Reinvestment at 3%
FIGURE 2.3 Effect of reinvestment rates on total return.
rate is equal to Y. The Y in the equation is yield, and it is the same value in
each term of the equation. This is equivalent to saying that we expect each
coupon cash flow (except the last two, coupon and principal) to be reinvested
for the remaining life of the underlying security at the yield level prevailing
when the security was originally purchased. Accordingly, the price of a 6 per-
cent coupon-bearing two-year Treasury with a 6 percent yield is $1,000 as
shown in the next equation.
6%>22 1 6%>22 2
$60>2 $60>2 & $1,000
6%>22 3 6%>22 4
If yield should happen to drop to 5 percent after initial launch, the
coupon rate remains at 6 percent and the price increases to $1,018.81. And
if the yield should happen to rise to 7 percent after launch, the price drops
to 981.63. Hence, price and yield move inversely to one another. Moreover,
by virtue of priceâ€™s sensitivity to yield levels (and, hence, reinvestment rates),
a coupon-bearing securityâ€™s unhedged total return at maturity is impossible
to pin down at time of purchase. Figure 2.4 confirms this.
Figure 2.5 plots the identical yields from the last equation after revers-
ing the order in which the individual terms are presented. This order rever-
Cash 6 months later 12 months later 18 months later 24 months later
outflow â€“ Coupon payment â€“ Coupon payment â€“ Coupon payment â€“ Coupon and principal
To be reinvested for To be reinvested for To be reinvested for
18 months 12 months 6 months
All reinvestments assumed to be for the remaining
life of the bond and at the yield that prevailed at the
time of the bondâ€™s purchase.
FIGURE 2.4 Reinvestment requirements of a 2-year coupon-bearing Treasury bond.
22 PRODUCTS, CASH FLOWS, AND CREDIT
($60/2)&$1,000 + $60/2 + $60/2 + $60/2 = $1,000
4 3 (1 + 6%/2)2 1
(1 + 6%/2) (1 + 6%/2) (1 + 6%/2)
0 6 12 18 Reinvestment
FIGURE 2.5 Reinvestment patterns for cash flows of a 2-year coupon-bearing
sal is done simply to achieve a chronological pairing between the timing of
when cash flows are paid and the length of time they are reinvested. Note
how the resulting term structure (a plotting of yields by respective dates) is
Note too that when a reinvestment of a coupon cash flow is made, the
new security that is purchased also may be a coupon-bearing security. As
such, it will embody reinvestment risk. Figure 2.6 illustrates this.
Let us now add another layer of uncertainty, called the uncertainty of
credit quality (the uncertainty that a credit may drift to a lower rating or go
into default). Instead of assuming that we have a two-year security issued
by the U.S. Treasury, let us now assume that we have a two-year bond issued
by a U.S. corporation. Unless we are willing to assume that the corporationâ€™s
bond carries the same credit quality as the U.S. government, there are a cou-
ple of things we will want to address. First, we will probably want to change
the value of Y in our equation and make it a higher value to correspond with
the greater risk we are taking on as an investor. And what exactly is that
greater risk? To be blunt, it is the risk that we as investors may not receive
complete (something less than 100 percent), and/or timely payments (pay-
ments made on a date other than formally promised) of all the cash flows
that we have coming to us. In short, there is a risk that the company debt
will become a victim of a distressed or default-related event.
Clearly there are many shades of real and potential credit risks, and these
risks are examined in much more detail in Chapter 3. For the time being,
FIGURE 2.6 How coupon cash flows of a 2-year Treasury bond give rise to additional
we must accept the notion that we can assign credit-linked probabilities to
each of the expected cash flows of any bond. For a two-year Treasury note,
each cash flow can be assigned a 100 percent probability for the high like-
lihood of full and timely payments. For any nongovernmental security, the
24 PRODUCTS, CASH FLOWS, AND CREDIT
probabilities may range between zero and 100 percent. Zero percent? Yes.
In fact, some firms specialize in the trading of so-called distressed debt, which
can include securities with a remaining term to maturity but with little or
no likelihood of making any coupon or principal payments of any kind. A
firm specializing in distressed situations might buy the bad debt (the down-
graded or defaulted securities) with an eye to squeezing some value from the
seizure of the companyâ€™s assets. Bad debt buyers also might be able to
reschedule a portion of the outstanding sums owed under terms acceptable
to all those involved.
If we go back to the formula for pricing a two-year Treasury note, we
will most certainly want to make some adjustments to identify the price of
a two-year non-Treasury issue. To compensate for the added risk associated
with a non-Treasury bond we will want a higher coupon paid out to usâ€”
we will want a coupon payment above C. And since a coupon rate is equal
to Y at the time a bond is first sold, a higher coupon means that we are
demanding a higher Y as well.
To transform the formula for a two-year Treasuryâ€”
Y>22 1 Y>22 2
11 11 Y>22 4
from something that is Treasury-specific into something that is relevant for
non-Treasury bonds, we can say that Yi represents the yield of a like-matu-
rity Treasury bond plus some incremental yield (and hence coupon) that a
non-Treasury bond will have to pay so as to provide the proper incentive to
purchase it. In the bond market, the difference between this incremental yield
and a corresponding Treasury yield is called a yield spread. Rewriting the
price formula, we have:
11 Yi>22 1 11 Yi>22 2
11 Yi>22 3 11 Yi>22 4
Since the same number of added basis points that are now included in
Yi are included in C, the price of the non-Treasury bond will still be par at
the time of original issueâ€”at least when it first comes to the marketplace.
Afterwards things change; yield levels are free to rise and fall, and real and
perceived credit risks can become greater or lesser over time. With regard
to credit risks, greater ones will be associated with higher values of Yi and
lower ones will translate into lower values of Yi.
So far, we have uncovered three uncertainties pertaining to pricing:
1. Uncertainty of price beyond time of original issue.
2. Uncertainty of reinvestment of coupons.
3. Uncertainty of credit quality.
To understand the layering effect, consider Figure 2.7. The first layer,
uncertainty of price, is common to any fixed income security that is sold
prior to maturity. The second layer, uncertainty of reinvestment, is applica-
ble only to coupon-bearing bonds that pay a coupon prior to sale or matu-
rity. And the third layer, uncertainty of credit quality, generally is unique to
those bond issuers that do not have the luxury of legally printing money (i.e.,
that are not a government entity; for more on this, see Chapter 3).