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# P1 Focus Review: 3rd of 8 (Quantitative)

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
P1 Focus Review, 3rd of 8 (Quantitative Methods--continued): Videos, Practice Questions and Learning Spreadsheets
• The 3rd (of 8) Part 1 Focus Review video (Quantitative Methods, continued) is located here.
• Associated practice question sets:
• P1.T2.Stock, Chapters 2-7 [reviewed in P1.FR2]
• P1.T2. Rachev, Chapters 2 & 3
• P1.T2. Joroin, Chapter 12
• P1.T2. Hull, Chapter 2
• The learning spreadsheets will soon be published into a single consolidates (T2) workbook
Concepts:
• Rachev's Distributions
• Monte Carlo Simulation
• Volatility (in VaR)
Rachev's Distributions
In the FRM's introduction to econometrics, we were primarily concerned with the so-called sampling distributions (i.e., normal and student's t for test of sample mean, chi-square for test of sample variance, and F distribution for joint regression coefficient and variance comparisons). Here, the two Rachev chapters are very short. The idea is to introduce some common non-normal distributions. Among this catalog of distributions, I think the following are most relevant exam-wise:
• The three discrete distributions: Bernouill, binomial and Poisson.
You should, at this point, be able to illustrate a financial use-case for each of these.
• The normal (of course). Already reviewed in Stock & Watson.
• Exponential. Primarily because, if given a hazard rate (instantaneous probability of default) of, say, 1.0%, the cumulative n-year PD is 1 - exp(-1.0%*n).
• Beta. You may just want to superficially register the beta as the most common distribution used to model loss given default (LGD) due to its flexibility. It's function will not be quizzed.
• Lognormal. Because this is the distribution used to model equities in T3 & T4 (e.g., BSM); i.e., log returns are normal such that price levels are lognormal.
• Normal mixture. I included in the FR because it's uniquely flexible.
Please note that, in regard to most of these distributions, you do NOT need to memorize their PDF/CDF functions. That is unrealistic and would be a misuse of your time! This includes the following distributional functions that you do not need to quantitatively memorize (pdf, CDF): Weibull, Chi-Square, Gamma, Beta, Logistic, EVT distributions (GEV and GPD), and skewed normal.

Monte Carlo Simulation
In the FR, I shared a couple of examples of the most common exam-type question: figuring the next step or two in a discrete GBM. The other concepts you should be familiar with:
• Inverse transformation of a random uniform variable [0,1] into a distributional deviate
• The (qualitative) idea that, in general, reducing the standard error by a multiple of 1/X (i.e., improving the MCS accuracy) requires X^2 additional trials. Increases in accuracy have a quadratic, not linear, expense.
Volatility (in VaR)
Hull Chapter 22 (volatility) is one of the more important quantitative readings. As reviewed in the FR, make sure you practice solving for an estimate of current volatility under both EWMA and GARCH. Exam questions here will be quantitative. There is a ton of GARCH(1,1) theory but you probably will just need to solve for a GARCH estimate given the lagged variance and return, and the key params. So, most of the theory you will not need.

Other thoughts:
• What's the key difference between moving average volatility versus ARCH? between EWMA and GARCH? (you should understand the differences between the three volatility models sufficiently that you can express them BRIEFLY)
• GARP loves to quiz the long-run (unconditional variance) using omega/(1 - alpha - beta)
• I probably would memorize the GARCH volatility FORECAST
• GARP has asked about EWMA for correlations (Hull 22.7) in a previous exam, which is otherwise unexpected

#### RiskNoob

##### Active Member
Thank you David for sharing summary of FR which is a part of tier 2 product. Do you have any plan to publish summaries for other FRs as well? It's like reading an great novel without an end...

RiskNoob

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi RiskNoob, yes, I will definitely furnish write-ups for all the focus reviews, just trying to keep up! ... thank you for finding them useful!

#### paiva85

##### New Member
Hi David,

On slide 26 of the video I believe there is a discrepancy in the value used for volatility in the answer vs. the question. The question states 0.12 and answer uses 0.14. Using 0.12 I came to an answer of 99.74.

These focus reviews you have created are EXTREMELY useful. I would have been all over the place with my studying during this last week if it were not for these and the BT mock exams. Much appreciated!

Devin

#### Quant_Noob

##### New Member
Hi David or anyone else - I am struggling with applying the formula on slide 15 of this Focus Review "1 - [Calculate for no questions correct, 1 question correct, and 2 questions correct: [C(10,0) + C(10,1) + C(10,2)]*0.5^10]", to 2012 GARP Practice exam Q 14, which talks about 5 choices ( implying 20% probability of picking the right answer) instead of 2 (50% probability) here in Question 6.

Are the 2 questions different in any other form other than the # of choices? Both have 10 questions and are looking for the probability of at least 3 questions correct? GARP has given the answer as 1- 67.8%=32.2% which I am unable to get, which reflects very poorly on my high school math. I am not sure if I am allowed to post the entire question here due to copyright protection and am hoping that you have access to GARP Practice questions.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
paiva85 you are totally correct: we somehow copied the question with the wrong volatility. FRM handbook example 4.1 assumes sigma = 0.14, consistent with the answer given. Apologies, thanks for your attention to detail, and for liking the Focus Reviews!

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi Quant_Noob, I'm really glad you brought this up , the commonality shows how the FRM really likes the binomial so it's a good spend of time ("fair use" allows me to copy the questions as I am adding commentary).

Question 2011.P1.6:
"Suppose that a quiz consists of 10 true-false questions. A student has not studied for the exam and just randomly guesses the answers. What is the probability that the student will get at least three questions correct?"

Question 2012.P1.14:
"A multiple choice exam has ten questions, with five choices per question. If you need at least three correct answers to pass the exam, what is the probability that you will pass simply by guessing?"

As is often the case, the hard part is maybe formulating the question. Both are binomial distribution: each question is a Bernoulli (i.e,. the student either answers correctly or incorrectly; 1 or 0) and a "test" is a series of Bernouillis (binomial = series of i.i.d. Bernouillis). Both tests are 10 questions (n = 10). Both tests want the probability that X >= 3 (i.e., passing) which is the same as 1 - Pr[not passing] = 1 - Pr[0 correct] + Pr[1 correct] + Pr[2 correct]. So both are essentially similar in asking, what is the Pr [X = 0 | X = 1 | X = 2] if distribution is binomial with n = 10. The only difference is probability of a correct answer (p), which is 50% on T/F and 20% with five choices. So,
• Pr[0 correct] = C[10,0]*0.5^0*0.5^10 for T/F where p = 0.5; or Pr[0 correct] = C[10,0]*0.2^0*0.8^10 for T/F where p = 0.20
• Pr[1 correct] = C[10,1]*0.5^1*0.5^9 for T/F where p = 0.5; or Pr[1 correct] = C[10,1]*0.2^1*0.8^9 for T/F where p = 0.20
• Pr[2 correct] = C[10,2]*0.5^2*0.5^8 for T/F where p = 0.5; or Pr[2 correct] = C[10,2]*0.2^2*0.8^8 for T/F where p = 0.20

#### Oliver Carson

##### New Member
Hi David, thanks for the explanation. One thing I am not clear on, is if we are looking for the probability of 3 correct answers out of 10 why then do we sum the probabilities of p(0), p(1) and p(2) as opposed to p(1), p(2) and p(3)? And then why is the answer one minus the sum of these outcomes?
Thanks.

Hi Quant_Noob, I'm really glad you brought this up , the commonality shows how the FRM really likes the binomial so it's a good spend of time ("fair use" allows me to copy the questions as I am adding commentary).

Question 2011.P1.6:

Question 2012.P1.14:

As is often the case, the hard part is maybe formulating the question. Both are binomial distribution: each question is a Bernoulli (i.e,. the student either answers correctly or incorrectly; 1 or 0) and a "test" is a series of Bernouillis (binomial = series of i.i.d. Bernouillis). Both tests are 10 questions (n = 10). Both tests want the probability that X >= 3 (i.e., passing) which is the same as 1 - Pr[not passing] = 1 - Pr[0 correct] + Pr[1 correct] + Pr[2 correct]. So both are essentially similar in asking, what is the Pr [X = 0 | X = 1 | X = 2] if distribution is binomial with n = 10. The only difference is probability of a correct answer (p), which is 50% on T/F and 20% with five choices. So,
• Pr[0 correct] = C[10,0]*0.5^0*0.5^10 for T/F where p = 0.5; or Pr[0 correct] = C[10,0]*0.2^0*0.8^10 for T/F where p = 0.20
• Pr[1 correct] = C[10,1]*0.5^1*0.5^9 for T/F where p = 0.5; or Pr[1 correct] = C[10,1]*0.2^1*0.8^9 for T/F where p = 0.20
• Pr[2 correct] = C[10,2]*0.5^2*0.5^8 for T/F where p = 0.5; or Pr[2 correct] = C[10,2]*0.2^2*0.8^8 for T/F where p = 0.20
th

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @Oliver Carson Because of the question that was asked: "What is the probability that the student will get at least three questions correct?" This question wants us to divide the distribution of outcomes into two pieces:
• Correct of two or fewer, Pr[ correct ≤ 2] which with discrete, whole numbers (binomial) is the same as Pr[ correct < 3]
• Correct of three or more, Pr [correct ≥ 3] which with discrete, whole numbers (binomial) is the same as Pr [correct > 2]
If we divide the entire distribution into these two pieces, since by definition of a probability distribution the sum of probabilities is 1.0 or 100%, we can find Pr [correct ≥ 3] = 100% - Pr[ correct ≤ 2] = 1 - Pr[ correct ≤ 2]

Re: p(1), p(2) and p(3): we need to include p(0): zero correct is a possible outcome. Thanks,

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