226 FINANCIAL ENGINEERING, RISK MANAGEMENT, AND MARKET ENVIRONMENT

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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227

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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229

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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230 FINANCIAL ENGINEERING, RISK MANAGEMENT, AND MARKET ENVIRONMENT

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