FRM一级备考公式表.pdf

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1、FOUNDATIONS OF RISK MANAGEMENT Types of Risk Key classes of risk include marker risk,credir risk,liquidity risk,operarional risk,legal and regulatory risk,business risk,srraregic risk,and repuracion risk.Market risk includes interest race risk,equity price risk,foreign exchange risk,and commodity pr

2、ice risk.Credit risk includes default risk,bankruptcy risk,downgrade risk,and sctdcmcnt risk.Liquidity risk includes funding liquidiry risk and crading liquidity risk.Enterprise Risk Management(ERM)Comprehensive and integraced framework for managing firm risks in order co meec business objeccives,mi

3、nimize unexpecred earnings volacility,and maximize firm value.Benefits include(I)increased organizarional effecciveness,(2)beccer risk reporting,and(3)improved business performance.Determining Optimal Risk Exposure Target certain default probability or specific credit rating-.high credit racing may

4、have opporcunity coses(e.g.,forego risky/proficable projeccs).Sensitivity or scenario analysis:examine adverse impaccs on value from specific shocks.Diversifiable and Systematic Risk The pare of the volacility of a single securitys recurns chac is uncorrelaced wich che volatility of the markec porcf

5、olio is chat securicys diversifiable risk.The pare of an individual securicys risk char arises because of the posirive covariance of thac securicys recurns with overall marker recurns is called its systematic risk.A standardized measure of systematic risk is beta:beta=Cov(R;.RM)I 2 OM Capital Asset

6、Pricing Model(CAPM)In equilibrium,all investors hold a porcfolio of risky assecs thac has the same weigh rs as rhe market porcfolio.The CAPM is expressed in che equacion of the security market line(SML).For any single security or portfolio of securicies i,the expected return in equilibrium,is:E(R;)=

7、Ri=+beca;E(RM)-RF)CAPM Assumptions Investors seek to maximize the expected utility of their wealth at the end of the period,and all investors have the same investment horizon.Investors are risk averse.Investors only consider the mean and standard deviation of returns(which implicicly assumes the ass

8、et returns are normally distributed).Investors can borrow and lend at the same risk-free rate.Investors have the same expectations concerning returns.There are neither raxes nor transactions costs,and assets are infinitely divisible.This is often referred to as perfect markets.Arbitrage Pricing Theo

9、ry(APT)The APT describes expecced recurns as a linear function of exposures to common risk factors:E(R)=R,.+G;iRP,+G;iRPl+.+0,kRPk where:0,i=/factor beta for stock i RPi=risk premium associated with risk factor j The APT defines the scruccure of rerurns but does noc define which faccors should be us

10、ed in the model.The CAPM is a special case of APT with only one factor exposure-che market risk premium.The Fama-French three-factor model describes recurns as a linear funccion of che markec index recurn,firm size,and book-co-markec faccors.Measures of Performance The Treynor measure is equal co ch

11、e risk premium divided by beta,or systemacic risk:Treynor measure-E(Rp)-RF(3p The Sharpe measure is equal co che risk premium divided by che standard deviation,or coral risk:Sh E(Rp)-RF arpe measure-Op The Jensen measure(a.k.a.Jensens alpha or jusc alpha),is the assets excess return over the return

12、predicred by the CAPM:Jensen measure-o.p=E(Rp)-Ri=+13pE(RM)-RF)The information ratio is essentially the alpha of the managed porcfolio relative co its benchmark divided by che cracking error.IR=E(Rp)-E(Rs)crackmg error The Sortino ratio is similar co the Sharpe ratio excepc we replace the risk-free

13、race wich a minimum acceptable return,denoted Rm,.and we replace the scandard deviarion wich a cype of semi-srandard deviation.Sortino racio-_ E_(_R_,_p_)_-_R_ .,_mir,_1-semi-standard deviation Financial Disasters Drysdale Securities:borrowed$300 million in unsecured funds from Chase Manhaccan by ex

14、ploiting a Raw in che syscem for compucing che value of collateral.Kit.Ukr Peabody:Joseph Jett reporced subscancial arcificial profits;afcer the fake profics were dececced,$350 million in previously reporced gains had co be reversed.Barinf(s:rogue crader,Nick Leeson,cook speculative derivative posic

15、ions(Nikkei 225 fucures)in an actempc co cover crading losses;Leeson had dual responsibilicies of crading and supervising settlement operacions,allowing him co hide crading losses;lessons include separacion of ducies and managemenc oversighc.Allied Irish Bank:currency crader,John Rusnak,hid$691 mill

16、ion in losses;Rusnak bullied backoffice workers inco not following-up on crade confirmations for fake trades.UBS:equicy derivacives business lose millions due co incorrecc modeling of long-daced opcions and ics srake in Long-Term Capical Managemenc.Sociite Genemle:junior crader,Jerome Kerviel,parcic

17、ipaced in unauthorized crading accivicy and hid accivicy with fake ofTseccing cransaccions;fraud resulred in losses of$7.I billion.Metal!gesellscha.ft:shorc-cerm futures concracts used co hedge long-cerm exposure in che pecroleum markecs;scack-and-roll hedging scrategy;marking co markec on fucures c

18、aused huge cash Row problems.Long-Term Capital Management:hedge fund that used relative value stracegies with enormous amouncs of leverage;when Russia defaulced on ics debt in 1998,the increase in yield spreads caused huge losses and enormous cash Row problems from realizing marking co market losses

19、;lessons include lack of diversificacion,model risk,leverage,and funding and crading liquidity risks.Bankers Trust:developed derivacive scruccures that were incencionally complex;in caped phone conversations,staff bragged abouc how badly chey fooled clients.JPMorgan and Citigroup:main councerparcies

20、 in Enrons derivatives transaccions;agreed to pay a$286 million fine for assiscing wich fraud against Enron investors.Role of Risk Management I.Assess all risks faced by che firm.2.Communicace these risks co risk-caking decision makers.3.Monicor and manage these risks.Objeccive of risk managemenc is

21、 co recognize chat large losses are possible and co develop contingency plans that deal with such losses if they should occur.Risk Data Aggregation Defining,gathering,and processing risk daca for measuring performance againsc risk colerance.Benefics of effeccive risk daca aggregacion and reporcing s

22、ystems:Increases abiliry to anticipate problems.Identifies rouces to financial health.Improves resolvabilicy in event of bank stress.Increases efficiency,reduces chance of loss,and increases profitability.GARP Code of Conduct Secs forth principles relaced co echical behavior wirhin che risk manageme

23、nc profession.It scresses ethical behavior in che following areas:Principles Professional integrity and cchical conduct Con A ices of interest Confidentiality Professional Standards Fundamental responsibilities Adherence to best practices Violations of the Code of Conduct may result in tempor:iry np

24、enion or permanent removal from GARP membership.In addition,violations could lead to a revocation of the right to use the FRM designation.QUANTITATIVE ANALYSIS Probabilities Unconditional probability(marginal probability)is the probability of an event occurring.Gmditiona/probability,P(A J B),is the

25、probability of an event A occurring given that event B has occurred.Bayes Theorem Updates the prior probability for an event in response to the arrival of new information.P(IIO)=P(OJI)xP(I)P(O)Expected V alue Weighted average of the possible outcomes of a random variable,where the weights are the pr

26、obabilities that the outcomes will occur.E(X)=EP(xi)Xi=P(x1)x1+P(x2)x2+.+P(x0)x0 Variance Provides a measure of the extent of the dispersion in the values of the random variable around the mean.The square root of the variance is called the standard deviation.variance(X)=EHX-)2 Covariance Expected va

27、lue of the product of the deviations of two random variables from their respective expected values.Cov(Ri,Rj)=ERi-E(Ri)x Rj-E(Rj)Correlation Measures the strength of the linear relationship between two random variables.It ranges from-1 to+l.()-Cov(Ri,Rj)Corr Ri,Rj-()o(Ri)o Rj Sums of Random Variable

28、s If X and Y are any random variables:E(X+Y)=E(X)+E(Y)If X and Y are independent random variables:Var(X+Y)=Var(X)+Var(Y)If X and Y are NOT independent:Var(X+Y)=Var(X)+Var(Y)+2 x Cov(X,Y)Skewness and Kurtosis Skewness,or skew,refers to the extent to which a distribution is not symmetrical.The skewnes

29、s of a normal distribution is equal to zero.A positively skewed distribution is characterized by many outliers in the upper region,or right tail.A negatively skewed distribution has a disproportionately large amount of outliers that fall within its lower(left)tail.Kurtosis is a measure of the degree

30、 to which a distribution is more or less peaked than a normal distribution.Excess kurtosis=kurtosis-3.Leptolwnic describes a distribution chat is more peaked than a normal ditrihution.Platykunic refers to a distribution chat is less peaked,or flatter,than a normal distribution.Desirable Properties o

31、f an Estimator An unbiased estimator is one for which the expected value of the estimator is equal to the parameter you are trying to estimate.An unbiased estimator is also efficient if the variance of its sampling distribution is smaller than all the ocher unbiased estimators of the parameter you a

32、re trying to estimate.A consistent estimator is one for which the accuracy of the parameter estimate increases as the sample size increases.A point estimate should be a linear estimator when it can be used as a linear function of sample data.Continuous Uniform Distribution Distribution where the pro

33、bability of X occurring in a possible range is the length of the range relative to the total of all possible values.Letting a and b be the lower and upper limits of the uniform distribution,respectively,then for a x1 is b:()(x2-xi)P x1 Xx2=-(b-a)Binomial Distribution Evaluates a random variable with

34、 two possible outcomes over a series of n trials.The probability ofsuccess on each trial equals:p(x)=(number of ways to choose x from n)p(l-p)n-For a binomial random variable:expected value=np variance=np(l-p)Poisson Distribution Poisson random variable X refers to the number of successes per unit.T

35、he parameter lambda(X)refers to the average number of successes per unit.For the distribution,both its mean and variance are equal to the parameter,X.Axe-.P(X=x)=-x!Normal Distribution Normal distrihurion i complerely de.crihed hy its mean and variance.68%of observations fall within ls.90%of observa

36、tions fall within l.65s.95%of observations fall within l.96s.99%of observations fall w ithin 2.58s.Standardized Random V ariables A standardi:ud random variable is normalized so that it has a mean of zero and a standard deviation of 1.z-scort:represents number of standard deviations a given observat

37、ion is from a population mean.observation-population mean x-z=-standard deviation CJ Central Limit Theorem When selecting simple random samples of size n from a population with mean and finite variance CJ2,the sampling distribution of sample means approaches the normal probability distribution with

38、mean and variance equal to CJ2/n as the sample size becomes large.Population and Sample Mean The population mean sums all observed values in the population and divides by the number of observations in the population,N.N Exi=i=l N The sample mean is the sum of all values in a sample of a population,E

39、X,divided by the number of observations in the sample,n.It is used to make informces about the population mean.Population and Sample Variance The population variance is defined as the average of the squared deviations from the mean.The population standard deviation is the square root of the populati

40、on variance.N E(xi-)2 c?=i=l-N The sample variance,r,is the measure of dispersion that applies when we are evaluating a sample of n observations from a population.Using n-1 instead of n in the denominator improves the statistical properties of i2 as an estimator of CJ2 n -2 L-(Xi-X)s2=i=l _ _ n-1 Sa

41、mple Covariance.En(X-X)(Y-Y)covariance=1 1 n-1 i=l Standard Error The standard error of the sample mean is the standard deviation of the distribution of the sample means.When the standard deviation of the population,CJ,is known,the standard error of the sample mean is calculated as:CJ CJx=Fa_ Confid

42、ence Interval If the population has a normal distribution with a known variance,a confidence interval for the population mean is:-CJ X Zo./2 Fa_ zll=1.65 for 90%confidence intervals(significance level 10%,5%in each tail)za12=1.96 for 95%confidence intervals(significance level 5%,2.5%in each tail)zll

43、=2.58 for 99%confidence intervals(significance level 1%,0.5%in each tail)Hypothesis Testing Null hypothesis(HJ:hypothesis the researcher wants to reject;hypothesis that is actually tested;the basis for selection of the test statistics.Al.ternatiVt:hypothesis(HA):what is concluded if there is signifi

44、cant evidence to reject the null hypothesis.One-tailed test:tests whether value is greater than or less than another value.For example:H0:0 versus HA:110 Two-tailed test:tests whether value is different from another value.For example:H0:=0 versus HA:0 T-Distribution The t-distribution is a bell-shap

45、ed probability distribution that is symmetrical about its mean.It is the appropriate distribution to use when constructing confidence intervals based on small samples from populations with unknown variance and a normal,or approximately normal,distribution.t-test:t=x-.st.In Chi-Square Distribution Th

46、e chi-square test is used for hypothesis tests concerning the variance of a normally distributed population.2(n-l)s2 chi-square test:X=F-Distribution The F-test is used for hypotheses tests concerning the equality of the variances of two populations.s2 F-test:F=1-s2 Simple Linear Regression Yi=B0+B1

47、 x Xi+Ei where:Yi=dependent or explained variable =independent or explanatory variable B0=intercept coefficient B1=slope cocfficicnc Ei=error term Total Sum of Squares For the dependent variable in a regression model,there is a total sum of squares(TSS)around the sample mean.total sum of squares=exp

48、lained sum of squares+sum of squared residuals TSS=ESS+SSR Coefficient of Detennination Represented by R 2,it is a measure of the goodness of fit of the regression.R 2=ESS=l _ SSR TSS TSS In a simple two-variable regression,the square root of R 2 is the correlation coefficient(r)between X and Y,If t

49、he relationship is positive,then:r=JR2 Standard Error of the Regression(SER)Measures the degree of variability of the actual Y-values relative to the estimated Y-values from a regression equation.The SER gauges the fit of the regression line.The smaller the standard error,the better the fit.Linear R

50、egression Assumptions A linear relationship exists between the dependent and the independent variable.The independent variable is uncorrelated with the error terms.The expected value of the error term is zero.The variance of the error term is constant for all independent variables.No serial correlat