to be a risky strategy.
b. Valuation Analysis: An attempt to select bonds based on their intrinsic value
by accurately gauging the cost of the relevant characteristics (e.g., credit
quality) of the bond.
c. Credit Analysis: This involves assessing the default risk of an issuer and
trading the bond when your perception differs from that of the market.
3 - 64
Overview of Bond Portfolio Strategies (cont.)
II. Active management strategies (cont.)
d. Yield Spread Analysis: A variation of interest rate anticipation, this strategy
attempts to predict when the credit spread for a certain credit class will
widen or narrow in anticipation of changing economic conditions.
e. Bond Swaps: This involves liquidating a current position and
simultaneously buying a different issue in its place with similar attributes,
but a chance for improved return. Notable examples of this approach
include pure yield pickup swaps, substitution swaps, and tax swaps.
While these transactions can involve anything from little risk (yield pickup
swaps) to great risk (rate anticipation swaps), they all are based on the
premise that it is possible to improve portfolio performance as a result of
changing market conditions.
3 - 65
Overview of Bond Portfolio Strategies (cont.)
II. Active management strategies (cont.)
Ronald Layard-Liesching of Pareto Partners assess the likelihood of adding
alpha to a fixed-income portfolio with active strategies as follows:
Source
Duration
Yield Curve
Sector Allocation
Country Allocation
Security Selection
Optionality
Prepayment
Anomaly Capture
Credit Risk
Liquidity
Currency
Scale
High
Low
High
High
Low
Medium
Medium
Low
High
Low
High
Sustainable
Very Weak
Very Weak
Strong
Strong
Medium
Medium
Medium
Weak
Strong
Strong
Medium
Information
Ratio*
1
3
6
5
5
7
6
7
8
3
2
Extreme
Values
Yes
No
No
No
No
Yes
Yes
Yes
Yes
Yes
Yes
* Range is from 1 (low) to 10 (high)
3 - 66
Examples of Typical Yield Curve Shifts
3 - 67
Active Bond Trades: Examples
For each of the following bond trades, identify the reason(s) investors may have had in making
each swap. [Note: Assume that each swap was executed in 1982.]
Action
Call
Price
YTM
108.24
75.625
15.71%
105.20
51.125
15.39%
101.50
96.125
14.02%
NC
102.200
13.83%
NC
25.250
14.37%
Buy: Aa-1 Chase Manhattan Notes
Floating Rate due 2009
103.90
90.250
---
(d) Sell: A-1 Texas Oil & Gas 1st Mtg.
8.25% due 1997
105.75
60.000
15.09%
NC
65.600
12.98%
103.90
62.750
10.83%
109.86
73.000
16.26%
(a) Sell: Baa-1 Georgia Pwr 1st Mtg.
11.625% due 2000
Buy: Baa-1 Georgia Pwr 1st Mtg.
7.325% due 2001
(b) Sell: Aaa Amer. Tel. & Tel. Notes
13.25% due 1991
Buy: U.S. Treasury Notes
14.25% due 1991
(c) Sell: Aa-1 Chase Manhattan Notes
0% due 1992
Buy: U.S. Treasury Bond
8.25% due 2005
(e) Sell: A-1 K Mart Convertible Deb.
6% due 1999
Buy: A-2 Lucky Stores S.F. Deb.
11.75% due 2005
3 - 68
Bond Swaps

Another type of active trade is a bond swap. This
involves liquidating a current position and simultaneously
buying a different issue in its place with similar attributes,
but a chance of improved returns.

Notable examples of bond swaps include:



Pure Yield Pickup Swaps: Swapping out of a low-coupon bond
into a comparable higher-coupon bond to realize an automatic
and instantaneous increase in current yield and yield to maturity.
Substitution Swaps: Swapping comparable bonds that are
trading at different yields; based on the premise that the credit
market is temporarily out of balance.
Tax Swaps: Trades motivated by prevailing tax codes and
accumulated capital gains in a portfolio (e.g., selling a bond with
a capital loss to offset one with a capital gain).
3 - 69
Bond Swap Example

Evaluate the following pure yield pickup swap: You are
currently holding a 20-year, Aa-rated, 9.0 percent coupon
bond priced to yield 11.0 percent.

As a swap candidate, you are considering a 20-year, Aarated, 11.0 percent coupon bond priced to yield 11.5
percent

You can assume that all cash flows are reinvested at
11.5 percent.
3 - 70
Bond Swap Example: Solution
3 - 71
Overview of Bond Portfolio Strategies (cont.)
III. Matched Funding Techniques
Generally, matched funding techniques are asset/liability management procedures,
whereby the bond portfolio is managed with respect to a specific set of future
liabilities.
a. Cash Matching/Dedicated: This approach designing a bond portfolio so that
it delivers cash in the exact amount and timing as it is needed to pay off a set
of liabilities. This can be done strictly or with reinvestment of excess cash
flows from prior periods to enhance returns.
b. Duration Matching/Classical Immunization: This strategy involves holding a
portfolio whose duration is equivalent to the duration of the underlying
liabilities. In this manner, it is possible to eliminate interest rate risk as that
exposure’s two components—price and reinvestment risk—will offset each
other. More detailed notes on this process are given on the following pages.
c. Horizon Matching: This is a combination of cash-matching dedication and
immunization, based on a division of the liability cash flows into two
segments involve short- and long-term time horizons.
3 - 72
3 - 73
The Mechanics of Bond Immunization
Suppose that an investor wants to invest money in the bond market for a period of three years. The
interest rate is currently 10% but it falls to 8% as soon as the initial investment is made. What is
the actual rate of return that this investor enjoys in each of the following scenarios?
(a) Purchase a 10-year bond paying a 9% annual coupon and sell it in three years;
(b) Purchase three 1-year "pure discount" (i.e., zero-coupon) bonds;
(c) Purchase a 3-year, pure discount bond;
(d) Purchase a 4-year bond paying a 34.85% annual coupon and sell it in three
years.
Bond A: Initial Investment:
10
P0 =
90
1000
t +
.10)
(1 + .10)10
 (1 +
t =1
= $938.55
Terminal (i.e., Year 3) Value:
(i) Sale of Bond:
7
P3 =
90
 (1 + .08)
t= 1
t
+
1000
(1 + .08)7
= $1052.06
(ii) Coupon Payments:
90(1 + .08)2 + 90(1 + .08) + 90 = $292.18
so, Total Terminal Value = $1344.24 and, thus, the annual realized rate of return is:
1344.24
ARR = 3
- 1 = 12.72%
938.55
3 - 74
The Mechanics of Bond Immunization (cont.)
Bond B: Invest $1000 initially at 10% and reinvest total proceeds for two more years at
8%:
Year 1: (1000.00)(1 + .10) = $1100.00
Year 2: (1100.00)(1 + .08) = $1188.00
Year 3: (1188.00)(1 + .08) = $1283.04
so, the realized rate of return is:
1283.04
ARR = 3
- 1 = 8.66%
1000
Bond C: Current Price (for $1000 face value): P0 = (1000) ÷ (1 + .10)3 = $751.31
so, the realized rate of return is:
1000.00
ARR = 3
- 1 = 10.00%
751.31
Bond D: Initial Investment:
4
P0 =
348.50
1000
+
+ .10)t
(1 + .10)4
 (1
t =1
= $1787.71
Terminal (i.e., Year 3) Value:
(i) Sale of Bond:
P3 = (1000 + 348.50) ÷ (1 + .08) = $1248.61
(ii) Coupon Payments:
348.50(1 + .08)2 + 348.50(1 + .08) + 348.50 = $1131.37
so, Total Terminal Value = $2379.98 and, thus, the realized rate of return is:
2379.98
ARR = 3
- 1 = 10.00%
1787.71
3 - 75
The Mechanics of Bond Immunization (cont.)
Notice that only for Bonds C and D does the yield to maturity (i.e., the expected return) equal the
realized rate of return (ARR) over the three investment horizon. To see why this is the case,
consider the duration of each of these bonds:
Bond A:
Bond B:
Bond C:
Bond D:
6.89 years
1.00 year (per bond)
3.00 years
3.00 years
Because the investor's planning horizon was three years, the only bonds that actually produced the
expected yield of 10% were the two that had a duration of three years. Put another way, by
investing in a bond that pays out the "average" cash flow at precisely the time it is desired, it is
possible to completely offset the effects of a subsequent change in interest rates. If interest rates fall
(rise) after purchase, the bond price will rise (fall) by exactly enough to offset the decline (increase)
in income from reinvested coupons. This selection process is known as immunization.
Given that bond risk caused by changing interest rates can be dichotomized into price risk and
reinvestment risk, the following general statements can be made:
If Duration > Planning Horizon, the investor faces Net Price Risk (i.e., Bond A)
If Duration < Planning Horizon, the investor faces Net Reinvestment Risk (i.e., Bond B)
If Duration = Planning Horizon, the investor is immunized (i.e., Bonds C and D)
3 - 76
Overview of Bond Portfolio Strategies (cont.)
IV. Derivative-Linked & Contingent Procedures
Contingent management procedures are hybrid strategies that attempt to marry the
“best practices” from both passive and active strategies with the risk control
mechanisms implied by dedicated strategies.
a. Contingent Immunization: This approach combines classical immunization and
active management by accepting a lower “lock in” yield than prevails in the market on
the immunized portion of the portfolio in exchange for some potential upside through
active strategies.
b. Enhanced Indexing: Also called “indexing plus”, this strategy supplements an
indexed position with enough active position bets. An enhanced indexing strategy
hopes to exceed the total return performance of the manager’s designated benchmark
on a enough consistent basis to cover the costs of active management. Under this
strategy, the total return on the index becomes the minimum objective rather than the
target itself.
3 - 77
Overview of Bond Portfolio Strategies (cont.)
3 - 78
Overview of Bond Portfolio Strategies (cont.)
IV. Derivative-Linked & Contingent Procedures (cont.)
c. “Core-Plus” Management: Similar to enhanced indexing, a core-plus mandate
attempts to add performance alpha by supplementing a fairly passive approach to
managing the main fixed-income assets (i.e., the core) relative to the designated
benchmark with “plus” investing in sectors not covered in the benchmark. A typical
core portfolio might include government, agency, and investment-grade sectors, with
the “plus” sectors then including high-yield and emerging market bonds.
d.
Derivative-Linked Investing:
Derivatives—such as futures contracts, swap
agreements, or options—are used to transform the cash flows and/or risk profile of a
“plain vanilla” bond offering it something other than the original structure. Although
derivatives represent a cost-effective way of implement this sort of financial “reengineering”, the point using derivatives in the fixed-income portfolio is usually to
create something synthetically that: (i) the manager cannot do directly, or (ii) does not
exist otherwise.
3 - 79
Overview of Bond Portfolio Strategies (cont.)
3 - 80
An Overview of Equity Alternatives

As we have seen, debt and equity securities are the fundamental
cornerstones of the capital markets. They represent the most prevalent
securities that companies use to raise external funds and that investors
purchase to hold in their portfolios.

Often, however, there will be cases when either investors or issuers will
want to do a transaction involving securities with an equity-like payoff
structure, but they may choose not (or otherwise be unable) to use “plain
vanilla” equity directly. Some reasons why conventional stock shares may
not be appropriate even when an equity payoff is desired include:

A corporation seeking to raise additional capital may find the market for its
common stock to be unreceptive, perhaps due to other recent issuances.
 An institutional investor may be restricted from holding equity directly but can
purchase a debt instrument with a equity-like principal payoff at maturity.
 A company may be able to lower the present cost of a debt financing by
structuring a bond contract that allows investors the right to convert the debt into
common equity at a future date.

We will look at two alternative forms of equity along these lines: (i)
convertible securities, and (ii) structured notes
3 - 81
Notion of Convertible Bonds

A convertible bond can be viewed as a pre-packaged portfolio containing
two distinct securities: (i) a regular bond and (i) an option to exchange the
bond for a pre-specified number of shares of the issuing firm’s common
stock. Thus, a convertible bond represents a hybrid investment involving
elements of both the debt and equity markets.

The option involved can be viewed as either a put (i.e., the investor has the
right to sell the bond back to the issuer and receive a fixed number of
shares) or a call (i.e., the investor can buy a fixed number of shares from
the issuing company, paid for with the bond).

From the investor’s standpoint, there are both advantages and
disadvantages to this packaging. Specifically, while buyer receives equitylike returns with a “guaranteed” terminal payoff equal to the bond’s face
value, he or she must also pay the option premium, which is embedded in
the price of the security.

Conversely, the issuer of a convertible bond increases the company’s
leverage while providing a potential source of equity financing in the future.
This arrangement may be particularly useful as a means for low-rated
issuers to borrow money more cheaply in the present than with a “straight”
debt issue while creating a potential demand for their shares if future
conditions are favorable.
3 - 82
Convertible Bond Example: Cypress Semiconductor

As an example of how one such issue is structured and priced,
consider the 4.00 percent coupon convertible subordinated notes
(“sub cv nt”) maturing in February of 2005 issued by a NYSE-traded
company, Cypress Semiconductor Corporation (CY). Cypress
Semiconductor designs, develops, manufactures and markets a
broad line of high-performance digital and mixed-signal integrated
circuits for a range of markets, including data communications,
telecommunications, computers and instrumentation systems.

The Bloomberg screen on the next slide shows the issue’s CUSIP
identifier, contract terms and default rating, (i.e., B1), and indicates
that this bond pays interest semi-annually on February 1 and August
1. The bond issue has $283 million outstanding and is callable at
101 percent of par.

At the time of this report (i.e. February 2001), the listed price of the
convertible was 92 percent of par and the price of Cypress
Semiconductor common stock was 27.375 per share.
3 - 83
CY Convertible Bond Example (cont.)
3 - 84
CY Convertible Bond Example (cont.)

As spelled out at the top of this display, each $1,000 face value of
this bond can be converted into 21.6216 shares of Cypress
Semiconductor common stock. This statistic is called the
instrument’s conversion ratio. At the current share price of
$27.375, an investor exercising her conversion option would have
received only $591.89 (= $27.375  21.6216) worth of stock, an
amount considerably below the current market value of the bond.

In fact, the conversion parity price (i.e. the common stock price at
which immediate conversion would make sense) is equal to $42.55,
which is the bond price of $920 divided by the conversion ratio of
21.6216. The prevailing market price of 27.375 is far below this
parity level, meaning that the conversion option is currently out of
the money. Of course, if the conversion parity price ever fell below
the market price for the common stock, an astute investor could buy
the bond and immediately exchange it into stock with a greater
market value.
3 - 85
CY Convertible Bond Example (cont.)

Most convertible bonds are also callable by the issuer. Of course, a
firm will never call a bond selling for less than its call price (which is
the case with the Cypress Semiconductor note). In fact, firms often
wait until the bond is selling for significantly more than its call price
before calling it. If the company calls the bond under these
conditions, investors will have an incentive to convert the bond into
the stock that is worth more than they would receive from the call
price; this situation is referred to as forcing conversion.

Two other factors also increase the investors’ incentive to convert
their bonds. First, some instruments have conversion prices that
step up over time according to a predetermined schedule. Since a
stepped up conversion price leads to a lower number of shares
received, it becomes more likely that investors will exercise their
option just before the conversion price increases. Second, a firm can
help to encourage conversion by increasing the dividends on the
stock, thereby making the income generated by the shares more
attractive relative to the income from the bond.
3 - 86
CY Convertible Bond Example (cont.)

Another important characteristic when evaluating convertible bonds
is the payback or break-even time, which measures how long the
higher interest income from the convertible bond (compared to the
dividend income from the common stock) must persist to make up
for the difference between the price of the bond and its conversion
value (i.e., the conversion premium). The calculation is as follows:
Payback =

Bond Price – Conversion Value
.
Bond Income – Income from Equal Investment in Common Stock
For instance, the annual coupon yield payment on the Cypress
Semiconductor convertible bond is $40, while the firm’s dividend
yield is zero. Thus, assuming you sold the bond for 920 and used
the proceeds to purchase 33.607 shares (= $920/$27.375) of
Cypress Semiconductor stock, the payback period would be:
$920.00 – $591.89
 8.20 years.
$40.00 – $0.00
3 - 87
CY Convertible Bond Example (cont.)

It is also possible to calculate the combined value of the investor’s conversion option
and issuer’s call feature that are embedded in the note. In the Cypress
Semiconductor example, with a market price of $920, the convertible’s yield-tomaturity can be calculated as the solution to:
8
$920  
t 1

20
1000

(1  y / 2) t (1  y / 2) 8
or y = 6.29 percent. This computation assumes 8 semi-annual coupon payments of
$20 (= 40  2). Since the yield on a Cypress Semiconductor debt issue with no
embedded options and the same (B1) credit rating and maturity was 8.5 percent, the
present value of a “straight” fixed-income security with the same cash flows would be:
8
$850.06  
t 1

20
1000

(1  0.0425) t (1  0.0425) 8
This means that the net value of the combined options is $69.94, or $920 minus
$850.06. Using the Black-Scholes valuation model, it is easily confirmed that a fouryear call option to buy one share of Cypress Semiconductor stock – which does not
pay a dividend – at an exercise price of $42.55 (i.e. the conversion parity value) is
equal to $6.35. Thus the value of the investor’s conversion option – which allows for
the acquisition of 21.6216 shares – must be $137.26 (= 21.6216  $6.35). This
means that the value of the issuer’s call feature under these conditions must be
$67.32 (= $137.26 – $69.94).
3 - 88
Illustrating Convertible Bond Valuation
3 - 89
Notion of Structured Notes

Generally speaking, structured notes are debt issues that have their
principal or coupon payments linked to some other underlying variable.
Examples would include bonds whose coupons are tied to the appreciation
of an equity index such as the S&P 500 or a zero-coupon bond with a
principal amount tied to the appreciation of an oil price index.

There are several common features that distinguish structured notes from
regular fixed-income securities, two of which are important for our
discussion. First, structured notes are designed for (are targeted to) a
specific investor with a very particular need. That is, these are not "generic"
instruments, but products tailored to address an investor's special
constraints, which are often themselves created by tax, regulatory, or
institutional policy restrictions.

Second, after structuring the financing to meet the investor's needs, the
issuer will typically hedge that unique exposure with swaps or exchangetraded derivatives. Inasmuch as the structured note itself most likely
required an embedded derivative to create the desired payoff structure for
the investor, this unwinding of the derivative position by the issuer
generates an additional source profit opportunity for the bond underwriter.
3 - 90
Overview of the Structured Note Market
3 - 91
Equity Index-Linked Note Example: MITTS

In July of 1992, Merrill Lynch & Co. raised USD 77,500,000 by
issuing 7,750,000 units of an S&P 500 Market Index Target-Term
Security, or "MITTS" for short, at a price of USD 10 per unit. These
MITTS units had a maturity date of August 29, 1997, making them
comparable in form to a five-year bond even though they traded on
the New York Stock Exchange. Indeed, Merrill Lynch issued them
as a series of Senior Debt Securities making no coupon payments
prior to maturity.

At maturity, a unit holder received the original issue price plus a
"supplemental redemption amount," the value of which depended on
where the Standard & Poor's 500 index settled relative to a
predetermined initial level. Given that this supplemental amount
could not be less than zero, the total payout to the investor at
maturity can be written:
10 + M ax

  Final S
0 , 
10 x




& P Value
- I nitial S & P Value
I nitial S & P Value
x




1. 15 
where the initial S&P value was specified as 412.08.
3 - 92
MITTS Example (cont.)

From the preceding description, recognize that the MITTS structure
combines a five-year, zero-coupon bond with an S&P index call
option, both of which were issued by Merrill Lynch. Thus, the MITTS
investor essentially owns a "portfolio" that is: (i) long in a bond and
(ii) long in an index call option position.

This particular security was designed primarily for those investors
who wanted to participate in the equity market but, for regulatory or
taxation reasons, were not permitted to do so directly. For example,
the manager of a fixed-income mutual fund might be able to
enhance her return performance by purchasing this "bond" and then
hoping for an appreciating stock market.

Notice that the use of the call option in this design makes it fairly
easy for Merrill Lynch to market to its institutional customers in that it
is a "no lose" proposition; the worst-case scenario for the investor in
that she simply gets her money back without interest in five years.
(Of course, the customer does carry the company's credit risk for
this period.) Thus, at origination the MITTS issue had no downside
exposure to stock price declines.
3 - 93
MITTS Example (cont.)

The call option embedded in this structure is actually a
partial position. To see this, we can rewrite the option
portion of the note's redemption value as:
Max
 
0 , 
10

 
x
Fin al S

& P - 412 . 08
412 . 08
( 0 . 0279 )

x

1. 15
Max
 0 ,

 =

Max

0,


 11 . 50 
Fin al S

 412 . 08 
& P - 412 . 08



Fi nal S & P - 412 . 08 . 
Thus, given that a regular index call option would have a
terminal payoff of Max[0, (Final S&P – X)], where X is the
strike price, the derivative in the MITTS represents
2.79% of this amount.
3 - 94
MITTS Example (cont.)

On February 28, 1996, the closing price for the MITTS issue was
USD 15.625, while the S&P 500 closed at 644.75. Further, the
semi-annually compounded yield of a zero-coupon (i.e., "stripped")
Treasury bond on this date was 5.35%.

Assuming a credit spread of 30 basis points to be appropriate for
Merrill Lynch's credit rating (i.e., A+ and A1 by Standard & Poor's
and Moody's, respectively) and the remaining time to maturity (i.e.,
one-and-a-half years, or three half-years), the bond portion of the
MITTS issue should be worth:
MITTS Bond Value =

10
.0565

1 +


2 
3
= 9.20.
This means that the investor is paying $6.43 (= 15.63 - 9.20) for the
embedded index call.
3 - 95
MITTS Example (cont.)

Without reproducing the full calculations, it is interesting
to note that the theoretical value on February 28, 1996 of
an S&P index call option expiring on August 29, 1997
with an exercise price of 412.08 is $243.19.

Thus, since the MITTS option feature represents 0.0279
of this amount, the call option embedded in the MITTS
issue is worth $6.78 (= 243.19 x 0.0279). Thus, on this
particular date the MITTS issue was priced in the market
below its theoretical value, presenting investors with a
potential buying opportunity depending on their
transaction costs. In fact, the embedded call is actually
priced below the index option’s intrinsic value of $6.49 (=
[644.75 – 412.08] x 0.0279), making the issue that much
more attractive to investors.
3 - 96
MITTS Example (cont.)

This MITTS transaction can be illustrated as follows:
Max(0,% SPX Rtn)
$10
August 1992
February 1996
August 1997
Zero-Coupon Bond
$9.20
$10
SPX Index Call Option
$6.43
3 - 97
Additional Structured Note Examples
3 - 98
Additional Structured Note Examples (cont.)
3 - 99