Thanks David
20 Nov 2008
Jan
21
FRM 2008 - Early Bird Episode #2
by David Harper, CFA, FRM, CIPM
Contents
- Learning Objectives
- Video tutorial
- Practice Questions (4)
Learning Objectives
For week 2 of the FRM 2008 early bird, I recorded a 30-minute tutorial that introduces three basic distributions:
- Binomial distribution (discrete)
- Poisson distribution (discrete)
- Normal distribution (continuous)
How important are distributions? Kevin Dowd says,
“Any risk measure at its most basic level involves an attempt to capture or summarize the shape of an underlying density function…” - Kevin Dowd, Market Risk
Ideas in the tutorial include:
- Two ways to look at distribution functions: 1. discrete versus continuous; and 2. density (pdf) versus cumulative
- Binomial is discrete with two params: p = probability of success and n = number of trials
- Poisson is discrete with only one param: lambda = mean = variance.
- Key normal properties; e.g., only two params (mean and standard deviation), symmetrical, not fat-tailed.
Video Tutorial
The 30-minute video tutorial is located here (with a table of contents).
If you are a paid member, you can also access this in the member section (where you will also find the downloadable slides, if you would like to view those. As well as an ipod format file.)
Practice Questions
Here are four practice questions. I wrote these especially to help you get familiar with the distributions. I hope you find time to work them!
Question #1 (binomial)
See image below. Let's assume a four-step binomial tree for the random walk of a stock that starts at price S(0). At each node, the stock can jump up (u) or down (d). This is a recombining binomial tree. That means, for example, the middle node (ud) at the second step is reached in two ways, either: an up jump (u) followed by a down jump (d), or a down jump (d) followed by an up jump (d). In other words, ud = du. Similarly, the node (uud) at the third step can be reached three ways: uud = udu = duu.
Assume S(0) = $10, p = 50%, u = 1.1 and d = 0.9.
That is, today's stock price, S(0), is $10. The probability (p) of an up jump is 50%. The magnitude of an up jump (u) is 1.1 and the magnitude of a down jump (d) is 0.9. For example, at the first step, the price of S(1) will be either $11.00 ($10 x 1.1 = $11, in the case of an up jump) or $9.00 ($10 x 0.9 = $9, in the case of a down jump).
(i) How many total paths are possible through the tree (separately counting even those paths that redundantly reach the same final nodes)?
(ii) How many paths can be taken to reach the middle node at S(4). That is, the node reached with two ups and two down jumps (e.g., uudd)? (Please try to express as a combination)
(iii) At the end of the tree are five possible events. Which of the five is most likely and what is its probability?
(iv) What is the stock’s expected value at S(4)?
(v) The final distribution at S(4) is a discrete binomial distribution, what is the variance of this binomial distribution?
(vi) Tough (unfair) question: assume we are not given p, u, or d. Rather, assume that each step is one year, that the annualized volatility of the stock is 20%, that the (annual) riskless rate is 5%, and that each step is three months; i.e., S(4) is one year. What are p, u, and d?
(vii) Tougher still: Now assume the annualized volatility is 20% and the (annual) riskless rate is 5%, but instead assume each time step is three months (0.25) such that S(4) is reached in one year. What are p, u, and d?
Question #2 (Poisson)
A company produces widgets but with a 2% defect rate (i.e., two widgets out of 100 widgets are defective, on average). The next production run will contain 100 widgets. Assume the Poisson distribution is a good fit.
(i) What is the probability that exactly two (2) defects will be produced?
(ii) What is the probability that no more than two (2) defects will be produced (i.e., less than three defects)?
(iii) A defect is considered a "rare event" (i.e., small probability and large number of trials). In this case, the binomial distribution closely approximates the binomial. Under the binomial distribution, what is the probability of exactly two (2) defects?
(iv) What is the mean and variance of this Poisson distribution?
(v) Can the normal distribution approximate this Poisson distribution? If not, under what circumstances could the normal approximate this Poisson?
Question #3 (Normal)
Assume mutual fund period returns are normally distributed with mean expected return of 8% and standard deviation of 10%.
(i) If a particular return is +15%, how many standard units is that from the mean?
(ii) Approximately what percentage of funds should return more than +15%? (simplistic single period assumption, nothing fancy)
(iii) What is the 95% parametric value at risk (VaR), in return terms?
(iv) Actual equity returns tend to violate normality. Cite at least one key difference between the empirical tendencies of equity returns and normality?
Question #4a (Normal) - difficult
Assume a stock with a current price of $10; S(0) = $10. Further, the stock has an expected return of 8%, continuously compounded; with annualized volatility of 20%. Over a single one-year period, what is the 90% confidence interval (i.e., from lowest price to highest price) for the future stock price in one year? Note this is a question about the future price level.
Question #4b (Normal) - difficult
Assume the same stock with current price of $10; S(0) = $10. And, again, the (continuously compounded) expected rate of return on the stock is 8% with annual volatility of 20%.
We estimate the company will default if the price in one year drops to $6.80 or lower. What is approximately the probability of default (PD) under this simple one-period assumption?
Answers to practice questions
That's all for this week. See you next week!
David Harper, CFA, FRM, CIPM
Founder
www.bionicturtle.com
Comments
I want to pay up for 2008. How do I go about?
Hi David
There is some problem downloading this episode.
Thanks
Neha
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