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scenario. As shown, the forward leaps over the three separate coupon cash
flows; the purchaser does not receive these cash flows since he does not actu-
ally take ownership of the underlying spot until the forward expires. And
since the holder of the forward will not receive these intervening cash flows,
he ought not to pay for them. As discussed, the spot price of a coupon-bear-
ing bond embodies an expectation of the coupon actually being paid.
Accordingly, when calculating the forward value of a security that generates
cash flows, it is necessary to adjust for the value of any cash flows that are
paid and reinvested over the life of the forward itself.
Bonds are unique relative to equities and currencies (and all other types
of assets) since they are priced both in terms of dollar prices and in terms
of yields (or yield spreads). Now, we must discuss how a forward yield of a
bond is calculated. To do this, let us use a real-world scenario. Let us assume
that an investor is trying to decide between (a) buying two consecutive six-
month Treasury bills and (b) buying one 12-month Treasury bill. Both
investments involve a 12-month horizon, and we assume that our investor
intends to hold any purchased securities until they mature. Should our
investor pick strategy (a) or strategy (b)? To answer this, the investor prob-

Cash flows

The purchaser of a forward does not receive
the cash flows paid over the life of the
forward and ought not to pay for them.


Date forward Date forward expires and
is purchased previously agreed forward
price is paid for forward™s
underlying spot

FIGURE 2.14 Relationship between forwards and ownership of intervening cash flows.

Cash Flows

ably will want some indication of when and how strategy (a) will break even
relative to strategy (b). That is, when and how does the investor become
indifferent between strategy (a) and (b) in terms of their respective returns?
Calculating a single forward rate can help us to answer this question.
To ignore, just for a moment, the consideration of compounding, assume
that the yield on a one-year Treasury bill is 5 percent and that the yield on
a six-month Treasury bill is 4.75 percent. Since we want to know what the
yield on the second six-month Treasury bill will have to be to earn an equiv-
alent of 5 percent, we can simply solve for x with

5% (4.75% + x)/2.

Rearranging, we have

x 10% 4.75% 5.25%.

Therefore, to be indifferent between two successive six-month Treasury
bills or one 12-month Treasury bill, the second six-month Treasury bill
would have to yield at least 5.25 percent. Sometimes this yield is referred to
as a hurdle rate, because a reinvestment at a rate less than this will not be
as rewarding as a 12-month Treasury bill. Now let™s see how the calculation
looks with a more formal forward calculation where compounding is con-

11 Y2>22 2
cc d 1d
11 Y1>22 1
F6,6 2

11 0.05>22 2
cc d 1d
F6,6 2
0.0475>22 1

The formula for F6,6 (the first 6 refers to the maturity of the future
Treasury bill in months and the second 6 tells us the forward expiration date
in months) tells us the following: For investors to be indifferent between buy-
ing two consecutive six-month Treasury bills or one 12-month Treasury bill,
they will need to buy the second six-month Treasury bill at a minimum yield
of 5.25 percent. Will six-month Treasury bill yields be at 5.25 percent in six
months™ time? Who knows? But investors may have a particular view on the
matter. For example, if monetary authorities (central bank officials) are in
an easing mode with monetary policy and short-term interest rates are
expected to fall (such that a six-month Treasury bill yield of less than 5.25
percent looks likely), then a 12-month Treasury bill investment would


appear to be the better bet. Yet, the world is an uncertain place, and the for-
ward rate simply helps with thinking about what the world would have to
look like in the future to be indifferent between two (or more) investments.
To take this a step further, let us consider the scenario where investors
would have to be indifferent between buying four six-month Treasury bills
or one two-year coupon-bearing Treasury bond. We already know that the
first six-month Treasury bill is yielding 4.75 percent, and that the forward
rate on the second six-month Treasury bill is 5.25 percent. Thus, we still need
to calculate a 12-month and an 18-month forward rate on a six-month
Treasury bill. If we assume spot rates for 18 and 24 months are 5.30 per-
cent and 5.50 percent, respectively, then our calculations are:

11 0.053>22 3
cc d 1d
F6,12 2
0.05>22 2
5.90%, and

11 0.055>22 4
cc d 1d
F6,18 2
0.053>22 3

For investors to be indifferent between buying a two-year Treasury bond
at 5.5 percent and successive six-month Treasury bills (assuming that the
coupon cash flows of the two-year Treasury bond are reinvested at 5.5 per-
cent every six months), the successive six-month Treasury bills must yield a
minimum of:

5.25 percent 6 months after initial trade
5.90 percent 12 months after initial trade
6.10 percent 18 months after initial trade

Note that 4.75% .25 5.25% .25 5.9% .25 6.1% .25 5.5%.
Again, 5.5 percent is the yield-to-maturity of an existing two-year
Treasury bond.
Each successive calculation of a forward rate explicitly incorporates the
yield of the previous calculation. To emphasize this point, Figure 2.15 repeats
the three calculations.
In brief, in stark contrast to the nominal yield calculations earlier in this
chapter, where the same yield value was used in each and every denomina-
tor where a new cash flow was being discounted (reduced to a present value),
with forward yield calculations a new and different yield is used for every
cash flow. This looping effect, sometimes called bootstrapping, differentiates
a forward yield calculation from a nominal yield calculation.

Cash Flows

(1 + 0.05/2)2
F6,6 = “1 2
(1 + 0.0475/2)

= 5.25%

(1 + 0.053/2)3 “1
F6,12 = 2
(1 + 0.05/2)

= 5.90%, and

(1 + 0.055/2)4 “1
F6,18 = 2
(1 + 0.053/2)

= 6.10%.

FIGURE 2.15 Bootstrapping methodology for building forward rates.

Because a single forward yield can be said to embody all of the forward
yields preceding it (stemming from the bootstrapping effect), forward yields
sometimes are said to embody an entire yield curve. The previous equations
show why this is the case.
Table 2.2 constructs three different forward yield curves relative to three
spot curves. Observe that forward rates trade above spot rates when the spot
rate curve is normal or upward sloping; forward rates trade below spot rate
when the spot rate curve is inverted; and the spot curve is equal to the for-
ward curve when the spot rate curve is flat.
The section on bonds and spot discussed nominal yield spreads. In the
context of spot yield spreads, there is obviously no point in calculating the
spread of a benchmark against itself. That is, if a Treasury yield is the bench-
mark yield for calculating yield spreads, a Treasury should not be spread
against itself; the result will always be zero. However, a Treasury forward
spread can be calculated as the forward yield difference between two
Treasuries. Why might such a thing be done?
Again, when a nominal yield spread is calculated, a single yield point on
a par bond curve (as with a 10-year Treasury yield) is subtracted from the
same maturity yield of the security being compared. In sum, two indepen-
dent and comparable points from two nominal yield curves are being com-
pared. In the vernacular of the marketplace, this spread might be referred to
as “the spread to the 10-year Treasury.” However, with a forward curve, if
the underlying spot curve has any shape to it at all (meaning if it is anything
other than flat), the shape of the forward curve will differ from the shape of
the par bond curve. Further, the creation of a forward curve involves a


TABLE 2.2 Table Forward Rates under Various Spot Rate Scenarios
Scenario A Scenario B Scenario C
Forward Expiration Spot Forward Spot Forward Spot Forward

6 Month 8.00 /8.00 8.00 /8.00 8.00 /8.00
12 Month 8.25 /8.50 7.75 /7.50 8.00 /8.00
18 Month 8.50 /9.00 7.50 /7.00 8.00 /8.00
24 Month 8.75 /9.50 7.25 /6.50 8.00 /8.00
30 Month 9.00 /10.00 7.00 /6.00 8.00 /8.00
Scenario A: Normal slope spot curve shape (upward sloping)
Scenario B: Inverted slope spot curve
Scenario C: Flat spot curve

process whereby successive yields are dependent on previous yield calcula-
tions; a single forward yield value explicitly incorporates some portion of an
entire par bond yield curve. As such, when a forward yield spread is calcu-
lated between two forward yields, it is not entirely accurate to think of it as
being a spread between two independent points as can be said in a nominal
yield spread calculation. By its very construction, the forward yield embod-
ies the yields all along the relevant portion of a spot curve.
Figure 2.16 presents this discussion graphically. As shown, the bench-
mark reference value for a nominal yield spread calculation is simply taken
from a single point on the curve. The benchmark reference value for a for-
ward yield spread calculation is mathematically derived from points all along
the relevant par bond curve.
If a par bond Treasury curve is used to construct a Treasury forward curve,
then a zero spread value will result when one of the forward yields of a par
bond curve security is spread against its own forward yield level. However,
when a non-par bond Treasury security has its forward yield spread calculated
in reference to forward yield of a par bond issue, the spread difference will likely
be positive.10 Therefore, one reason why a forward spread might be calculated
between two Treasuries is that this spread gives a measure of the difference
between the forward structure of the par bond Treasury curve versus non-par
bond Treasury issues. This particular spreading of Treasury securities can be
referred to as a measure of a given Treasury yield™s liquidity premium, that is,

One reason why non-par bond Treasury issues usually trade at higher forward
yields is that non-par securities are off-the-run securities. An on-the-run Treasury is
the most recently auctioned Treasury security; as such, typically it is the most
liquid and most actively traded. When an on-the-run issue is replaced by some
other newly auctioned Treasury, it becomes an off-the-run security and generally
takes on some kind of liquidity premium. As it becomes increasingly off-the-run,
its liquidity premium tends to grow.

Cash Flows


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