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.

8

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.

TLFeBOOK

36 PRODUCTS, CASH FLOWS, AND CREDIT

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.

Forwards

& futures

Bonds

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

TLFeBOOK

37

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

Coupon

accretes on a daily basis and is

value

called “accrued interest.”

Time

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

same

coupon rate divided by 2, and T is time (equal

immediately after

to one-half of 6 months).

payment at

$1,000.

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

9

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

referred to as the full price.

TLFeBOOK

38 PRODUCTS, CASH FLOWS, AND CREDIT

TABLE 2.1

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,

where

Fd the full or dirty price of the forward (clean price plus accrued

interest)

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

TLFeBOOK

39

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%)

100

101.5 “ 102.5 = “1.0

100.0 “ 1.0 = 99.0 = F,

where F is the clean

forward price

Time

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.

TLFeBOOK

40 PRODUCTS, CASH FLOWS, AND CREDIT

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