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181
Risk Management



TABLE 5.5 True versus Estimated Price Values Generated by Present Value and
Modified Duration and Convexity, 7.75% 30-year Treasury Bond

Price plus Price plus
Change in Accrued Interest, Accrued Interest,
Yield Level Present Value Duration and
(basis points) Equation Convexity Equation Difference

400 76.1448 76.2541 (0.1090)
300 81.0724 80.8378 0.2350
200 86.4398 86.0067 0.4330
100 92.2917 91.7606 0.5311
0 98.0996 98.0996 0.0000
100 105.6525 105.0237 0.6290
200 113.2777 112.5328 0.7449
300 121.6210 120.6270 0.9440
400 130.7582 129.3063 1.4519


Price & accrued
interest
140


Modified duration
120
Modified duration &
convexity
100


80

500 400 300 200 100 0 +100 +200 +300 +400 +500 Change in yield
(basis points)

FIGURE 5.3 A comparison of price/yield relationships, duration versus duration and
convexity.



To summarize, duration and convexity are important risk-measuring
variables for bonds. While duration might be sufficient for scenarios where
only small changes in yield are involved, both duration and convexity gen-
erally are required to capture the full effect of a price change in most fixed
income securities.




TLFeBOOK
182 FINANCIAL ENGINEERING, RISK MANAGEMENT, AND MARKET ENVIRONMENT




Quantifying
risk
Equities




EQUITY PRICE RISK: BETA
The concepts of duration and convexity can be difficult to apply to equities.
The single most difficult obstacle to overcome is the fact that equities do not
have final maturity dates, although the issue that an equity™s price is thus
unconstrained in contrast to bonds (where at least we know it will mature
at par if it is held until then) can be overcome.2
One variable that can come close to the concept of duration for equi-
ties is beta. Duration can be defined as measuring a bond™s price sensitivity
to a change in interest rates; beta can be defined as an equity™s price sensi-
tivity to a change in the S&P 500. As a rather simplistic way of testing this
interrelationship, let us calculate beta for a five-year Treasury bond. But
instead of calculating beta against the S&P 500, we calculate it against a
generic U.S. bond index (comprising government, mortgage-backed securi-
ties, and investment-grade [triple-B and higher] corporate securities). Doing
this, we arrive at a beta of 0.78.3 Hence, in the same way that duration can
give us a measure of a single bond™s price sensitivity to interest rates, a beta
calculation (which requires two series of data) can give us a measure of a
bond™s price sensitivity in relation to another series (e.g., bond index).
Accordingly, two interest rate”sensitive series can be linked and quantified
using a beta measure.


2
One way to arrive at a sort of proxy of duration for an equity is to calculate a
correlation for the equity versus a series of bonds sharing a comparable credit risk
profile. If it is possible to identify a reasonable pairing of an equity to a bond that
generates a correlation coefficient of close to 1.0, then it could be said that the
equity has a quasi-duration measure that™s roughly comparable to the duration of
the bond it is paired against. All else being equal, such strong correlation
coefficients are strongest for companies with a particular sensitivity to interest rates
(as are finance companies or real estate ventures or firms with large debt burdens).
3
A five-year Treasury was selected since it has a modified duration that is close to
the modified duration of the generic index we used for this calculation. We used
monthly data over a particular three-year period where there was an up, down,
and steady pattern in the market overall.



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183
Risk Management



As already stated, beta is a statistical measure of the expected increase
in the value of one variable for a one-unit increase in the value of another
variable. The formula4 for beta is

cov(a,b) / 2 (b)
cov(a,b) (a,b) (a) (b),
2
where Sigma squared (variance); standard deviation squared
r Rho, correlation coefficient
Sigma, standard deviation

Sigma is a standard variable in finance that quantifies the variability or
volatility of a series. Its formula is simply

1x xt 2 2
_
T

1B n 1
t

where x Mean (average) of the series
xt Each of the individual observations within the series
n Total number of observations in the series

A correlation coefficient is a statistical measure of the relationship
between two variables. A correlation coefficient can range in value between
positive 1 and negative 1. A positive correlation coefficient with a value near
1 suggests that the two variables are closely related and tend to move in tan-
dem. A negative correlation coefficient with a value near 1 suggests that two
variables are closely related and tend to move opposite one another. A cor-
relation coefficient with a value near zero, regardless of its sign, suggests that
the two variables have little in common and tend to behave independently
of one another. Figure 5.4 provides a graphical representation of positive,
negative, and zero correlations.
Figure 5.5 presents a conceptual perspective of beta in the context of equi-
ties. There are three categories: betas equal to 1, betas greater than 1, and
betas less than 1. Each of the betas was calculated for individual equities rel-
ative to the S&P 500. A beta equal to 1 suggests that the individual equity
has a price sensitivity in line with the S&P 500, a beta of greater than 1sug-
gests an equity with a price sensitivity greater than the S&P 500, and a beta



4
A beta can be calculated with an ordinary least squares (OLS) regression.
Consistent with the central limit theorem, any OLS regression ought to have a
minimum of about 30 observations per series. Further, an investor ought to be
aware of the assumptions inherent in any OLS regression analysis. These
assumptions, predominantly concerned with randomness, are provided in any basic
statistics text.



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184 FINANCIAL ENGINEERING, RISK MANAGEMENT, AND MARKET ENVIRONMENT



Positive correlation Negative correlation Zero correlation
A A A




B B B
There is no pattern in the
Larger values of A are
Larger values of A are
relationship between A
associated with smaller
associated with larger
and B
values of B
values of B

FIGURE 5.4 Positive, negative, and zero correlations.

of less than 1 suggests an equity with a price sensitivity that is less than the
S&P 500. After calculating betas for individual equities and then grouping
those individual companies into their respective industry categories, industry
averages were calculated.5 As shown, an industry with a particularly low beta
value is water utilities, an industry with a particularly high beta value is semi-
conductors, and an industry type with a beta of unity is tires.
To the experienced market professional, there is nothing new or shocking
to the results. Water utilities tend to be highly regulated businesses and are
often thought fairly well insulated from credit risk since they are typically
linked with government entities. Indeed, some investors believe that holding
water utility equities is nearly equivalent in risk terms to holding utility bonds.
Of course, this is not a hard-and-fast rule, and works best when evaluated
on a case-by-case basis. At the very least, this low beta value suggests that
water utility equity prices may be more sensitive to some other variable”
perhaps interest rates. In support of this, many utilities do carry significant
debt, and debt is most certainly sensitive to interest rate dynamics.
On the other end of the continuum are semiconductors at 2.06. Again,
market professionals would not be surprised to see a technology-sector equity
with a market risk factor appreciably above the market average. Quite sim-
ply, technology equities have been a volatile sector, as they are relatively new
and untested”at least relative to, say, autos (sporting a beta of 0.95) or
broadcasting (with a beta of 1.05).
And what can we say about tires? In good times and bad, people drive
their cars and tires become worn. The industry sector is not considered to
be particularly speculative, and the market players are generally well known.
In a sense, the S&P 500 serves as a line in the sand as a risk manage-
ment tool. That is, we are picking a neutral market measure (the S&P 500)


5
“Using Target Return on Equity and Cost of Equity,” Parker Center, Cornell
University, May 1999.



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Risk Management



Semiconductors
Water utilities
Beta = 2.06
Beta = 0.37
Industry code 1033
Industry code 1209
Beta < 1 Beta > 1



Beta = 1


Tires
Beta = 1.00
Industry code 0936

FIGURE 5.5 Beta by industry types.

and are essentially saying: Equities with a risk profile above this norm (at
least as measured by standard deviation) are riskier and equities below this
norm are less risky. But such a high-level breakdown of risk has all the flaws
of using a five-year Treasury duration as a line in the sand and saying that
any bond with duration above the five-year Treasury™s is riskier and anything
below it is less risky. However, since equity betas are calculated using price,
and to the extent that an equity™s price can embody and reflect the risks inher-
ent in a particular company (at least to the extent that those risks can be pub-
licly communicated and, hence, incorporated into the company™s valuation),
then equity beta calculations can be said to be of some value as a relative risk
measure. The hard work of absolute risk measurement (digging through a
company™s financial statements) can certainly result in unique insights as well.
Finally, just as beta or duration can be calculated for individual equi-
ties and bonds, betas and durations can be calculated for entire portfolios.
For an equity portfolio, a beta can be derived using the daily price history
of the portfolio and the daily price history of the S&P 500. For a bond port-
folio, individual security durations can be aggregated into a single portfolio
duration by simply weighting the individual durations by their market value
contribution to the portfolio.

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