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Forward curve

Par bond curve

Ten years

FIGURE 2.16 Distinctions between points on and point along par bond and forward curves.

the risk associated with trading in a non-par bond Treasury that may not always
be as readily available in the market as a par bond issue.
To calculate a forward spread for a non-Treasury security (i.e., a secu-
rity that is not regarded as risk free), a Treasury par bond curve typically is
used as the reference curve to construct a forward curve. The resulting for-
ward spread embodies both a measure of a non-Treasury liquidity premium
and the non-Treasury credit risk.
We conclude this section with Figure 2.17.

Two formulaic modifications are required when going from a bond™s for-
ward price calculation to its futures price calculation. The first key differ-
ence is the incorporation of a bond™s conversion factor. Unlike gold, which
is a standard commodity type, bonds come in many flavors. Some bonds
have shorter maturities than others, higher coupons than others, or fewer
bells and whistles than others, even among Treasury issues (which are the
most actively traded of bond futures). Therefore, a conversion factor is an
attempt to apply a standardized variable to the calculation of all candidates™
spot prices.11 As shown in the equation on page 46, the clean forward price

A conversion factor is simply a modified forward price for a bond that is eligible
to be an underlying security within a futures contract. As with any bond price, the
necessary variables are price (or yield), coupon, maturity date, and settlement date.
However, the settlement date is assumed to be first day of the month that the
contract is set to expire; the maturity date is assumed to be the first day of the
month that the bond is set to mature rounded down to the nearest quarter (March,
June, September, or December); and the yield is assumed to be 8 percent regardless
of what it may actually be. The dirty price that results is then divided by 100 and
rounded up at the fourth decimal place.


. . . is used to construct a
A par bond curve
forward yield curve.
of spot yields . . .
If the par bond curve
is flat, or if T=0
(settlement is
immediate), then the
. . . is identical to a par
forward curve . . .
Spot Forwards
bond curve.

FIGURE 2.17 Spot versus forward yield curves.

of a contract-eligible bond is simply divided by its relevant conversion fac-
tor. When the one bond is flagged as the relevant underlying spot security
for the futures contract (via a process described in Chapter 4), its conver-
sion-adjusted forward price becomes the contract™s price.
The second formula modification required when going from a forward
price calculation to a futures price calculation concerns the fact that a bond
futures contract comes with delivery options. That is, when a bond futures
contract comes to its expiration month, investors who are short the contract
face a number of choices. Recall that at the expiration of a forward or future,
some predetermined amount of an asset is exchanged for cash. Investors who
are long the forward or future pay cash and accept delivery (take owner-
ship) of the asset. Investors who are short the forward or future receive cash
and make delivery (convey ownership) of the asset. With a bond futures con-
tract, the delivery process can take place on any business day of the desig-
nated delivery month, and investors who are short the contract can choose
when delivery is made during that month. This choice (along with others
embedded in the forward contract) has value, as does any asymmetrical deci-
sion-making consideration, and it ought to be incorporated into a bond
future™s price calculation. Chapter 4 discusses the other choices embedded
in a bond futures contract and how these options can be valued.
A bond futures price can be defined as:

3S 11 T 1R Yc 2 2 Od 4 >CF
Fd Af

where Od the embedded delivery options
CF the conversion factor

Cash Flows

A minus sign appears in front of Od since the delivery options are of
benefit to investors who are short the bond future. Again, more on all this
in Chapter 4.

& futures

To calculate the forward price of an equity, let us consider IBM at $80.25 a
share. If IBM were not to pay dividends as a matter of corporate policy, then
to calculate a one-year forward price, we would simply multiply the number
of shares being purchased by $80.25 and adjust this by the cost of money for
one year. The formula would be F S (1 RT), exactly as with gold or
Treasury bills. However, IBM™s equity does pay a dividend, so the forward price
for IBM must reflect the fact that these dividends are received over the com-
ing year. The formula really does not look that different from what we use for
a coupon-bearing bond; in fact, except for one variable, it is the same. It is
S 11 T 1R Yd 2 2
where Yd dividend yield calculated as the sum of expected dividends in
the coming year divided by the underlying equity™s market price.
Precisely how dividends are treated in a forward calculation depends on
such considerations as who the owner of record is at the time that the inten-
tion of declaring a dividend is formally made by the issuer. There is not a
straight-line accretion calculation with equities as there is with coupon-
bearing bonds, and conventions can vary across markets. Nonetheless, in
cases where the dividend is declared and the owner of record is determined,
and this all transpires over a forward™s life span, the accrued dividend fac-
tor is easily accommodated.

As with bonds, there are also equity futures. However, unlike bond futures,
which have physical settlement, equity index futures are cash-settled. Physical
settlement of a futures contract means that an actual underlying instrument
(spot) is delivered by investors who are short the contract to investors who
are long the contract, and investors who are long pay for the instrument. When


a futures contract is cash-settled, the changing cash value of the underlying
instrument is all that is exchanged, and this is done via the daily marking-to-
market mechanism. In the case of the Standard & Poor™s (S&P) 500 futures
contract, which is composed of 500 individual stocks, the aggregated cash
value of these underlying securities is referenced with daily marks-to-market.
Just as dividend yields may be calculated for individual equities, they
also may be calculated for equity indices. Accordingly, the formula for an
equity index future may be expressed as
S 11 T 1R Yd 2 2
where S and Yd market capitalization values (stock price times out-
standing shares) for the equity prices and dividend yields of the com-
panies within the index.

Since dividends for most index futures generally are ignored, there is typ-
ically no price adjustment required for reinvestment cash flow considerations.
Equity futures contracts typically have prices that are rich to (above)
their underlying spot index. One rationale for this is that it would cost
investors a lot of money in commissions to purchase each of the 500 equi-
ties in the S&P 500 individually. Since the S&P future embodies an instan-
taneous portfolio of securities, it commands a premium to its underlying
portfolio of spot instruments. Another consideration is that the futures con-
tract also must reflect relevant costs of carry.
Finally, just as there are delivery options embedded in bond futures con-
tracts that may be exercised by investors who are short the bond future,
unique choices unilaterally accrue to investors who are short certain equity
index futures contracts. Again, just as with bond futures, the S&P 500 equity
future provides investors who are short the contract with choices as to when
a delivery is made during the contract™s delivery month, and these choices
have value. Contributing to the delivery option™s value is the fact that
investors who are short the future can pick the delivery day during the deliv-
ery month. Depending on the marketplace, futures often continue to trade
after the underlying spot market has closed (and may even reopen again in
after-hours trading).

& futures

The calculation for the forward value of an exchange rate is again a mere

Cash Flows

variation on a theme that we have already seen, and may be expressed as

S 11 T 1Rh Ro 2 2

where Rh the home country risk-free rate
Ro the other currency™s risk-free rate

For example, if the dollar-euro exchange rate is 0.8613, the three-month
dollar Libor rate (London Inter-bank Offer Rate, or the relevant rate among
banks exchanging euro dollars) is 3.76 percent, and the three-month euro
Libor rate is 4.49 percent, then the three-month forward dollar-euro
exchange rate would be calculated as 0.8597. Observe the change in the dol-
lar versus the euro (of 0.0016) in this time span; this is entirely consistent
with the notion of interest rate parity introduced in Chapter 1. That is, for
a transaction executed on a fully hedged basis, the interest rate gain by invest-
ing in the higher-yielding euro market is offset by the currency loss of
exchanging euros for dollars at the relevant forward rate.
If a Eurorate (not the rate on the euro currency, but the rate on a Libor-
type rate) differential between a given Eurodollar rate and any other euro
rate is positive, then the nondollar currency is said to be a premium currency.
If the Eurorate differential between a given Eurodollar rate and any other
Eurorate is negative, then the nondollar currency is said to be a discount cur-
rency. Table 2.3 shows that at one point, both the pound sterling and
Canadian dollar were discount currencies to the U.S. dollar. Subtracting
Canadian and sterling Eurorates from respective Eurodollar rates gives neg-
ative values.
There is an active forward market in foreign exchange, and it is com-
monly used for hedging purposes. When investors engage in a forward trans-
action, they generally buy or sell a given exchange rate forward. In the last
example, the investor sells forward Canadian dollars for U.S dollars. A for-

TABLE 2.3 Rates from May 1991
Country 3 Month (%) 6 Month (%) 12 Month (%)

United States 6.0625 6.1875 6.2650
Canada 9.1875 9.2500 9.3750
United Kingdom 11.5625 11.3750 11.2500



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