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between a forward and a future is how the cash flows work between the
time the trade is first executed and when it expires, that is, the daily mark-
to-market dynamic with the futures contracts. The net effect of receiving a
security at some later day but at an earlier agreed-on price is exactly the same
for these two instrument types.
Chapter 6 will delve into more detail of when and why certain types of
investors might prefer using futures instead of forwards.
Figure 2.11 shows the relationship between spots and forwards explic-
itly. Spot (S) is a key variable for calculating a forward (F) value. In fact,
spot is so important that when S 0, F 0; a forward is nothing without
some value for spot. Further, if there were no opportunity cost to money
(meaning that money is borrowed and lent at an interest rate of zero), then
F S. And if there is no forward time horizon (T 0; there is immediate
settlement), then F S. In short, a key difference between spots and for-
wards is the SRT term. RT is sometimes called cost of carry or simply carry.
For a forward settlement on a Treasury bill, the same logic applies that
was just used for gold.

If the market moves against a futures investor in a big way (as with a large decline
in the price of gold in this example), the futures exchange might ask the investor to
post margin. The investor is required to deposit money into the margin account to
assure the exchange that she has the financial resources to make good on her
commitments for future purchases at the agreed-on terms.


Spot Forwards

S F = S(1 + RT)

FIGURE 2.11 Relationship between spots and forwards.

For example, let us say we have a three-month Treasury bill with 90 days
to maturity and with a dollar price of 98.7875 (from a rate of discount of
4.85 percent). Let us further assume that the prevailing forces of supply and
demand for our Treasury bill in the securities lending market (or repurchase
(repo) market for bonds), dictates a financing rate of 5.5 percent for a period
of 30 days. A securities lending market is where specific financial instruments
are borrowed (lent) by (to) investors over predetermined periods of time, and
at an agreed-upon rate of financing. To calculate this Treasury bill™s forward
rate, we do the following calculation:

F S(1 RT)
99.2403 98.7875 98.7875 (1 5.5% 30/360).

The difference between the cash price of 98.7875 dollars and the for-
ward price of 99.2403 dollars, or 45.28 cents, is often referred to as the
carry, the forward drop, or simply the drop.

& futures

The calculation is pretty much the same to perform a forward price calcu-
lation with a coupon-bearing Treasury. Since coupon-bearing U.S. Treasuries
pay coupons on a semiannual basis, and knowing that coupon payments key

Cash Flows

off of maturity dates, let us assume we have a Treasury that matures on
November 15 in five years. Accordingly, this Treasury will pay a coupon
every November 15 and May 15 until maturity. If today happens to be
October 1 and we want to calculate a forward price for 30 days from now
(October 31), our forward price formula will need to consider that the bond
will be accruing (accumulating) coupon value over those 30 days. For
coupon-bearing securities, prices often are referred to as being either clean
or dirty.9 If an investor is being quoted a bond™s dirty price, the price includes
any accrued coupon value; if it is quoted as clean, the price does not include
any accrued coupon value. Figure 2.12 clarifies this point.
For a coupon-bearing bond, S must be defined in terms of both price
and coupon dynamics. In particular, this added dimension of the coupon
component gives rise to the need for inserting an additional rate in the for-
ward calculation. This second rate is current yield.
In the case of bonds, current yield is defined as a bond™s coupon divided by
its current price, and it provides a measure of annual percentage coupon return.
As shown in Table 2.1, for a security whose price is par, its yield-to-maturity

In the period between coupon
payments, a coupon™s value
accretes on a daily basis and is
called “accrued interest.”

6 months later
Coupon payment another coupon
is made payment is made

At the halfway point between coupon
A par bond™s
payments, a semiannual coupon-bearing par
clean price and
bond™s clean price is still $1,000, but its dirty
dirty price are the
price is equal to $1,000 (1 + CT), where C is
coupon rate divided by 2, and T is time (equal
immediately after
to one-half of 6 months).
payment at

FIGURE 2.12 Accrued interest, dirty prices, and clean prices.

The clean price is also referred to as the flat price, and the dirty price is also
referred to as the full price.


Comparisons of Yield-to-Maturity and Current Yield for
a Semiannual 6% Coupon 2-Year Bond

Price Yield-to-Maturity (%) Current Yield (%)

102 4.94 5.88
100 6.00 6.00
98 7.08 6.12

and current yield are identical. Further, current yield does not have nearly the
price sensitivity as yield to maturity. Again, this is explained by current yield™s
focus on just the coupon return component of a bond. Since current yield does
not require any assumptions pertaining to the ultimate maturity of the security
in question, it is readily applied to a variety of nonfixed income securities.
Let us pause here to consider the simple case of a six-month forward
on a five-year par bond. Assume that the forward begins one day after a
coupon has been paid and ends the day a coupon is to be paid. Figure 2.13
illustrates the different roles of a risk-free rate (R) and current yield (Yc).
As shown, one trajectory is generated with R and another with Yc.
Clearly, the purchaser of the forward ought not to be required to pay the
seller™s opportunity cost (calculated with R) on top of the full price (clean
price plus accrued interest) of the underlying spot security. Accordingly, Yc
is subtracted from R, and the resulting price formula becomes:

Yc 2 2
F S11 T1R

for a forward clean price calculation.
For a forward dirty price calculation, we have:

Fd Sd(1 T (R Yc)) + Af,

Fd the full or dirty price of the forward (clean price plus accrued
Sd the full or dirty price of the underlying spot (clean price plus
accrued interest)
Af the accrued interest on the forward at expiration of the forward

The equation bears a very close resemblance to the forward formula pre-
sented earlier as F S (1 RT). Indeed, with the simplifying assumption that
T 0, Fd reduces to Sd Af. In other words, if settlement is immediate rather

Cash Flows

Of course, these
particular prices
may or may not
actually prevail in
Price 6 months™ time¦

102.5 = 100 + 100 * 5% * 1/2
Yc trajectory (5%)

101.5 = 100 + 100 * 3% * 1/2

R trajectory (3%)
101.5 “ 102.5 = “1.0
100.0 “ 1.0 = 99.0 = F,
where F is the clean
forward price


Coupon 6-month forward Coupon payment
payment date is purchased date and forward
expiration date

FIGURE 2.13 Relationship between Yc and R over time.

than sometime in the future, Fd Sd since Af is nothing more than the accrued
interest (if any) associated with an immediate purchase and settlement.
Inserting values from Figure 2.13 into the equation, we have:

Fd 100 (1 1/2 (3% 5%)) 5% 100 1/2 101.5,

and 101.5 represents an annualized 3 percent rate of return (opportunity
cost) for the seller of the forward.
Clearly it is the relationship between Yc and R that determines if F S,
F S, or F S (where F and S denote respective clean prices). We already
know that when there are no intervening cash flows F is simply S (1 RT),
and we would generally expect F S since we expect S, R, and T to be pos-
itive values. But for securities that pay intervening cash flows, S will be equal
to F when Yc R; F will be less than S when Yc R; and F will be greater
than S only when R Yc. In the vernacular of the marketplace, the case of
Yc R is termed positive carry and the case of Yc R is termed negative
carry. Since R is the short-term rate of financing and Yc is a longer-term yield
associated with a bond, positive carry generally prevails when the yield curve
has a positive or upward slope, as it historically has exhibited.


For the case where the term of a forward lasts over a series of coupon
payments, it may be easier to see why Yc is subtracted from R. Since a for-
ward involves the commitment to purchase a security at a future point in
time, a forward “leaps” over a span of time defined as the difference
between the date the forward is purchased and the date it expires. When the
forward expires, its purchaser takes ownership of any underlying spot secu-
rity and pays the previously agreed forward price. Figure 2.14 depicts this


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