financial risk manager handbook with test bank 6th edition

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Sixth Edition

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Copyright 2011 by Philippe Jorion, ex.cepr for FR.M sample quesr:ionsi which are copyrighr l 997-2011 by GARP. The FR.M designation is a GARP trade1nark. All rights reserved.

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For general information on our other produces and services or for ten Data:

Jorion, Philippe. Financial risk nianager handbook plus tesr bank : FR?\1 Part I/Parr JI, 6th Edition I Philippe ]orion. - 6rh ed.

p. cn1. Includes index. ISBN 978-0-470-90401-5 (p1per/online)

1. Financial risk 1nanagen1enr. 2. Risk 111anagen1enr. 3. Corpor.arions-Finance. 1. Tide. HD61.J67 2009 6 "8 1 -i --d ")') .'h . .:> .) c __

Printed in the United States of An1erica

10 9 8 7 6 5 4 3 2 l

2010047263

Preface iX

About the Author XI

AboutGARP Xiii

Introduction xv

PART ONE ~lmr.-~~t:~~~~~~~~'l'i:~z~)>i~~~M&~~~J:W~~m-.m~

QuantitatiV8 Analysis 25 CHAPIER2

Fundamentals ot ProbabDity 27

CHAPTBl8 Fundamentals o1 Statisttcs 81

CHAPTBl4 Monte Cirlo Meth4)ds. 83

CHAPTER 5 Modeling Risk Factors 103

PART THREE ~~~~~~tmC~~f~.::.~~r.s~~~~tt'V~~~~~~~~~~trn;:~u

Financial Markets and Products 125 .

CHAPTER 8. Bond Fundamentals 127

CHAPTBl7 Introduction to. Derivatives 157

CHAPTERS OpUon Markets 177

v

vi CONTENTS

CHAPTBIB Axed-Income Securities 207

CHAPTER 10 Axed-Income Derivatives 231

CHAPTBI 11 Equity, Currency, and Commodity Markets 255

PART FOUR Valuation and Risk Models 281

CHAPTBl12 Introduction to Risk Models 283

CHAPTBI 13 Managing unear Risk 811

CHAPTER 14 NonDnear (Option) Risk Models 331

PARTRVE MarkQt Risk Management 355

CHAPTBI 15 Advanced Risk Models: Univariate 357

CHAPTBI 18 Advanced Risk Models: Multivariate 375

CHAPTBI 17 Managing Volatllity Risk 405

CHAPTfl18 Mortgage-Backed Securities Risk 427

PARTSIX Credit Risk Management

CHAPTBI 19 tntrodUctian to Credit Risk

CHAPTBl20 Measuring Actuarial Delault Risk

CHAPTBl21 Measuring Default Risk from Market Prices

CHAPTBl22 Credit Exposure

449

451

471

501

523

Contents

CHAPTER23 Creart Derivatives and Structured Products

CHAPTBl24 Managing Credit Risk

PART SEVEN Operational and lnteared Risk Management

CHAPIBl25 : .

Operational .Rlik

CHAPTBl28 liquidity Rlslc

CHAPTBl27 Rrmwtde Risk l\llanlueinent

CHAPJBl28 The.Basel AccoPd

PARTBGHT lnvesbnent Risk Management

CHAPTBl29 Portfolio Risk Management

CHAPTER SO Hedge Fund Risk Management

Index

' .

VII

555

587

811

813

839

857 .

885

725

727

749

779

'

The .Fina11cial Risk M:c:inager Handbook Plus Test Bank provides the core body of knowledge for financial r.isk n1anagers . . Risk management has rapidly evolve.cl over the past decade and has beco1ne an indispensable function in ma11y institutions.

This H4ndbook was or~ginally \vritten to provide support for candid.ates tak-ing the F.RM exa,mination. administere.d by GARP. ,,:\s such, it reviews a wide variety 9 ,practical topics i.11 : a consistent and systematic fasl1ion. It covers q.uan.; titative methods, niajor financial products, as well as n1arket, credit, operational, and integrated risk management. it also discusses investine11t risk management issues essential for risk professionals.

This editio11 l1as been thorc>t1ghly updated to reflect recent develop.ments in financial markets a.nd changes in. th.e structt1re of tl1e FRM. program .. The book is now structur.ed to correspond to the t\vo .l.evels of the FRM exams. All of the chapters have been updated to accou.nt for recent developments in financial n1arkets. and regulations. In particular, current issues are integr~ted iri the second part of the book. New chap.ters have been added, including chapters that deal with advanced univariate and multivariate n1t)dels, as welJ as advanced option. modefs. Finally, this Ha1ulbook incorprates rl1e latest questions from the FRM examinations.

Modern risk man.agemerit syste.ms cut across the entire organization. This breadt11 is reflected in the subjects covered in this Hatiibook. Th.e Handbook was designed t.o be self-contained, b'ut only f

Philippe Jorion is a Professor of Finance. at the Paul Merage Schc>ol of Business at the University of California at Irvine. He l1as also bee11 a professor at Col.u1nbia University, Northwestern University; the University of Chicago, and the University of British Columbia. In addition, he taught the risk managen1ent class in the Master .of Financial E11gineering programs at the Unjversity of California at Berl

Founded in 1996, the Global Association o.f Risk Professionals (GARI') is the leading 11ot-for-profit ass CERTIFICATION The. bencl1mark FR1\t1 desig11ation is the globally accepted risk n1anagemen.t certifi-cation for fi11ancial risk profe.ssio11als. The FRM. objectively measures competency in th.e. risk ma11age1nenr profession based on. globally accepted standards. \Xlith a compound annual growth rate of 25(X> over the past seven years, tl1e FRM program has experie11ced significant growth in every financial center around the world. No'v 16,000+ individuals 11c>ld the FRM des.ignatiil in ()Ver 90 C()ttn-tries. In ad.ditiori, organizations with five c>r more FRM registrants grew fro1n 105 in 2003 to 424 in 2008, further demo11strati.ng the FR.l\.1 prgy

Xiii

xiv ABOUT GARP

(IT), legal, compliance, and sales, acknov~'1edging that everyone in the orga11ization is a risk manager!

Energy Risk Professional Program The Energy Risk Professional (ERP@) ) program is designed to measure a candi-date's knowledge of tl1e inajor energy markets and gauge their ability to manage the physical a11d financial risks ir1herent i11 the complex world of energy. This progra1n is valuable for anyone working .in or servicin.g the energy field, requiring an understanding of the physical and financial markets, how they interrelate, and the risks involved.

GARP DIGITAL LIBRARY As the world's largest digital library dedicated to financial risk management, the GARP Digital Library (GDL) .is the hub for ris.k .management education and research nlaterial. The library's tinique iReadings TM allow users to download in-dividual chapters of books, saving both time and money. There are over 1,000 readings available from 1.2 different publishers. The GDL collection offers read-ings to meet the needs of anyone interested in risk management.

For more information, please visit W\V\v.garpdigitallibrary.org.

GARP EVENTS AND NETWORKING GARP l1osts major conventions throt1ghout the world, where risk professionals come together to share knowledge, 11etwork, and learn from leading experts in the field. Conventions are bookended \Vith interactive workshops that provide practical insights and case studies presented by the industry's leading practitioners.

GARP .regional chapters provide an opportunity for financial risk profession-als to net\vork a11d share new trends and discoveries in risk management. Eacl1 of our 52 chapters holds several meetings each year, in some locations more of-. ten, foct1sing on. issues of importance to the risk n1a11agement community, either globally or locally.

\

G~RP's formal mission is to be the le~ding profession.al as~ociation for financial risk managers, managed by and for its 1ne1nbers and dedicated to the advance-ment of tl1e risk professiri thrc>ugh education, training, and the promotio.n of best practices.globally. A/$ a part of delivering C)n that n1ission, GAR.P has again teamed with Philippe Jorion .to prodt1ce the sixth edition o.f the Financial Risk Manager Handbook PltlS Test Bank.

The Handbook follows GARP's FR.~1 Com1nittee's publis.hed F.R.M. Study Guide, whicl1 sets forth primary topics and subtopics cove.red in the FRM exam. The topics are selected by tl1e FRM. Con1n1ittee as being representative of tl1e theories and con~epts utilized by risk .n1anagement prc>fessionaJs .as they address current 1ssu.es.

Over the years th~ Study Guide has taken. on a11 importan.ce far exceeding its initial intent of providing guidanc

xvi INTRODUCTION

Developing credibility and global acceptance for a professional certification program is a le11gthy and complicated process. Whe11 GARP first administered its FR.N1 exa1n. in 1997, the concept of a professiona] risk n1anager and a global certification relati11g to that persc>n's skill set \Vas more theory tl1an reality. That has no\v cornpletely changed, as the number of current FRM holders exceeds 16,000.

The FRM is now the benchmark for a financial risk manager anywhere arou11d tl1e world. Professinal risk managers having.earned the FRM credential are glob-ally recognized as having achieved a level of professional competency and a demon-strated ability to dy11amically measure and manage financial risk in a real-world setting in accordance with global standards.

Gi\RP is proud to continue to make this Ha,,idboc>k available to financial risk professionals around the world. Philippe Jorin, a preeminent risk professional, has again con1piled an exceptional reference book. Supplemented by an interactive Test Bank, this Handbook is a requirement for any risk professiona1's library.

The Test Bank is a preparatory revie\v for anyone studying for tl1e FRM exam and for risk professionals interested in self-study to review and i1nprove their knowledge of market, credit, and operational risk management. The Test Bank co11tains hundreds of multiple-choice questions fron1 tl1e 2007, 2008, and 2009 FRN1 exams, witl1 a11swers and solutions provided. Tl1e Test Bank can be do,vnloaded fc>llowi11g the instructions on the FRM Test Bank Do\vnload page at the e11d of tl1is book.

Global 1\ssociation of Risk Professionals October 2010

ne

Foundations ol Risk Management

Risk Management

... Financial risk management is the pr.ocess by whicil financial r~sks are identified, assessed, measured, and managed in order to cr~ate ecQnomtc value. Some risks ca.n be measured reasonably well. For those, risk can be quantified

using statistical tools to generate a probability .distribution of profits and losses. Other risks are .not anJenable to. for.i11al measurement but are nonetheless impor-tant~ The function of the risk manager is to evaluate financial risks using both quantitative tools and judgment.

As .financial markets have expanded over recent decades, the risk management function has become more important. "Risk can never be entirely avoided. Mo.re generally, the goal is not to minimize risk; it is to takesmart risks.

Risk that can be measured can be managed better .. Investors assume risk only because they expect to be compensated for it in the form of higher returns. To decide how to balance risk against return, however, .requires risk .measurement.

Centralized risk management tools such as value at risk (VAR) were

4 FOUNDATIONS OF RISK MANAGEMENT

1.1 RISK MfASUREMENT

1.1.1 Example The first step in risk managen1ent is the r11easurement of risk. To illustrate, consider a portfolio \Vith $100 million invested in U.S. equities. Presun1ably, the investor undertook the position becat1se of an expectation for profit, or invest1nent gro\vth. This portfolio is also risky, 11owever.

The key issue is whether the expected profit for this portfolio \varrants the assumed risk. Thus a trade-off is ir1volved, as in most economic problen1s. To help answer this question, the risk manager should construct the distribution of potential profits and losses on this investment. This shows how much the portfolio can lose, thus enabling the investor to make investment decisions.

Define li P as the profit or loss for the portfolio over a fixed horizon, say the coming month. This must be measu.red in a risk currency, such as the dollar. This is also the product of the initial investment value P and the future rate of return Rp. The latter is a random variable, \Vhich should be described using its probability density function. Using historical data over a Jong period, for example, the risk manager produces Figure. 1.1.

Tl1is graph is based on the actual distribution of total returns on the S&P 500 index since 1925. The line is a s.moothed histogram and does not assume a simplified model sucl1 as the norm.al distribution.

The vertical axis represents the frequency, or probability, of a gain or loss of a size indicated on the horizontal axis. The entire area under the curve covers all of the possible realizations, so should add up to a total probability of 1.

Most of the weight is i11 the center of the distribution. This shows that it is most likely that the return will be s1nall, wl1ether positive or negative. Tl1e tails have less weight, indicating that large retur11s are less Jikely. This is. a typical characteristic of retu.r11s on fina.ncial assets. So far, this pattern resembles the bell-shaped curve for a normal distribution.

Probability ---- --- .. -;-..-------- ---

' :

. ' '

,

' , ' .. -------

-30 - 20 -10

j j Mean= 1o/o

Standard -+---.-.\deviation= 5.5

On the downside, however, there is a substantial probability of losing 1 Q


To compare these two approaches, take the case of an active equity portfolio inanager who is given the task of beating a benchmark. In the first year, the active portfolio returns -6% but the benchmark drc>ps by -10o/o. So, the excess return is psitive: e = -6% - (-10o/o) = 4%. In relative terms, the portfolio has done welJ even though the absolute performance is negative. 1n the second year, the portfolio returns +6%, which is good using absolute measures, but not so good if the benchmark goes up by +10%.

EXAMPLE 1.1: ABSOLUTE AND RELATIVE RISK

An investment manager is given tl1e task of beating a benchmark. Hence the risk should be measured in terms of

a. Loss relative to the initial investment b. Loss relative to the expected portfolio value c. Loss relative to the benchmark d. Loss attributed to the benchmark

1.2 EVALUATION OF THE RISK MEASUREMENT PROCESS

A major function of the risk measurement process is to estimate the distribntio.n of future profits and losses. The first part of tl1is assignment is easy. The scale of the dollar returns should be proportional to tl1e initial investment. In other words, given the distribution in Figure 1.1, an investment of $100 million should have a standard deviation of u(A P) = $100 x 5.So/o = $5.5 million. Scaling the current position by a factor of 2 should increase this risk to $11 .million.

The second part of the assignment, which consists of constructing the distri-bt1tion of future rates of return, is much harder. In Figure 1.1, we have taken the historical distribution and assumed that this provides a good representation of future risks. Because we have a long history of returns over many different cycles_, this is a reasonable approach.

This is not always th.e case, however. The return may have been constant over its recent history. This does not mean that it could not change in the future. For example, the price of gold was fixed to $35 per ounce from 1934 to 1967 by the U.S. government. As a result, using a historical distribution over the 30 years ending i11 1967 would have shown no risk. Instead, gold prices started to fluctuate wildly thereafter. By 2008, gold prices had reached $1,000. Thus, the responsibility of the risk manager is to judge whether the history is directly relevant.

How do we evaluate the quality of a risk measur~ment process? The occurrence fa large loss does not mean that risk management has failed. This could be simply due to bad luck. An investment in stocks would have lost 17% in October 2008. While this is a grievous Joss, Figure 1.1 shows that it \Vas not inconceivable. For

Risk Management 7

exan1ple, the stock market lost 30% in September 1931and22/o on October 19, 1987, before recovering. So, the risk rr1anager could have done a perfect job of forecasting the distribution of returns. How can we tell whether thjs loss is due to bad luck or a fla\v i11 the risk m.odel?

1.2.1 Known Knowns To l1elp answer this question, it is useful to classify risks into variot1s categories, wl1ich \.Ve can call (1) kno\>VO know11s, (2) known unknowns, and (3) unl\vns, includes model weaknesses that are known or shot1ld be known to exist but are not properly measured by risk n1an-agers. For example, the risk manager could have ignored important kno\vn risk factors. Second, the distribution of risk factors, including volatilities and correla-tions, could be measured inaccu.rately. Third, the mapping process, \vhich consists of replacing positions with exposures


risk of these investments, these losses ca11 be viewed as a failure of risk manage-n1ent. Even so, the UBS report indicates that the growth strategy undertaken by top managetnent was a "contri.buting factor to the buildup of UBS's subprime positions which subsequently incurred losses." In other W

Risk Management

Managing thse risks if given the authority to do so 111 Commu11icating these risks to the decision n1ake.rs

9

A large loss is not necessarily an indication of a risk 1nanageme11t failure. It could have been \Vithin tl1e scope of known knowns a11d properly communicated to the firn1, in \Vhich case it reflects bad lt1ck. After all, the objective of risk managen1ent is not to preve11t losses.

Otherwise, risk managen1ent can fail if any of these tasks has not been met. Some risks could go unrecognized. Mismeast1rement of ris.k can occur due to model risk, due to liquidity risk, or if distributions are not adequ.ately measured. Risk limits could not have been enforced. Finally, risk .management fails when it does not communicate risks effectively.

EXAMPLE 1.2: FRM EXAM 2008-QUESTION 111

Based on tl1e risk assessment of the CRO, Bank United's CEO decided to make a large investment in a levered portfolio of CDOs. The CRO had estimated that the portfolio had a 1 o/o chance of losing $1 bil\ion or more over one year, a loss that would make the bank insolvent. At the end of the first year the portfolio l1as lost $2 billion and the bank was closed by regulators.

Which of the following statements is correct?

a. The outco.1ne demonstrates a risk management failure because the bank did not eliminate the possibility of financial distress.

b. The outcome demonstrates a risk managen1ent failure because the fact tl1at an extremely unlikely outcome occurred means that the probability of the outcome\vas poorly estimated.

c. The outco1ne den1onst.rates a risk ina.nagement failure because the CRO failed to go to regulators t stop the shutdown.

d. Based on the information provided, one cannot deter1nine whether it was a risk management failure.

1.3 PORTFOLIO CONSTRUCTION

1.3.1 Comparing Multiple Assets We now turn to the portfolio construction .process, which involves co1nbi11ing expected return and risk. Assume that a11other choice is to invest in long-term U.S. governn1ent bonds.

Over the sa1ne period, the monthly average return of this other asset class was 0.47%. This is half that of equities. Tl1e monthly sta.ndard deviation \Vas 2.3%,


TABLE 1.1 Risk and Expected Return on Two Assets

A veragc Volatility Correlation

Equiries 11.2% L l d - 6()' ong-tenn )011 s .) . lo ,.., . , " ..... "';ftlil ,

19 JO' - 70

8.1% 0.13

again lower than for equities. To make the nun1bers more intuitive, monthly returns have been converted to an11ualized terms, as shown in Table 1.1.

Here, our investor is .faced \.Vith a typical trade-off, which is to choose between these t\vo alternatives. Neither dominates the other, as shown in Figure 1.2.

This graph describes a simple investn1ent decision. More generally, it also represents more complex business decisions that involve risk. For instance, a bank must decide how rnuch leverage to assume, as defined by the amount of assets divided by the amount of equity on its balance sheet. The horizontal axis could then represent the bank's credit rating. On one hand, higher leverage involves higher risk a11d accordingly a lower credit rating. In Figure 1.2, this corresponds to a move to the right. On the

Risk Management 11

The simplest metric is the Sharpe ratio (SR), which is the ratio of the average rate of return, (Rp), in excess of the risk-free rate Rf, to the abso]ute risk:

SR= [(Rp) - RF] CJ'{ Rp) ( l .3)

The Sharpe ratio focuses on total risk measured in absolute terms. l"'his ap-proach can be extended to include VAR in the denominator instead of the volatility of returns.

Figure 1.3 compares the SR for the two investment choices. Assume that \Ve have a risk-free asset, cash, with a .return of 3%. The SR is the slope of the line from cash to each asset. This line represents a portfolio mix between casl1 and each asset. In this case, stocks have a higher SR than bonds. This means that a mix of cash and stocks could be chosen with the same volatility as bonds but with higher returns.

This can be extended to relative risk measures. The information ratio {IR) is the ratio of the average rate of return of portfolio P in excess of the benchn1ark B to the TEV:

IR= [(Rp) - (RB)] o-(Rp - RB) (1.4)

Table 1.2 presents an illustration. The risk-free interest rare is RF = 3o/o and the portfolio average return is -6%, with volatility of 25o/o. Hence, the Sharpe ratio of the portfolio is SR= [(-6%) - (3%)]/25% = -0.36. Because this is neg-ative, the absolute performance is poor.

Assume no\v tha_t the be11chmark returned -10% over the same period and that the tracking error volatility \Vas 8%. Hence, the informatic>ri ratio is IR = [(-6%) - (-10%)]/8% = 0.50, which is _positive. The relative perfor111ance is gc;>od even though the absolute performance is poor.

Expected return (0/o pa) 15 -r----------------------------------,

~ 10 -j I

~ l 1

5 -j -i

Stocks

~::;....__--IG-

l '

0 -'1--...--....-....,.---~-...,-- -r--,--- -r----......---..--..,......---r--i-- '. 0 10 20 30

Volatility (% pa) FIGURE 1.3 Comparing Sharpe Ratios


TABLE 1.2 Absolute and .Relative Performance

Averag~ Volatility Performance

Cash "IQ/ ,) / O 0"/ :to Portfolio P -6o/i) 2S% SR= -0.36 Benchmark B - 1 0~/o 20% SR= - 0.65

Deviation e 4% 8% IR= 0.50

The tracking error volatility can be derived from the volatilities a p and a 8 of the portfolio and tl1e benchmark as well as their correlation p. Chapter 2 shows that the variance of a sum of random variables can be expressed in terms of the sum of the individual variances plus twice a covariance term. In terms of difference, the variance is

w2 =a~ - 2pa pa B + cr~ (1.5)

In this case, if c:r p = 25%, Cf B = 20%, a11d p = 0.961, we have w2 = 25'X>2 - 2 x 0.961 x 25% x 20% + 20%2 = 0.0064, giving w = 8%.

The IR has become commonly used to co1npare active managers in the same peer group. It is a pure measure of active management skill that is scaled for active risk. Consider, for exan1ple, two rr1anagers. Manager A has TEV of 2% per annum and excess return of 1 %. Manager B has TEV of 6o/o per annum and excess return of 2%. Manager A has .lower excess return but a higher information ratio, 1/ 2 = 0.50, vs. 2/6 = 0.33. As a result, it has better management skills. For example, Manager A could be asked to amplify its tracking error by a factor of 3_, which would lead to an excess return of 3 %, thus beating Manager B with the same level of tracking error of 6%. A11 information ratio of 0.50 is typical of the performance of the top 25th percentile of money managers and is considered "good. " 6

One of the drawbacks of the information ratio is that the TEV does not adjust for average returns. For instance, a portfolio could be systematically above its bencl1mark by 0.1 Oo/o per month. In this case, the tracking error has an average of 0.10% and a standard deviation close to zero. This leads to a very 11igh information ratio, which is not realistic if the active risk cannot be scaled easily.

1.3.3 Mixing Assets The analysis has so far considered a discrete choice to invest in either asset. M.ore generally, a portfolio can be divided between the two assets. Define w1 as the weight placed on asset i. With full investn1ent, we must l1ave L ;'\1 1 w; = 1. where N is the total number of assets . .In other vvords, the portfolio \veights must sum to 1.

6 G.rinold, Richard and Ronald Kahn, Active Portfolio Ma-nage7nent (New York: ~1cGra\v-Hill, 2000).

/

Risk Management 13

Start with a portfolio fulJy invested in bonds, represented by w1 = 1.00, w 2 = 0.00. As we shift the weight to stocks, \Ve can trace a line that represents the risk and expected rett1rn of the portft)lio 1nix. Eventually, this n1oves to the stock-only position, w1 = 0.00, 1n2 = 1.00. Figure 1.4 shows the line that describes all of these portfolios. This is an example of an asset allocation problen1, where the investor has to decide how 1nuch to aJlocate across asset classes.

The shape of this line depends on the correlation coefficient, p, which n1ea-sures the extent to which the two assets con1ove with each other. The correlation coefficient is scaled so that it mt1st be between -1 a.nd + 1. If p = 1, the two assets move perfectly .Proportionately ro each other a11d the line becomes a straight line.

More generally, the line is curved. This leads to an i11teresting observation. The line contains a portfolio with the same level of risk as that of bonds but with a higher return. Thus, a diversified portfolio can dominate one of the assets.

To demonstrate this point, consider the con1putation of the expected return and volatility for a portfolio. The portfolio variance depends on the weights, the individual asset variances, as well as the correlatio11:

(1.6)

Hence.the _portfolio volatility, which is the square root of the variance, is a nonlin-ear function f the weights. 111 contrast, the port.folio expected return is a simple linear average:

(1.7}

Consider, for insra11ce, the mix of 77o/o bonds and 23% stocks. The portfolio mean is easy to compute. Using data from Table 1.1, this is

f.Lr> = 0.77 x 5.6 + 0 .23 x 11.2 = 6.9%

Expected return (o/o pa) 15 -.-~~~~~~~~~~~----~~~~--.

I l

1 -i

10 ~

5

j

Portfolio

0-_i.i ~ '

-;--:----!--~, ~

0 10 Volatility (0/o pa)

FIGURE 1.4 Mixing Two Assets

Stocks

20 30


When p = 1, the portfolio variance is, using Equation (1.6):

CJ'~= 0.7728.1.2 +2 x 0.77 x 0.23(1x8.1x19.2) + 0.23219.22 = 113.49

The portfolio volatility is 10.65. But in chis case, this is also a linear average of the two volatilities a p = 0.77 x 8.1+0.23 x 19.2 = 10.65. Hence this point is on a straight line between the two assets.

Consider next the case where the correlation is the observed value of p = 0.13, shown in Table 1.1. The portfolio mix then has a volatility of 8.1 %. This portfolio has the same volatility as bonds, yet better performance. The mix has expected return of 6.9%, against the 5.6% for bonds, which is an improvement of 1.4%. This demonstrates the power of diversification.

Finally, consider a hypothetical case where the correlation is p = -1. In this case, the variance drops to 3.32 and the volatility to 1.8%. This illustrates the important point that low correlations help to decrease portfolio risk (at least when the portfolio weights are positive).

1.3.4 EfDcient Portfolios Consider now a 1nore general problem, which is that of diversification across a large number of stocks, for example, N = 500 stocks within the S&P 500 index. This seems an insurmountable problem because there are so many different combinations of these stocks. Yet Markowitz showed how the problem can be reduced to a much narrower choice.

The starting point is the assumption that all assets follow a jointly normal distribution. If so, the entire distribution of portfolio returns can be summarized by two paran1eters only, the mean and the varjance.

To solve the diversification problem, all that is needed is the identification of the efficient set in this mean-variance space (more precisely, mean-standard devi-ation). This is the locus of points that represent portfolio mixes with the best risk-return characteristics. More formally, each portfolio, defined by a set of weights {w], is such that, for a specified value of expected return JJ.p, the risk is minimized:

M. 2 lflwO' p (1.8)

subject to a number of conditions: (1) the portfolio return is equal to a specified value k, and (2) the portfolio weights sum to 1. Changing this specified value k traces out the efficient set.

When there are no short-sale restrictions on portfolio weights, a closed-form solution exists for the efficient set. Any portfolio is a linear combination of two portfolios. The first is the global minimum-variance portfolio, which has the lowest volatility across all fully invested portfolios. The second is the portfolio with the highest Sharpe ratio.

This framework can be generalized to value at risk, which is especially valu-able when return distributions have fat tails. In the general case, no closed-form solution exists, however.

Risk Management 16

1.4 ASSET PRICING THEORIES

1.4.1 Capital Asset Pricing Model (CAPM) These insights have Jed to the capital asset pricing n1odel (CAPM), developed by WiJ1ia1n Sharpe (1964).7 Sharpe's .first step \Vas t si1nplify the covariance structure of stocks v:ith a one-factor model. Define R.t as the return on stock i during month t, RF,t as the risk-free rate, and .R.\11.t as the market return. Tl1en run a regression of the excess return for i on the market across time:

(1.9)

The slope coefficient, 131, measures the exposure of i to the market factor and is also .known as systematic risk. The intercept is ai . For an actively managed portfolio ex is a measure of management skill. Finally, e; is the residual, which has mean zero and is uncorrelated to RM by construction and to all other residuals by

.

assun1pt1011. This is a one-factor model because any interactions between stocks are due to

their exposure to the market. To simplify, ig11ore tl1e risk-free rate and a., which is co11stant any\vay. We now examine the covariance between two stocks, i and ;. This is a measure of the comovements bet,veen t\VO random variables, and is explained further in Chapter 2.

Cov( ~, Ri) = Cov(f3; R,.1 + e;, f3; R1v1 + e i) = 13if3;Cov(RJ\1,RM) + l3iCov(RM,E;) + l3;Cov(RA-1,Ei) + Cov(e;~e;) = 13; 13 i


issued shares of equities). The total demand can be aggregated because investors are assun1ed to have homogeneous expectations about the distribution

Risk Management 17

This required rate of return on equity can also be vie\ved as the cost of equity capital. I11 other \.Vords, the CY.i term in Equation (1.9) should all be zero. This explains why for an actively managed portfolio a is a comm{)I11y used term to .measure skill.

This theory has profound consequences. Ir says that investors can. easily diver-sify their portfolios. As a result, most idiosyncratic risk washes a\vay. In contrast, systematic risk cannot be diversified. This explains why Equation (l .11) only con-tains (', as a risk factr for stock i. Note that volatility a; never appears directly in the pricing of risk.

This leads to an a)ternative measure of .Performance, which is the Treynor ratio (TR), defined as

(1.12)

If the CAPM holds, rl1is ratio should be the same for all assets. Equatic>n (1.11) indeed sho\VS t11at the ratio of(( R;) - RF ]/a( Ri) should l'e constant.

The TR penalizes for 11igh J3, as opposed to the SR, which penalizes for high a. For an investor with a portfolio similar to the market, J3 n1easures the contributio11 to the risk of the portfolio. Hence, this is a better performance measure for well-diversified .portfc>lios. In contrast, the SR can be used to adjust the perfor111ance of undiversified portfolios. Therefore, a major drawback of the SR and IR is that they do 11ot pe11alize for systematic risk.

This leads to a11other performance ineasure, which directly derives frm the CAPM. Suppose an active investment manager claims to add value for portfo-

lio P. The observed. average excess return is ,( Rp) - Rp. So111e of this, how-ever, may be due to market beta. The proper n1easure of perforrnance is then Jensen's alpha:

("l.13) '

For a fixed portfolio, if all stocks were priced according to the CAPM, the port-folio alpha should be zero (actually it should be negative if one acC()Unts for manageme11t fees and other costs).

In summary, the CAP~1 assumes that (1) i11vestors have ho1nogeneous ex-pectations, (2) distributions are normal, (3) capital markets are perfect, anti (4) markets are in equilibriu1n. The difficulty is that in theory, the market portfolio should contain all investible assets \vorld\vide. As a result, it is not l)bservable.

1.4.2 Arbitrage Pricing Theory (APT) The CAP.M specification starts fron1 a single-factor model. This can be generalized tstulate a risk structure wl1ere movements . in asset returns are due to multiple sources of risk .

. For instance, in the stock market, smal1 fir1ns behave differently from large fir1ns. This could be a second factor in addition to the market. Other possible


factors are energy prices, interest rates, and so on. With K factors, Equation (1.9) can be ge11eralized to

R; = O'.j + '3;1 Y1 + ... + '3;KYK + Ei (l.14)

Here again, the residuals E are assun1ed to be uncorrelated with tbe factors by con-struction and with each other by assumption. The risk decomposition in Eq11ation (1.10) can be ge11eralized in the sa111e fashion.

The arbitrage pricing theory (APT) developed by Stephe11 Ross (1976) relies on such a factor str11cture and the assumption that there is .no arbitrage in fi-nancial markets.8 Formally, portfolios can be co11structed that are well-diversified and have little risk. To prevent arbitrage opportunities, these portfolios should have zero expected returns. TJ1ese conditions force a linear relationship between expected returns and the factor exposures:

K

E[R;] = RF+ L f3ikAk (1.15) k=1

where Ak is the market price of factor k. Consider, for example, the simplest case where th.ere is one factor only. We

have three stocks, A, B, and C, with betas of 0.5, 1.0, and 1 .5, respectively. Assume nO\V that their expected returns are 6%, 8o/o, and 12%. \Ve can then construct a portfolio long 50% in A, long 50o/o in C, and short 100o/o in B. The portfolio beta is 50% x 0.5 + 50% x 1.5 - 100% x 1.0 = 0. This portfolio has no initial investment but no risk and therefore should have expected retur11 of zero. Computing the expected return gives 50% x 6% + 50% x 12 % - 100% x 8o/o = + 1 %, however. This would create an arbitrage opportunity, which must be ru1ed out. These three expected returns are h1consistent with the APT. From A and C, we have RF = 3o/o and A.1 = 6%. As a result, the APT expected return on B shot1ld be, by Equation (1.15), E[R;J =RF+ (3;1A1=3%+1.0 x 6o/o = 9%.

Note that the APT expected return is very similar to the CAP~'!, Equation (1.11), in the case of a single factor. The interpretation, ho\.vever, is totally dif-ferent. The APT does not rely 011 equilibrium but sin1ply 011 the assumption that there should be no arbitrage opportunities it1 ca.pita! markets, a much weake.i; requiren1ent. It does not even need the factor model to hold strictly. Instead, it requires only that the residual risk is very small. This must be the case if a.sufficient number of com1nc>n factors is identified and in a well-diversified portfolio, also ca11ed highly granular.

The APT model does not require the market to be identified, \vhich is an adva11tage. Unfortunately, tests of this model are ambiguous because the theory provides 110 guida11ce as to what the fact

Risk Management 19

example, factors such as value, size> and momentum have been 'videly used to explain expected stock returns. The second is a statistical approach that extracts the factors from the observed data. For exan1ple, principal con1ponent analysis (PCA) is a tech11ique that .provides the best fit to the correlation matrix of asset returns. The first PC is a linear combinatio11 of asset retur.ns that provides the best approximation to the diagonal o.f this matrix. The seco.nd PC is a linear combination of asset returns that is orthogonal to the first and provides the next best approximation. The analysis can continue until the remajning factors are no longer significant.

Such an appracl1 is very imprtant for risk management when it is a large-scale problem, as for a ]arge portfolio. For example, the risk manager migl1t be able to reduce the dime11siona1ity of a portfolio of 100 stocks using perhaps 10 risk factors. This reduction is partjcularly importa11t for Monte Carlo simulation, as it cuts down the C(>mputing time.

EXAMPLE 1.3: FRM EXAM 2009-QUESTION 1 4

An analyst at CAPM Research Inc. is projecting a return of 21 % on Portflio A. The market risk premiun1 is 11. o/c), the volatility of the market portfolio is 1.4o/o, and the risk-free rate is 4.5cYo. Portfolio A has a beta of 1.5. According to the capital asset 11ricing model, wl1ich of the following statements is true?

a. The expected return of Portfolio A is greater tl1a11 tl1e expected return of the market portfolio.

b. The expected return of Portfolio A is less than the expected return of the market portfolio.

c. The return of Portfolio A has lower volatility tl1an the market l)Ortfolio. d. Tl1e ex_pected return of 'Portfolio A is equal to the expected return of the

market portfolio.

EXAMPLE 1.4: FRM EXAM 2009-QUESTION 1-8

Suppose Port.folio .A has a11 expected retur11 of 8%, volatility of 20o/o, and beta of 0.5. Suppose the t.narket has an expected retur11of10% and volatility of 25%. Finally, suppose the risk-free rate is 5%. What is Jensen's alpha for Portfolio A?

a. 10.0% b. 1.0o/c, c. 0.5%, d. 15o/o

\



W.hich of the following state111e.nts about the Sharpe ratio is false?

a. The Sharpe ratio considers both the systematic and unsyste1narjc risks of a portfolio.

b. The Sharpe ratio is equa] to the excess retur11 of a portfolio over tl1e risk-free rate div.ided by the total risk of the portfolio.

c. The Sharpe ratio cannot be used to evaluate relative performance of undiversified portfolios.

d. The Sharpe ratio is derived from the capital nlarket li11e.

EXAMPLE 1.8: SHARPE AND INFORMATION RATIOS

A portfolio manager returns 1 Oo/o with a volatility of 20%. The benchmark returns 8% with risk of 14%. Tl1e correlation between the two is 0.98. The risk-free rate is 3 %. Wl1ich of the fc>llowi11g statements is correct?

a. The _portfolio has higher SR than the benchmark. b. The portfolio has negative IR. c. The IR is 0.35. d. The IR is 0.29.

1.5 VALUE OF RISK MANAGEMENT

The previous sections have sllO\Vn rhat .most i11vestment or business dec.isions re-quire information about the risks of alter11ative choices. The asset pricing theories also give us a frame of reference to think about hovv risk 1nanage1nent can increase economic value.

1.5.1 Irrelevance of Risk Management The CAPM emphasizes that investors dislike sysr.ematic risk. They can diversify other, no11systematic risks on their O\.vn. Hence, systen1atic risk is the only risk that needs to be priced.

Companies could use fina11cial derivatives to hedge tl1eir volatility. If this changes the volatility but not the 1narker beta, however, the company's cost of capita) and hence valuation cannot be affected. Such a result holds only under the perfect capital market assun1ptions that u11derlie the CAP.M.

Risk Management

As an example, suppose that a con1pany hedges Svvever, for investors in the company's stock to hedge such oil exposure on their O\vn if they so desired. For instance, they could use oil futures, \Vhich can be eas.ily traded. In this sitt1ation, the risk management practice by this compa11y should not add economic value.

In this abstract world, risk management is irrelevant. This is an application of the classic Modigliani and Miller (.NfM) theoren1, which says that the value of a firm cannot depend on its financial policies. The intuition for this result is that any financing action undertake11 by a corporation that its investors could easily undertake on their ow.n should not add value. \Vorse, if risk management practices are costly, they could damage the firm's econon1ic value.

1.5.2 Relevance of Risk Management Tl1e MM theore1n, however, is based on a number of assu111ptio11s: Tl1ere are no frictions such as financial distress costs, taxes, and access to capital markets; there is no asymmetry of information between financiaJ n1arket participants.

In practice, risk management can add value if some of these assumptions do not hold true.

Hedging should i11crease value if it helps avoid a large cost of fina11cial dis-tress. Companies that go bankrupt often experience a large drop in value due to forced sales or the legal costs of the bankruptcy process. Consider, for ex-ample, the Lehn1an Brothers bankruptcy. Lehman had around $130 billion of outstanding bonds. The value of these bonds dropped to 9.75 cents on the dollar after September 2008. This jmplied a huge drop in the firm value due to the bankruptcy event.

Corporate income taxes can be viewed as a form of friction. Assuming no carry-over provisions, the income tax in a very good year is high and is not offset by a tax reft1nd in a year with Losses. A tax is paid on income if pos-itive, wl1ich is akin to a 1011g pSitin in an option, with a similar convexity effect. Therefore, by stabilizing earnings, corporations reduce the average tax payment over time, \vhich should increase their value.

Other frictions arise \vhen external financing is more costly than internally .. generated funds. A company could decide not to hedge its financial risks, which leads to more volatile earnings. In good years, projects are financed internally. In bad years, it is al\vays possible to finance projects by borrowing from capital markets. If t11e external borrowing cost is toe> high, however> some wortl1y projects will not be funded i11 bad years. Hedging helps avoid this underinvestment problem, \vhich sl1ould increase firm value.

A forin of infor111ation asym111etry is due to agency costs of managerial discre-tion. Investors hire ma11agers to serve as their agents, and give them discretion to run the con.1pany. Good and bad inanagers, however, are not always easy to identify. Without hedging, earnings fluctuate due to outside forces. This


makes it difficult to identify the perforn1ance f management. With hedging, there is less room for excuses. Bad managers can be identified more easily and fired, \vl1ic:h should increase fir1n value.

Another form of information asym1netry arises \.vhen a large shareholder has expertise i11 the firm;>s busi11ess. This expertise_, in addition to the ability to monitor management more effectively tl1an others, 1nay increase firtn value. Typically, such investors have a large fraction of their wealth invested in this firm. Because they are not diversified, they may be more willing to invest in the com_pa11y and hence add value when the company lowers its risk.

In practice, there is empirical evidence that firms that engage in risk manage-me11t progra1ns tend to have higher valuation tl1an others. This type of anal-ysis is fraught with difficulties, however. Researchers do not have access to identical firms that solely differ by thejr hedging program. Other confound-ing effects might be at work. Hedging might be correlated with the qual-ity of 1nanagement, which does have an effect on firm valtte. For example, firms witl1 risk. manageme11t programs are more likely to en1ploy financial risk managers (FRMs).


In perfect inarkets, risk managen1e11t expenditures aimed at reducing a firm's diversifiable risk serve to

a. Make the firm more attractive to shareholders as long as costs of risk management are reasonable

b. I11crease the firm's value by lowering its cost of equity c. Decrease tl1e firm's value vv11enever the costs of such risk management

are positive d. Has no impact on fir.m value


By red11cing tl1e risk of financial distress and bankruptcy, a firm's use of derivatives co11tracts to hedge its casl1 flo\v uncertainty will

a. Lo,ver its value due to the transaction costs of derivatives trading b. E11hance its value since investors cannot hedge such risks by then1selves c. Have no impact on its value as investors can costlessly diversify this risk d. Have no impact as only systematic risks can be hedged with derivatives

Risk Management

1.8 IMPORTANT FORMULAS

Absolute risk: cr(il P) =


Example 1.5: FRM Exam 2007-Question 182 c. The SR considers total risk, which includes systematic and unsyste1natic risks, so a. and b. are correct statements, and i11cor.rect answers. Sin1ilarly, the SR is derived from the C:NIL,, which states that the market is mean-variance efficient and hence has the highest Sharpe ratio of a11y feasible portfolio. FinaHy, the SR can be used to evaluate undiversified portfolios, pre~isely because it includes idiosyncratic risk.

Example 1.8: Sharpe and Inf ormatlon Ratios d. The Sharpe ratios of the portfolio and benchmark are (lOo/o - 3o/o)/20/o = 0.35 and. (8% - 3%)/14% = 0.36, respectively. So the SR of the portfolio is lower tha11 that of the benchmark; answer a. is incorrect. The TEV is the square root of 20o/o2 + 14%2 - 2 x 0.98 x 20% x 14%, which is J0.00472 = 6.87%. So, the IR of the portfolio is (10% - 8%)/6.87% = 0.29. This is positive, so answer b. is incorrect. Answer c. is the SR of the portfolio, not the IR, so it is . incorrect.

Example 1.7: FRM Exam 2008-Question 1-8 c. In per.feet markets, risk rnanagement actions tl1at lo\ver the firtn's diversifiable risk should not affect its cost of capital, and hence will not increase value. Further, if these activities are costly, the firm value should decrease.

Example 1.8: FRM Exam 2008-Questlon 1 2 b. The cost of fi11a11cial distress is a market imper.fection, or deadweight cost. By hedging, firms will lower this cost_, which should increase the economic value of the firm.

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28 QUANTITATIVE ANALYSIS

2.1.1 Univariate Distribution functions .. :\random variable Xis characterized by a distribution function,

F(x) = P(X < x) (2.1)

which is tl1e probability that the reaJizati

Fundamentals of Probability 28

Probability density function

1 -- - --------------------- .. --------. l ! I I L ' ~x) , . . ... ---- :

"

-------------=~,.,,,I Cumulative distribution function

.F(x) ----- -- -

FIGURE 2.1 Density and Distribution Functions

A gambler \vants to characterize the probability density functio11 of the outcomes from a pair of dice. Because each has six faces, there are 62 = 36 possible throw co111binarions. Out of these, there is one occurrence of an utcorne of two (each die sho\ving one). So, the frequency of an ot1tcome of two is one. We can have two occurrences of a three (a one and a t\VO and vice versa)., a11d so on.

The gambler co111piles the freque11cy of each value, from 2 to 12, as shown in Table 2.1. From this, the gambler can con1pute the probability of each outcome. For instance, the probability of observing three is equal to 2, the frequency n(x), divided by the total number of outcon1es, of 36, which gives 0.0556. We can verify that all the probabilities indeed add up to 1., since all occurrences mt1st be accounted for. From the table, we see that the probability of an outco1ne of three or less is 8.33 o/o.

TABLE 2.1 Probability Density Function

Cumulative Outcome Frequency Probability Probability X; n(x) f (x) F(x)

? -

1 l /36 0.0278 0.0278 3 2 2136 0.0556 0.0833 4 3 3136 0.0833 0.1667 5 4 4136 0.1 l 11 0.2778 6 5 5/36 0.1389 0.4167 I 6 6136 0.1667 0.5833 8 5 5136 0.1389 0.7222 9 4 4136 0.1111 0.8333

10 3 3136 0.0833 0.9167 11 2 2136 0.0556 0.9722 1.2 1 1136 0.0278 1.0000 Sun1 36 1 1.0000


2.1.2 Moments A randrn variable is cl1aracterized by its distribution ft111cti


Probability density function ,----------,----------- -~

t(x) I I i I

!

-------------

l I VAR

--- --

Cumulative distribution function F(x) I

I i

5 /o ~ .:::..:.. ... "

FIGURE 2.2 V.AR as a Quantile

The standard deviatio11 is 1nore convenient to use as it has the sa1ne units as the original variable X:

SD(X) =er= )V(X) (2.11)

Next, tl1e scaled third moment is tl1e skewness, which describes departures fron1 svmmetrv. It is defined as , ,

~ = (J:'O [x - E(X)]3 f{x)dx) / o-3 (2.12) Negative skewness indicates that the distribution has a long left tail, \vhich indi-cates a high probability of observing large negative valt1es. If this represents the distribution of profits and .losses for a portfolio, this is a dangerous situation. Figure 2.3 displays distributions \Vi th various signs for the ske\vness.

The scaled fourth moment is the kurtosis, whicl1 describes the degree of flatness of a distributio11, or \vidtl1 of its tails. It is defined as

(2.13)

Because of the fourth po\ver, large observations i11 the tail \vilJ have a large weight and hence create large kurtosis. Such a distributio11 is called Jeptokurtic, or fat-tailed . This paran1eter is very in1p


Probability density function -----

f ~-~~-~-----~--,

! ;

I

FIGURE 2.3 Effecr of Skewness

Probability density function ~-------------i l

. ,

Zero skewness

Positive skewness

Negative skewness

j I ! !

I j

Fat tailsi1

1

Thin tails, \l(urtosis :> 3), (kurtosis< 3) ->-\ r

' " .. .. ~-.. ~- --... --

--

FIGURE 2.4 Effect of Kurtosis

Our ga111bler wants to know the expected value of tl1e outcome of throwing two dice. He con1putes the prc>duct of each outcome and associated probability, as shown in Table 2.2. For instance, the first entry is xf.(x) = 2 x 0.0278 = 0.0556, and so on. Sumn1ing across all events, the mean is = 7.000. 1his is also tl1e median, sir1ce the distribution is perfectly symn1etrical.

Nex.t, \Ve can use Equation (2.10) t


TABLE 2.2 Con1puring Ivlornents of a Distribution --- ---

Outcome Prob. 1\1ean Variance Skewness I


2.2.1 Joint Distributions We ca11 extend Equation (2.1) to

(2.14)

which defines a joint bivariate distribution function. In the continuous case, this is also

(2.15)

where f(ui , u2) is now the joint density. In general, adding random variables con-siderably complicates the characterization of the density or distribution functions.

The analysis simplifies considerably if the variables are independent. In this case, the joint density separates out into the product of the densities:

(2.16)

and tl1e integral reduces to

(2.17)

This is very convenient because \Ve only need to know the individual densities to reconstruct the joint density. For example, a credit loss can be viewed as a combination of ( 1) default, which is a random variable with a value of one. for default and zero otherwise, and (2) the exposure, which is a random variable representing the amount at risk, for instance the positive market value of a swa_p. If the two variables are independent, we can construct t11e distribution of the credit loss easily. In the case of the two dice, the events are indeed independent. As a result, the probability of a joint event is simply the product of probabilities. For instance, the probability of throwing two ones is equal to 1/6 x 1/6 = 1/36.

It .is also useful to characterize the distribution of x1 abstracting from x2. By integrating over all values of x2, we obtain the marginal density:

(2.18)

and similarly for x2 We can then defi11e the conditional de11sity as

(2.19)

Here, we keep x2 fixed and divide the joint density by the n1arginal probability of x2. T11is normalization is necessary to ensure that the conditional density is a proper density function that integrates to one. This relationship is also known as Bayes' rule.


2.2.2 covariances and Correlations Wl1en dealing \Vith two ra11don1 variables, the comoven1ent can be described l)y the covariance

(2.20)

It is often useful to scale the covarian.ce into a unitless nun1ber, called the correla-tion coefficient, obtained as

X X ) Cov(X1 , X2) p( J , 2 = -----CT J


is given by f1 (x1) = f12(x1 ~ x2 = -10) + f12(x1, x2 = + 10) = 0.30 + 0.20 = 0.50. The marginal probability of x1 = +5 inust be 0.50 as \veil. Tabl.e 2.3b shows that the n1eans and standard deviations are, respectively, XJ = 0.0, cr 1 = 5.0, and X2 = 1.0,



The joint probability distribution of random variables X and Y is given by f (x, y) = k x x x y for x = 1. 2, 3, y = 1, 2, 3, and k is a positive constant. What is the probability that X + Y will exceed 5?

a. 1/9 b. 1/4 c. 1136 d. Cannot be determined

2.3 FUNCTIONS OF RANDOM VARIABLES

Risk manage1nent is about uncovering th.e distribution of portfolio values. Con-sider a secl1rity that depends on a unique sou.rce of risk, such as a bond. The risk manager could model the change in the bond price as a randon1 variable ('RV) directly. The problem with this choice is that the distribution of the bond price is not stationary, because the price converges to the face value at expiration.

Instead, the practice is to mode) the change i11 yields as a random variable because its distribution is better behaved. The next step is to use the relationship between the bo.nd price and the yield to uncover the distribution of the bond price.

This illustrates a general principle of risk n1anagen1ent, whicl1 .is to model the risk factor first, then to derive the distribution o.f the instrument fron1 i11formation about the function that links the instrument value to the risk factor. This may not be easy to do, unfortunately, if the relationship is highly nonlinear. In what follo\vs, we first focus on tl1e n1ean and variance of sin1ple transformations of random variables.

2.3 .. 1 Linear Translormation ot Random Variables Consider a transformation that multiplies the origi11al random variable by a con-stant and add a fixed amount, Y =a + bX. The expectation of Y is

E(a + bX) =a.+ bE(X) (2.22)

and its variance is

V(a + bX) = b2 V(X) (2.23)

Note that adding a constant never affects the variance since the computation involves the differe1ice between the variable and its mean. The stan.dard deviation is

SD{a + bX) .= bSD(X) (2.24}


A dollar-based investor has a portfolio consisting of $1 million in cash plus a position in 1,000 mil1ion Japanese yen. The distribution of the dollar/yen exchange rate Xhas a mean of E(X) = 1/100 = 0.01 and volatility of SD(X) = 0.10/100 = 0.001.

The portfolio value can be \vritten as Y = a + bX, witl1 fixed parameters (in millions) a = $1 a11d b = Yl,000. Therefore, the portfolio expected value is E(Y) = $1 + Yl,000 x 1/100 = $11 million, and the standard deviation is SD( Y) = Yl,000 x 0.001 = $1 million.

2.3.2 Sum of Random Variables Another useful transformation is the summation of two random variables. A portfolio, for instance, could contain one share of Intel plus one share of Microsoft. The rate f rett1rn on eacl1 stock behaves as a random variable.

The expectation of the sum Y = X1 + X2 can be written as

(2.25)

and its variance is

(2.26)

When the variables are uncorrelated, tl1e variance of the sum reduces to the sum of variances. Otherwise, we have to account for the cross-product term.

~ $ . . .

2.3.3 Portfolios of Random Variables More generally, consider a linear con1bination of a number of random variables. This could be a portfolio witl1 fixed weights, for whicl1 the rate of return is

N Y= Lw;X; (2.27)

i=1

where N is the number of assets, Xi is the rate of return on asset i, and w; its weight.


To shorten notation, this can be written in matrix notation, replacing a string of numbers by a single vector:

'X =w (2.28)

where w' represents the transposed vector (i.e., horizontal) of weights and Xis the vertical vector containing individual asset returns. Appendix A provides a brief review of matrix multiplication.

The portfolio expected return is now

N E(Y) = ,P = L w;.i (2.29)

i:::1

which is a weigl1ted average of the expected returns.; = E(X; ). The variance is

N N N N N N

V(Y) =


Follovving Equation (2.31 ), \Ve \vrite

(J"~ = [0.60 15] [0.60] = [O 60 99 0.40 . 0 40] [25 x 0.60 + 15 x 0.40] . 15 x 0.60 + 99 x 0.40

ai = [0.60 0.40) [!~:~~] = 0.60 x 21.00 + 0.40 x 48.60 = 32.04 Therefore, the portfolio volatility is a p = J32.04 = 5 .66. Note that this is hardly higher tl1an the volatility of the Canadian dollar alone, even though the risk of the euro is much higher. The portfolio risk l1as been kept low due to a diversification effect_, or low correlation between the t\VO assets.

!.8.4 Product ol Random Variables Some risks result from the prodt1ct of two rand.om variables. A credit loss, fo.r instance, arises from the product of the occurrence of default and the loss given default.

Using Eqt1ation (2.20}, the expectation of the product Y = X1 X2 can be .

written as

(2.32)

When the variables are independent, this reduces to the product of the means. The variance is more cmplex to evaluate. With independence, it reduces to:

(2.33}

2.3.5 Distributions of Translormations of Random Variables The preceding results focus on the mean and variance of simple transformations only. They do not fully describe the distribution of the transformed variable Y = g(X). This, unfortunately, is usually complicated for all b.ut the simplest transforn1ations g( ) and densities f ( X).

Even if there is no closed-forn1 s0Jt1tion for the density, we can describe the cum11lative distributio11 functio11 of Y 'vhen g( X) is a one-to-one transformation .from X into Y. This i1nplies that the function can be inverted, or that for a given y, we can find x such that x = g-1(y). We can then write

(2.34)

where F ( ) is the cun1u]ative distribution function of X. Here, \Ve assumed the relationship is positive. Otl1er\vise, the right-hand term is changed to 1 - Fx(g- 1(y)).


This a11ows us to derive the quantile of, say, the bond price from information about the probability distribution of the yield. Suppose \.Ve consider a zero-coupon bond, for vv-l1ich tl1e market value V is

'100 V= ---(1 + r)T (2.35)

where r is the yield. Tl1is equatio11 describes V as a function of r, or .Y = g( X). Using r = 6o/o and T = 30 years, the current price is V = $17.41. The inverse function X = g- 1 ( Y) is

r = (100/ V) 11T - l (2.36)

We wish to estimate the probability that the bond price could fall belo\v a cut-off price V = $15. We invert the price-yield function and compute the associated yield level, g- 1 (y) = (100/$15)1i 30 - 1 = 6.528%. Lower prices are associated with higher yield levels. Using Equation (2.34 ), tl1e probability is given by

P[V < $15] = P[r > 6.528%]

Assuming the yield change is norm.al with volatility 0.8%, tl1is gives a proba-bility of 25.5%.1 Even though \Ve do not k11ow the density of tl1e bo11d price, this method allows us to trace out its cumulative distribution by changing the cutoff price of $15. TakiI1g the derivative, we can recover the density function of the bond price. Figure 2.5 shows that this p.d.f. is skewed to the right.

Probability density function

$5 $10 $15 $20 Bond price

FIGURE 2.5 Densitv Function for the Bond Price ,

$25 $30 $35

1 We will see later that this is obtained from the standard nor1nal variable z = (6.528 - 6.000)/ 0.80 = 0.660. Using standard norn1al tables, or the NORMSDIS~f(-0.660} Excel function, this gives 25.So/o.


On. the extreme right, if the yield falls to zero, the bond price will go to $"100. On the extreme left, if the yield goes to infinity, the bond p.rice will fall to, but not go below, zero. Relative to the current value of $17.41, there is a greater likelihood of large movements up than down.

This nlethod, unfortunately, cannot be easily extended. For general density functions and transformations, risk managers turn to numerical methods, espe-cially when the number of random. variables is large. This is why credit risk 111odels, for instance, all describe the distribution of credit losses through simulations.


Suppose that A and B are random variables, each follows a standard .nor-mal distribution, and the covariance bet\veen A and B is 0.35. What is the variance of f3A+ 2B)?

a. 14.47 b. 17.20 c. 9.20 d. 15.10


Given that x and y are random variables and a, b, c, and d are constants, which one of the following defi11itions is wrong?

a. E(ax +by+ c) =a E(x) + bE(y) + c. if x and y are correlated. b. V(ax +by+ c) = V(ax +by)+ c, if x and y are correlated. c. Cov(ax + by, ex+ dy) = ac V(x) + bdV(y) + (ad+ bc)Cov(x, y), if x

and y are correlated. d. V(x - y) = V(x + y) = V(x) + V(y), if x and y are uncorrelated.

2.4 IMPORT ANT DISTRIBUTION FUNCTIONS

2.4.1 Unilorm Distribution The simplest continuous distribution function is the uniform distribution. This is defined over a range of values for x, a ~ x < b. The density function is

1 f (x} = (b _a), a < x < b (2.37}


Frequency ___ ...................... ,, __ ,,_ , .. , ..... _____ ,._ -

I .

a b Realization of the uniform random variable

FIGURE 2.8 Uniform Density .Function

which is constant and indeed integrates to unity. This distribution puts the same \Veight on each observation within the allo\vable ra11ge, as shown i11 Figure 2.6. We denote this distribution as U(a. b).

Its mean and variance are given by

E(X)=a+b 2

V(.X) = (b - a)2 12

(2.38)

(2.39)

The uniform distribution U(O, 1) is \videly used as a starti11g distribution for generati11g random variables from any distribution F ( Y) in simulations. We need to have analytical formulas for tl1e p.d.. f ( Y) and its cun1ula_tive distribution F ( Y}. As any cumt1lative distribution function ranges fro111 zero to unity, we .first draw X from U(O, 1) and then co1npute )' = p - 1 (x). The random variable Y will then have the desired distribution f ( Y).


The randon1 variable X\vith density function f (x} = l j(b - a) for a < x < b. and 0 otherwise, is said to have a uniform distributic>11 over (a. b). C:alculate its mea11.

a. (a+ b)/2 b. a - b/2 c. a+ b/4 d. a - b/4


2.4.2 Normal Distribution Perhaps the most important continuous distribution is the normal distribution, \vhich represents adequately many random processes. This 11as a bell-like shape \Vith inore weight i11 the ce11ter and tails tape.ring off to zero. The daily .rate of return in a stock price, for instance, has a distributio.n sin1ilar to the i1or1.naJ p.d.f.

The normal distribution can be characterized by its first two moments only, the mean and variance cr2 The first parameter represents the location; the second, the dispersion. The normal density function .has the following expression:

1 1 f(x) = exp[- 2 1 (x - )

2] J2'Trcr2 a-

(2.40)

Its mean is E[XJ = and variance V[X] = cr2 We denote this distribution as N(, cr2 ). Because the function ca11 be fully specified by these two parameters, it is called a parametric function.

Instead of having to deal with different paran1eters, it is ofte11 more con-venient to use a standard normal variable as e, which has been standardized, or normalized, so that E(e) = 0, V(e) = cr(e) = 1. Figure 2.7 plots the standard normal density. Appendix B describes the standard normal distribt1tion.

First, note tl1at the function is symmetrica1 around the mean. Its mean of zero is the same as its mode (which is also tl1e most likely, or highest, point on this curve) and median (which is sucl1 that the area to the left is a 50o/o probability). Tl1e skewness of a normal distribution is 0, which indicates chat it is symmetrical around the mean. The kurtosis of a normal distribution is 3. Distributions with fatter tails have a greater kurtosis coefficient.

About 95 percent of the distribution is -contained between values of t = -2 and e2 = +2, and 68 percent of the distribution falls bet\veen values of Et = -1 and 2 = + 1. Table. 2.4 gives the values that correspond to right-tail probabilities, such that

1: f(E)dE = C (2.41) Frequency .,... ~ ,._ - .... ~ ...... ., .. ___ ._ .....

distribution is between

-1 and +1 . ' ~ '4---- - '

' .

'

' ' '

' ' ' 95~ is ' ' ' ' between ' ' '

' -2and +2 ' '

' '

r-----r---- -4 -3 -2 - 1 0 1 2 4

Realization of the standard normal random variable

FIGURE 2.7 Nonnal Density Function


TABLE 2.4 Lo\ver (~uantiles of the Standardized Normal Distribution ---~OllQOltl:ltl$ll:ll~I(~'-----------------~---------

Confidence Level ( o/o) --- --------

c 99.99 99.9 99 97.72 97.5 95 90 84.13 50 ( -0'.) -3.715 -3.090 - 2.326 - 2.000 -1.960 -1.645 -- 1'8' ..... - -1.000 -0.000

For instance, -a == - 1.645 is the quantile that correspo11ds to a 95o/o probability (see Table 2.4).2

This distribution plays a central role in finance because it represents adequately the behavior of many financial variables. It enters, for instance, tl1e Black-Scholes option pricing formula vvhere the function N( ) represents the cumulative stan-dardized normal distribution function.

The distributic>n of any normal variable can then be recovered fro1n that of the sta11dard nor1nal, by defining

X = + ECJ' (2.42)

Using Equations (2.22) and (2.23 ), we can sho\v that X has indeed the desired 1noments, as E(X) = + E(e)cr =.and V(.X) = V(e)cr2 = 1n variable as tlie change in the dollar value of a portfolio. The ex1,ected value is E( X) = . To find the quantile of X at the specified confidence level c, \Ve replace e by -ex in Equation (2.42). This gives Q(X, c) = - cxcr. Using Equation (2.9), we ca.n co1npt1te V J\R as

VAR = E ( X) - Q( X, c} = - ( - au) = acr (2.43)

For exam.pJe, a portfolio with a standard deviation of $10 million \VOuld have a VAR, or potential downsjde lc>ss, of $16.45 mj}Jion at the 95o/o confidence level.

KEY CONCEPT. ~ . . . .. , , "

'~ ' . . . . , ~

.. , .. ; . . . , . ' . . . ... " , . . , . ~ ... "' .. .

~1tl~ normal di~tri.bution$, .the VAR:of- a ,p.o~tfol19 .. is obtained from ti1e .. , ' , ; , ' , ) ; .. ' ff ' $ 't'

p_roduct of the portfoJio sta.ndard deyi~tion. and a standa~d n~trrial deviate fact9r that reflects the-cbnfidence level, for i!1st~nce 1.645' a~ the 9s% l~vel.

An in1portant property f the normal distribution is that it is one of the fe\v distributions t11at is stable under addition. In other words, a linear coml,ination of joi11tly 11


is extremely useful because we need to k11ow only the mean and variance of the .Portfolio to reconstruct its whole distribution.

';' .

l(El-:.i:ONCEPT . . ,, , . , . '

.,

C' ... . , ( "J; ..... ,, , t , ' ' , I '.

. ,. ' .. , ..... .. ~-~ ~ ' , . " ' . . A linear to.mbinatio~ of j.ointly :np,rql'~r-variables has a normaJ dist.ri'bt~~n.~

Fundamentals of Probability


Which type of distribution produces the lowest probability for a variable t exceed a specified extre1ne value that is greater than the mean, assun1ing the distributions all have the sa1ne mean and variance?

a. A leptokurtic distribution with a kurtosis of 4 b. A leptokurtic distribution \Vith a kurtosis of 8 c. A normal distribution d. A platykurtic distribution

2.4.3 Lognormal Distribution

47

The normal distribution is a good approximation for many financial variables, such as the rate of return on a stock, r = ( P1 - Po)/ Po, where Po and P1 are the stock prices at time 0 and 1.

Strictly speaking, this is inconsistent \Vith reality since a normal variable has infinite tails 011 both sides. In theory, r could end up below -1, \vhich implies P1 < 0. In reality, due to the limited liability of corporations, stock prices cannt turn negative. In many situations, ho,vever, this is a11 excellent approxi111ation. For instance, \Vith short horizons or small price moves, the probability of having a negative price is so small that it is negligible. If this is not the case, we need to resort to other distributions that prevent prices fro1n going negative. One such distributio11 is the lognormal.

A random variable Xis said to have a lognormal distribution if its logarith1n Y = ln(X) is normally distributed. Define 11ere X = (P1/ Po). Because the argument X in the logarithm function must be positive, the price P1 can never go below zero.

The lognormal density function has the following expression:

1 . [ 1 . J f (x) = exp -2 2

(ln(x) - )2 , xJ2'ITCT2 CT

x>O (2.45)

Note that this is more complex than .sin1ply plugging ln{x) into Equation (2.40), because x also appears in the denominator. Its mean is

(2.46)

and variance V[X] = exp[2 + 2a2 ] - exp[2 + cr2]. The paran1eters were cho-sen to correspond to those of the normal variable, E(YJ = E[ln( X).I = and V[Y] = V[Jn( X)] = rr2

Conversely, if we set E[XJ = exp[r ], the 1nean of the assc>ciated no.rmal vari-able is E[Y] = E{ln(X)] = (r - u 2 /2). We will see later that this adjustment is also


Frequency ----- - - .. ---"- .. - ... -----~ f :. ! ; ~ ! : \ ~ ! \ ' I\ ' \ i I i ' ' . \ : ~ ' I \ ; J). ' ? \ . ' ' I : t ' : ; ~ ; f \ 1 I / ! l ' ; l ' . l \ ;, f ~ ' ! ~ \ ~ f ' i

f J f

I I i

r................ -------~., -~-----:

! - Sigma= 1 I I ~ , Sigma = 1.2 !

! --- Sigma = 0.6 !..._ .... , .. , ___ ,....... i

0 1 2 3 4 5 6 7 8 9 10 Realization of the lognormal random variable

FIGURE 2.8 Lognormal Density Function

used in the Black-Scholes option valuation model, wl1ere the for1nula involves a trend in (r - cr2 /2) for the log-price ratio.

Figure 2.8 depicts the lognormal density function with . = 0, and various values cr = 1.0. 1.2, 0.6. Note that the distribution is skewed to the right. The tail increases for greater values f

Fundamentals of Probability


The skew of a lognor.maJ distribution is always

a. Positive b. Negative c. 0 d. 3

EXAMPLE 2.12: f RM EXAM 2002-QUESTION 125

Consider a. stock with an initial price of $100. lts price one year from now is given by S = 100 x exp(r), where the rate of return r is nor1nally distribt1ted with a mean of 0.1 and a standard deviation of 0.2. With 95o/o confidence, after rounding, Swill be between

a. $67.57 and $147.99 b. $70.80 and $149.20 c. $74.68 and $163.56 d. $102.18 and $119.53


For a lognormal variable X, we kno\V that ln(X) has a no.rmal distribution with a mean of zero and a standard deviation of 0.5. Wl1at are the expected value and the variance of X?

a. 1.025 and 0.187 b. 1.126 and 0.217 c. 1.133 and 0.365 d. 1.203 and 0.399

2.4.4 Student's t Distribution

49

Another im_portant distribution is the Student's t distribution. T.his arises in hy-pothesis testing, because it describes the distribution of the ratio of the estimated coefficient to its standard error.


This distribution is characterized by a parameter k known as the degrees of freedom. Its density is

f[(k + 1)/2] 1 1 f(x ) = f(k/ 2) ,Jk1T (1 + ,_:2; k) (k+t l/2 (2.48)

wl1ere r is the gamma function, defi11ed as f (k) = J; xk-te-xdx. Ask increases, this function converges to the normal p.d.f.

The distribution is symmetrical with mean zero and variance

provided k > 2. Its kurtosis is

V[X] = k k-2

6 8 = 3+k - 4

(2.49)

(2.50)

provided k > 4. It 11as fatter tails. than the normal distribution, which often pro-vides a better representation of typical financial variables. Typical esti1nated values of k are around four to six for stock returns. Figure 2.9 displays the density for k = 4 and k. = SO. The latter is close to the normal. With k = 4, however, the p.d.f. has fatter tails. As was done for the normal density, we can also use the Student's t to compute VAR as a function of the volatility

(2.51)

where the multi.plier now depends on the degrees of freedom k.

Frequency

/ .l.

, '

I .i i ! i

1 L ~ k=4 1 1! _ :::::::::- . . -~ ~-.--. k = 5CJ- ,-. ..... I == --,-- -r-- .. - --1

-4 -3 - 2 -1 0 1 2 3 4 Realization of the Student's t random variable

'

FIGURE 2.9 Student's t Density "Function


Another distribution derived from the nor1nal is the chi-square distribu-tion, which can be viewed as the sum of independent squared standard normal variables

(2.52)

where k is also called the degrees of freedom. Its 1nean is E[X] = k and vari-ance V[X] = 2k. For values of k sufficiently large, x2(k) converges to a normal distribution N(k, 2k). This distribution describes the sample variance.

Finally, another associated distribution is the F distribution, which can be viewed as the ratio of independent chi-square variables divided by their degrees of freedom

x2 (a)/a F (a, b) = xi(b)/b

This distribution appears in joint tests of regression coefficients.


(2.53)

Which of the following statements is the most accurate about the relationship between a normal distribution a11d a Student>s t distribution that have the same mean and standard deviation?

a. They have the same skewness and the same kurtosis. b. The Student's t distribution has larger skewness and larger_ kurtosis. c. The kurtosis of a Student's t distribution converges to that of the normal

distribution as the number of degrees of freedom increases. d. The normal distribution is a good approximation for the Student's

t distribution when the number of degrees of freedom is small.

2.4.5 Binomial Distribution Consider now a random variable that can take discrete values between zero and n. This could be, for instance, the number of times VAR is exceeded over the last year, also called the number of exceptions. Thus, the binomial distribution plays an important role for the backtesting of VAR models.

A binomial variable can be viewed as the result of n independent Bernoulli trials, where each trial results in an outcome of y = 0 or y = 1. This applies, for example, to credit risk. In case of default, \Ve have y = 1, otherwise y = 0. Each Bernoulli variable has expected value of E[Y] = p and variance V[Y] = p(1 - p).


A random variable is defined to have a bin.omial distribution if the discrete density function is given by

f (x) = (:) /T(l - p)"-x. x = 0, 1, ... , n (2.54)

where (:) is the number of combinations of n things taken x at a time, or

(n) n! x - x!(n - x)! (2.55) and the parameter p is between zero and one. This distribution also represents the total number of successes inn repeated experiments where each success has a probability of p.

The bino1nial variable has mean and variance

E[X] = p1i (2.56)

V[X] = p(l - p)n (2.57)

It is described in Iigure 2.10 in the case where p = 0.25 and n = 10. The proba-bility of observing X = 0, 1, 2 ... is 5.6o/o, 18.8%, 28.1/o, and so on.

For instance, we want to kno\v what is the probability of observing x = 0 exceptions out of a sa1nple of n = 250 observations when the true probability is 1 o/o. We should expect to observe 2.5 exceptions on average across many such

Frequency 0.3 ~

0.2 ~ 4

0.1

0 ' ' .

0 1 2 3 4 5 6 7 8 9 10 Realization of the binomial random variable

FIGURE 2.10 Binomial Density Function \Vith p = 0.25, 11 = 10

, . '


samples. There \vill be, however, son1e samples \Vith .no exceptions at all. This probability is

' 2 ... 0, f (X = 0) = J'l. px( 1 - fJ) 11- x = .) . 0.010.99250 = 0.081 x!(1i - x)! 1 x 250!

So, we would expect to observe 8.1 % of san1ples with zerc> exceptions, under the null hypotl1esis. We can repeat this calculation \Vitl1 different values for x. For example, tl1e probability of observing eight exceptions is f (X = 8) = 0.20o/o only. We can use this information to test the null hypothesis. Because tl1is probability is so low, observing eight exceptions would make 11s question whether the true probability is 1 % .

A related distribution is the 11egative binomial distribution. Instead of a fixed number of trials, this involves repeating the experiment u11til a fixed 11umber of failures r has occurred. Th.e density is

f (X) = pX(l - p)', ( x+r -1) 1' - 1 x=0, 1i (2.58) In other \Vords, this is the distribution f the number of successes before the rth failure of a Bernoulli process. Its mean is E[ X] = r (l ~Pl . Also note that x does not have an upper lin1it, t1nlike the variable in the usual bino1nial density.


On a n1ultiple-choice exa1n with four choices for each of six questions, what is the probability that a student gets fewer than. two questions correct simply by guessing?

a. 0.46o/o . b. 23.73%

c. 35.60% d. 53.39%

2.4.8 Poisson Distribution The Poisson distribution is a discrete distribution, which typically is used to de-scribe the number f events occurring over a fixed period of time, assun1ing eve11ts are indepe11dent of each other. It is defined as

e- >- A:'" f (x)= , x=0,1, ...

x! (2.59)

( .


where X. is a positive number representing the average arrival rate during the period. This distribution, for example, is widely used to represent the frequency, or number of occurrences, of operational losses over a year.

The parameter .A represents th~ expected value of X and also its variance

E[X] =.A (2.60)

V[XJ =.A (2.61)

The Poisson distribution is the Jimitjng case of the binomial distribution, as n goes to infinity and p goes to zero, while np = .A remains fixed. In addition, whe11 X. .is large the Poisson distribution is well approximated by the norn1al distribution with mean and variance of .A, through the central .limit theorem.

If the number of arrivals follows a Poisson distribution, then the tin1e period between arrivals follows an exponential distribution with mean 1/X.. The latter has density taking the form f(x) = x.e-Ax, for x > 0. For example, if \Ve expect X. = 12 losses per year, the average time interval between losses should be one year divided by 12, or one month.


When can you use the normal distribution to approximate the Poisson dis-tribution, assuming you haven independent trials, each with a probability of success of p?

a. When the mean of the Poisson distribution is very small b. When the variance of the Poisson distribution is very small c. When the number of observations is very large and the success rate is

close to 1 d. When the number of observations is very large and the success rate is

close to 0

2.5 DISTRIBUTION OF AVERAGES

The nor1nal distribution is extremely important because of the central limit theo-rem (CLT), which states that the mean of n independent and identically distributed (i.i.d.) vai-iables converges to a normal distribution as the number of observations n increases. This very powerful result is valid for any underlying distribution, as lc>ng as the realizations are independent. For instance, the distribution of total credit losses converges to a normal distribution as the number of loans increases to a large value, assuming defat1lts a.re always independent of each other.


Define .X as the mean ,11 L,;1 t X, where each variable has mean a11d standard

deviation u. We have

Standardizing the variable, we can \vrite

X- ..jii ~ N (0, 1) (u I }

(2.62)

(2.63)

Thus, the normal distribution is the lirniting distribution of the average, which explains why it has such a prominent place in statistics.

As an exan1_ple, consider the bino1ni.al variable, \vhich is rl1e sum of independent Bernoulli trials. When n is large, we can use the CL T and approximate the bi11c>n1ial distr.ibi1tion by the normal distribution. Using Equation (2.63) fr the su111, \Ve have

x-pn z = ~ N(O, 1)

,! p(1 - p)n {2.64)

which is much easier to evaluate tha11 the binomiaJ distribution. Consider, for example, the issue of whether the number of exceptions x we

observe is co1npatible with a 99'Yo V ... t\R .. For our example, the 111ean and va.riance of x are E[XJ = 0.01 x 250 = 2.5 and V[X] = 0.01(1 - 0.01) x 2~50 = 2.475. We observe x = 8, which gives z = (8 - 2.5)/J2.475 = 3.50. We can now co111pare this number to the standard normal distribution. Say, for in.stance, that \Ve decide to reject the hypothesis that VAR is co.rrect if the statistic .faJls outside a 95% t\.vo-tailed confidence band.4 This interval is (-1.96~ +1.96) for the standardized normal distribution. Here, the value of 3.50 is muc.h higher than the cutoff point of+ 1.96. As a result, we would reject the null hypothesis that the true probability of observing an exception is 1 'X) only. In other words, there are simply too n1an'y exceptions to be explained by bad luck. Ii is n1ore likely that the VAR model underestimates risk.

2.8 IMPORTANT FORMULAS

Probability density function: f (x) = Prob(X = x) (Cumulative) distribution function: F (x) = j~X> f (1e)di1 Mea11: .E(X) = = f xf(x)dx

Variance: V(X) = cr2 = j'[,,"C - ] 2 f(x)d.,"C

4 Nore that the choice of this confidence level has nothing to do \Virh the VAR confidence level. .l-lere, the 95% level represents the rare at \vhich the decision rule will com1nit the errorof falsely-rejecting a correct model.

56

Skewness: -y = (j'[x - .] 3 f (x}dx) /a3 Kt1rtosis: 8 = (f [x - J 4 f (x)dx) /a 4 Quantile, VAR: VAR= E(X) - Q(X. c) = cxcr Independentjoii1tdensities: f12(x1x2) = f1(x1) x f2(x2) :Niarginal densities: f1(x1) = }' lt2{x1 , u2)di1.2, Conditional densities: f1.1(x1 I x1) = fizixi.-~l

- .. {z (>."'z,)

Covariance: cr12 = f1 f1[x1 - 1][x2 - J.L2l f12dx1dx2 Correlation: Pi2 = a12/(cr10"2)

QUANTITATIVE ANALYSIS

.Linear tr

financial risk manager handbook with test bank 6th edition

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