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A resource for the Professional Risk Manager (PRM) exam candidate. Sample PRM exam questions, Excel models, discussion forum and more for the risk professional.

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Exam 1 (Finance Theory, Financial Instruments and Markets)

Exam 1 covers basic financial theory, securities and markets. If you are looking for free sample questions, click this link: Sample practice questions for PRM Exam I

The link 'practice all questions' (paid access only) below will allow you to review all questions and answers sequentially. At any point, you can check your answer by clicking the 'check answer' tab to the left. Or you can complete as many questions as you wish to in one sitting, and review your responses together.

The link 'simulated exam' (paid access only) launches 36 questions based upon the topic weights that PRMIA uses. For each topic, the questions are selected at random from Riskprep's question pool relating to that topic.

Practice All Questions: Exam 1
Our Question bank currently includes 287 practice questions . This link will let you review all questions.

Simulated Test: Exam 1
Review randomly selected 36 questions weighted according to the PRM syllabus.

Tutorials:

	Stochastic processes	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.

	Risk adjusted performance measures	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.

	Introduction to vanilla options	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.

	Valuing an option	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.

	Understanding option Greeks	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.

	Option strategies	A brief discussion of option strategies relevant to the PRM exam

	Volatility, returns and the behavior of stock prices	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: Continuous returns and discrete returns models of stock prices, An understanding of how volatility affects the net realized returns obtained by an investor, and how the 'average' of discrete returns over a period of time can be misleading, How the true returns obtained by an investor are driven lower by the volatility of the discrete returns, The difference between log returns and discrete returns, and how the two are related, and Why is the formula for expected stock prices in the future and not e^(mu+sigma^2 /2)

	Modeling interest rate changes	Modeling the behavior of short term interest rates We have seen how we expect stock prices to behave – returns being normally distributed and consequently prices distributed lognormally. If you recall, the process for the behavior of stock prices is explained as follows The same process however cannot be applied to interest rates. This is because while stock prices may follow a random walk, interest rates are generally considered mean reverting. Mean reverting means they tend to come back to some long term average, and can’t increase or decrease indefinitely. 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. What should you know? Well, in all likelihood you will not be asked a question on this topic, but if you have 20 minutes, have a read and you will be slightly better prepared. The models of short term interest rates help determine the shape of the yield curve, and option pricing on bond options. There are two broad categories of models of the short rate – Equilibrium models, and No-arbitrage models. Equilibrium models The Vasicek model In the Vasicek model, interest rates can be modeled using the following equation: dr = a(b – r)dt + σ dz where dr is the change in the rate, a is the ‘speed of reversion’ to the mean, b is the long term mean for the rate, σ is the volatility of the rate, and dz is a weiner process. (Recall that for our practical purposes a ‘weiner process’ is nothing but a random drawing from a normal distribution). Interest rate volatility is constant at σ. With the Vasicek model, rates will get pulled to the long term mean ‘b’ because of the first term in the formula above. With the equation dr = a(b – r)dt + σ dz it is possible (with some more mathematics, and with known values of a, b and r) to determine the price of a zero coupon bond at any time t in the future, and thereby construct an yield curve. This yield curve will likely be different from the real one in the markets at the present time, and this divergence from reality is a big limitation of the Vasicek model. One way to overcome this limitation is to choose a, b and r in such a way that we get an yield curve close to the current yield curve, but that is often not a practical solution as such ‘fitting’ still leaves large errors. The Cox, Ingersoll, and Ross model According to the Cox, Ingersoll and Ross model, the risk neutral process for the short rate r is as follows: dr = a(b – r) dt + σ √r dz The only difference between the Vasicek model and this is that the change in the short rate is also determined by r, in the form of √ r being incorporated in the formula. This has the effect of increasing the standard deviation of the rate as r goes up. Both the Vasicek and the Cox, Ingersoll and Ross models are single factor models, dependent only upon the value of r as the single factor driving changes to short rates. No-arbitrage models over this limitation by providing the presently known yield curve as an input into the model. The ‘drift’ of the interest rate is determined from the yield curve – implying that if the yield curve is sloping downwards, then the drift is negative, zero if the yield curve is flat, and positive if the yield curve is rising. The drift has ‘time’ associated with it, ie the rise or fall of the rate per unit of time. The model therefore ends up predicting the range of short term interest rates around the known yield curve. No-arbitrage models The yield curves predicted by the equilibrium models are generally different from what are being observed at the current time in the markets. It is possible to tweak the values of a and b to arrive to a close fit, but still large errors often remain. This problem is solved by using the current term structure as an input into the model so the model ends up being exactly consistent with the today’s term structure. The Ho-Lee model Under the Ho-Lee model, dr = Θ (t) dt + σ dz All variables have their usual meaning, but Θ(t) is a function of time selected to fit the initial term structure. Θ(t) can be analytically calculated using a formula (probably not relevant for the exam), suffice it to say it is a function time t. The Hull White model The Hull White model is an extension of the Vasicek model with the difference that it can be made to fit the current term structure. Under the Hull White model: dr = [Θ(t} – ar]dt + σ dz dr = a[Θ(t}/a– r]dt + σ dz The mean reversion happens at the rate a. At time t, the rate shows a mean reversion to Θ(t)/a at rate a. The Black-Derman-Toy model Under the Ho-Lee and Hull-White models, interest rates can become negative. The BDT model allows only positive interest rates, and is as follows: d ln⁡r= [Θ(t)–σ^'/σ ln⁡(r) ]dt + σ(t)dz Where Θ(t) is a function of time, and σ is the volatility of the short rate. The models of short rates are used to price fixed income derivatives and bonds. 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. What should you know? Well, in all likelihood you will not be asked a question on this topic, but if you have 20 minutes, have a read and you will be slightly better prepared. Recognize the formulae for the 5 models mentioned here, and try to remember some of the differences between the no-arbitrage and equlibrium models. Or you could entirely skip this as well. I will be putting a few questions in on this subject, but will clearly mark them as unlikely.

Modeling the behavior of short term interest rates