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securities will be exchanged for cash on the next business day (since settle-
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

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¦

0 Time

2 months 3 months
1 month
later later

Purchase date.
Cash flow known
with 100% certainty.

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

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.


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


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

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

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.

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.


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:
Price at time of purchase
Y 2) 1 Y 2) 2
(1 (1

11 11 Y 22 4
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

Reinvestment at 9%

Reinvestment at 6%
Passage of
1 2 3 4 5
“2 time


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