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Having now addressed uncertainties associated with credit and rein-
vestment of cash flows, let us now consider uncertainties related to timing
and payment of coupon and principal as with pass-through securities. As
shown in Figure 5.29, credit risk fades as a concern with pass-through secu-
rities, though risks associated with the timing and amounts of cash flows
step into the picture. We use the same key for designating cash flow char-
acteristics as we used in Chapter 2.
The cash flows of an equity can be illustrated as in Figure 5.30.
As the figure confirms, there is a much greater degree of uncertainty
related to an equity™s cash flow profile than to that of a bond. Accordingly,
it ought not come as any surprise that the price risk of equities (typically
measured in terms of price volatility) is generally greater than that of bonds.
Further, and consistent with risk-reward trade-offs, historically a basket of




Denotes actual payment or receipt of cash for a cash flow value that™s known at
time of initial trade (as with a purchase price or a coupon or principal payment).

Denotes that a cash flow™s value cannot be known at time of initial trade and that
an exchange of cash may or may not take place.

Of course, a product may be be sold prior to actual maturity/expiration at a gain,
loss, or break even.




Uncertainties:
• Reinvestment of coupon income
• Timing and amounts of coupon and principal payments
• Total return prior to maturity

Cash flows Prepayment risk; cash flows may include coupon and principal

+ Reinvestment risk




O
Time





FIGURE 5.29 15-year pass-thru security.




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Risk Management



Uncertainties:
• Reinvestment of dividends
• Amount of dividends
• Credit drift and default
• Total return prior to end of investment horizon
• Price at any time
Price risk
Cash flows

+ Reinvestment risk




O
Time





FIGURE 5.30 Equity.




diversified equities will generate higher returns relative to a basket of diver-
sified bonds over long stretches of time (say five years or more).
Next we describe a hierarchy or ranking of probabilities for cash flows.
The three principal types of cash flows are spot, forwards and futures, and
options. At first pass it may be tempting to assert that a derivative of a spot
(i.e., its forward or option) at the very least embodies all the risks embed-
ded within the underlying spot. This is not necessarily the case. For exam-
ple, with a spot purchase of a coupon-bearing bond, there is a reinvestment
risk with the coupons that are paid over time. If an 8 percent coupon-bear-
ing bond is purchased at par and held to maturity, its total return will be
less than 8 percent if coupons are reinvested at rates under 8 percent.
However, with a forward on an 8 percent coupon-bearing bond, the holder
of a forward contract receives no coupons, so there are no coupons to be
reinvested. To be sure, the value of all relevant coupons is embedded in a
forward contract™s price at time of purchase, and it is this locking in of the
coupon™s value (inclusive of reinvested income) that allows the holder of the
forward contract to dispense with the reinvestment risk associated with the
underlying spot. The same is true for an option on the underlying spot.
Figure 5.31 repeats the illustrations for spot, forwards and futures, and
options from Chapter 2.




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228 FINANCIAL ENGINEERING, RISK MANAGEMENT, AND MARKET ENVIRONMENT



Spot
+
2-year Treasury




O






Forward
+
2-year Treasury
one year forward


O





The fact that the forward does not require an
upfront payment and that the option costs a
fraction of the upfront cost of spot is what
contributes to forwards and options being
referred to as leveraged cash flows.
Option
+
At-the-money one year
expiration on a 2-year
Treasury

O





Denotes actual payment or receipt of cash for a cash flow value that is known at
time of initial trade (as with a purchase price or a coupon or principal payment)

Denotes a reference to payment or receipt amount that is known at the time of
initial trade, but with no exchange of cash taking place

Denotes that a cash flow™s value cannot be known at time of initial trade and that
an exchange of cash may or may not take place

Of course, any product may be sold prior to actual maturity/expiration at a gain,
loss, or break even.


FIGURE 5.31 Spot, forwards and futures, and options.



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Risk Management



However, although a forward or option might save an investor from
directly confronting the matter of actually reinvesting coupon cash flows,8
other unique risks do surface with forwards and options. To see how, sim-
ply consider the following variables and formulas below.

S Spot
F S (1 RT), Forward (for non“cash-flow paying securities)
Oc F X V, Option (call)

As shown, F is differentiated from S with RT (cost-of-carry), and Oc is
differentiated from F with V (volatility value). Since both cost-of-carry and
volatility value are functions of time (T), they will shrink in value until they
have a value of zero at the expiration of the forward or option. Thus, if the
investment horizon of relevance is the expiration date, then there may be
no risk to speak of for either carry or volatility, since both are zero at that
juncture. However, if the horizon of relevance is a point in time prior to
expiration, then carry and volatility values will likely be non-zero. And since
their precise value cannot be known with certainty at the time a forward
or option contract is purchased, it is not possible to know total return at
time of purchase.
In the base case scenario involving a Treasury bill, we know its total
return at time of purchase if the Treasury bill is held to maturity. In this sim-
ple case, the probability of knowing the Treasury bill™s total return at time
of purchase is 100 percent (ptb 100%). It is 100 percent since there is no
reinvestment risk of coupon payments and no credit risk, and we know that
the Treasury bill will mature at par. If the Treasury bill is not held to matu-
rity, the probability of knowing its total return at time of purchase is less
than 100 percent. However, we can say that any uncertainty associated with
a 12-month-maturity Treasury bill will be less than the uncertainty associ-
ated with a 12-month coupon-bearing Treasury. Why? Because the 12-month
coupon-bearing Treasury carries reinvestment risk.
Accordingly, if not held to maturity, we can say that ptb p1t (where p1t
is the probability of knowing total return at time of purchase for a one-year



8
While a forward or option on a bond might “save an investor from directly
confronting the matter of actually reinvesting coupon cash flows,” this may or may
not be desirable. If reinvestment rates become more favorable relative to when the
forward contract was purchased, then it is an undesirable development. However,
reinvestment rates could become less favorable, and in any event, it is not
something that holders of a forward contract can control in the way they can if
they were holding the underlying bond.




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coupon-bearing Treasury, and ptb involves the same type of probability esti-
mate for a 12-month Treasury bill). Further, with the added component of
carry with a forward, we could say that ptb p1t p1tf (where p1tf is the prob-
ability of knowing total return at time of purchase for a forward contract
on a one-year coupon-bearing Treasury). And with the added components
of both carry and volatility values embedded in an option, we could say that
ptb p1t p1tf p1to (where p1to is the probability of knowing total return
at time of purchase for an option on a one-year coupon-bearing Treasury).
We conclude this section with a series of charts that provide another per-
spective of the varying risk characteristics of equities, bonds, and currencies.
Beginning with bonds, Figure 5.32 presents a price cone for a five-year-
maturity coupon-bearing Treasury bond. The cone was created by shocking
the Treasury with interest rate changes of both plus and minus 300 basis
points at the end of each year from origination to maturity. As shown, as
the maturity date draws near, the pull to par becomes quite strong.
Figure 5.33 is a price cone for both the previous five-year Treasury and
a one-year Treasury bill. Among other considerations, the cone of the
Treasury bill relationship to price is not centered symmetrically around par.
The simple reason for this is that unlike the five-year Treasury, the Treasury
bill is a discount instrument and thus has no coupon. Accordingly, this price
cone helps to demonstrate the price dynamics of a zero coupon security.




Price
120
Price trajectory for “300 bps
changes in par bond yield
Maturity
110

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