This chapter is similar to Chapter 9 of the third edition, but in keeping with its growing
importance there is more discussion of expected shortfall (ES). The chapter requires 2 to
3 hours of classroom time. It starts by explaining value at risk (VaR). It compares VaR
to ES and shows that the latter has better theoretical properties. (To use the technical
term, it is more “coherent.”) It discusses the choice of parameters for VaR and ES, and
the impact of autocorrelation on estimates.
One thing I found after I wrote the first edition of the book is that many students have
difficulty calculating VaR and expected shortfall when the probability distribution of losses
is discrete. Consequently, I included several extra examples illustrating the calculations.
I find it useful to go through most of these examples in class. Many financial institutions
like the properties of expected shortfall and use it internally as one of their risk measures.
The Fundamental Review of the Trading Book (see Chapter 17) indicates that ES will
soon become the regulator’s measure of choice for market risk even though it is considered
more difficult to backtest than VaR.
Sections 12.7 to 12.9 explain marginal VaR, incremental VaR, and component VaR.
They explain the role of Euler’s theorem in the allocation of VaR and give a way of
aggregating VaRs. The material on back-testing in Section 12.10 gives some standard
tests of statistical significance and includes a relatively powerful two-tailed test proposed
by Kupiec.
Any of Further Questions can be used as hand-in assignments. I usually use 12.13.
Chapter 13: Historical Simulation and Extreme Value Theory
This chapter is similar to Chapter 14 of the third edition. It has more emphasis on
expected shortfall. The sequence of chapters has been improved for the fourth edition with
the calculation of VaR and ES coming immediately after the introduction of the measures
in Chapter 12. This chapter requires about 1.5 hours of classroom time. The advantage
of the historical simulation approach is that it requires no assumptions about probability
distributions and correlations. It assumes that percentage changes in all market variables
over the next day are a random sample from the last Ndays (or in the case of stressed
VaR a random sample from consecutive days during a period of particular stress for the
portfolio). Results are presented for a simple portfolio involving investments in four indices.
Section 13.1 explains the basic historical simulation approach while Section 13.2 ex-
plains how to calculate a standard error for VaR. (The standard error is quite large and
would be even larger if the assumption that the joint distribution of daily changes in market
variables is stationary through time could be relaxed.) Section 13.3 describes some exten-
sions of the historical simulation approach. These involve alternative weighting schemes,
procedures involving the volatility updating procedures of Chapter 10, and the use of
the bootstrap method to determine a confidence interval for VaR. (New material in the
fourth edition includes a simpler application of volatility updating.) Section 15.4 covers
the delta/gamma approximation. Section 13.5 and 13.6 present material on extreme value
theory which extends the material on the power law in Chapter 10. It provides a scientific
way of “smoothing the tails” of an empirically observed distribution.
Excel worksheets that go with the four-index example that is covered in the chapter
can be downloaded from my web site.
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Problem 13.13 works well as an assignment question. Problem 13.12 can also be used
as a short assignment question. Problem 13.17 is similar to Problem 13.13, but a little
more difficult as it is not quite so easy for students to use the worksheets that have been
created for students. If the worksheets for the four-index example are displayed in class,
Problems 13.14, 13.15, and 13.16 can be used to illustrate how they are manipulated.
Chapter 14: Model-Building Approach
This chapter is similar to Chapter 15 of the third edition. Like chapters 12 and 13, it
has more emphasis on expected shortfall. This chapter covers the model-building approach
for calculating market-risk VaR, which is the main alternative to historical simulation. It
explains the relationship of VaR to the Markowitz results and also shows how covariance
matrices can be used. The chapter uses the same four-index example as Chapter 13 and
worksheets for the use of the model building approach for the example are on the author’s
website. The example shows that volatilities and correlations increased during the stressed
market conditions of September 2008.
The chapter first explains how the model building approach can be used for the situ-
ation where the value of the portfolio is linearly dependent on the values of the underlying
market variables. (This includes a discussion of how interest rates can be handled with
cash flow mapping.) After that it moves on to consider what can be done in the situation
where the portfolio is not linearly dependent on the underlying variables. The alternatives
here are a) use a linear approximation (delta), b) use a quadratic approximation (delta +
gamma), and c) use Monte Carlo simulation.
This chapter requires a good understanding of the Greek letters and Taylor Series
expansions (covered in Chapter 8 and Appendix G). I generally spend about two hours
on the material. It should be noted that most banks now use the historical simulation
approach because of the problems mentioned in the text at the end of Section 14.10.
Any of Problems 14.16 to 14.23 can be used as assignment questions. 14.21 is a more
challenging than the others. 14.23 is based on the Excel spreadsheets for the four-index
example that is in the chapter and can be covered in class if the spreadsheets for the
example are displayed in class.
Chapter 15: Basel I, Basel II, and Solvency II
This is similar to Chapter 12 in the third edition. In order to understand the way
in which banks are currently regulated it is essential to have an understanding of the
history of regulation since 1988. For example, the concepts underlying many of the current
regulations (e.g., risk weighted assets) have their origins in Basel I. This chapter explains
how banks were regulated prior to the crisis. It also covers the Solvency I and Solvency II
regulations for insurance companies in Europe.
I usually start by asking students why banks are regulated and other companies en-
gaged in manufacturing and retailing are not. The answer is that banks are allowed to
take deposits from consumers and a stable banking system is an essential part of a healthy
economy. Many governments have deposit insurance systems and to avoid large payouts
they need to ensure that a bank’s capital is sufficient for the risks it is taking.
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