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# L1.T2.201. Random variables

#### David Harper CFA FRM

##### David Harper CFA FRM
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Today I start a fresh batch of questions devoted to Econometrics per the Stock & Watson assignment. This is really the start of fresh 2012 material. Since I have years of exposure to the econometrics, I hope (I think) I can bring you even higher-density questions. What I mean is, the questions below are a notch more difficult or time-consuming than the average exam question; however, that's often because I am cross-referencing, or touching on multiple concepts, or really asking 2/3 questions in one question ... so I would like to thinks it's a more effective use of your practice time - David

Define random variables, and distinguish between continuous and discrete random variables. Define the probability of an event. Define, calculate, and interpret the mean, standard deviation, and variance of a random variable.

Questions:

201.1. Which of the following is most likely to be characterized by a DISCRETE random variable, and consequently, a discrete probability distribution (aka, probability mass function, PMF) and/or a discrete CDF?
a. The future price of a stock under the lognormal assumption (geometric Brownian motion, GBM) that underlies the Black-Scholes-Merton (BSM)
b. The extreme loss tail under extreme value theory (EVT; i.e., GEV or GPD)
c. The empirical losses under the simple historical simulation (HS) approach to value at risk (VaR)
d. The sampling distribution of the sample variance

201.2. A model of the frequency of losses (L) per day, for a certain key operational process, assumes the following discrete distribution: zero loss (events per day) with probability (p) = 20%; one loss with p = 30%; two losses with p = 30%; three losses with p = 10%; and four losses with p = 10%. What are, respectively, the expected (average) number of loss events per day, E(L), and the standard deviation of the number of loss events per day, StdDev(L)?
a. E(L) = 1.20 and StdDev(L) = 1.44
b. E(L) = 1.60 and StdDev(L) = 1.20
c. E(L) = 1.80 and StdDev(L) = 2.33
d. E(L) = 2.20 and StdDev(L) = 9.60

201.3. A volatile portfolio produced the following daily returns over the prior five days (in percentage terms, %, for convenience): +5.0, -3.0, +6.0, -1.0, +3.0. Although this is a tiny sample, we have two ways to calculate the daily volatility. The first is to compute a technically proper daily volatility as an unbiased sample standard deviation. The second, a common practice for short-period/daily returns, is to make two simplifying assumptions: assume the mean return is zero since these are daily periods, and divide the sum of squared returns by (n) rather than (n-1). For this sample of only five daily returns, what is respectively (i) the sample daily volatility and (ii) the simplified daily volatility?
a. 1.65 (sample) and 2.55 (simplified)
b. 2.96 (sample) and 3.00 (simplified)
c. 4.11 (sample) and 3.65 (simplified)
d. 3.87 (sample) and 4.00 (simplified)

201.4. Consider the following five random variables:
• A standard normal random variable; no parameters needed.
• A student's t distribution with 10 degrees of freedom; df = 10.
• A Bernoulli variable that characterizes the probability of default (PD), where PD = 4%; p = 0.040
• A Poisson distribution that characterizes the frequency of operational losses during the day, where lambda = 5.0
• A binomial variable that characterizes the number of defaults in a basket credit default swap (CDS) of 50 bonds, each with PD = 2%; n = 50, p = 2%
Which of the above has, respectively, the lowest value and highest value as its variance among the set?
a. Standard normal (lowest) and Bernoulli (highest)
b. Binomial (lowest) and Student's t (highest)
c. Bernoulli (lowest) and Poisson (highest)
d. Poisson (lowest) and Binomial (highest)

Answers: