ment does not occur on weekends or market holidays). Thus, for an agree-

ment on a Friday to exchange $1,000 dollars for euros at a rate of 1.10 using

next day settlement, the $1,000 would not be physically exchanged for the

1,100 until the following Monday.

Generally speaking, a settlement day is quoted relative to the day that

the trade takes place. Accordingly, a settlement agreement of T plus 3 means

three business days following trade date. There are different conventions for

how settlement is treated depending on where the trade is done (geograph-

ically) and the particular product types concerned.

Pretty easy going thus far if we are willing to accept that the market™s

judgment of a particular asset™s spot price is also its value or true worth (val-

uation above or below the market price of an asset). Yes, there is a distinc-

tion to be made here, and it is an important one. In a nutshell, just because

the market says that the price of an asset is “X” does not have to mean that

we agree that the asset is actually worth that. If we do happen to agree, then

fine; we can step up and buy the asset. In this instance we can say that for

us the market™s price is also the worth of the asset. If we do not happen to

agree with the market, that is fine too; we can sell short the asset if we believe

that its value is above its current price, or we can buy the asset if we believe

its value is below its market price. In either event, we can follow meaning-

ful strategies even when (perhaps especially when) our sense of value is not

precisely in line with the market™s sense of value.

Expanding on these two notions of price and worth, let us now exam-

ine a few of the ways that market practitioners might try to evaluate each.

Broadly speaking, price can be said to be definitional, meaning that it

is devoid of judgment and simply represents the logical outcome of an equa-

tion or market process of supply and demand.

Let us begin with the bond market and with the most basic of financial

instruments, the Treasury bill. If we should happen to purchase a Treasury

bill with three months to maturity, then there is a grand total of two cash

flows: an outflow of cash when we are required to pay for the Treasury bill

at the settlement date and an inflow of cash when we choose to sell the

Treasury bill or when the Treasury bill matures. As long as the sale price or

price at maturity is greater than the price at the time of purchase, we have

made a profit.

A nice property of most fixed income securities is that they mature at

par, meaning a nice round number typically expressed as some multiple of

$1,000. Hence, with the three-month Treasury bill, we know with 100 per-

cent certainty the price we pay for the asset, and if we hold the bill to matu-

rity, we know with 100 percent certainty the amount of money we will get

in three months™ time. We assume here that we are 100 percent confident

TLFeBOOK

17

Cash Flows

that the U.S. federal government will not go into default in the next three

months and renege on its debts.1 If we did in fact believe there was a chance

that the U.S. government might not make good on its obligations, then we

would have to adjust downward our 100 percent recovery assumption at

maturity. But since we are comfortable for the moment with assigning 100

percent probabilities to both of our Treasury bill cash flows, it is possible

for us to state with 100 percent certainty what the total return on our

Treasury bill investment will be.

If we know for some reason that we are not likely to hold the three-

month Treasury bill to maturity (perhaps we will need to sell it after two

months to generate cash for another investment), we can no longer assume

that we can know the value of the second cash flow (the sale price) with 100

percent certainty; the sale price will likely be something other than par, but

what exactly it will be is anyone™s guess. Accordingly, we cannot say with

100 percent certainty what a Treasury bill™s total return will be at the time

of purchase if the bill is going to be sold anytime prior to its maturity date.

Figure 2.1 illustrates this point.

Certainly, if we were to consider what the price of our three-month

Treasury bill were to be one day prior to expiration, we could be pretty con-

fident that its price would be extremely close to par. And in all likelihood

Maturity date.

Cash flow known

3-month Treasury bill

with 100% certainty.

Precise cash flow value in between time of

purchase and maturity date cannot be known

Cash with certainty at time of purchase¦

inflow

0 Time

Cash

outflow

2 months 3 months

1 month

later later

later

Purchase date.

Cash flow known

with 100% certainty.

FIGURE 2.1 Cash flows of a 3-month Treasury bill.

1

If the government were not to make good on its obligations, there would be the

opportunity in the extreme case to explore the sale of government assets or

securing some kind of monetary aid or assistance.

TLFeBOOK

18 PRODUCTS, CASH FLOWS, AND CREDIT

the price of the Treasury bill one day after purchase will be quite close to

the price of the previous day. But the point is that using words like “close”

or “likelihood” simply underscores that we are ultimately talking about

something that is not 100 percent certain. This particular uncertainty is

called the uncertainty of price.

Now let us add another layer of uncertainty regarding bonds. In a

coupon-bearing security with two years to maturity, we will call our uncer-

tainty the uncertainty of reinvestment, that is, the uncertainty of knowing

the interest rate at which coupon cash flows will be reinvested. As Figure

2.2 shows, instead of having a Treasury security with just two cash flows,

we now have six.

As shown, there is a cash outlay at time of purchase, coupons paid at

regular six-month intervals, and the receipt of par and a coupon payment

at maturity; these cash flows can be valued with 100 percent certainty at the

time of purchase, and we assume that this two-year security is held to matu-

rity. But even though we know with certainty what the dollar amount of the

intervening coupon cash flows will be, this is not enough to state at time of

purchase what the overall total return will be with 100 percent certainty. To

better understand why this is the case, let us look at some formulas.

First, for our three-month Treasury bill, the annualized total return is

calculated as follows if the Treasury bill is held to maturity:

Cash out cash in 365

Annualized total return

Cash in 90

Accordingly, for a three-month Treasury bill purchased for $989.20, its

annualized total return is 4.43 percent. The second term, 365/90, is the

annualization term. We assume 365 days in a year (366 for a leap year), and

Cash

inflow

0 Time

Cash 6 months later 12 months later 18 months later 24 months later

outflow “ Coupon payment “ Coupon payment “ Coupon payment “ Coupon and principal

payments

Purchase

FIGURE 2.2 Cash flows of a 2-year coupon-bearing Treasury bond.

TLFeBOOK

19

Cash Flows

90 days corresponds to the three-month period from the time of purchase

to the maturity date. It is entirely possible to know at the time of purchase

what the total return will be on our Treasury bill. However, if we no longer

assume that the Treasury bill will be held to maturity, the “cash-out” value

is no longer par but must be something else. Since it is not possible to know

with complete certainty what the future price of the Treasury bill will be on

any day prior to its maturity, this uncertainty prevents us from being able

to state a certain total return value prior to the sale date.

What makes the formula a bit more difficult to manage with a two-year

security is that there are more cash flows involved and they all have a time

value that has to be considered. It is material indeed to the matter of total

return how we assume that the coupon received at the six-month point is

treated. When that coupon payment is received, is it stuffed into a mattress,

used to reinvest in a new two-year security, or what? The market™s conven-

tion, rightly or wrongly, is to assume that any coupon cash flows paid prior

to maturity are reinvested for the remaining term to maturity of the under-

lying security and that the coupon is reinvested in an instrument of the same

issuer profile. The term “issuer profile” primarily refers to the quality and

financial standing of the issuer. It also is assumed that the security being pur-

chased with the coupon proceeds has a yield identical to the underlying secu-

rity™s at the time the underlying security was purchased,2 and has an identical

compounding frequency. “Compounding” refers to the reinvestment of cash

flows and “frequency” refers to how many times per year a coupon-bear-

ing security actually pays a coupon. All coupon-bearing Treasuries pay

coupons on a semiannual basis. The last couple of lines of text give four

explicit assumptions pertaining to how a two-year security is priced by the

market. Obviously, this is no longer the simple and comfortable world of

Treasury bills.

Coupon payments prior to maturity are assumed to be:

1. Reinvested.

2. Reinvested for a term equal to the remaining life of the underlying bond.

3. Reinvested in an identical security type (e.g., Treasury-bill).

4. Reinvested at a yield equal to the yield of the underlying security at the

time it was originally purchased.

2

It would also be acceptable if the cash flow“weighted average of different yields

used for reinvestment were equal the yield of the underlying bond at time of

purchase. In this case, some reinvestment yields could be higher than at time of

original purchase and some could be lower.

TLFeBOOK

20 PRODUCTS, CASH FLOWS, AND CREDIT

To help reinforce the notion of just how important reinvested coupons

can be, consider Figure 2.3, which shows a five-year, 6 percent coupon-

bearing bond. Three different reinvestment rates are assumed: 9 percent, 6

percent, and 3 percent. When reinvestment occurs at 6 percent (equal to the

coupon rate), a zero contribution is made to the overall total return.

However, if cash flows can be reinvested at 9 percent, then at the end of five

years an additional 7.6 points ($76 per $1,000 face) of cumulative dollar

value above the 6 percent base case scenarios is returned. By contrast, if rates

are reinvested at 3 percent, then at the end of five years, 6.7 points ($67 per

$1,000 face) of cumulative dollar value is lost relative to the 6 percent base

case scenario.

Figure 2.3 portrays the assumptions being made.

The mathematical expression for the Figure 2.4 is:

C C

Price at time of purchase

Y 2) 1 Y 2) 2

(1 (1

C C&F

11 11 Y 22 4

$1,000

Y 22 3

The C in the equation is the dollar amount of coupon, and it is equal

to the face amount (F) of the bond times the coupon rate divided by its com-

pounding frequency. The face amount of a bond is the same as the par value

received at maturity. In fact, when a bond first comes to market, face, price,

and par values are all identical because when a bond is launched, the coupon

Cumulative point values of

reinvested coupon income

relative to 6% base case

8

Reinvestment at 9%

6

4

2

Reinvestment at 6%

0

Passage of

1 2 3 4 5

“2 time