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.

TLFeBOOK

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.

TLFeBOOK

185

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.