FRM study guide 2017 part 2

COVERS
EVERY LEARNING
OBJECTIVE ON
THE EXAM

FRM

EXAM REVIEW

STUDY GUIDE:

RT
BITE-SIZED
LESSON
FORMAT

WILEY



Wiley FRM Exam Review Study Guide 2017
Fart II



Wiley FRM Exam Review Study Guide 2017
Part II
Market Risk Measurement and Management,
Credit Risk Measurement and Management,
Operational and Integrated Risk Management,
Risk Management and Investment Management,

Current Issues in Financial Markets

Christian H. Cooper, CFA, FRM

W il

ey


Cover image: Loewy Design
Cover design: Loewy Design
Copyright © 2017 by John Wiley & Sons, Inc. All rights reserved.
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Contents
How to Study for the Exam

xi

About the Author

xii

Market Risk Measurement and Management
Lesson: Kevin Dowd, Measuring Market Risk, 2nd Edition (West Sussex, England:
John Wiley & Sons, 2005). Chapter 3. Estimating Market Risk Measures:
An Introduction and Overview

3

Lesson: Kevin Dowd, Measuring Market Risk, 2nd Edition (West Sussex, England:
John Wiley & Sons, 2005). Chapter 4. Non-Parametric Approaches

11

Lesson: Philippe Jorion, Value-at-Risk: The New Benchmark for Managing Financial Risk,
3rd Edition (New York: McGraw-Hill, 2007). Chapter 6. Backtesting VaR

15

Lesson: Philippe Jorion, Vaiue-at-Risk: The New Benchmark for Managing Financial Risk,
3rd Edition (New York: McGraw-Hill, 2007). Chapter 11. VaR Mapping

19

Lesson: "Messages from the Academic Literature on Risk Measurement for the Trading Book,"
Basel Committee on Banking Supervision, Working Paper No. 19, January 2011.

25

Lesson: Gunter Meissner, Correlation Risk Modeling and Management
(Hoboken, NJ: John Wiley & Sons, 2014). Chapter 1. Some Correlation Basics:
Properties, Motivation, Terminology

31

Lesson: Gunter Meissner, Correlation Risk Modeling and Management (Hoboken, NJ:
John Wiley & Sons, 2014). Chapter 2. Empirical Properties of Correlation:

How Do Correlations Behave in the Real World?

37

Lesson: Gunter Meissner, Correlation Risk Modeling and Management (New York:
John Wiley & Sons, 2014). Chapter 3. Statistical Correlation Models— Can We Apply
Them to Finance?

41

Lesson: Gunter Meissner, Correlation Risk Modeling and Management (Hoboken, NJ:
John Wiley & Sons, 2014). Chapter 4. Financial Correlation Modeling— Bottom-Up Approaches
(Sections 4.3.0 (intro), 4.3.1, and 4.3.2 only)

43

©2017 Wiley


CONTENTS

Lesson: M c e T u c k m n , Fixed Income Securities, M Edition (Hoboken, NJ: John Wiley & Sons, 2011).
Chapter 6. Empirical Approaches to Risk Metrics and Hedges

49

Lesson: Bruce Tuckman, Fixed Income Securities, 3rd Edition (Hoboken, NJ: John Wiley & Sons, 2011).
Chapter 7. The Science of Term Structure Models
53
Lesson: BruceTuckman, Fixed Income Securities, 3rd Edition (Hoboken, NJ: John Wiley & Sons, 2011).

Chapter 8. The Evolution of Short Rates and the Shape of the Term Structure

63

Lesson: BruceTuckman, Fixed Income Securities, 3rd Edition (Hoboken, NJ: John Wiley & Sons, 2011).
Chapter 9. The Art of Term Structure Models: Drift

67

Lesson: BruceTuckman, Fixed Income Securities, 3rd Edition (Hoboken, NJ: John Wiley & Sons, 2011).
Chapter 10. The Art of Term Structure Models: Volatility and Distribution

77

Lesson: John Hull, Options, Futures, and Other Derivatives, 9th Edition (New York:
Pearson Prentice Hall, 2014). Chapter 9. OIS Discounting, Credit Issues, and Funding Costs

81

Lesson: John Hull, Options, Futures, and Other Derivatives, 9th Edition (New York:
Pearson Prentice Hall, 2014). Chapter 20. Volatility Smiles

85

Credit Risk Measurement and Management

Vi

Lesson: Jonathan Golin and Philippe Delhaise, The Bank Credit Analysis Handbook
(Hoboken, NJ: John Wiley & Sons, 2013). Chapter 1. The Credit Decision

91

Lesson: Jonathan Golin and Philippe Delhaise, The Bank Credit Analysis Handbook
(Hoboken, NJ: John Wiley & Sons, 2013). Chapter 2. The Credit Analyst

95

Lesson: Giacomo De Laurentis, Renato Maino, and Luca Molteni, Developing,
Validating and Using Internal Ratings (West Sussex, United Kingdom: John Wiley & Sons, 2010).
Chapter 2. Classifications and Key Concepts of Credit Risk

97

Lesson: Giacomo De Laurentis, Renato Maino, and Luca Molteni, Developing,
Validating and Using Internal Ratings (West Sussex, United Kingdom: John Wiley & Sons, 2010).
Chapter 3. Ratings Assignment Methodologies

105

Lesson: Rene Stulz, Risk Management & Derivatives (Florence, KY: Thomson South-Western, 2002).
Chapter 18. Credit Risks and Credit Derivatives

119

Lesson: Allan Malz, Financial Risk Management: Models, History, and Institutions
(Hoboken, NJ: John Wiley & Sons, 2011). Chapter 7. Spread Risk and Default Intensity Models

127

Lesson: Allan Malz, Financial Risk Management: Models, History, and Institutions (Hoboken, NJ:
John Wiley & Sons, 2011). Chapter 8. Portfolio Credit Risk (Sections 8.1,8.2,8.3 only)

135

Lesson: Allan Malz, Financial Risk Management: Models, History, and Institutions (Hoboken, NJ:
John Wiley & Sons, 2011). Chapter 9. Structured Credit Risk

139

©2017 Wiley


Lesson: Jon Gregory, Counterparty Credit Risk and Credit Value Adjustment: A Continuing Challenge
for Global Financial Markets, 2nd Edition (West Sussex, UK: John Wiley & Sons, 2012).
Chapter 3. Defining Counterparty Credit Risk

147

Lesson: Jon Gregory, Counterparty Credit Risk and Credit Value Adjustment: A Continuing Challenge
for Global Financial Markets, 2nd Edition (West Sussex, UK: John Wiley & Sons, 2012).
Chapter 4. Netting, Compression, Resets, and Termination Features

151

Lesson: Jon Gregory, Counterparty Credit Risk and Credit Value Adjustment: A Continuing
Challenge for Global Financial Markets, 2nd Edition (West Sussex, UK: John Wiley & Sons, 2012).
Chapter 5. Collateral

153

Lesson: Jon Gregory, Counterparty Credit Risk and Credit Value Adjustment: A Continuing
Challenge for Global Financial Markets, 2nd Edition (West Sussex, UK: John Wiley & Sons, 2012).
Chapter 7. Central Counterparties

159

Lesson: Jon Gregory, Counterparty Credit Risk and Credit Value Adjustment: A Continuing
Challenge for Global Financial Markets, 2nd Edition (West Sussex, UK: John Wiley & Sons, 2012).
Chapter 8. Credit Exposure

165

Lesson: Jon Gregory, Counterparty Credit Risk and Credit Value Adjustment: A Continuing
Challenge for Global Financial Markets, 2nd Edition (West Sussex, UK: John Wiley & Sons, 2012).
Chapter 10. Default Probability, Credit Spreads, and Credit Derivatives

171

Lesson: Jon Gregory, Counterparty Credit Risk and Credit Value Adjustment: A Continuing
Challenge for Global Financial Markets, 2nd Edition (West Sussex, UK: John Wiley & Sons, 2012).
Chapter 12. Credit Value Adjustment

177

Lesson: Jon Gregory, Counterparty Credit Risk and Credit Value Adjustment: A Continuing
Challenge for Global Financial Markets, 2nd Edition (West Sussex, UK: John Wiley & Sons, 2012).
Chapter 15. Wrong-Way Risk

181

Stress Testing: Approaches, Methods, and Applications, Edited by Akhtar Siddique and
Iftekhar Hasan (London: Risk Books, 2013). Chapter 4. The Evolution of Stress Testing
Counterparty Exposures

183

Lesson: Michel Crouhy, Dan Galai, and Robert Mark, The Essentials o f Risk Management, 2nd Edition
(New York: McGraw-Hill, 2014). Chapter 9. Credit Scoring and Retail Credit Risk Management
191
Lesson: Michel Crouhy, Dan Galai, and Robert Mark, The Essentials o f Risk Management, 2nd Edition
(New York: McGraw-Hill, 2014). Chapter 12. The Credit Transfer Markets— and Their Implications
197
Lesson: Moorad Choudhry, Structured Credit Products: Credit Derivatives & Synthetic Securitization,
2nd Edition (Hoboken, NJ: John Wiley & Sons, 2010). Chapter 12. An Introduction to Securitization

205

Lesson: Adam Ashcraft and Til Schuermann/'Understanding the Securitization of Subprime
Mortgage Credit," Federal Reserve Bank of New York Staff Reports, No. 318 (March 2008)

215

©2017 Wiley


CONTENTS

Operational and Integrated Risk Management
Lesson: "Principles for the Sound Management of Operational Risk" (Basel Committee on

Banking Supervision Publication, June 2011)

221

Lesson: Brian Noccoand Rene Stulz,"Enterprise Risk Management: Theory and Practice,
Journal o f Applied Corporate Finance 18, no. 4 (2006): 8 -2 0

229

Lesson: "Observations on Developments in Risk Appetite Frameworks and IT Infrastructure,"
Senior Supervisors Group, December 2010

233

Lesson: Anthony Tarantino and Deborah Cernauskas, Risk Management in Finance: Six Sigma
and Other Next Generation Techniques (Hoboken, NJ: John Wiley & Sons, 2009).
Chapter 3. Information Risk and Data Quality Management

239

Lesson: Marcelo G. Cruz, Gareth W. Peters, and Pavel V. Shevchenko, Fundamental Aspects
o f Operational Risk and Insurance Analytics: A Handbook o f Operational Risk (Hoboken,
NJ: John Wiley & Sons, 2015). Chapter 2. OpRisk Data and Governance

241

Lesson: Philippa X. Girling, Operational Risk Management: A Complete Guide to a
Successful Operational Risk Framework (Hoboken, NJ: John Wiley & Sons, 2013). Chapter 8.
External Loss Data

247

Lesson: Philippa X. Girling, Operational Risk Management: A Complete Guide to a
Successful Operational Risk Framework (Hoboken, NJ: John Wiley & Sons, 2013).
Chapter 12. Capital Modeling

251

Lesson: "Standardised Measurement Approach for Operational Risk— Consultative
Document" (Basel Committee on Banking Supervision Publication, March 2016).

255

Lesson: Kevin Dowd, Measuring Market Risk, 2nd Edition (West Sussex, UK: John
Wiley & Sons, 2005). Chapter 7. Parametric Approaches (II): Extreme Value

259

Lesson: Giacomo De Laurentis, Renato Maino, and Luca Molteni, Developing,
Validating, and Using Internal Ratings (Hoboken, NJ: John Wiley & Sons, 2010.
Chapter 5. Validating Rating Models

263

Lesson: Michel Crouhy, Dan Galai, and Robert Mark, The Essentials o f Risk Management,
2nd Edition (New York: McGraw-Hill, 2014). Chapter 15. Model Risk

267

Lesson: Michel Crouhy, Dan Galai, and Robert Mark, The Essentials o f Risk Management,

2nd Edition (New York: McGraw-Hill, 2014). Chapter 17. Risk Capital Attribution and
Risk-Adjusted Performance Measurement

271

Lesson: "Range of Practices and Issues in Economic Capital Frameworks" (Basel
Committee on Banking Supervision Publication, March 2009)

275

Lesson: "Capital Planning at Large Bank Holding Companies: Supervisory Expectations and
Range of Current Practice," Board of Governors of the Federal Reserve System, August 2013

285

©2017 Wiley


CONTENTS

Lesson: Bruce Tuckman and Angel Serrat, Fixed Income Securities: Tools for Today's
Markets, 3rd Edition (Hoboken, NJ: John Wiley & Sons, 2011). Chapter 12. Repurchase
Agreements and Financing

291

Lesson: Kevin Dowd, Measuring Market Risk, 2nd Edition (West Sussex,
UK: John Wiley & Sons, 2005). Chapter 14. Estimating Liquidity Risks

299

Lesson: Allan Malz, Financial Risk Management: Models, History, and Institutions
(Hoboken, NJ: John Wiley & Sons, 2011). Chapter 11. Section 11.1, Assessing the
Quality of Risk Measures

303

Lesson: Allan Malz, Financial Risk Management: Models, History, and Institutions
(Hoboken, NJ: John Wiley & Sons, 2011). Chapter 12 Liquidity and Leverage

307

Lesson: Darrell Duffie,"The Failure Mechanics of Dealer Banks,"
Journal o f Economic Perspectives 24, no. 1 (2010): 51-72

317

Lesson: Til Schuermann, "Stress Testing Banks "prepared for the Committee on
Capital Market Regulation, Wharton Financial Institutions Center, April 2012

321

Lesson: "Guidance on Managing Outsourcing Risk," Board of Governors of the
Federal Reserve System, December 2013

325

Lesson: John Hull, Risk Management and Financial Institutions, 4th Edition
(Hoboken, NJ: John Wiley & Sons, 2015). Chapter 15. Basel I, Basel II, and Solvency II

329

Lesson: John Hull, Risk Management and Financial Institutions, 4th Edition
(Hoboken, NJ: John Wiley & Sons, 2015). Chapter 16. Basel II.5, Basel III,
and Other Post-Crisis Changes

339

Lesson: John Hull, Risk Management and Financial Institutions, 4th Edition
(Hoboken, NJ: John Wiley & Sons, 2015). Chapter 17. Fundamental Review of the Trading Book

345

Regulatory Readings

349

Risk Management and Investment Management
Lesson: Andrew Ang, Asset Management: A Systematic Approach to Factor Investing
(New York: Oxford University Press, 2014). Chapter 6. Factor Theory

353

Lesson: Andrew Ang, Asset Management: A Systematic Approach to Factor Investing
(New York: Oxford University Press, 2014). Chapter 7. Factors

363

Lesson: Andrew Ang, Asset Management: A Systematic Approach to Factor Investing
(New York: Oxford University Press, 2014). Chapter 10. Alpha (and the Low-Risk Anomaly)

369

Lesson: Andrew Ang, Asset Management: A Systematic Approach to Factor Investing
(New York: Oxford University Press, 2014). Chapter 13. Illiquid Assets (excluding
section 13.5, Portfolio Choice with Illiquid Assets)

379

©2017 Wiley

©


CONTENTS

Lesson: Richard Grinold and Ronald Kahn, Active Portfolio Management: A
Quantitative Approach for Producing Superior Returns and Controlling Risk, 2nd Edition
(New York: McGraw-Hill, 2000). Chapter 14. Portfolio Construction

383

Lesson: Philippe Jorion, Value a t Risk: The New Benchmark for Managing Financial Risk,
3rd Edition (New York: McGraw-Hill, 2007). Chapter 7. Portfolio Risk: Analytical Methods

389

Lesson: Philippe Jorion, Value a t Risk: The New Benchmark for Managing Financial Risk,
3rd Edition (New York: McGraw-Hill, 2007). Chapter 17. VaR and Risk Budgeting in
Investment Management

395

Lesson: Robert Litterman and the Quantitative Resources Group, Modern Investment
Management: An Eguilibrium Approach (Hoboken, NJ: John Wiley & Sons, 2003).
Chapter 17. Risk Monitoring and Performance Measurement

399

Lesson: Zvi Bodie, Alex Kane, and Alan J. Marcus, Investments, 10th Edition
(New York: McGraw-Hill, 2013). Chapter 24. Portfolio Performance Evaluation

405

Lesson: G. Constantinides, M. Harris, and R. Stulz, eds., Handbook o f the Economics o f
Finance, Volume 2B (Oxford, UK: Elsevier, 2013). Chapter 17. Hedge Funds, by William
Fung and David Hsieh

415

Lesson: Kevin R. Mirabile, Hedge Fund Investing: A Practical Approach to Understanding
Investor Motivation, Manager Profits, and Fund Performance (Hoboken, NJ: Wiley
Finance, 2013). Chapter 11. Performing Due Diligence on Specific Managers and Funds

421

Current Issues in Financial Markets
Lesson: Rainer Bohme, Nicolas Christin, Benjamin Edelman, and Tyler Moor, "Bitcoin:
Economics, Technology, and Governance!'Journal o f Economic Perspectives,
Vol. 29, No. 2, Spring 2015

427

Lesson: Dudley, William C., "Market and Funding Liquidity— An Overview," Remarks at the
Federal Reserve Bank of Atlanta 2016 Financial Markets Conference, Fernandina Beach,
Florida, May 1,2016

431

Lesson: "Chapter 2: Market Liquidity - Resilient or Fleeting?" International Monetary
Fund, Global Financial Stability Report, October 2015

433

Lesson: "Algorithmic Trading Briefing Note," Federal Reserve Bank of New York, April 2015

437

Lesson: Morten Bech, Anamaria Hies, Ulf Lewrick, and Andreas Schrimpf,"Hanging Up
the Phone— Electronic Trading in Fixed Income Markets and Its Implications,''BIS Quarterly
Review, 2016

441

Lesson: Morten Linnemann Bech and Aytek Malkhozov,"How Have Central Banks
Implemented Negative Policy Rates?"/?/5 Quarterly Review, March 6,2016

445

Lesson: "Corporate Debt in Emerging Economies: A Threat to Financial Stability?"Committee

on International Economic Policy and Reform, Brookings Institution, September 2015

449

©2017 Wiley


How to Study for the Exam
The FRM Exam Part II curriculum covers the tools used to assess financial risk:
•
•
•
•
•

Market risk measurement and management—25 %
Credit risk measurement and management—25%
Operational and integrated risk management—25 %
Risk management and investment management— 15%
Current issues in financial risk management— 10%

It is important to focus only on the learning objectives as you are asked about them and
pay close attention to the percentages of each section. That is the core of my focus through
the text, the online lecture sessions, and with the practice questions. A study hour doesn’t
count unless you are laser focused on specifically how GARP asks a learning objective.
Consistency is also key. Making a regular weekly study time is going to be important
to staying on track. There is a reason only -50% of the candidates pass the exam every
year. It’s a tough exam. It also tests intuition, not just memorization. That is why I
attempt at every opportunity to connect the dots across readings and teach how changing
environments change both markets and the models we use to model them, as well as

helping you with the questions GARP specifically wants you to calculate an outcome for.
Calculator policy:
It is best to begin your study with one of the approved calculators. You will not be admitted
to the exam without one of these approved calculators!
•
•
•
•
•

Hewlett Packard 12C (including the HP 12C Platinum and the Anniversary Edition)
Hewlett Packard 10B II
Hewlett Packard 10B 11+
Hewlett Packard 20B
Texas Instruments BA II Plus (including the BA II Plus Professional)

Every year, candidates are turned away from the exam site because of wrong calculators.
Make sure you aren’t one of them.

©2017 Wiley

xi


ABOUT THE AUTHOR
Christian H. Cooper is an author and trader based in New York City. He initially created
the FRM program because, as a candidate, he was frustrated with the quality of study
programs available. Writing from a practitioner’s point of view, Christian drew on his
experience as a trader across fixed income and equity markets, most recently as head of
derivatives trading at a bank in New York, to create a program that is very focused on exam

results while connecting the dots across topics to increase intuition and understanding.
Christian is active with the Aspen Institute; he is a Truman National Security Fellow, and a
term member at the Council on Foreign Relations.

xii

©2017 Wiley


M a r k e t Ri s k M e a s u r e m e n t
M a n a g e m e n t (MR)

a nd

The broad areas of knowledge covered in readings related to Market Risk Measurement
and Management include the following:
•

•
•
•
•

VaR and other risk measures:
o Parametric and nonparametric methods of estimation
o VaR mapping
o Backtesting VaR
o Expected shortfall (ES) and other coherent risk measures
o Extreme value theory (EVT)
Modeling dependence: correlations and copulas

Term structure models of interest rates
Discount rate selection
Volatility: smiles and term structures

©2017 Wiley

©



Do w d , Ch

a pt e r

3

Kevin Dowd, Measuring Market Risk, 2nd Edition (West Sussex, UK:
John Wiley & Sons, 2005). Chapter 3. Estimating Market Risk Measures:
An Introduction and Overview
After completing this reading you should be able to:
•
•
•
•
•
•
•

Estimate VaR using a historical simulation approach.
Estimate VaR using a parametric approach for both normal and lognormal return

distributions.
Estimate the expected shortfall given P/L or return data.
Define coherent risk measures.
Estimate risk measures by estimating quantiles.
Evaluate estimators of risk measures by estimating their standard errors.
Interpret QQ plots to identify the characteristics of a distribution.

Reading notes: This is a very dense reading about 20 pages long with a lot of VaR
“estimate” questions. This is probably one of the most important readings in the Market
Risk Measurement and Management section of the exam. Make sure you can calculate
these answers and be able to qualitatively discuss these topics. However, don’t get lost
in the technical material that asks you to “describe.” The assigned readings go into deep
detail that isn’t required for the exam. Make sure you are paying attention to what you are
actually asked.

Learning objective: Estimate VaR using a historical simulation approach.

First recognize that as flawed as VaR is, using VaR based on historical data is even worse.
The idea here is to order the historical losses observed in a portfolio, ordered by the actual
amount of the loss. So, if we have 1,000 loss observations ordered from smallest to largest
and we are interested in the 97% confidence level, that implies that 30 observations will be
in the tail of the distribution that will define the VaR of this portfolio.
So “estimate” here really means “observe,” and in the example the 31st loss observation
would be the amount of the expected VaR of the portfolio based on the losses the portfolio
has historically experienced.
VaR is flawed when used incorrectly, and VaR based on historical observation is really
flawed. There is no insight into the risk the portfolio has, no idea how that risk may change
in the future, and so on. There are almost too many problems here to mention and even
more problems when used with a portfolio that has embedded options.

Learning objective: Estimate VaR using a parametric approach for both
normal and lognormal return distributions.
I will move forward on the assumption you are very comfortable with the lognormal and
normal distributions; take a break here if you need a review.
©2017 Wiley

©


MARKET RISK MEASUREMENT AND MANAGEMENT (MR)

Starting first with the assumption of a normal distribution, we have to define the mean and
standard deviation of the portfolio in order to calculate the VaR. Likely this will be given
to you on the exam, but in the real world this will be estimated parameters based on the
risk manager’s expectations.
Since we want to interpret VaR in terms of lost money, our formula for losses is:
VaR - - | i + oZ a
where negative mu is expected losses, sigma is the standard deviation of the loss, and Z is
the standard normal variate corresponding to the level of significance we are looking for.
Recall that the standard normal variate at the 95% level of significance is 1.645.
If profits and losses over some period are normally distributed with mean of 20 and
standard deviation of 30, the 95% VaR is calculated as:
VaR = -20 + 30(1.645) = -29.35
This is straightforward enough, but the learning objective is asking about a normal and
lognormal distribution of returns, not dollars gained or lost.
When we transition to returns, we have to add additional parameters to describe the
starting dollar value and ending dollar value so we can arrive at the return estimate.
Furthermore, we have to establish some critical value r* so we can compare our expected
probability to our critical value return where that probability is equal to our level of
confidence.

Taking the previous equation:
V a R - - j i + oZ a
I am going to modify it to:
r* = - f t + c Z a

This will be our return critical value that we need to establish a confidence interval around
the return of our portfolio at any given time, rt
Also, since any given return is the starting value relative to ending value, we can define
rt as:
Pt - Pt-i

Don’t let all the equations fool you; we are defining return as you normally would: My
return is my portfolio value at time t minus what it was before, divided by the starting
value.

4

©2017 Wiley


DOWD, CHAPTER 3

However, since we are thinking about VaR at risk, we are only concerned with the losses.
We can extend the relationship to include VaR and recover our critical value:
* _ fi - Pm _ VaR
1

Pm

Pm

Now, substitute into this equation:
r* = - |i + c Z a
to arrive at:

and last,
VaR = (-(I + oZ a )Pt_!
The only difference between the normal and lognormal version is going to be the critical
values.
Let’s learn a quick calculation. If I know returns are normally distributed at 12% with
standard deviation of 18% and I want the 95% level VaR:
-12% +18% (1.645) = 0.1761 (converting from percentages)
On the exam, you may be given estimates of profits and losses, which would mean you
don’t use the population parameters of mu and sigma, but the math and order are the same.

Lognormal Version
The real dividing issue between normal and lognormal that should be clear by now is
that a normal distribution is symmetric and assigns an equal probability of going up or
down in price. This implies that prices can be negative if you use the normal distribution.
Now for stocks, this doesn’t make sense since they have a zero floor. That is why when
we talk about financial assets we often refer to the distribution of returns being normally
distributed, not the asset prices themselves, which, of course, is the lognormal distribution.
This is why we need the lognormal.
So how do we calculate VaR when lognormally distributed? From the discussion of
normally distributed VaR, we know we are dealing with potential returns instead of dollar
values of potential loss (price moves) under the normal distribution, so we have the extra
step of converting our lognormally derived VaR back into dollar terms.
You do not have to memorize the derivation of this formula, but you will use the very last
line in the calculation.

©2017 Wiley

5


MARKET RISK MEASUREMENT AND MANAGEMENT (MR)

As in the normal, we are going to use this formula from before, but let’s call VaR a new
random variable that will correspond to a loss equal to our VaR. Ultimately, this is what we
want to know.
Our original equation:
VaR = - |i + oZ a
becomes:
X* = - p + a Z a
Since we are using geometric returns, we insert a term that describes the price at some
period right now (assuming a loss) relative to some time in the past to establish our return.
It sounds complicated, but this is the same as a $50 stock that goes to $45 so the loss is
10%. 45/50 = .90 & (1 - 0.90) = 0.10. We are going to do the exact same thing here.
X* = ln(Pricenow/Pricepast)
Reordering terms and using the properties of logarithms so the ratio of logarithms can be
expressed as the difference of two logarithms:
X* = ln(Pricenow) - ln(Pricepast)
.^.

0

0

0

0

0

Remember, X is our critical value that ultimately becomes our VaR as some certainty, alpha.
Reordering terms,
ln(Pricenow) = X* + ln(Pricepast)
Distribute the logarithm across: This goes back to early calculus, but don’t worry—you
don’t need to know it; just remember the last step.
(Pricenow) - (Pricepast)ex
Insert original equation for X*:
(Pricenow) = (Price past )e_tl+aZa
Skipping a few messy steps that are unnecessary to know for the exam, we arrive at
what you need to know for VaR under a lognormal distribution of returns, and that is this
relationship:
VaR = Ppast[l - exp(p - oZ)]

6

©2017 Wiley


DOWD, CHAPTER 3

To bring this all together, and since this is an “estimate” question, you may be asked to
calculate/evaluate this on exam day:
Let’s assume returns are normally distributed with mean 10% and standard deviation 18%,
and our portfolio is $100.
We can express the lognormal VaR at the 95% level as:
1 - exp(0.10 -0 .1 8 * 1.645) = 0.26 or $26

Learning objective: Estimate the expected shortfall given P/L or return data.

Risk as Shortfall
Another way to specify a risk objective is by means of the expected portfolio standard
deviation, which is the square root of the expected portfolio variance. For example, the
S&P 500 might have an annual portfolio standard deviation of 23%. Using this figure
and either a normal or lognormal distribution, we can quantify the probability that a
given portfolio will have a loss, or a return below a certain minimum requirement (this
probability is called a shortfall risk).
As might be expected, the focus of this criterion is minimizing the chances that a specified
minimum return (say RL) is not achieved. If the portfolio return (call it RP) is normally
distributed, then it is not very difficult to calculate the probability that RP will fall short
of Rl —in other words, P(RP < RL)—or to calculate the number of standard deviations RL
falls below the expected portfolio return, E(RP).
Note that the portfolio that maximizes E(RP) - RL will at the same time minimize P(RP < RL).
If we divide the expression E(RP) - RL by
the standard deviation of the portfolio, we
get the safety margin in units of portfolio standard deviation. This ratio is called Roy’s
safety-first criterion (SFRatio):

SFRatio = EaP
Assuming returns are normally distributed, the safety-first ratio will be maximized by
choosing the optimal portfolio, using the three steps of Roy’s safety-first criterion:
1.
2.
3.

Calculate the SFRatio for the portfolio.

Calculate P(RP < Rl ) by looking up N(-SFRatio) = 1 - N(SFRatio).
Choose the portfolio that results in the lowest probability for step 2.

Example: You work for the investment advisor of a major educational institution
with a $2 billion endowment. The school wishes to be able to use $50 million of
the endowment’s investment income annually for operational expenses, but does
not wish to invade the endowment principal. Also, the school intends to place the

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MARKET RISK MEASUREMENT AND MANAGEMENT (MR)

endowment in one of three investment pools, which have the following return and risk
characteristics:

Expected return
Standard deviation of return

P o o ll

Pool 2

Pool 3

25%
35%

15%
25%

10%
15%

Find the shortfall level of return, RL. Next, according to Roy’s safety-first criterion, which
investment pool is best suited to the school’s objectives? Last, find the probability that the
optimal investment pool will fail to generate a return equal to or greater than RL. Assume
returns are normally distributed.
Solution: If the school wishes to spend $50,000,000 of the endowment investment
income annually, it would imply that the shortfall return level is equal to
$50,000,000/$2,000,000,000 = 2.5%.
Next, to figure out which investment pool is optimal for the school, we must calculate the
SFRatio for each pool.
Pool 1: SFRatio = (0.25 - 0.025)/0.35 = 0.6429
Pool 2: SFRatio = (0.15 - 0.025)/0.25 = 0.5000
Pool 3: SFRatio = (0.10 - 0.025)/0.15 = 0.5000
Pool 1 has the highest SFRatio, so if returns are normally distributed, then it is the optimal
portfolio, and it will minimize the probability that returns will fall short of RL.
The probability that Pool 1 will generate a return less than RL is found by looking up:
N(-SFRatio) = 1 - N(SFRatio) = 1 - N(0.6429) = 1 - 0.7389 = 0.2611 = 26.11%.
Regardless of which investment criterion comes first, risk and return objectives are
essential ingredients in setting a strategic asset allocation.

Learning objective: Define coherent risk measures.

A coherent risk measure satisfies the following five properties, all of which are highly
theoretical and some of which are hard to visualize.
First, a portfolio that has no holdings has no risk (normalized property); second, the lowerreturn portfolio has less risk (monotonicity property); third, risk doesn’t increase when

adding two portfolios together, also known as diversification (subaddivity property); fourth,
an identical portfolio of double the notional should have the potential for double the notional
loss (positive homogeneity property); and fifth, the addition of risk-free assets dilutes the
potential notional loss percentage (translation invariance property). This last one needs an
example: Pretend you have a $100 portfolio with a $1 loss potential for a 1% potential loss.

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DOWD, CHAPTER 3

Adding $100 of cash, pretending cash is risk free, increases the portfolio size to $200 with
the same potential for a $1 loss, so the loss percentage is reduced to 50 basis points.

Learning objective: Estimate risk measures by estimating quantiles.

This portion of the reading gets very technical, but stay focused on what you are asked.
Any measure of risk is only as good as its precision. Consider shortfall risk: How sure are
we that we have estimated either the probability or the degree (amount) of loss? VaR is
fairly precise, but that isn’t the only measure of risk.
So when we are talking about quantiles (a quantile comes from statistics, and it is basically
just a regularly spaced interval that divides up a probability distribution into equal-sized
parts—5%, 10%, or half of a distribution), we are talking about setting up a confidence
indicator around the risk measure to indicate the precision of that particular risk measure.
So a quantile by definition is just a sample of a defined size (e.g., top 5%) of a given
distribution, and, just like any other sampling, we can define the standard error of the
sample (quantile).
Now the relationship you need to understand is that the standard error for any sample
size falls as the sample size gets larger. Also, the standard error rises as the estimator
gets farther into the tail. So the standard error of a one-in-a-million event is much wider

than the standard error of a one standard deviation estimation. Stated differently, the
more extreme the probability of an event, the less precise we can be about its estimation.
This has huge implications for risk management using VaR and is one of the key reasons
VaR creates a false sense of security among those who don’t understand the quantitative
limitations of the model itself.
So how do we apply this to expected shortfall (ES) and VaR?
When comparing VaR and ES, both have similar standard errors for normal distributions,
but expected shortfall has much bigger standard errors in distributions with heavy tails,
also known as reality. By definition, ES estimates are less accurate than VaR when
considering distributions with heavy tails.

Learning objective: Evaluate estimators of risk measures by estimating
their standard errors.

The method of estimating I think will most likely be covered on the exam is one that is
called “bootstrapping” and shouldn’t be confused with the method of yield curve creation
by the same name. In this case we take a large sample of estimators from a particular
distribution and then estimate the standard error of that sample. This is convoluted for
sure, but just know for the exam that there is this method called bootstrapping and it
means to create a standard error estimate for a sample of estimators from a particular
distribution.

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MARKET RISK MEASUREMENT AND MANAGEMENT (MR)

Learning objective: Interpret QQ plots to identify the characteristics of a

distribution.

Recall from the Part I material that a distribution is just a special equation that describes
the distribution of some data set. The probability density function is the name of the
unique equation that describes what we call the bell curve. However, we can define any
equation that fits a set of data as a density function as long as it fits the data and satisfies
a few requirements. A QQ plot is useful to find that special equation that describes a set
of data. QQ stands for quantile-quantile rank and plots the empirical (observed) data in a
particular quantile to what is predicted in the distribution we are testing.
What you need to know for the exam is if a QQ plot is linear, then the empirical data
matches the distribution we are testing, but if it is nonlinear, our empirical data does not
match the distribution we are testing.
Stated differently, if the plot is not linear at a 45-degree angle, the empirical data and the
distribution belong to two different families of distributions.

10

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Do w d , Ch

a pt e r

4

Kevin Dowd, Measuring Market Risk, 2nd Edition (West Sussex, England: John Wiley &
Sons, 2005). Chapter 4. Non-Parametric Approaches
After completing this reading you should be able to:
•

•
•
•

Apply the bootstrap historical simulation approach to estimate coherent risk
measures.
Describe historical simulation using nonparametric density estimation.
Compare and contrast the age-weighted, the volatility-weighted, the correlationweighted, and the filtered historical simulation approaches.
Identify advantages and disadvantages of nonparametric estimation methods.

Reading notes: This is a relatively long reading of 30 pages with no “calculate” questions.
You can waste a lot of time here if you aren’t careful. Most of the assigned reading covers
material not actually on the FRM exam. Recall that nonparametric means a way to model
data without actually using the parameters of a model— so this is using historic data or
other number crunching to measure market risk without the use of probability density
functions.

Learning objective: Apply the bootstrap historical simulation approach to
estimate coherent risk measures.

This is a little misleading: a coherent risk measure we defined in the first reading. Recall
that Var is not a coherent risk measure because of subadditivity, and all you need to know
here is how the bootstrap method actually works.
Bootstrapping is another term for resampling from our distribution with replacement. We
resample many times, replacing each time, and create a large number of samples.
We can chart each of these samples on a histogram and then look at any alpha, or degree
of certainty, we want, and that is our bootstrapped VaR. One thing to know for the exam is
that the bootstrap method won’t tell us much about how precise these sample estimates of
risk measures are, but this is how the method works.

Learning objective: Describe historical simulation using nonparametric
density estimation.

Remember, all density estimation refers to is an attempt to build a probability distribution
using past observed profit and loss data. It requires no model parameters since no model
is actually used. We are simply looking at the past and trying to explain the future from it.
Historical simulation has many, many problems, but it is a widely used method.

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