FRM学习资料七：FRM Handbook 5th Edition E-Book
金融风险管理师（FRM）学习资料：FRM Handbook 5th Edition E-Book-非扫描版，超清晰PDF电子书
About the Author xi
About GARP xiii
Bond Fundamentals 3
Fundamentals of Probability 31
Fundamentals of Statistics 67
Monte Carlo Methods 89
Introduction to Derivatives 111
Fixed-Income Securities 161
Fixed-Income Derivatives 195
Equity, Currency, and Commodity Markets 217
Market Risk Management
Introduction to Market Risk 247
Sources of Market Risk 273
Hedging Linear Risk 297
Nonlinear Risk: Options 315
Modeling Risk Factors 341
VAR Methods 359
Investment Risk Management
Portfolio Management 383
Hedge Fund Risk Management 401
Credit Risk Management
Introduction to Credit Risk 431
Measuring Actuarial Default Risk 451
Measuring Default Risk from Market Prices 479
Credit Exposure 499
Credit Derivatives and Structured Products 531
Managing Credit Risk 561
Legal, Operational, and Integrated Risk Management
Operational Risk 587
Liquidity Risk 607
Firm-Wide Risk Management 623
Legal Issues 643
Regulation and Compliance
Regulation of Financial Institutions 657
The Basel Accord 667
The Basel Market Risk Charge 699
About the CD-ROM 715
he Financial Risk Manager Handbook provides the core body of knowledge
for ﬁnancial risk managers. Risk management has evolved rapidly over the past
decade and has become an indispensable function in many institutions.
This Handbook was originally written to provide support for candidates tak-
ing the FRM examination administered by GARP. As such, it reviews a wide
variety of practical topics in a consistent and systematic fashion. It covers quan-
titative methods and capital markets, as well as market, credit, operational, and
integrated risk management. It also discusses regulatory and legal issues essential
to risk professionals.
This edition has been thoroughly updated to reﬂect recent developments in
ﬁnancial markets. The unprecedented losses incurred by many institutions have
raised questions about risk management practices. These issues are now addressed
in various parts of the book, which also include lessons from recent regulatory
reports. The securitization process and structured credit products are critically
examined. A new chapter on liquidity risk has been added, given the importance
of this risk during the recent crisis. Finally, this Handbook incorporates the latest
questions from the FRM examinations.
Modern risk management systems cut across the entire organization. This
breadth is reﬂected in the subjects covered in this Handbook. The book was de-
signed to be self-contained, but only for readers who already have some exposure
to ﬁnancial markets. To reap maximum beneﬁt from this book, readers should
have taken the equivalent of an MBA-level class on investments.
Finally, I want to acknowledge the help received in writing this Handbook.
In particular, I thank the numerous readers who shared comments on previous
editions. Any comment or suggestion for improvement will be welcome. This
feedback will help us to maintain the high quality of the FRM designation.
When successive returns are uncorrelated, the volatility increases as the hori-
zon extends following the square root of time.
More generally, the variance can be added up from different values across
different periods. For instance, the variance over the next year can be computed as
the average monthly variance over the ﬁrst three months, multiplied by 3, plus the
average variance over the last nine months, multiplied by 9. This type of analysisP1: ABC/ABC P2: c/d QC: e/f T1: g
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70 QUANTITATIVE ANALYSIS
is routinely used to construct a term structure of implied volatilities, which are
derived from option data for different maturities.
It should be emphasized that this holds only if returns have constant parame-
ters across time and are uncorrelated. When there is non-zero correlation across
days, the two-day variance is
V(R2) = V(R1) + V(R1) + 2ρV(R1) = 2V(R1)(1 + ρ) (3.8)
Because we are considering correlations in the time series of the same variable, ρ
is called the autocorrelation coefﬁcient,orthe serial autocorrelation coefﬁcient.A
positive value for ρ implies that a movement in one direction in one day is likely to
be followed by another movement in the same direction the next day. A positive
autocorrelation signals the existence of a trend. In this case, Equation (3.8) shows
that the two-day variance is greater than the one obtained by the square root of
A negative value for ρ implies that a movement in one direction in one day
is likely to be followed by a movement in the other direction the next day. So,
prices tend to revert back to a mean value. A negative autocorrelation signals
EXAMPLE 3.1: FRM EXAM 1999—QUESTION 4
A fundamental assumption of the random walk hypothesis of market returns
is that returns from one time period to the next are statistically independent.
This assumption implies
a. Returns from one time period to the next can never be equal.
b. Returns from one time period to the next are uncorrelated.
c. Knowledge of the returns from one time period does not help in predict-
ing returns from the next time period.
d. Both b) and c) are true.
EXAMPLE 3.2: FRM EXAM 2002—QUESTION 3
Consider a stock with daily returns that follow a random walk. The annual-
ized volatility is 34%. Estimate the weekly volatility of this stock assuming
that the year has 52 weeks.
d. 4.71%P1: ABC/ABC P2: c/d QC: e/f T1: g
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Fundamentals of Statistics 71
EXAMPLE 3.3: FRM EXAM 2002—QUESTION 2
Assume we calculate a one-week VAR for a natural gas position by rescal-
ing the daily VAR using the square-root rule. Let us now assume that we
determine the true gas price process to be mean-reverting and recalculate the
Which of the following statements is true?
a. The recalculated VAR will be less than the original VAR.
b. The recalculated VAR will be equal to the original VAR.
c. The recalculated VAR will be greater than the original VAR.
d. There is no necessary relation between the recalculated VAR and the
mean reversion. In this case, the two-day variance is less than the one obtained by
the square root of time rule.
3.1.3 Portfolio Aggregation
Let us now turn to aggregation of returns across assets. Consider, for example, an
equity portfolio consisting of investments in N shares. Deﬁne the number of each
share held as qi with unit price Si . The portfolio value at time t is then
Thus, derivatives valuation focuses on the discounted center of the distribution,
while VAR focuses on the quantile on the target date.
Monte Carlo simulations have been long used to price derivatives. As will
be seen in a later chapter, pricing derivatives can be done by assuming that the
underlying asset grows at the risk-free rate r (assuming no income payment).
For instance, pricing an option on a stock with expected return of 20% is done
assuming that (1) the stock grows at the risk-free rate of 10%and (2) we discount
at the same risk-free rate. This is called the risk-neutral approach.
In contrast, riskmeasurement deals with actual distributions, sometimes called
physical distributions. For measuring VAR, the risk manager must simulate asset
growth using the actual expected return µ of 20%. Therefore, risk management
uses physical distributions, whereas pricingmethods use risk-neutral distributions.
It should be noted that simulation methods are not applicable to all types
of options. These methods assume that the value of the derivative instrument at
expiration can be priced solely as a function of the end-of-period price ST,and
perhaps of its sample path. This is the case, for instance, with an Asian option,
where the payoff is a function of the price averaged over the sample path. Such an
Simulation methods, however, are inadequate to price American options, be-
cause such options can be exercised early. The optimal exercise decision, however,
is complex to model because it should take into account future values of the op-
tion. This cannot be done with regular simulation methods, which only consider
present and past information. Instead, valuing American options requires a back-
ward recursion, for example with binomial trees. This method examines whether
the option should be exercised or not, starting fromthe end andworking backward
in time until the starting time.
Finally, we shouldmention the effect of sampling variability. Unless K is extremely
large, the empirical distribution of ST will only be an approximation of the trueP1: ABC/ABC P2: c/d QC: e/f T1: g
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100 QUANTITATIVE ANALYSIS
distribution. There will be some natural variation in statistics measured from
Monte Carlo simulations. Since Monte Carlo simulations involve independent
draws, one can show that the standard error of statistics is inversely related to the
square root of K. Thus more simulations will increase precision, but at a slow
rate. For example, accuracy is increased by a factor of ten going from K = 10
to K = 1,000, but then requires going from K = 1,000 to K = 100,000 for the
same factor of 10.
This accuracy issue is worse for risk management than for pricing, because
the quantiles are estimated less precisely than the average. For VAR measures,
the precision is also a function of the selected conﬁdence level. Higher conﬁ-
dence levels generate fewer observations in the left tail and hence less-precise
VAR measures. A 99% VAR using 1,000 replications should be expected to have
only 10 observations in the left tail, which is not a large number. The VAR
estimate is derived from the tenth and eleventh sorted number. In contrast, a
95% VAR is measured from the ﬁftieth and ﬁfty-ﬁrst sorted numbers, which is
more precise. In addition, the precision of the estimated quantile depends on the
shape of the distribution. Relative to a symmetric distribution, a short option
position has negative skewness, or a long left tail. The observations in the left
tail therefore will be more dispersed, making is more difﬁcult to estimate VAR
Various methods are available to speed up convergence:
Antithetic Variable Technique. This technique uses twice the same sequence
of random draws from t to T. It takes the original sequence and changes the
sign of all their values. This creates twice the number of points in the ﬁnal
distribution of FT without running twice the number of simulations.
Control Variate Technique. This technique is used to price options with trees
when a similar option has an analytical solution. Say that fE is a European
option with an analytical solution. Going through the tree yields the values
of an American and European option, FA and FE. We then assume that the
error in FA isthesameasthatin FE, which is known. The adjusted value is