fixed income securities-portfolio

while recognizing the limitations of some price quotations in the STRIPS
market, Figure 1.4 does suggest that shorter-term C-STRIPS traded rich,
longer-term C-STRIPS traded cheap, and P-STRIPS traded closer to fair.
Some P-STRIPS, like the longest three shown in Figure 1.4, traded rich
because the bonds associated with those STRIPS traded rich (that these
particular bonds trade rich will be discussed in Chapter 4). The 10- and
30-year P-STRIPS, cut off by the scale of the vertical axis in Figure 1.4,
traded extremely rich because the associated bonds enjoyed financing advantages and were particularly liquid. These factors will be discussed in
Chapter 15.
The previous section shows how to construct a replicating portfolio
and price a bond by arbitrage. The construction of a portfolio of STRIPS
that replicates a coupon bond is a particularly simple example of this procedure. To replicate 100 face value of the 5
3
/4s of August 15, 2003, for example, buy
5.75
/2 or 2.875 face value of each STRIPS in Table 1.5 to
replicate the coupon payments and buy an additional 100 face value of August 15, 2003, STRIPS to replicate the principal payment. Since one may
choose between a C- and a P-STRIPS on each cash flow date, there are
many ways to replicate the 53
/4s of August 15, 2003, and, therefore, to
compute its arbitrage price. Using only P-STRIPS, for example, the arbitrage price is
(1.3)
This is below the 102.020 market price of the 53
/4s of August 15, 2003. So,
in theory, if the prices in Table 1.5 were executable and if transaction costs
were small enough, one could profitably arbitrage this price difference by
buying the P-STRIPS and selling the 5
3
/4s of August 15, 2003.
While most market players cannot profit from the price differences between P-STRIPS, C-STRIPS, and coupon bonds, some make a business of
it. At times these professionals find it profitable to buy coupon bonds, strip
them, and then sell the STRIPS. At other times these professionals find it
profitable to buy the STRIPS, reconstitute them, and then sell the bonds.
On the other hand, a small investor wanting a STRIPS of a particular maturity would never find it profitable to buy a coupon bond and have it
stripped, for the investor would then have to sell the rest of the newly created STRIPS. Similarly, a small investor wanting to sell a particular STRIPS
would never find it profitable to buy the remaining set of required STRIPS,
reconstitute a coupon bond, and then sell the whole bond.
Given that investors find zeros useful relative to coupon bonds, it is
not surprising some professionals can profit from differences between the
two. Since most investors cannot cost-effectively strip and reconstitute by
themselves, they are presumably willing to pay something for having it
done for them. Therefore, when investors want more zeros they are willing
to pay a premium for zeros over coupon bonds, and professionals will find
it profitable to strip b

onds. Similarly, when investors want fewer zeros,
they are willing to pay a premium for coupon bonds over zeros and professionals will find it profitable to reconstitute bonds.APPENDIX 1A
DERIVING THE REPLICATING PORTFOLIO
Four bonds are required to replicate the cash flows of the 10
3
/4s of February 15, 2003. Let F
i
be the face amount of bond i used in the replicating
portfolio where the bonds are ordered as in Table 1.4. In problems of this
structure, it is most convenient to start from the last date. In order for the
portfolio to replicate the payment of 105.375 made on February 15, 2003,
by the 10
3
/4s, it must be the case thatThe face amounts of the first three bonds are multiplied by zero because
these bonds make no payments on February 15, 2003. The advantage of
starting from the end becomes apparent as equation (1.4) is easily solved:
(1.5)
Intuitively, one needs to buy more than 100 face value of the 6
1
/4s of February 15, 2003, to replicate 100 face of the 10
3
/4s of February 15, 2003,
because the coupon of the 6
1
/4s is smaller. But, since equation (1.4) matches
the principal plus coupon payments of the two bonds, the coupon payments alone do not match. On any date before maturity, 100 of the 103
/4s
F4 102 182 = .
FFFF 1234
1
4
000 1006
2
105 375 ×+ ×+ ×+ × + ?
?
?
?
?
?
?
?
?
?
?
?= %.
APPENDIX 1A Deriving the Replicating Portfolio 17
makes a payment of 5.375, while the 102.182 face of the 61
/4s makes a
payment of 102.182×6
1
/4%/2 or 3.193. Therefore, the other bonds listed in
Table 1.4 are required to raise the intermediate payments of the replicating
portfolio to the required 5.375. And, of course, the only way to raise these
intermediate payments is to buy other bonds.
Having matched the February 15, 2003, cash flow of the replicating
portfolio, proceed to the August 15, 2002, cash flow. The governing equation here is
(1.6)
Since F
4
is already known, equation (1.6) can be solved showing that
F
3
=–2.114. Continuing in this fashion, the next equations, for the cash
flows on February 15, 2002, and August 15, 2001, respectively, are
(1.7)
(1.8)
When solving equation (1.7), F
3
andF
4
are already known. When solving
equation (1.8), F
2
is also known.
Note that if the derivation had started by matching the first cash payment on August 15, 2001, the first equation to be solved would have been
(1.8). This is not possible, of course, since there are four unknowns. Therefore, one would have to solve equations (1.4), (1.6), (1.7), and (1.8) as a
system of four equations and four unknowns. There is nothing wrong with
proceeding in this way, but, if solving for the replicating portfolio by hand,
starting from the end proves simpler.
APPENDIX 1B