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).

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191

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

Investment

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

matures.

an underlying 6- 6-month Treasury

month Treasury bill. bill.

Convergence

between spot and

forward rates.

FIGURE 5.9 Convergence between forward and spot yields.

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192 FINANCIAL ENGINEERING, RISK MANAGEMENT, AND MARKET ENVIRONMENT

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-

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193

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

yields

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.

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194 FINANCIAL ENGINEERING, RISK MANAGEMENT, AND MARKET ENVIRONMENT

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

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195

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