Calculus/Integration

[Serdar7891]

Hi David,
Apologies if this question is non-sensible or has been asked already (please tag a link if this question has been asked) but I just started preparing for an exam and I have one question so I know what to expect when I am learning it. Integration and calculus sort of questions as far as I know are not really asked on the exam. However, learning how to integrate will be beneficial on the exam, so my question is, how exactly?
I.e you have a question on P1.T2.301. Miller's probability matrix:

301.1. A random variable is given by the discrete probability function f(x) = P[X = x(i)] = a*X^3 such that x(i) is a member of {1, 2, 3} and (a) is a constant. That is, X has only three discrete outcomes. What is the probability that X will be greater than its mean? (bonus: what is the distribution's variance?)

Knowing this and how to perform calculation on example above, can you just give me one FRM question where the knowledge of solving above example would help me solve other related questions?

Thank you very much in advance.

 

[David Harper CFA FRM]

Hi @Serdar7891 as the exam was Saturday, and my support schedule was relentless, I am taking a bit of a pause in the forum ... if it's okay with you, I will respond to your (very good actually ...) question next week? Thanks!

 

[Serdar7891]

Hi David,
Of course, no worries.

Thank you.

 

[David Harper CFA FRM]

Hi @Serdar7891 Thank you for your patience. First, in case you haven't seen Bill May's reply to my previous calculus question. Bill May is GARP's SVP, FRM and he is basically the top guy in charge of the FRM. I previously shared his reply over here but is copied below (new emphasis is mine); i.e.,

@PortoMarco79 I received a response to my question ....

Subject: Common question asked about P1.T2. Quantitative Methods (Miller)
Hi Bill,
Does GARP have any "official guidance" on the role of calculus in the actual exam?
Especially because the Miller reading utilizes (basic) calculus, we are often asked "Do I need to know any calculus?"

fwiw, i typically explain that from a practical standpoint, time limitations preclude the tedium implied by (eg) differentiating or integrating and, further, the historical record of questions generally does not DIRECTLY query calculus, but that an understanding of basic calculus is recommended. For example, it's very helpful to understand that a distributional pdf integrates to its CDF. Thanks for any insight! Cheers, David

Bill May replied:

David,

This is a not uncommon question for us as well. Let me respond in two parts:

1)The FRM Candidate Guide describes the general quantitative level of the Exam thusly:

Q. How quantitative is the FRM Exam?

A. The FRM Exam has a quantitative component, but these quantitative concepts are presented in the context of questions that approximate real-world situations that risk managers would face. The level of mathematical rigor of the FRM Exam is consistent with an advanced undergraduate or introductory graduate level finance course at most universities.

2) Specifically regarding calculus, the FRM Exam does not explicitly require an understanding of calculus and, as a perusal of the learning objectives will indicate, does not test directly on differentiation, integration, or other aspects of calculus. That being said, an understanding of calculus, not to mention matrix manipulation that is typically associated with linear algebra, would likely prove beneficial to candidates as they prepare for the Exam. Many of tools and techniques related to probability, statistics, modelling, and estimation draw on these concepts and the better candidates understand these topics the better prepared they are likely to be.

I hope that helps.

Bill

The question of mine that you've quoted (i.e., 301.1) is actually consistent with questions asked by the assigned author's text, Miller's Math and Statistics for FRM. I mention this to "defend it" on the grounds that it's typical in the way it stays within the assigned text. I have definitely been accused of going "too deep," but my questions rarely stray from strict adherence to the assigned text(s) and learning objective(s). Also, by way of defense, please note that my calculus questions do not go (much, if at all) beyond application of the power rule, which is the really just the entry point in calculus (it's basic calculus). I don't mean to be brusk, but if I need to defend the utility of the power rule, then I'd almost rather not bother

Re: can you just give me one FRM question where the knowledge of solving above example would help me solve other related questions?
No, I cannot give you a direct example (nor would searching one out be a good use of my time candidly)

The relevance, in my opinion, is indirect. But strongly indirect. Here are three reasons why I think that knowledge of the most basic ("entry level") calculus, and/or my question above, are relevant to an FRM candidate:

1. Several key concepts indirectly apply such math. For example, expected shortfall and, importantly, linear approximation VaR concepts (e.g., Portfolio VaR). It's really hard to grok marginal (portfolio) VaR and component VaR with words, but really easy if you are fluent with first partial derivatives. Most importantly, perhaps, the first partial derivative is arguably the single most important common idea in analytical VaR: via Taylor Series, it underlies option delta, bond duration, and Portfolio VaR (marginal VaR) among others. These concepts are almost trivial if you understand (even just basic) differentiation, but somewhat difficult if you must approach them only textually.
2. It develops what is called "mathematical thinking." When the density function is expressed by f(x) = P[X = x(i)] = a*X^3, we are unequivocally tested to see if we really do understand what exactly is meant by a discrete probability function. The use of the constant, a, actually has been tested (albeit not the integration), as my template for this question was Gujarati; so, you could say that "about half" of this question has appeared on the exam.
3. Related to my 2nd point, understanding the math (and arithmetic) greatly increases our qualitative understanding. To illustrate, one of the EPPs entered the FRM space with author(s) who skipped the mathematical rigor and it's no secret that it's been a disaster (because you can't skip math in the FRM ...). Here is a recent example/discussion at https://www.analystforum.com/forums/frm-forum/91368353 ... the author probably never calculated kurtosis. Does the exam make you calculate kurtosis? Probably not. But the calculation is not difficult, and knowing the calculation enables you to speak intelligently and accurately about kurtosis. This EPP also refers in a video to beta as "volatility;" but if you know the calculation, then you know that beta is "correlation multiplied by cross-volatility." I have lost count at how many examples have been sent to me w.r.t this EPP: where they've made simple mistakes because they skipped the math. My point here is about indirect relevance: while the affirmative calculation itself may not be tested, familiarity with the calculation in many cases enables you to be accurate even in qualitative statements. I can make this argument easily for power rule (entry level) calculus but hopefully I've made the point ...

Please let me know if that is a satisfactory defense? Thank you!

 

[Serdar7891]

Hi Dawid,
Yes, it is. Thank you for such a detailed answer. I appreciate it.

Serdar