ñòð. 35 |

equities, bonds, options, and foreign currencies. Looking for the arbi-

trage opportunities â€˜â€˜across the boardâ€™â€™ is technically more challenging

but potentially rewarding.

Some academic research on efficiency of the arbitrage trading

strategies can be found in [9â€“12] and references therein. Note that

the research methodology in this field is itself a non-trivial problem

[13].

10.5 REFERENCES FOR FURTHER READING

A good introduction into the finance theory, including CAPM, is

given in [1]. For a description of the portfolio theory and investment

science with an increasing level of technical detail, see [5, 14].

10.6 EXERCISES

1. Consider a portfolio with two assets having the following

returns and standard deviations: E[R1 ] Â¼ 0:15, E[R2 ] Â¼ 0:1,

s1 Â¼ 0:2, s2 Â¼ 0:15. The proportion of asset 1 in the portfolio

g Â¼ 0:5. Calculate the portfolio return and standard deviation.

The correlation coefficient between assets is (a) 0.5; (b) Ã€0.5.

2. Consider returns of stock A and the market portfolio M in three

years:

A Ã€7% 12% 26%

M Ã€5% 9% 18%

Assuming the risk-free rate is 5%, (a) calculate b of stock A; and

(b) verify if CAPM describes pricing of stock A.

3. Providing the stock returns follow the two-factor APT:

Ri (t) Â¼ ai Ã¾ bi1 f1 Ã¾ bi2 f2 Ã¾ ei (t), construct a portfolio with

three stocks (i.e., define w1 , w2 , and w3 Â¼ 1 Ã€ w1 Ã€ w2 ) that

yields return equal to that of the risk-free asset.

4. Providing the stock returns follow the two-factor simple APT,

derive the values of the risk premiums. Assume the expected

returns of two stocks and the risk-free rate are equal to R1 , R2 ,

and Rf , respectively.

Chapter 11

Market Risk Measurement

The widely used risk measure, value at risk (VaR), is discussed in

Section 11.1. Furthermore, the notion of the coherent risk measure is

introduced and one such popular measure, namely expected tail losses

(ETL), is described. In Section 11.2, various approaches to calculating

risk measures are discussed.

11.1 RISK MEASURES

There are several possible causes of financial losses. First, there is

market risk that results from unexpected changes in the market prices,

interest rates, or foreign exchange rates. Other types of risk relevant

to financial risk management include liquidity risk, credit risk, and

operational risk [1]. The liquidity risk closely related to market risk is

determined by a finite number of assets available at a given price (see

discussion in Section 2.1). Another form of liquidity risk (so-called

cash-flow risk) refers to the inability to pay off a debt in time. Credit

risk arises when one of the counterparts involved in a financial

transaction does not fulfill its obligation. Finally, operational risk is

a generic notion for unforeseen human and technical problems, such

as fraud, accidents, and so on. Here we shall focus exclusively on

measurement of the market risk.

In Chapter 10, we discussed risk measures such as the asset return

variance and the CAPM beta. Several risk factors used in APT were

121

122 Market Risk Measurement

also mentioned. At present, arguably the most widely used risk meas-

ure is value at risk (VaR) [1]. In short, VaR refers to the maximum

amount of an asset that is likely to be lost over a given period at a

specific confidence level. This implies that the probability density

function for profits and losses (P=L)1 is known. In the simplest case,

this distribution is normal

1

PN (x) Â¼ pï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒ exp [Ã€(x Ã€ m)2 =2s2 ] (11:1:1)

2ps

where m and s are the mean and standard deviation, respectively.

Then for the chosen confidence level a,

VAR(a) Â¼ Ã€sza Ã€ m (11:1:2)

The value of za can be determined from the cumulative distribution

function for the standard normal distribution (3.2.10)

Ã°

za

1

pï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒ exp [Ã€z2 =2]dz Â¼ 1 Ã€ a

Pr(Z < za ) Â¼ (11:1:3)

2p

Ã€1

Since za < 0 at a > 50%, the definition (11.1.2) implies that positive

values of VaR point to losses. In general, VaR(a) grows with the

confidence level a. Sufficiently high values of the mean

P=L (m > Ã€sza ) for given a move VaR(a) into the negative region,

which implies profits rather than losses. Examples of za for typical

values of a Â¼ 95% and a Â¼ 99% are given in Figure 11.1. Note that

the return variance s corresponds to za Â¼ Ã€1 and yields a % 84%.

The advantages of VaR are well known. VaR is a simple and

universal measure that can be used for determining risks of different

financial assets and entire portfolios. Still, VaR has some drawbacks

[2]. First, accuracy of VaR is determined by the model assumptions

and is rather sensitive to implementation. Also, VaR provides an

estimate for losses within a given confidence interval a but says

nothing about possible outcomes outside this interval. A somewhat

paradoxical feature of VaR is that it can discourage investment

diversification. Indeed, adding volatile assets to a portfolio may

move VaR above the chosen risk threshold. Another problem with

VaR is that it can violate the sub-additivity rule for portfolio risk.

According to this rule, the risk measure r must satisfy the condition

123

Market Risk Measurement

0.45

0.4

0.35

VaR at 84%

z = âˆ’1

0.3

0.25

VaR at 95%

z = âˆ’1.64

0.2

0.15

VaR at 99%

z = âˆ’2.33

0.1

0.05

0

âˆ’5 âˆ’4 âˆ’3 âˆ’2 âˆ’1 0 1 2 3 4 5

Figure 11.1 VaR for the standard normal probability distribution of P/L.

r(A Ã¾ B) r(A) Ã¾ r(B) (11:1:4)

which means the risk of owning the sum of two assets must not be

higher than the sum of the individual risks of these assets. The

ñòð. 35 |