分类： 金融风险管理师（FRM）考试

金融风险管理师（FRM）考试学习笔记（整理的很清晰）

市场风险
期货
1. 市场风险重要的 5 个原因：1、management information （将风险暴露和资本相比较）
2、设定限额3、resoure allocation 4、performance evaluation 5、监管
2. 巴塞尔协议对市场风险的计量包括标准方法（固定收益、外汇、权益等）和内部评级法。
3. 成功期货合约的三个性质是标的资产的深度市场，资产价格要有足够的波动性以及风险
控制不能以直接的方式进行。
4. 含有 carrying cost 的 forward price： ，I 就是期间产生的现金流
rt
eISF )( 00 −=
5. forward contract 的定价： （连续 cash flow 支付） ，没有现金流
的话 .S为 spot price，K为执行价格。
rt qt
KeeSV − −
−= 0
rt
KeSV −
−= 0
6. 股指期货的 beta 调整策略（比如说完全对冲系统风险） ： A
P N )(
*
ββ −= ,其中前一
个β是对冲后的，后面的是对冲前的β，P 是组合的价值，A 是对冲资产的价值，一般
是单位标的资产的价值×乘子。
7. 对冲权益组合所需的股指期货的份数＝ 期货乘数 期货价格
组合价值
×
× portfolio β ， portfolio β 是
组合相对于基准的β，如果股指期货本身也有β的话，则所需份数＝
期货乘数 期货价格
组合价值
× ×
×
future
portfolio
β
β ，这时要和上面的 beta调整策略区分开。

12. 对于利率期货，用基于久期的对冲公式如下（对于利率衍生品，一般都用久期对冲来平
衡，注意欧洲美元期货也是利率期货而不是外汇期货，类似于 3 个月到期的 FRA，而且
是是柜台交易，标准合约规模为 100 万。FRA 是OTC 的（98试题） ）：
N=－
FC
P
DF
DP
*
*

13. 其中，P 为组合的价值， 为期货的价值，两个 D 分别为组合的久期和期货的久期。
负号表示期货的头寸和组合中的头寸是相反的。
C F
14. 远期汇率的计算公式为 = 2/1 F
1
2
2/1
1
1
r
r
X
+
×
＋ ，注意多期远期汇率的计算公式：
15. 当 cash price 和期货价格之间有很强的正相关性的时候，就可以进行有效的对冲。
16. 当被对冲的头寸和标的资产没有完全相关的时候，就会存在基点风险。所以当标的资产
和对冲资产不一样、相关性不唯一已经到期日不一样的时候就会产生基点风险。
17. 从收益率曲线上读取到的远期利率称之为隐含的远期利率（implied forward rate） 。
18. 期货和远期的价格只有在当利率不变（costant）和确定（distermintic）时才会相等,
因为期货和远期的区别是一个是盯市，一个不是。
19. 计算欧洲美元期货合约的凸性调整（convexity Adjustment） 。由于期货合约每日盯市
的特征导致实际的远期利率（期货的利率）和隐含的远期利率会发生差别，凸性调整就
是要降低这方面的差别（也就是调整期货利率和远期合约利率之间的差别）：
实际远期利率（期货利率）=隐含远期利率－0.5* 21
2
** tt σ

1 t 指的是期货合约的到期日， 指的是标的资产利率的到期日。从上可知，期货的
利率是要低于远期的利率的，所以长期而言，期货的价格是要高于远期的价格的（利率和价
格成反比，00 试题） 。
2 t

互换
22. Vanilla 互换的现金流和定价：
现金流 floating=L*浮动利率*期限 （注：利率是以年利率计算的）
现金流 fix=L*固定利率*期限
定价时，需要将一个互换看成是一个浮动债券和一个固定债券的组合。从一个例子来看
利率互换的定价。一个面值是 1m 的互换，pay6 月的 libor,收取 6%的固定利率。互换
2

全英文国际金融风险管理师（FRM）培训课程

FRM Training07-A.ppt

FRM Training07-B.ppt

FRM Training07-C.ppt

FRM Training07-D.ppt

FRM Training07-E.ppt

FRM Training07-F1.ppt

FRM Training07-F2.ppt

FRM Training07-F3.ppt

FRM Training07-F4.ppt

Outline of This Lecture
What is a swap
Interest rate swaps
Mechanics of interest rate swaps
Pricing interest rate swaps
Currency swaps
Mechanics of currency swaps
Pricing currency swaps
1. What is a swap? Example
Company A buys electricity from wholesalers and provides power service to consumers in California

The electricity price selling to consumers is fixed, while the purchasing price from wholesalers is variable

For A, the cash inflows are quite stable, but the outflows are uncertain

In case such as the high oil price, the electricity price in wholesale market become very high, and the company may bankrupt.

To manage the risk, the company may want to exchange their floating (variable) outflows with a fixed outflow or to exchange their fixed inflows with a floating inflows.

The company can do it by using a swap.
What is a swap?
A swap is an OTC agreement between two parties (called counterparties) to exchange a series of cash flows over a period of time.

Four major types of swaps:
Interest rate swaps (IRS)
Currency swaps
Equity swaps
Commodity swaps
Some terminologies about swaps
Notional principal: the amount of money used to determine the payments or the sizes of swap contracts. It may or may not be exchanged

Initially, the value of a swap is zero. Thus, no payment is needed at the beginning

Settlement date: the date on which a payment occurs.

Settlement period: the period between two consecutive settlement dates

Tenor: time to maturity
Features of the swaps market
Pros:
Privacy: only the counterparties know your position

Virtually no government regulation in U.S.
Industrial self-regulation. Major regulator:
ISDA: The International Swap and Derivatives Association

Three concerns or limitations
Difficult to find counterparties
Now swap dealers make markets, and this problem has been solved

Difficult to close before maturity: liquidity risk

Counterparty default risk:
This is an important concern in dealing with swaps.
Swap rates depend on credit ratings
Only deal with large firms
2. Interest rate swaps (IRS)
IRS are swaps to exchange interest payments in the same currency

The most popular IRS is fixed-for-floating swap, also called plain vanilla interest rate swap

1) Mechanics of IRSs
In a plain vanilla interest rate swap, there are two counterparties: A and B.
A: agrees to pay B a sequence of interest rate payments based to a fixed rate and a “principal”, called the notional principal.
A is called the fixed payer

B: agrees to pay A a sequence of interest rate payments based to the market rates, or floating-rates and the notional principal.
B is called the floating payer

No fund is exchanged initially.
The fixed-rate is predetermined, called the swap rate.
The floating rate in many IRS is LIBOR rate.
Example: A Plain Vanilla IRS
Company A entered an agreement with Bank B initiated on Sept.1, 2000

Company A:
Pays Bank B a fixed rate of 5% per annum every 6 months for 3 years on a notional principal of $100 million.
Receives 6-month LIBOR every 6 months for 3 years on a notional principal of $100 million.

The notional principal is not exchanged.

There is no fund exchange at t=0.

The actual payments are the net payments.

Tenor: (the time to maturity): three years
Plain Vanilla IRS
Example: A Plain Vanilla IRS
Cash flow to company A:

Date LIBOR Inflow Outflow Net Cash Flow
receive-floating pay-fixed

09/01/2000 4.2%
03/01/2001 4.8% 2.10 -2.50 -0.40
09/01/2001 5.3% 2.40 -2.50 -0.10
03/01/2002 5.5% 2.65 -2.50 +0.15
09/01/2002 5.6% 2.75 -2.50 +0.25
03/01/2003 5.9% 2.80 -2.50 +0.30
09/01/2003 6.4% 2.95 -2.50 +0.45

On Sept. 1(time t=0): the first LIBOR rate was known, and so the first cash flow was known. The subsequent cash flows are unknown, because the LIBOR rates are uncertain.
Gain or loss: Depend on the difference between the floating rates and the fixed-rate.
Zero-sum game.
Note:
The fixed-rate in a plain vanilla IRS is set so that the initial value of the agreement is zero

Swap pricing is referred to determine the fair value of a fixed rate, called a swap rate.

In this example, the floating payment is determined in advance and paid in arrears. Most swaps are of this type.

e.g.:
The payment paid on Mar. 01, 2003, $2.80 million, is determined on Sept. 01, 2002, based on the spot 6-month LIBOR rate quoted on Sept. 1, 2002, 5.6%.

Some swaps are paid in advance, called in-advance swaps
Some motivations of using an IRS
Converting a liability:
From fixed rate to floating rate
From floating rate to fixed rate

Converting an investment or asset
From fixed rate to floating rate
From floating rate to fixed rate

Taking comparative advantages
Example 1: Converting a liability
Company A has a floating rate liability to pay a LIBOR+0.8% interest rate on a loan, and wishes to convert it into a fixed rate loan.

A can enter into a plain vanilla IRS with a bank to pay a fixed rate of 5.5% in exchange of receiving LIBOR rate.

Thus, in net, A pays a fixed rate of 6%.

Converting a liability
Example 2: Converting an asset
Mr. Li has an investment which pays a fixed rate of 5.2%. Li wishes to receive a market rate.

Li can enter into a plain vanilla IRS with a bank to pay a fixed rate of 5.5% in exchange of receiving LIBOR rate.

Thus, in net, A receives LIBOR-0.3%.

Converting a liability
Example 3. Comparative advantages 
Company A wants to borrow $10 million for 5 years at a
floating rate

Company B wants to borrow $10 million at fixed rate for 5
years

The interest rates offered by banks:
Comparative advantages
The difference between the two fixed rates is:
11.20-10.00 =1.20%

The difference between the two floating rate is
(LIBOR+1.00%)-(LIBOR+0.30%) =0.70%

Though B has always to pay higher rate than A, B could pay a relatively lower rate at floating than at fixed, comparing to the rates A pays.

That is, though A has absolute advantages in two markets, B has comparative advantage in floating.

The comparative advantage is 1.20-0.70=0.50%

A should borrow at fixed and B should borrow at floating, then, they exchange the loans to meet their needs.
Comparative advantages
To take the comparative advantage:

A borrows at the fixed rate, paying: 10%

B borrows at the floating rate, paying:

LIBOR + 1%

A and B enter a swap in which A pays LIBOR to B, and B pays fixed 9.95% to A.

FRM学习资料一：固定收益债券定价理论PDF电子书

固定收益债券定价理论PDF电子书扫描

FRM学习资料三：词汇表词典和英汉证券期货及财务用语汇编

金融风险管理师（FRM）学习资料：词汇表词典、英汉证券期货及财务用语汇编电子书、常用金融词汇列表.doc、金融学英语词典.doc、实用金融词汇.doc

classification process归类过程
clawback (用附加税)填补（福利开支）
client 顾客
clinic 诊所
collateral agent 副代理人
collateral tracking system抵押物跟踪制度
collateral value 抵押物价值
collateralized by third party medical receivables due以第三方到期医疗应收款作
为质押
collateral抵押物
collect and disburse收取和支付
collectibility可收回程度
collection 托收
comfort level 方便程度
commerce clause 商务条款
commercial risk商业风险
commercial terms商业条款
commissions 佣金
commitment 承诺
common carriage通用车队
common law country（英美等）海洋法系国家
common trust 共同信托
commonality通用性
compensate 补偿

competitive risk 竞争风险
competitor 竞争者
complex finance leases 复杂的融资租赁
comprehensive income 综合收入
comptroller 审计官
computer 计算机
conceptual difference概念上的差别
concession period 持有特许权的期间
concession 让步、特许
conclusion 结论
conditional sales agreement附条件销售协议
conditions of usage使用条件
conduit structure管道结构(的公司)
confidentiality保密性
configure改装
conflict冲突
connectivility（信息传递中的）可连通性
consensual or non-consensual lien同意或非经同意的留置权
consent 同意

technological orientation技术上的定位
telecommunications电信
teledensity电信密度（指每百个居民拥有的通信线路数）
tender 提交
terminate 终止
termination penalty提前结束时的罚金
terminology 术语
terrestrial and satellite wireless system 地面及卫星无线系统
the four pillars（支撑租赁交易的）四大支柱
the six phase of leasing cycles 租赁周期的六个阶段
theocratic legal system神权法制
therapeutic equipment 治疗设备
third-party logistics第三方后勤
threshold门坎
time pattern时间模式
time to market（一个新产品从构思到实际推入市场所用的时间）上市时间
titleholder 所有权人
title 所有权
titling trust 产权信托
toll（路桥隧道等的）通行费
total gross investment 毛出资总额
total solution全面解决方案
TRAC（Terminal Rental Adjustment Clause） leases 期末租金调整条款租赁协
议
track 卡车

trade discount 贸易折扣
trade names商品牌号
trade tax add-backs贸易税附加返还
trade tax贸易税
trade-in回购
traditional contract of hire 传统的租借合同
traditional rental 传统出租
training培训
TRALA（美国）载重汽车出租及租赁协会
tranches组别
tranching分组
transaction costs 交易成本
transaction privilege(sale)tax交易特许（销售）税
transaction tax交易税
transactions交易
transferors 出让人
transfer转移
transit district tax地区通过税

transportation 运输
treasury locks 库存锁定
treasury securities 国库券
treatment equipment 治疗设备
trigger events 触发器事件
trip leases 铁路车辆的短期租赁
triple net三方网络
truck 卡车
true leases 真实租赁协议
true lease 真实租赁
trust account 信托账户
Trust Indenture Act信托契约法
trustee 受托人
tunnels隧道
turnaround time 周转时间
type 类型
UCC(Uniform Commercial Code)（美国）统一商法典
ultimate useful life 最终可用寿命
unamortized residual value（经营租赁中）未摊销的余值
uncertainty 不确定性
uncollectible lease payments receivable应收未收租赁付款
undersecured creditor未被全额担保的债权人
understated被少报的
underwater 缩水
underwriting and credit policies保险和信贷政策
underwriting commission 承销佣金

underwriting standard 保险标准
underwriting 保险
unearned finance income未实现财务收益
unguaranteed residual value无担保的残值
Unidroit Conventions on International Financial Leasing 国际统一私法协会国际
融资租赁公约
unit price单价
universal lease documentation 租赁协议通用文本
unrated无信用等级的
unreimbursable 不能回收的
unsecured creditor 无担保的债权人
unsecuritizable不可证券化的
upgrade升级
ups and collars上下限
US federal income taxes 美国联邦所得税
US Internal Revenue Service code 美国国家税务局法规

usage leases使用权租赁（对经营租赁的形容）
use tax 使用税
used equipment leasing用过的设备的租赁
useful life 有用寿命
utility 设施
utilization leases（美国铁路上采用的）轨道使用
valuation估值
value ratio价值比
value-added services增值服务
valuing定值

value at risk（风险价值—金融风险管理新标准）下载

FRM学习资料六：FRM公式表、信用风险课件

金融风险管理师（FRM）学习资料：FRM公式表frm09年quicksheet，扫描的清晰版.pdf、frm公式表，简洁实用.pdf、09frm信用风险PDF电子书、09年Credit Risk.pdf

教育FRM全景班讲义
Credit Risk Analysis
The deviations from the mean11

Measuring credit risk
Credit risk diversification
A portfolio of loans is less risky than single loans
The most important feature of credit risk management is the ability to
diversify across defaults
Diversification12

Measuring credit risk
Question (1)13

Measuring credit risk
Question (2)14

Measuring actuarial default risk15

Credit event
Credit event
A credit event is a discrete state
Either it happens or not
Definition of ISDA
Bankruptcy
Failure to pay
Obligation/cross default
Obligation/cross acceleration
Repudiation/moratorium
Restructuring
Downgrade
Currency inconvertibility
Governmental action
Overview16

Credit event
Question17

Default Rates
Credit ratings
The rating is an “evaluation of
creditworthiness” issued by a
rating agency
represent actuarial probabilities
of default
Moody’s definition
an opinion of the future ability,
legal obligation, and willingness
of a bond issuer or other obligor
to make full and timely
payments on principal and
interest due to investors
Credit ratings (1)18

Default Rates
Accounting ratios
Leverage
Cash flow coverage
MDA
Z-score model
Working capital over total assets
Retained earnings over total assets
EBIT over total assets
Market value of equity over total
liabilities
Net sales over total assets
Credit ratings (2)
0.4 0.9 114 CCC
1.2 1.9 76 B
2.5 3.5 54 BB
4.7 6.5 43 BBB
8.0 10.2 38 A
19.5 24.6 28 AA
23.8 25.5 12 AAA
EBIT/I EBITDA/I D/C Rating
Cash flow coverage leverage19

Credit ratings
Question (1)20

Credit ratings
Question (2)21

Default Rates
How to understand the historical default rate ?
The proportion of firms that default, which is a statistical estimate of the
true default probability
Historical default rate
Higher ratings are associated with lower default rates
For an initial credit rating, credit risk increases sharply with the horizon
For investment-grade credits, the increase is more than proportional with
the horizon
For speculative-grade credits, the increase is less than proportional with
the horizon
Low sample size
In non-U.S. markets
When the true p is changing over time
Historical default rate

Cumulative and Marginal default rates
Sequential default process
We define
is the number of issuers rated R at the end of year that
default in T= t + N
is the number of issuers rated R at the end of year that
have not defaulted by the beginning of year t + N
Default process (1)
[|()] nt N Rt +
[ | ()] mt N Rt +23

Cumulative and Marginal default rates
Five important rates
Marginal Default Rate during Year T
Survival Rate
Marginal Default Rate from Start to Year T
Cumulative Default Rate
Average Default Rate

Why use the market prices
Infer credit risk from corporate bond prices
Infer credit risk from equity prices
conclusion
Contents34

Why use the market prices35

Why use market prices
Credit risk ratings
External ratings focus on forecasting credit losses from historical default
rates and recovery rates
Market prices method
Credit risk can be measured by market price of securities whose value are
affected by default
These securities include corporate bond, equity, and other derivatives
Market price method can provide more up-to-date and accurate measures
of credit risk, because financial markets have access to a large amount of
information
External rating and market price36

Infer credit risk from bond prices38
Spreads and credit risk
Default bond
Consider a bond which makes only one payment of $100 in one period, its
market price is , we can get the market-determined yield
This bond also can be describe as a simple default process
Using risk neutral pricing, we get
Default bond pricing

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

Preface ix
About the Author xi
About GARP xiii
Introduction xv
PART ONE
Quantitative Analysis
CHAPTER 1
Bond Fundamentals 3
CHAPTER 2
Fundamentals of Probability 31
CHAPTER 3
Fundamentals of Statistics 67
CHAPTER 4
Monte Carlo Methods 89
PART TWO
Capital Markets
CHAPTER 5
Introduction to Derivatives 111
CHAPTER 6
Options 127
CHAPTER 7
Fixed-Income Securities 161
CHAPTER 8
Fixed-Income Derivatives 195
CHAPTER 9
Equity, Currency, and Commodity Markets 217
PART THREE
Market Risk Management
CHAPTER 10
Introduction to Market Risk 247
CHAPTER 11
Sources of Market Risk 273
CHAPTER 12
Hedging Linear Risk 297
CHAPTER 13
Nonlinear Risk: Options 315
CHAPTER 14
Modeling Risk Factors 341
CHAPTER 15
VAR Methods 359
PART FOUR
Investment Risk Management
CHAPTER 16
Portfolio Management 383
CHAPTER 17
Hedge Fund Risk Management 401
PART FIVE
Credit Risk Management
CHAPTER 18
Introduction to Credit Risk 431
CHAPTER 19
Measuring Actuarial Default Risk 451
CHAPTER 20
Measuring Default Risk from Market Prices 479
CHAPTER 21
Credit Exposure 499
CHAPTER 22
Credit Derivatives and Structured Products 531
CHAPTER 23
Managing Credit Risk 561
PART SIX
Legal, Operational, and Integrated Risk Management
CHAPTER 24
Operational Risk 587
CHAPTER 25
Liquidity Risk 607
CHAPTER 26
Firm-Wide Risk Management 623
CHAPTER 27
Legal Issues 643
PART SEVEN
Regulation and Compliance
CHAPTER 28
Regulation of Financial Institutions 657
CHAPTER 29
The Basel Accord 667
CHAPTER 30
The Basel Market Risk Charge 699
About the CD-ROM 715
Index 717

Preface
T
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.
Philippe Jorion
February 2009

KEY CONCEPT
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
c03 JWBT102-Jorian April 1, 2009 10:5 Printer Name: Courier Westford
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
time rule.
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.
a. 6.80%
b. 5.83%
c. 4.85%
d. 4.71%P1: ABC/ABC P2: c/d QC: e/f T1: g
c03 JWBT102-Jorian April 1, 2009 10:5 Printer Name: Courier Westford
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
VAR.
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
original VAR.
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
optionissaidtobe path-dependent.
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.
4.2.3 Accuracy
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
c04 JWBT102-Jorian April 1, 2009 10:6 Printer Name: Courier Westford
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
precisely.
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

FRM学习资料八：2010 FRM Examination Part I AIM Statements

金融风险管理师（FRM）学习资料：2010 FRM Examination Part I AIM Statements PDF电子书

5th HANDBOOK 中FRM一级需要看的内容

PART ONE Quantitative Analysis
Chapter1—4（全部）

PART TWO Capital Markets
Chapter5—9（6.4 奇异期权除外 7.6证券化除外）

PART THREE Market Risk Management
Chapter 10、12、13、14、15

PART FOUR Investment Risk Management
Chapter 16 中的 16.1和16.2

PART FIVE Credit Risk Management
Chapter 19 中的 19.1 19.2.1 19.2.4 19.4.2 19.4.3

2010 FRM Examination Part I AIM Statements

AIMS – Candidates, after completing this reading, should be able to:
Describe the responsibility of each GARP member with respect to professional integrity,
ethical conduct, conflicts of interest, confidentiality of information and adherence to
generally accepted practices in risk management.
Describe the potential consequences of violating the GARP Code of Conduct.

         
AIM Statements, 2010 FRM Part I Page 9 of 39
2010 by Global Association of Risk Professionals, Inc.

Quantitative Analysis
Part I Exam Weight: 20%
Probability distributions
Mean, standard deviation, correlation, skewness, and kurtosis
Estimating parameters of distributions
Linear regression
Statistical inference and hypothesis testing
Estimating correlation and volatility: EWMA and GARCH Models
Maximum likelihood methods
Volatility term structures
Simulation methods
Readings for Quantitative Analysis
8. Damodar Gujarati, Essentials of Econometrics, 3rd
Edition (New York: McGraw〩ill,
2006).
Chapter 1 – The Nature and Scope of Econometrics
Chapter 2 – Review of Statistics: Probability and Probability Distributions
Chapter 3 – Characteristics of Probability Distributions
Chapter 4 – Some Important Probability Distributions
Chapter 5 – Statistical Inference: Estimation and Hypothesis Testing
Chapter 6 – Basic Ideas of Linear Regression: The Two-Variable Model
Chapter 7 – The Two-Variable Model: Hypothesis Testing
Chapter 8 – Multiple Regression: Estimation and Hypothesis Testing

9. Jorion, Value゛t㏑isk, 3rd Edition
Chapter 12- Monte Carlo Methods
10. John Hull, Options, Futures, and Other Derivatives, 7
th
Edition (New York: Pearson,
2009).
Chapter 21 – Estimating Volatilities and Correlations
11. Svetlozar Rachev, Christian Menn, and Frank Fabozzi, Fat㏕ailed and Skewed Asset
Return Distributions: Implications for Risk Management, Portfolio Selection and Option
Pricing (Hoboken, NJ: Wiley, 2005).
Chapter 2 – Discrete Probability Distributions
Chapter 3 – Continuous Probability Distributions
12. Linda Allen, Jacob Boudoukh and Anthony Saunders, Understanding Market, Credit and
Operational Risk: The Value at Risk Approach (Oxford: Blackwell Publishing, 2004).
Chapter 2 – Quantifying Volatility in VaR Models
AIM Statements, 2010 FRM Part I Page 10 of 39
2010 by Global Association of Risk Professionals, Inc.

Readings for Quantitative Analysis

FRM学习资料九：handbook（第五版）纠错汇总，FRM主要知识点梳理

金融风险管理师（FRM）学习资料：FRM handbook fifth edition错误汇总、FRM全景班讲义主要知识点（Handbook）梳理PDF电子书

资料9：handbook纠错，知识点梳理

FRM handbook fifth edition错误汇总、FRM全景班讲义主要知识点（Handbook）梳理

1． P120 Example 5.1
答案中原为F=1000*exp(0.03*1/12)/exp(-0.06*1/12) 将12均改为4，答案为1022.8
2． P132 Example 6.4
答案中-=10-15+90exp(0.05*5)=65.09的0.05改为-0.05
3. P140 Example 6.11
答案为50-42.379=7.621
4. P197 Example 8.2
答案中V=1000000*（3.75%-3.50%）*（2-1）*exp(-3.50%*2)=2331改为V=1000000*（3.75%-3.75%）*（2-1）*exp(-3.50%*1)=2331

Management: Insurance, Self-Insurance, Derivatives
Technical Risk & Model Risk
Technical Risk
Model Risk
Integrated Risk Management and ERM
Basel II
Three pillars of Basel II
Types of institutions that the Basel II Accord will be applied to
Describe the major risk categories covered by the Basel II Accord
Major Approaches to calculating credit risk, market risk and
operational risk
Define in the context of Basel II
Module IV: Operational and Integrated Risk25
Performance Analysis
CAPM, CML and SML
Market efficiency, equilibrium
Sharpe ratio and information ratio
Tracking error
Factor models and Arbitrage Pricing Theory Portfolio construction
Portfolio Risk
Risk Budgeting
Setting risk limits
Hedge Fund Risk Management
Risk-return metrics specific to hedge funds
Risks of specific strategies
Asset illiquidity, valuation, and risk measurement
The use of leverage and derivatives and the risks they create
Measuring exposures to risk factors and Pension fund risk management
Module V: Investment and Portfolio Risk
26
How to Use the 2009 AIM Statements
27
AIMS: Applying Instructional Materials Statements, are designed to serve
as an additional study resource only and will not in and of themselves fully
prepare a candidate for the FRM examination. They should be used as
guidance and support for the readings outlined in the Study Guide to help
identify key learning objectives for each core reading.
Study guide: The FRM Study Guide sets forth primary topics and subtopics
under the risk-related disciplines covered in the FRM exam. The topics were
selected by the FRM Committee as topics that risk managers who work in
practice today have to master. The topics are reviewed yearly to ensure the
FRM exam is kept timely and relevant.
AIMS is a explanation of the Topics and Readings in Study Guide, so
AIMS is more specific material for preparation of FRM exam.
Take the first reading of Foundations of Risk Management as an example.
Philippe Jorion, Value-at-Risk: The New Benchmark for Managing
Financial Risk, 3rd Edition (New York: McGraw〩ill, 2007). Chapter
1 The Need for Risk Management.
What is AIMS?
28
AIMS: After completing this reading, candidates should be able to:
Define risk and describe some of the major sources of risk
Differentiate between business and financial risks and give examples of
each
Relate significant market events of the past several decades to the
growth of the risk management industry
Describe the functions and purposes of financial institution as they relate
to financial risk management
Define what a derivative contract is and how it differs from a security
Describe the dual role leverage plays in derivatives and why it is
relevant to a risk manager
Define financial risk management
Define VaR and describe how it is used in risk management
–
How to use the AIMS: an example29
Define risk and describe some of the major sources of risk
Risk: the volatility of unexpected outcomes, which can represent the
value of assets, equity, or earnings, including business and financial risk.
majorsourcesofrisk
Human-created
Unforeseen natural phenomena
Long-term economic growth
technological innovations
Risk and the willingness to take risk are essential to the growth of our
economy
Accumulation of assets or savings—a cushion against income risk;
Personal loan—smoothing of consumption through borrowing;
Insurance—protect against accidents and other disasters;
Modern publicly held corp. —spread the risk of ownership in a
company
Welfare state create the “safety nets”—a risk-sharing institution
How to use the AIMS: an example
30
Differentiate between business and financial risks and give examples of
each
Business risk: relates to business decisions and business environment
Financial risk: relates to possible losses owing to financial market
activities
Relate significant market events of the past several decades to the growth of
the risk management industry
The recent growth of the risk management industry can be traced
directly to the increased volatility of financial market since the early
1970s.
Describe the functions and purposes of financial institution as they relate to
financial risk management
Function: to manage financial risk actively
Purpose: to assume, intermediate, or advise on financial risks.
Financial institutions must measure financial risk as precisely as
possible in order to control and price them properly.
How to use the AIMS: an example

Define what a derivative contract is and how it differs from a security
A derivative contract can be defined generally as a private contract
deriving its value from some underlying asset price, reference rate, or
index, such as a stock, bond, currency, or commodity.
Difference: securities such as bond and stock are issued to raise capital,
derivatives are contracts or private agreements between two parties.
Describe the dual role leverage plays in derivatives and why it is relevant to
a risk manager
Leverage: no (full-amount) upfront cash flow, involves borrowing, it is no
more risky than dealing the same notional amount in the underlying cash
market.
A double-edged sword
It makes derivatives an efficient instrument for hedging and
speculation owing to very low transaction costs.
It is more difficult to assess the potential downside risk.
How to use the AIMS: an example
32
Define financial risk management
A Financial risk management refers to the design and implementation of
procedures for identifying, measuring, and managing financial risks.
Define VaR and describe how it is used in risk management
Comparison of Risk Limits
VaR is a statistical risk measure of potential losses, combines the price-
yield relationship with the probability of an adverse market movement.
VaR summarizes the worst loss over a target horizon that will not be
exceeded with given level of confidence.
How to use the AIMS: an example33
How to calculate VaR
Definition
VaR is the maximum loss over a target horizon such that there is a
low, pre-specified probability that the actual loss will be larger.
直观定义：VaR是在一定的置信水平下和一定的目标期间内，预期的
最大可能损失。
Example
1.假定JP摩根公司在2004年置信水平为95%的日VaR值为1500万
美元，其含义指该公司可以以95%的把握保证，2004年某一特
定时点上的金融资产在未来24小时内，由于市场价格变动带来
的损失不会超过1500万美元。或者说，只有5%的可能损失超过
1500万美元。
2. 用下例得到VaR的步骤的思路。
How to use the AIMS: an example
34
How to use the AIMS: an example
35
How to calculate VaR
We simulate the 1-month return on $100 million worth of medium-term
notes investment from history data.
We can get monthly returns on 5-year US Treasury notes since 1953.
The sample size is 624 months.

FRM学习资料十：FRM数量部分备考必备－计量经济学精要

金融风险管理师（FRM）学习资料：FRM数量部分备考必备－计量经济学精要PDF电子书

FRM学习资料十：FRM数量部分备考必备－计量经济学精要

金融风险管理师（FRM）学习资料：FRM数量部分备考必备－计量经济学精要PDF电子书，343页！

在经济学、金融学、管理学、营销学以及一些相关学科的研究中，定量分析用得越来越多，
对于这些领域的初学者来说，掌握一至两门经济计量方面的课程是必要的—这个领域的研究
变得十分流行。本章的目的旨在给初学者一个经济计量学的概貌。
1 什么是经济计量学
简单地说，经济计量学(E c o n o m e t r i c s)就是经济的计量。虽然，对诸如国民生产总值( G N P)、
失业、通货膨胀、进口、出口等经济概念的定量分析十分重要，但从下面的定义中，我们不难
看出经济计量学的研究范围更为宽泛：
经济计量学是利用经济理论、数学、统计推断等工具对经济现象进行分析的一门社会科学。1
经济计量学运用数理统计知识分析经济数据，对构建于数理经济学基础之上的数学模型提
供经验支持，并得出数量结果。2
2 为什么要学习经济计量学
从上述定义我们知道经济计量学涉及经济理论、数理经济学、经济统计学(即经济数据)，
以及数理统计学等相关学科，但它是一门有其自己研究方向的一门独立学科。
从本质上说，经济理论所提出的命题和假说，多以定性描述为主。例如，微观经济理论中
提到的：在其他条件不变的情况下(经济学中著名的 Ceteris paribus从句)，一种商品价格的上升
会引起该商品需求量的减少。因而得出结论：商品的价格与该商品的需求量呈反方向变动—
这就是著名的向下倾斜的需求曲线，简称需求法则。但是，该理论本身却无法度量价格和需求
量这两个变量之间的数量关系，也就是说，它不能告诉我们商品的价格发生某一变动时，该商
品的需求量增加或减少了多少。经济计量学家的任务就是提供这样的数量估计。换一种说法，
经济计量学是依据观测和试验，对大多数经济理论给出经验的解释。如果在研究或试验中发现，
当每单位商品的价格上升一美元，引起该商品需求量的下降，比如说下降 1 0 0个单位，那么，
我们不仅验证了需求法则，而且还提供了价格和需求量这两个变量之间的数量估计。

数理经济学(mathematical economics)主要关心的是用数学公式或数学模型来描述经济理
论，而不考虑对经济理论的度量和经验解释。而经济计量学家感兴趣的却是对经济理论的经验
确认。下面我们将会讲到，经济计量学家通常采用数理经济学家提供的数学模型，但把它们用
于经验检验。经济统计学家主要关心的是收集、处理经济数据并将这些数据绘制成图表的形式。
这是经济统计学家的工作：他或她收集 G N P、失业、就业、价格等数据，这些数据就成为经济
计量分析的原始数据。但经济统计学家却不关心用这些收集到的数据来检验经济理论。
虽然，数理统计学提供了许多分析工具，但由于经济数据独特的性质，即许多数据的生成
并非可控制试验的结果，因此，经济计量学经常需要使用特殊的方法。类似于气象学，经济计
量学所依据的数据不能直接控制。所以，由公共和私人机构收集的消费、收入、投资、储蓄、
价格等方面的数据从本质上说是非试验性的。这就产生了数理统计学不能正常解决的一些特殊
问题。而且，这些数据很可能包含了测量的误差，或是遗漏数据或是丢失数据。这就要求经济
计量学家去运用特殊的方法来处理这些测量误差。
对于主修经济学和商业专业的学生来说，学习经济计量学有实用性。毕业以后，在其工作
中，或许被要求去预测销售量、利息率、货币供给量或是估计商品的需求函数、供给函数以及
价格弹性等等。在经济学家以专家的身份出现在联邦政府调节机构中之前，通常代表当事人或
公众。而汽油和电的价格是由政府调节机构规定的，因此，这就要求经济学家能估计提议的价
格的上涨对需求量(如用电量)的冲击。在这种情况下，经济学家需要建立一个关于用电量的需
求函数，并根据这个需求函数估计需求的价格弹性，即，价格变动的百分比所引起需求量改变
的百分比。掌握经济计量学知识对于估计这些需求函数是很有帮助的。
客观地说，在经济学和商科专业的学习与培训中，经济计量学已成为不可或缺的一部分。
1.3 经济计量学的方法论
一般说来，用经济计量方法研究经济问题可分为如下步骤：
(1) 理论或假说的陈述；
(2) 收集数据；
(3) 建立数学模型；
(4) 建立统计或经济计量模型；
(5) 经济计量模型参数的估计；
(6) 检查模型的准确性：模型的假设检验；
(7) 检验来自模型的假说；
(8) 运用模型进行预测。
为了阐明经济计量学的方法论，我们来考虑这样一个问题：经济形势会影响人们进入劳动
力市场的决定吗？也就是说，经济形势是否对人们的工作意愿有影响？假设用失业率
(Unemployment Rate, UNR)来度量经济形势，用劳动力参与率( Labor Forle Participation Rate,
L F P R)来度量劳动力的参与，U N R和LFPR 的数据由政府按时公布，那么，如何回答这个问题
呢？我们按上述步骤进行分析。
1.3.1 理论或假说的陈述
首先要了解经济理论对这一问题的阐述是怎样的。在劳动经济学中，关于经济形势对人们
工作意愿的影响有两个相对立的假说。一个是受挫-工人假说[discouraged-worker hypothesis
( e ff e c t ) ]，该假说提出当经济形势恶化时，表现为较高的失业率，许多失业工人放弃寻找工作

的愿望并退出劳动市场。另一个是增加-工人假说[added-worker hypothesis (eff e c t ) ]，该假说认
为当经济形势恶化时，许多目前并未进入劳动市场的二手工人(比如带孩子的母亲)可能会由于
养家的人失去工作而决定进入劳动市场，即使这些工作的报酬很低，只要可以弥补由于养家人
失去工作而造成的收入方面的一些损失就行。
劳动力参与率的增加或减少依赖于增加工人和受挫工人的力量对比。如果增加工人的影响
占主导地位，则L F P R将升高，即使是在失业率很高的情况下。相反地，如果是受挫工人的影
响占主导力量，那么L F P R将会下降。我们是如何发现这一结果的呢？这只是一个实践问题。
1.3.2 收集数据
由于实验的目的，我们需要这两个变量的数量信息。一般来说，有三种统计数据可用于实
践分析：
(1) 时间序列数据
(2) 横截面数据
(3) 合并数据(时间序列数据与横截面数据的联合)
1. 时间序列数据
这种数据是按时间序列排列收集得到的。比如G N P、失业、就业、货币供给、政府赤字等。
数据是按照一定的时间间隔收集的 —每日(比如股票)，每周(比如货币供给)，每月(比如失业
率)，每季度(比如G N P)，每年(比如政府预算)。这些数据可能是定量的( q u a n t i t a t i v e )(比如价格、
收入、货币供给等)，也可能是定性的( q u a l i t a t i v e )，(比如男或女，失业或就业，已婚或未婚，
白人或黑人等)。我们将会发现，定性的变量(又称为虚拟变量)与定量的变量同样重要。
2. 横截面数据
横截面数据(cross-sectional data)是指一个或多个变量在某一时点上的数据的集合。例如美国
人口调查局每1 0年进行的人口普查数据(最近的一次是在1 9 9 0年4月1日)，以及密执安大学进行的
夏季居民开支调查数据。这些民意调查的结果由G A l l u p、Harris 和其他的一些调查机构处理。
3. 合并数据
合并数据(pooled data)中既有时间序列数据又有横截面数据。例如，如果我们收集 2 0年间
1 0个国家有关失业率方面的数据，那么，这个数据集合就是一个合并数据，每个国家的 2 0年间
的失业率数据是时间序列数据，而2 0个不同国家每年的失业率数据又组成横截面数据。
在合并数据中有一类特殊的数据，称为 p a n e l数据(panel data)，又称纵向数据( o n g i t u d i n a l
or micropanel data)。即同一个横截面单位，比如说，一个家庭或一个公司，在不同时期的调查
数据。例如，美国商业局在一定时期间隔内对住房的调查。在每一时期的调查中，同样的(或居
住在同一地区的)家庭被调查，以观察自上一次调查以来，其住房和经济状况是否有变化。纵向
数据就是通过重复上述过程而得到的，它可对研究家庭行为的动态化提供非常有用的信息。
4. 数据来源
成功的经济计量研究需要大量高质量的数据。幸运的是国际互联网为我们提供了大量详实
的数据。附录1 A列出了一些网址，提供了各类微观和宏观的经济数据。学生必须熟悉这些网
站并学会下载数据。当然，这些数据会不断更新，因此可得到最新的数据。
为了便于分析，这里给出一组时间序列数据。表 1 – 1给出了美国1 9 8 0～1 9 9 6年间城市劳动
力参与率(Civilian Labor Force Participation Rate, CLFPR)和城市失业率(Civilian Unemployment
Rate, CUNR)数据。城市失业率是指城市失业人口占城市劳动力的百分比。 1
与物理学不同，许多收集的经济数据(比如 G N P、货币供给、道-琼斯指数、汽车销售量等)

1.3.7 检验来自模型的假设
模型最终确定之后，我们进行假设检验(hypothesis testing)。即验证估计的模型是否有经
济含义，以及用模型估计的结果是否与经济理论相符。例如，受挫工人假说假设劳动力参与与
失业率之间负相关。这个假说与结果相符吗？我们统计的结果与假说相一致，因为估计得到的
城市失业率系数为负。
然而，假设检验或许更复杂。在这个例子中，假设得知在先前的研究中，城市失业率的系
数约为-1，那么得到的结果还会与假设一致吗？如果以式 ( 1 – 3 )这个模型为基础，我们可能得到
一个结果，但是如果以式( 1 – 5 )模型为基础，则可能得到另一个结果。怎样解决这个问题呢？我
们会在适当的章节中利用一些必要的工具来解决诸如此类的问题，但是需要提醒注意的是：根
据某一特定的假说所得到的结果将依赖于最终所选择的模型。
还有一点，在回归分析中，我们不仅对模型参数的估计感兴趣，而且对检验来自于某个经
济理论(或先验经验)的假设感兴趣。