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FIGURE 5.8 The relationship between cost-of-carry and duration.

forward™s expiration. As a forward expiration date lengthens, carry will
become larger (via a larger T value), and carry™s positive or negative con-
tribution to overall price sensitivity of the forward will increase. Whether
the contribution to duration is positive or negative depends on whether carry
is positive or negative. If carry is zero, then the duration of the forward over
its life will be the duration of the underlying spot as calculated at the expi-
ration date of the forward agreement. Indeed, as expirations lengthen, the
importance of R and Yc™s contributions increases as well. Parenthetically, with
longer-dated options as with LEAPS (long-term equity anticipation securi-
ties), unit changes in R can make as important a contribution to the value
of the option as a unit change in the underlying spot.
In sum, and as shown in the figure, the duration of a forward is some-
thing less than the duration of its underlying spot. However, this lower level
of market risk (via duration) should not be construed to be an overall reduc-
tion in risk with the strategy in general. That is, do not forget that a for-
ward transaction means that payment is not exchanged for an asset until
some time in the future; it is hoped that the counterparty to the trade will
still be in business at that point in the future, but that is not 100 percent
certain. Thus, the reduction in market risk (via duration) is accompanied by
some element of credit risk (via delayed settlement).

Risk Management

Let us consider the forward duration value for an underlying security
that does not yet exist. For example, consider the forward duration of a six-
month Treasury bill 18 months forward. For relatively short financing hori-
zons, the duration of a forward will not be much greater than the duration
of the underlying spot security. Hence, the total forward duration of a six-
month Treasury bill will not be much different from six months. However,
it is appropriate to ask what yield the forward duration will be sensitive to
if we assume that the risk-free rate is relatively constant; will it be sensitive
to (a) changes in a generic 6-month Treasury bill spot yield, or (b) changes
in the forward curve (which, by construction, embodies a six-month spot
yield)? The answer is (b). Let us examine how and why this is the case.
Figure 5.9 shows that if this strategy is held to the expiration of the last
remaining component, the investment horizon will stretch over two years:
18 months for the length of the forward contract and then an additional 6
months once the forward expires and is exchanged for the spot six-month
Treasury bill.
Recall from Chapter 2 that we calculated an 18-month forward yield
on a six-month Treasury bill to be 6.10 percent. Recall also that the step-by-
step methodology used to arrive at that yield was such that a forward curve
is embedded within the yield. This yield value of 6.10 percent is certainly
not equal to the 4.75 percent spot yield value on a six-month Treasury bill
referred to in Chapter 2, nor is it equal to the 5.5 percent spot yield value
on the two-year Treasury bond cited there. In sum, despite the underlying
security of this forward transaction being a spot six-month Treasury bill, and
despite its having an investment horizon of two years, the relevant yield for
duration/risk management purposes is neither one of these; it is the 18-month
forward yield on an underlying six-month asset. Nonetheless, a fair question
to pose might be: Is there a meaningful statistical correlation between an 18-
month forward yield on an underlying six-month asset and the nominal yield

Forward yield is of relevance Spot yield is of relevance

O O horizon
24 months later
Trade date 18 months later
(6 months after
Investor goes long Forward contract
forward expires)
an 18-month expires and is
Treasury bill
forward contract on exchanged for spot
an underlying 6- 6-month Treasury
month Treasury bill. bill.
between spot and
forward rates.

FIGURE 5.9 Convergence between forward and spot yields.


of a two-year spot Treasury? Not surprisingly, the short answer is “It
depends.” A number of statistical studies have been performed over the years
to study the relationship between forward and spot yields and prices.
Generally speaking, the conclusions tend to be that forward values over
short-term horizons have strong correlations with spot values of short-life
assets (as with Treasury bills or shorter-dated Treasury bonds). Accordingly,
it would be a reasonably safe statistical bet that the correlation would be
strong between a two-year Treasury and an 18-month forward yield on an
underlying six-month asset. Why might this be of interest to a fixed-income
investor? Consider the following.
Let us assume that an investor believes that market volatility will
increase dramatically, but that for some reason she is precluded from exe-
cuting a volatility strategy with options. Perhaps the firm she works for has
internal or external constraints pertaining to the use of options. There is a
Treasury bill futures market, and the underlying spot Treasury bill tends to
have a three-month maturity. The futures are generally available in a string
of rolling 3-month contracts that can extend beyond a year. However, futures
on three-month Eurodollar instruments typically will extend well beyond the
forward horizon of Treasury bill contracts. The price of these futures con-
tracts is calculated as par minus the relevant forward yield of the underly-
ing spot instrument. Thus, if the relevant forward yield of the underlying
Treasury or Eurodollar spot is 6.0 percent then the price of the futures con-
tract is 100 6 94. A Treasury bill future typically involves a physical
settlement if held to expiration (where a physical exchange of cash and
Treasury bills takes place), while a Eurodollar future involves a cash settle-
ment (where there is no physical exchange, but simply a last marking-to-mar-
ket of final positions).
While there generally are no meaningful delivery options to speak of
with Treasury bill or Eurodollar futures, there is one interesting price char-
acteristic of these securities: Price changes are linked to a fixed predetermined
amount. Accordingly, each time the forward yield changes by one-half basis
point, the value of the futures contract changes by a fixed amount of $12.50
(or $25 per basis point). Why $25 per basis point? Simple. Earlier it was
said that the Macaulay duration of a zero coupon security is equal to its
maturity. The Macaulay duration of an underlying three-month asset is one-
quarter of a year, or 0.25. Therefore, with a notional contract value of $1
million, 1 basis point change translates into $25. Figure 5.8 showed scenarios
that might create a slight slope in the duration line of the forward as it
approached the duration of the underlying spot asset; this slope represents
carry™s contribution to duration.
In short, as purely convenience for itself and its investors, the futures
exchanges price the sensitivity of the underlying spot values of the Treasury
bills and Eurodollars at their spot duration value (three months). This con-

Risk Management

venience can create a unique volatility-capturing strategy. By going long both
Treasury bill futures and a spot two-year Treasury, we can attempt to repli-
cate the payoff profile shown in Figure 5.10. If the Macaulay duration of
the spot coupon-bearing two-year Treasury is 1.75 years, for every $1 mil-
lion face amount of the two-year Treasury that is purchased, we go long
seven Treasury bill futures with staggered expiration dates. Why seven?
Because 0.25 times seven is 1.75. Why staggered? So that the futures con-
tracts expire in line with the steady march to maturity of the spot two-year
Treasury. Thus, all else being equal, if the correlation is a strong one
between the spot yield on the two-year Treasury and the 21-month forward
yield on the underlying three-month Treasury bill, our strategy should be
close to delta-neutral. And as a result of being delta-neutral, we would expect
our strategy to be profitable if there are volatile changes in the market,
changes that would be captured by net exposure to volatility via our expo-
sure to convexity.
Figure 5.11 presents another perspective of the above strategy in a total
return context. As shown, return is zero for the volatility portion of this strat-
egy if yields do not move (higher or lower) from their starting point. Yet even
if the volatility portion of the strategy has a return of zero, it is possible that
the coupon income (and the income from reinvesting the coupon cash flows)
from the two-year Treasury will generate a positive overall return. Return

Price profile for a 3-month Treasury bill
21 months forward and leveraged seven times

Price level
Price profile for a spot 2-year Treasury

Starting point, and point of intersection
between spot and forward positions; also
corresponds to zero change in respective

This gap represents the
difference between
duration alone and
duration plus convexity;
the strategy is
increasingly profitable
as the market moves
appreciably higher or
Yields lower Yields higher
lower beyond its
starting point.
Changes in yield

FIGURE 5.10 A convexity strategy.


Total return This dip below zero (consistent with a slight
negative return) represents transactions costs
in the event that the market does not move
+ dramatically one way or the other.

0 O

Yields lower Yields higher

Changes in yield

FIGURE 5.11 Return profile of the “gap.”

can be positive when yields move appreciably from their starting point. If
all else is not equal, returns easily can turn negative if the correlation is not
a strong one between the spot yield on the two-year Treasury and the for-
ward yield on the Treasury bill position. The yields might move in opposite
directions, thus creating a situation where there is a loss from each leg of
the overall strategy. As time passes, the convexity value of the two-year
Treasury will shrink and the curvilinear profile will give way to the more
linear profile of the nonconvex futures contracts. Further, as time passes,
both lines will rotate counterclockwise into a flatter profile as consistent with
having less and less of price sensitivity to changes in yield levels.
Finally, while R and T (and sometimes Yc) are the two variables that dis-
tinguish spot from forward, there is not a great deal we can do about time;
time is simply going to decay one day at a time. However, R is more com-
plicated and deserves further comment.
It is a small miracle that R has not developed some kind of personality
disorder. Within finance theory, R is varyingly referred to as a risk-free rate
and a financing rate, and this text certainly alternates between both char-
acterizations. The idea behind referring to it as a risk-free rate is to highlight
that there is always an alternative investment vehicle. For example, the price
for a forward purchase of gold requires consideration of both gold™s spot
value and cost-of-carry. Although not mentioned explicitly in Chapter 2,
cost-of-carry can be thought of as an opportunity cost. It is a cost that the
purchaser of a forward agreement must pay to the seller. The rationale for
the cost is this: The forward seller of gold is agreeing not to be paid for the

Risk Management

gold until sometime in the future. The seller™s agreement to forgo an imme-
diate receipt of cash ought to be compensated. It is. The compensation is in
the form of the cost-of-carry embedded within the forward™s formula. Again,
the formula is F S (1 RT) S SRT, where SRT is cost-of-carry.
Accordingly, SRT represents the dollar (or other currency) amount that the
gold seller could have earned in a risk-free investment if he had received cash
immediately, that is, if there were an immediate settlement rather than a for-
ward settlement. R represents the risk-free rate he could have earned by
investing the cash in something like a Treasury bill. Why a Treasury bill?
Well, it is pretty much risk free. As a single cash flow security, it does not
have reinvestment risk, it does not have credit risk, and if it is held to matu-
rity, it does not pose any great price risks.
Why does R have to be risk free? Why can R not have some risk in it?
Why could SRT not be an amount earned on a short-term instrument that


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