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SPOT PRICING FOR BONDS
Unlike equities or currencies, bonds are often as likely to be priced in terms
of a dollar price as in terms of a yield. Thus, we need to differentiate among
a few different types of yields that are of relevance for bonds.
The examples provided earlier made rather generic references to “yield.”
To be more precise, when a yield is calculated for the spot (or present) value
of a bond, that yield commonly is referred to as yield-to-maturity, bond-
equivalent yield, or present yield. There are also current yields (the result of
dividing a bond™s coupon by its current price), and spot yields (yield on bonds
with no cash flows to be made until maturity). Thus, a spot yield could be
Rising uncertainty




Uncertainty of credit quality Generally when a security is a nongovernmental issue



Uncertainty of reinvestment When a coupon is paid prior to sale or maturity



Uncertainty of price For any fixed income security


FIGURE 2.7 Layers of uncertainty among various types of bonds.




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26 PRODUCTS, CASH FLOWS, AND CREDIT



a yield on a Treasury bill,3 a yield on a coupon-bearing bond with no remain-
ing coupons to be paid until maturity, or a yield on a zero-coupon bond. In
some instances even a yield on a coupon-bearing bond that has a price of
par may be said to have a spot yield.4 In fact, for a coupon-bearing bond
whose price is par, its yield is sometimes called a par bond yield. For all of
the yield types cited, annualizing according to U.S. convention is assumed
to occur on the basis of a 365-day year (except for a leap year). Finally, when
an entire yield curve is comprised of par bond yields, it is referred to as a
par bond curve. A yield curve is created when the dots are connected across
the yields of a particular issuer (or class of issuers) when its bonds are plot-
ted by maturity. Figure 2.8 shows a yield curve of Treasury bonds taken from
November 2002.
As shown, the Treasury bond yield curve is upward sloping. That is,
longer-maturity yields are higher than shorter-maturity yields. In fact, more




Yield



6


4


2

Time (years)
0.5 1 2 5 10 30

FIGURE 2.8 Normal upward-sloping yield curve.


3
As a money market instrument (a fixed income security with an original term to
maturity of 12 months or less), a Treasury bill also has unique calculations for its
yield that are called “rate of discount” and “money market yield.” A rate of
discount is calculated as price divided by par and then annualized on the basis of
a 360-day year, while a money market yield is calculated as par minus price
divided by par and then annualized on the basis of a 360-day year.
4
The reason why a coupon-bearing bond priced at par is said to have a yield
equivalent to a spot yield is simply a function of algebraic manipulation. Namely,
since a bond™s coupon rate is equal to its yield when the bond is priced at par, and
since its price and face value are equivalent when yield is equal to coupon rate,
letting C = Y and P = F and multiplying through a generic price/yield equation by
1/F (permissible by the distributive property of multiplication) we get 1 = Y/2/(1 +
Y/2)1 + Y/2 /(1 + Y/2)2 + ... In short, C drops away.




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Cash Flows



often than not, the Treasury bond yield curve typically reflects such a pos-
itive slope.
When a bond is being priced, the same yield value is used to discount
(reduce to a present value) every cash flow from the first coupon received
in six months™ time to the last coupon and face amount received in 2 or even
30 years™ time. Instead of discounting a bond™s cash flows with a single yield,
which would suggest that the market™s yield curve is perfectly flat, why not
discount a bond™s cash flows with more representative yields? Figure 2.9
shows how this might be done.
In actuality, many larger bond investors (e.g., bond funds and invest-
ment banks) make active use of this approach (or a variation thereof) to pric-
ing bonds to perform relative value (the value of Bond A to Bond B)
analysis. That is, if a bond™s market price (calculated by market convention
with a single yield throughout) was lower than its theoretical value (calcu-
lated from an actual yield curve), this would suggest that the bond is actu-
ally trading cheap in the marketplace.5



Price = C&F + C + C + C = $1,000
2 1
4 3
(1 + Y/2) (1 + Y/2) (1 + Y/2) (1 + Y/2)




Yield



6


4


2

Time (years)
0.5 1 2 5 10 30

FIGURE 2.9 Using actual yields from a yield curve to calculate a bond™s price.


5
It is important to note that it is theoretically possible for a given bond to remain
“cheap” (or rich) until the day it matures. A more likely scenario is that a bond™s
cheapness and richness will vary over time. Indeed, what many relative value
investors look for is a good amount of variability in a bond™s richness and
cheapness as a precondition for purchasing it on a relative value basis.




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While there is just one spot price in the world of bonds, there can be a
variety of yields for bonds. Sometimes these different terms for yield apply
to a single value. For example, for a coupon-bearing bond priced at par, its
yield-to-maturity, current yield, and par bond yield are all the same value.
As previously stated, a yield spread is the difference between the yield
of a nonbenchmark security and a benchmark security, and it is expressed
in basis points.


Yield of nonbenchmark Yield of benchmark Spread in basis points

Therefore, any one of the following things might cause a spread to nar-
row or become smaller (where the opposite event would cause it to widen
or become larger):

A. If the yield of the nonbenchmark (YNB) issue were to . . .
i. . . . decline while the yield of the benchmark (YB) issue were to
remain unchanged
ii. . . . rise while the YB rose by more
iii. . . . remain unchanged while the YB rose
B. If the yield of the benchmark issue (YB) were to . . .
i. . . . rise while the yield of the nonbenchmark (YNB) issue remained
unchanged
ii. . . . decline while the YNB fell by more
iii. . . . remain unchanged while the YNB fell

Thus, the driving force(s) behind a change in spread can be attributable
to the nonbenchmark, the benchmark, or a combination of both.
Accordingly, investors using spreads to identify relative value must keep these
contributory factors in mind.
Regarding spreads generally, while certainly of some value as a single sta-
tic measure, they are more typically regarded by fixed income investors as
having value in a dynamic context. At the very least, a single spread measure
communicates whether the nonbenchmark security is trading rich (at a lower
yield) or cheap (at a higher yield) to the benchmark security. Since the bench-
mark yield is usually subtracted from the nonbenchmark yield, a positive yield
spread suggests that the nonbenchmark is trading cheap to the benchmark
security, and a negative yield spread suggests that it is trading rich to the
benchmark security. To say much beyond this in a strategy-creation context
with the benefit of only one data point (the one spread value) is rather diffi-
cult. More could be said with the benefit of additional data points.
For example, if today™s spread value is 50 basis points (bps), and we
know that over the past four weeks the spread has ranged between 50 bps



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29
Cash Flows



and 82 bps, then we might say that the nonbenchmark security is at the
richer (narrower) end of the range of where it has traded relative to the
benchmark issue. If today™s spread value were 82 bps, then we might say
that the nonbenchmark security is at the cheaper (wider) end of the range
of where it has been trading. These types of observations can be of great
value to fixed income investors when trying to decipher market trends and
potential opportunities. Yet even for a measure as simple as the difference
between two yields, some basic analysis might very well be appropriate. A
spread might change from day to day for any number of reasons. Many bond
fund managers work to know when and how to trade around these various
changes.




Spot
Equities




Now we can begin listing similarities and differences between equities and
bonds. Equities differ from bonds since they have no predetermined matu-
rity. Equities are similar to bonds since many equities pay dividends, just as
most bonds pay coupons. However, dividends of equities generally tend to
be of lower dollar amounts relative to coupons of bonds, and dividend
amounts paid may vary over time in line with the company™s profitability
and dividend-paying philosophy; the terms of a bond™s coupon payments typ-
ically are set from the beginning. And while not typical, a company might
choose to skip a dividend payment on its equity and without legal conse-
quences, while a skipped coupon payment on a bond is generally sufficient
to initiate immediate concerns regarding a company™s ongoing viability.6
Occasionally a company may decide to skip a dividend payment altogether,
with the decision having nothing to do with the problems in the company;
there may simply be some accounting incentives for it, for example.
Otherwise, in the United States, bonds typically pay coupons on a semian-
nual basis while equities tend to pay dividends on a quarterly basis. Figure
2.10 is tailored to equities.

6
Terms and conditions of certain preferred equities may impose strict guidelines on
dividend policies that firms are expected to follow.




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Cash
inflow




0
3 months later 6 months later 9 months later 12 months later
Dividend payment Dividend payment Dividend payment Dividend payment

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