Quantifying Allocating

risk risk

Managing risk

This chapter examines ways that financial risks can be quantified, the

means by which risk can be allocated within an asset class or portfolio, and

the ways risk can be managed effectively.

Quantifying

risk

Generally speaking, “risk” in the financial markets essentially comes down

to a risk of adverse changes in price. What exactly is meant by the term

“adverse” varies by investor and strategy. An absolute return investor could

well have a higher tolerance for price variability than a relative return

investor. And for an investor who is short the market, a dramatic fall in prices

may not be seen as a risk event but as a boon to her portfolio. This chap-

ter does not attempt to pass judgment on what amount of risk is good or

bad; such a determination is a function of many things, many of which (like

risk appetite or level of understanding of complex strategies) are entirely

subject to particular contexts and individual competencies. Rather the text

highlights a few commonly applied risk management tools beginning with

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172 FINANCIAL ENGINEERING, RISK MANAGEMENT, AND MARKET ENVIRONMENT

products in the context of spot, then proceeding to options, forwards and

futures, and concluding with credit.

Quantifying

risk

Bonds

BOND PRICE RISK: DURATION AND CONVEXITY

In the fixed income world, interest rate risk is generally quantified in terms

of duration and convexity. Table 5.1 provides total return calculations for

three Treasury securities. Using a three-month investment horizon, it is clear

that return profiles are markedly different across securities.

The 30-year Treasury STRIPS1 offers the greatest potential return if

yields fall. However, at the same time, the 30-year Treasury STRIPS could

well suffer a dramatic loss if yields rise. At the other end of the spectrum,

the six-month Treasury bill provides the lowest potential return if yields fall

yet offers the greatest amount of protection if yields rise. In an attempt to

quantify these different risk/return profiles, many fixed income investors

evaluate the duration of respective securities.

Duration is a measure of a fixed income security™s price sensitivity to a

given change in yield. The larger a security™s duration, the more sensitive that

security™s price will be to a change in yield. A desirable quality of duration

is that it serves to standardize yield sensitivities across all cash fixed income

securities. This can be of particular value when attempting to quantify dif-

ferences across varying maturity dates, coupon values, and yields. The dura-

tion of a three-month Treasury bill, for example, can be evaluated on an

apples-to-apples basis against a 30-year Treasury STRIPS or any other

Treasury security.

The following equations provide duration calculations for a variety of

securities.

1

STRIPS is an acronym for Separately Traded Registered Interest and Principal

Security. It is a bond that pays no coupon. Its only cash flow consists of what it

pays at maturity.

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Risk Management

TABLE 5.1 Total Return Calculations for Three Treasury Securities

on a Bond-Equivalent Basis, 3-Month Horizon

Change in 7.75%

Yield Level Treasury Bill Treasury Note Treasury STRIPS

(basis points) (1 year) (%) (10 year) (%) (30 year) (%)

100 8.943 36.800 75.040

50 7.580 21.870 39.100

0 6.229 8.030 7.920

50 4.883 4.820 19.130

100 3.545 16.750 42.610

To calculate duration for a Treasury bill, we solve for:

P Tsm

Duration

P 365

where P Price

Tsm Time in days from settlement to maturity

The denominator of the second term is 365 because it is the market™s

convention to express duration on a bond-equivalent basis, and as presented

in Chapter 2, a bond-equivalent calculation assumes a 365-day year and

semiannual coupon payments.

To calculate duration for a Treasury STRIPS, we solve for:

P

Duration T

P sm

where Tsm Time from settlement to maturity in years.

It is a little more complex to calculate duration for a coupon security.

One popular method is to solve for the first derivative of the price/yield equa-

tion with respect to yield using a Taylor series expansion. We use a price/yield

equation as follows:

F C>2 F C>2 F11 C>22

11 11 11

Pd ...

Y>22 TSC>Tc Y>22 TSC>Tc TSC>TC

Y>22 N 1

where Pd Dirty price

F Face value (par)

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174 FINANCIAL ENGINEERING, RISK MANAGEMENT, AND MARKET ENVIRONMENT

C Coupon (annual %)

Y Bond-equivalent yield

Tsc Time in days from settlement to coupon payment

Tc Time in days from last coupon payment (or issue date) to next

coupon date

The solution for duration using calculus may be written as (dP™/dY)P™,

where P™ is dirty price. J. R. Hicks first proposed this method in 1939.

The price/yield equation can be greatly simplified with the Greek sym-

bol sigma, , which means summation. Rewriting the price/yield equation

using sigma, we have:

C't

T

11 Y>22 t

Pd

t 1

where Pd Dirty price

Summation

T Total number of cash flows in the life of a security

Ct Cash flows over the life of a security (cash flows include

coupons up to maturity, and coupons plus principal at maturity)

Y Bond-equivalent yield

t Time in days security is owned from one coupon period to the

next divided by time in days from last coupon paid (or issue date)

to next coupon date

Moving along then, another way to calculate duration is to solve for

C't t

T

11 Y>22 t

t 1

C't

T

11 Y>22 t

t 1

There is but a subtle difference between the formula for duration and the

price/yield formula. In particular, the numerator of the duration formula is

the same as the price/yield formula except that cash flows are a product of

time (t). The denominator of the duration formula is exactly the same as the

price/yield formula. Thus, it may be said that duration is a time-weighted

average value of cash flows.

Frederick Macaulay first proposed the calculation above. Macaulay™s

duration assumes continuous compounding while Treasury coupon securities

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Risk Management

are generally compounded on an actual/actual (or discrete) basis. To adjust

Macaulay™s duration to allow for discrete compounding, we solve for:

Dmac

11 Y>22

Dmod

where Dmod Modified duration

Dmac Macaulay™s duration

Y Bond-equivalent yield

This measure of duration is known as modified duration and is gener-

ally what is used in the marketplace. Hicks™s method to calculate duration

is consistent with the properties of modified duration. This text uses modi-

fied duration.

Table 5.2 calculates duration for a five-year Treasury note using

Macaulay™s methodology. The modified duration of this 5-year security is

4.0503 years.

For Treasury bills and Treasury STRIPS, Macaulay™s duration is noth-

ing more than time in years from settlement to maturity dates. For coupon

securities, Macaulay™s duration is the product of cash flows and time divided

by cash flows where cash flows are in present value terms.

Using the equations and Treasury securities from above, we calculate

Macaulay duration values to be:

1-year Treasury bill, 0.9205

7.75% 10-year Treasury note, 7.032

30-year Treasury STRIPS, 29.925

Modified durations on the same three Treasury securities are:

Treasury bill, 0.8927

Treasury note, 6.761

Treasury STRIPS, 28.786

The summation of column (D) gives us the value for the numerator of

the duration formula, and the summation of column (C) gives us the value

for the denominator of the duration formula. Note that the summation of

column (C) is also the dirty price of this Treasury note.

Dmac 833.5384/98.9690 8.4222 in half years

8.4222/2 4.2111 in years

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