Miller, Mathematics & Statistics for Financial Risk Management

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1 P1.T2. Quantitative Analysis Miller, Mathematics & Statistics fr Financial Risk Management Binic Turtle FRM Study Ntes Sample By David Harper, CFA FRM CIPM and Deepa Raju

2 Miller, Chapter 2: Prbabilities DESCRIBE AND DISTINGUISH BETWEEN CONTINUOUS AND DISCRETE RANDOM VARIABLES DEFINE AND DISTINGUISH BETWEEN THE PROBABILITY DENSITY FUNCTION, THE CUMULATIVE DISTRIBUTION FUNCTION (CDF) AND THE INVERSE CDF

3 Miller, Chapter 2: Prbabilities Describe and distinguish between cntinuus and discrete randm variables. Define and distinguish between the prbability density functin, the cumulative distributin functin and the inverse cumulative distributin functin, and calculate prbabilities based n each f these functins. Calculate the prbability f an event given a discrete prbability functin. Distinguish between independent and mutually exclusive events. Define jint prbability, describe a prbability matrix and calculate jint prbabilities using prbability matrices. Define and calculate a cnditinal prbability, and distinguish between cnditinal and uncnditinal prbabilities. Selected key terms: Statistical r randm experiment: An bservatin r measurement prcess with multiple but uncertain utcmes Ppulatin r sample space: Set f all pssible utcmes f an experiment. Sample pint: Each member r utcme f the sample space. Outcme: The result f a single trial. Fr example, if we rll tw dice, an utcme might be a three (3) and a fur (4); a different utcme might be a (5) and a (2). Event: The result that reflects nne, ne, r mre utcmes in the sample space. Events can be simple r cmpund. An event is a subset f the sample space. If we rll tw-dice, an example f an event might be rlling a seven (7) in ttal. Randm variable (r stchastic variable): A stchastic r randm variable (r.v.) is a variable whse value is determined by the utcme f an experiment. Discrete randm variable: A randm variable (r.v.) that can take a finite number f values (r cuntably infinite). Fr example, cin, six-sided die, bnd default (yes r n). Cntinuus randm variable: A randm variable (r.v.) that can take any value in sme interval; e.g., asset returns, time. Mutually exclusive events: Events which cannt simultaneusly ccur. If A and B are mutually exclusive, the prbability f (A and B) is zer. Put anther way, their intersectin is the null set. Cllectively exhaustive events (a.k.a., cumulatively exhaustive): Events that cumulatively describe all pssible utcmes. 3

Describe and distinguish between cntinuus and discrete randm variables. We characterize (describe) a randm variable with a prbability distributin.

A randm variable s value is determined by the utcme f an experiment (aka, stchastic variable). A randm variable is a numerical summary f a randm utcme.

4 Describe and distinguish between cntinuus and discrete randm variables. We characterize (describe) a randm variable with a prbability distributin. The randm variable can be discrete r cntinuus; in either the discrete r cntinuus case, the prbability can be lcal (pmf r pdf) r cumulative (CDF). A randm variable s value is determined by the utcme f an experiment (aka, stchastic variable). A randm variable is a numerical summary f a randm utcme. The number f times yur cmputer crashes while yu are writing a term paper is randm and takes n a numerical value, s it is a randm variable. Stck & Watsn Cntinuus randm variable A cntinuus randm variable (X) can take n an infinite number f values within an interval: ( < < ) = ( ) 4

5 Discrete randm variable A discrete randm variable (X) assumes a value amng a finite set including x 1, x 2, x 3 and s n. The prbability functin is expressed by: ( = ) = ( ) Cntinuus versus discrete randm variables Discrete randm variables can be cunted. Cntinuus randm variables must be measured. Examples f a discrete randm variable include: Cin tss (head r tails, nthing in between) r rll f the dice (1, 2, 3, 4, 5, 6) Tggle: did the fund beat the benchmark? (yes, n). In risk, cmmn discrete randm variables are default/n default (0/1) and lss frequency. Examples f a cntinuus randm variable include distance and time. A cmmn example f a cntinuus variable in risk is lss severity. Nte the similarity between the summatin ( ) under the discrete variable and the integral ( ) under the cntinuus variable. The summatin ( ) f all discrete utcmes must equal ne. Similarly, the integral ( ) captures the area under the cntinuus distributin functin. The ttal area under this curve, frm (- ) t ( ), must equal ne. All fur f the s-called sampling distributins each f which cnverges t the nrmal are cntinuus: nrmal, student s t, chi-square, and F distributin. 5

6 Cntinuus Are measured Infinite Distance, Time (e.g.) Severity f lss (e.g.) Asset returns (e.g.) Cmparisn Discrete Are cunted Finite Applicatins in Finance Default (1,0) (e.g.) Frequency f lss (e.g.) Fr example: Nrmal Bernulli (0/1) Student s t Binmial (series i.i.d. Bernullis) Chi-square Pissn F distributin Lgarithmic Lgnrmal Expnential The fur in range are sampling Gamma, Beta distributins : e.g., the student s t is EVT Distributins (GPD, GEV) used t test the sample mean Gamma, Beta EVT Distributins (GPD, GEV) Miller n discrete versus cntinuus: we will be wrking with bth discrete and cntinuus randm variables. Discrete randm variables can take n nly a cuntable number f values fr example, a cin, which can nly be heads r tails, r a bnd, which can nly have ne f several letter ratings (AAA, AA, A, BBB, etc.). In cntrast t a discrete randm variable, a cntinuus randm variable can take n any value within a given range. A gd example f a cntinuus randm variable is the return f a stck index. If the level f the index can be any real number between zer and infinity, then the return f the index can be any real number greater than 1. Even if the range that the cntinuus variable ccupies is finite, the number f values that it can take is infinite. Fr this reasn, fr a cntinuus variable, the prbability f any specific value ccurring is zer. Even thugh we cannt talk abut the prbability f a specific value ccurring, we can talk abut the prbability f a variable being within a certain range. Take, fr example, the return n a stck market index ver the next year. We can talk abut the prbability f the index return being between 6% and 7%, but talking abut the prbability f the return being exactly 6.001% is meaningless. Even between 6.0% and 7.0% there are an infinite number f pssible values. The prbability f any ne f thse infinite values ccurring is zer. 6

7 Define and distinguish between the prbability density functin, the cumulative distributin functin (CDF) and the inverse CDF. Prbability density functins (pdf): Tp rw in the graph belw The prbability density functin answers a lcal questin: If the randm variable is discrete, the pdf (a.k.a., prbability mass functin, pmf, if discrete) is the prbability the variable will assume an exact value f x; i.e., : ( = ). If the randm variable is cntinuus, the pdf tells us the likelihd f utcmes ccurring n an interval between any tw pints. Given ur cntinuus randm variable, X, with a prbability p f being between r 1 and r 2, we can define ur pdf, f(x), such that: : ( ) = The pdf functins are illustrated n the tp rw in the graph belw with cntinuus (left-hand) and discrete (right-hand) functins: [ < < ] = ( ) = ( ) = ( = ) Cumulative distributin functins (CDF): bttm rw in the abve graph The cumulative density functin (CDF) assciates with either a PMF r PDF (i.e., the CDF can apply t either a cntinuus r randm variable). The CDF gives the prbability the randm variable will be less than, r equal t, sme value. CDF: P (X x). The cntinuus (left-hand) and discrete (right-hand) cumulative distributin functins are: ( ) = ( ) = [ ] ( ) = ( ) 7

8 Inverse cumulative distributin functin If F(x) is a cumulative distributin functin, then we define F 1 (p), the inverse cumulative distributin, as fllws: ( ) = ( ) = s. t. 0 1 We can examine the inverse cumulative distributin functin by applying it t the standard nrmal distributin, N(0,1). Fr example, F -1 (95%) = because at standard deviatins t the right f mean in a standard nrmal distributin, 95% f the area is t the left (under the curve). The inverse cumulative distributin functin (CDF) is als called the quantile functin; see We can nw see why Dwd says that VaR is just a quantile [functin]. Fr example, the 95% VaR is the inverse CDF fr p = 5% r p = 95%; fr the standard nrmal, that is = NORM.S.INV(5%) = Here are cmmn inverse CDFs: Standard Nrmal Distributin p F -1 (p) 1.0% (2.326) 5.0% (1.645) 50.0% % % % Must knw 97.72% % Must knw Univariate versus multivariate prbability density functins A single variable (univariate) prbability distributin is cncerned with nly a single randm variable; e.g., rll f a die, default f a single bligr. A multivariate prbability density functin cncerns the utcme f an experiment with mre than ne randm variable. This includes the simplest case f tw variables (i.e., a bivariate distributin). Density Cumulative Univariate f(x)= P (X = x) F(x) = P (X x) Bivariate f(x)= P (X = x, Y =y) F(x) = P (X x, Y y) 8

P1.T1. Foundations of Risk Management Elton & Gruber, Modern Portfolio Theory & Investment Analysis Bionic Turtle FRM Study Notes Reading 7

P1.T1. Foundations of Risk Management Elton & Gruber, Modern Portfolio Theory & Investment Analysis Bionic Turtle FRM Study Notes Reading 7 P1.T1. Fundatins f Risk Management Eltn & Gruber, Mdern Prtfli Thery & Investment Analysis Binic Turtle FRM Study Ntes Reading 7 By David Harper, CFA FRM CIPM www.binicturtle.cm KEY IDEAS IN ELTON & GRUBER