Posts by Mukul Pareek

This article is a practical and real example of how value-at-risk, marginal VaR, betas, undiversified VaR and component VaRs are calculated.  Credit to Pawel Lachowicz for the idea for this article, he has done the same thing in Python here: http://www.quantatrisk.com/2015/01/18/applied-portfolio-value-at-risk-decomposition-1-marginal-and-component-var/

This article covers how Marginal and Component VaRs are calculated.  We follow up (in a separate article) with a real life example of how VaR, MVaR, undiversified VaR, Component VaR are calculated – based on actual price data pulled from Quandl, an open source market data website.

The topic of kurtosis can cause some confusion.  Think fat and thin tails, and “peakedness” and leptokurtosis.  How is it that having a ‘pointier’ peak means having fat tails, and how is that different from a normal distribution that just has a smaller standard deviation? And how does a t-distribution manage to have fatter tails when its peak can in fact be lower than that of the normal distribution?

This article attempts to provide an intuitive understanding of what PCA is, and what it can do.  PRMIA has been asking questions on PCA, but the way the subject is presented in the Handbook is not appropriate for someone who has not studied it before in the classroom. This article aims to provide an intuitive understanding of what PCA is so you can approach the material in the handbook with confidence.

Linear regression is an important concept in finance and practically all forms of research. It is also used extensively in the application of data mining techniques. This article provides an overview of linear regression, and more importantly, how to interpret the results provided by linear regression.

Modeling the behavior of short term interest rates

There are a number of ways that are used for modeling short term interest rates. These have not been covered in the PRMIA handbook, but they find a reference in one of their study guides. So just to be cautious, a bit of explanation for these is provided here in case there are questions in the exam relating to these concepts.

This article follows the earlier tutorial on stochastic processes.  It talks about a couple of related things, and tries to provide an intuitive understanding of some of the following concepts:

This is the final of five articles – each explaining at a high level one each of the five credit risk models in the PRMIA handbook.  This write-up deals with the actuarial, or the ‘CreditRisk+’ model.Credit Risk +, or the actuarial approach.

This is the fourth of five articles covering each of the main portfolio approaches to credit risk as explained in the handbook.  The idea is to provide a high level and concise explanation of each of the approaches so it may be easier to deal with the detail provided in the handbook.  I hope you find it useful.

This is the third of five articles covering credit risk – this one addresses the ‘structural approach’.

This is the second of five articles that discuss the various approaches to measuring credit risk in a portfolio.  This article covers CreditPortfolio view.  CreditPortfolio View is conceptually not too dissimilar from the Credit Metrics model described earlier, ie it relies upon a knowledge of the transition matrices between the different credit ratings.  The only difference is that the transition matrix itself has an adjustment applied to it for the business cycle.  But once this adjusted transition matrix has been obtained, the rest of the process works in the same way as for the Credit Metrics model.

The conceptual idea behind Black Scholes is rather simple – but as the argument advances beyond the initial idea, things become more complex with differential equations, risk-neutrality and log returns stepping in.  For the PRMIA exam, you will not be asked for a derivation of Black Scholes, so it may suffice to know just a couple of things.  This brief write-up aims to summarize just those few things.

This is a quick write up on eigenvectors, eigenvalues, orthogonality and the like. These topics have not been very well covered in the handbook, but are important from an examination point of view.

First and second derivatives are important in finance – in particular in measuring risk for fixed income and options.  In fixed income – the first and second derivatives are modified duration and convexity respectively, and for options, these are delta and gamma.  But what do these really mean – and what does one think about them when one sees a number?  The rest of this article attempts to provide an intuitive look at how price changes for a bond (or an option) are determined by the first and the second derivative, what they mean, and how they are to be interpreted.

This is the first of five articles that provide a high level understanding of the various portfolio models of credit risk covered in the PRMIA syllabus.  Being the first one, this discusses the credit migration framework.  (I am still working on the others.)  This article is intended to provide a conceptual understanding of the approach and I have not provided numerical examples for the reason that I don’t want to duplicate what is already there in the Handbook.  Once you have read this, the scattered explanation in the Handbook will hopefully make more sense.

When constructing portfolios we are often concerned with the return (ie the mean, or expected value), and the risk (ie the volatility, or standard deviation) of of combining positions or portfolios.  We may also be faced with situations where we need to know the risk and return if position sizes were to be scaled up or down in a linear way.  This brief article deals with how mean and variances for two different variables can be combined together, and how they react to being added or multiplied by constants.

This is a brief article on default correlations – what it means, and how to interpret it.  To keep things simple, let us consider only two securities– A and B.  Let us look at how default correlations are calculated, and then try to think about how to intuitively interpret a given default correlation number between two securities. 

This brief article intends to clarify the differences between some concepts relating to credit VaR.  One thing to note about credit risk is that you need to watch out whether you are inferring VaR from a distribution of the value of the portfolio, or from a distribution of the losses in the portfolio.  One is a mirror image of the other, they give the same results, but they are not identical and you should intuitively understand the difference.

The constituents of capital under Basel II

Basel II provides for three tiers of capital.  Tier 1 is the purest and most reliable form of capital.  The agreement provides limits on how much Tier 2 or Tier 3 capital can be relied upon for capital adequacy, the idea being to make sure that there is always sufficient Tier 1 capital available.  Of course, Tier 1 capital needs no limits, the more the better.

It is important to understand continuously compounded rates.  These rates are rarely encountered in day-to-day life, but are relevant to a finance professional.  You will never see, for example, a bank advertise ‘continuously compounded rates’ for its deposits.  (In fact, it may even be against the law to do so as they may be required to disclose easier to understand APRs).

This is a very brief article, perhaps unjust given what it covers.  I have tried to keep it very short, so as to be a practical reference to key statistical terms that are used throughout risk management.  This covers standard deviation, variance, covariance, correlation, regression and the famous ‘square root of time’ rule.  The PRMIA handbook has more stuff but this covers the key things you must know – almost by heart!

Value at risk is affected by tails and there is so much stuff in the PRMIA handbook about dealing with heavy tails.  This can be confusing as the handbook sort of presumes an understanding of how tails affect VaR – so here is a short tutorial to explain how heavy tails affect Value at Risk.

A lot of finance and risk management is about distributions.  For the PRMIA exam, you really need to understand the concepts underlying distributions, what different shapes mean, what the parameters are, what a cdf is (vs a pdf) and which to use when.  Of course, the most commonly used distribution assumption is that returns are normally distributed, so this article talks about the normal distribution and also other important distributions.  More importantly, I have provided spreadsheets that model each of the distributions so you can play around and see the behaviour of the distribution as you change the parameters underlying it.

A quick refresher on logarithms.  Just the basics, and not a whole lot more.  Explains what logarithm means, how it is related to e, how to add/subtract logs and convert bases.

Quick refresher on arithmetic and geometric progressions – straightforward, and lists just the formulae.  Not a whole lot.

If you are new to finance, or haven’t actually done much math in a while, the differences between discrete, compounded and continuously compounded interest rates can be quite confusing.  You may go through many chapters in the handbook while still having a nagging doubt as to if you really get the interest rate part – sometimes they use (1+r)^n, at other times it is exp(rn), what’s going on?  This brief article explains what continuously compounded interest rates are, how they work and how they are to be used. 

An easy hit in the PRMIA exam is getting the question based on covered interest parity right.  It will come with a couple of exchange rates, interest rates and dates, and there would be one thing missing that you will be required to calculate.  This brief write up attempts to provide an intuitive understanding of how and why covered interest parity works.  There are a number of questions relating to this that I have included in the question pool, and this article addresses the key concepts with some examples.

This article is about an Excel model for calculating portfolio variance.  When it comes to calculating portfolio variance with just two assets, life is simple.  But consider a situation when there are 10, 15, maybe hundreds of assets.  This brief article is a practical demonstration of how portfolio variance can be modeled in Excel – the underlying math, and an actual spreadsheet for your playing pleasure! Enjoy!

A brief discussion of option strategies relevant to the PRM exam

A ‘strategy’ is a combination of basic option positions. 

Option Greeks are tricky beasts with their complex formulae and intimidating names.  This article is not about the formulae at all. The idea is to discuss what each of the Greeks represent, and understand what drives each of them.  Often the PRMIA exam will ask a question about a Greek, and very likely that question will expect you to understand the relationship between the variable and the underlying asset, and how the Greeks can be used to measure and manage risks.

This rather brief article covers some familiar theory – how are options valued – but without repeating all the text you can find in a textbook.  Of course, we all know about the Black Scholes model, and anyone can punch it into Excel or use any number of online calculators available for free.  What this article tries to do is to provide an intuitive understanding of what drives option values, and also provides an Excel model incorporating the Black Scholes that you could use to play around with.

In this article we will discuss basic vanilla options, calls and puts, and understand payoff diagrams. We will also look at the put-call parity. The put-call parity is important to understand for the PRMIA exam as a number of questions, such as those relating to the relationship between call and put values, the additive nature of option Greeks is based upon the put-call parity.

Returns are the reward for taking risk: when there will be no risk, there will be no profits either.  This article discusses the Sharpe ratio, Treynor ratio, Information Ratio, Jensen’s alpha and the Kappa indices, which are all measures to evaluate risk adjusted performance.

Stochastic processes

In finance and risk, you will always be running into what are called ‘stochastic processes’.  Well, that is just a more complex way of saying that a variable is random.  A variable is considered ‘stochastic’ when its value is uncertain.  In finance, security returns are usually considered stochastic.  In this brief article , we look at some key concepts relating to stochastic variables, including the geometric Brownian motion process (borrowed from particle physics) which is often used to model asset returns.