# PRM Certification - Exam II: Mathematical Foundations of Risk Measurement v6.0

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Exam contains 135 questions

What is the total derivative of the function f(x,y) = ln(x+y), where ln() denotes the natural logarithmic function?

• A. 1 / (x+y)
• B. (x + y) / (x+y)
• C. -x/(x+y) - y/(x+y)
• D. ln(x+y) x + ln(x+y) y

A biased coin has a probability of getting heads equal to 0.3. If the coin is tossed 4 times, what is the probability of getting heads at least two times?

• A. 0.7367
• B. 0.3483
• C. 0.2646
• D. None of these

For a quadratic equation, which of the following is FALSE?

• A. If the discriminant is negative, there are no real solutions
• B. If the discriminant is zero, there is only one solution
• C. If the discriminant is negative there are two different real solutions
• D. If the discriminant is positive there are two different real solutions

Stress testing portfolios requires changing the asset volatilities and correlations to extreme values. Which of the following would lead to a non positive definite covariance matrix?

• A. Changing the volatilities to be greater than 100%
• B. Changing all the correlations to be unity
• C. Changing all the correlations to be zero
• D. All of the above

In a multiple linear regression, the significance of R2 can be tested using which distribution?

• A. Normal distribution
• B. Student's t distribution
• C. F-distribution
• D. Binomial distribution

What is the angle between the following two three dimensional vectors: a=(1,2,3), b=(-
4,2,0)?

• A. 90 degrees
• B. 180 degrees
• C. 57 degrees
• D. 45 degrees

Which of the following is a false statement concerning the probability density function and the cumulative distribution function of a random variable?

• A. the PDF is non-negative.
• B. the definite integral of the CDF from minus infinity to plus infinity is undefined.
• C. the CDF approaches 1 as its argument approaches infinity.
• D. the definite integral of the PDF from minus infinity to plus infinity is zero.

An operational risk analyst models the occurrence of computer failures as a Poisson process with an arrival rate of 2 events per year. According to this model, what is the probability of zero failures in one year?

• A. 0.02
• B. 0.14
• C. 0.25
• D. 0.50

In a portfolio there are 7 bonds: 2 AAA Corporate bonds, 2 AAA Agency bonds, 1 AA
Corporate and 2 AA Agency bonds. By an unexplained characteristic the probability of any specific AAA bond outperforming the others is twice the probability of any specific AA bond outperforming the others. What is the probability that an AA bond or a Corporate bond outperforms all of the others?

• A. 5/7
• B. 8/11
• C. 6/11
• D. None of these

If a random variable X has a normal distribution with mean zero and variance 4, approximately what proportion of realizations of X should lie between -4 and +4?

• A. 66.60%
• B. 90%
• C. 95%
• D. 99%

A linear regression gives the following output:
Figures in square brackets are estimated standard errors of the coefficient estimates.
Which of the following is an approximate 95% confidence interval for the true value of the coefficient of ?

• A. [0, 1.5]
• B. [1, 2]
• C. [0, 3]
• D. None of the above

A bond has modified duration 6 and convexity 30. Find the duration-convexity approximation to the percentage change in bond price when its yield increases by 5 basis points

• A. 10 basis point rise
• B. 24 basis fall
• C. 24 basis point rise
• D. 30 basis points fall.

Which of the following statements about skewness of an empirical probability distribution are correct?
1. When sampling returns from a time series of asset prices, discretely compounded returns exhibit higher skewness than continuously compounded returns
2. When the mean is significantly less than the median, this is an indication of negative skewness
3. Skewness is a sign of asymmetry in the dispersion of the data

• A. All three statements are correct
• B. Statements 1 and 2 are correct
• C. Statements 1 and 3 are correct
• D. Statements 2 and 3 are correct

You invest \$100 000 for 3 years at a continuously compounded rate of 3%. At the end of 3 years, you redeem the investment. Taxes of 22% are applied at the time of redemption.
What is your approximate after-tax profit from the investment, rounded to \$10?

• A. \$9420
• B. \$7350
• C. \$7230
• D. \$7100

Which of the following statements is not correct?

• A. Every linear function is also a quadratic function.
• B. A function is defined by its domain together with its action.
• C. For finite and small domains, the action of a function may be specified by a list.
• D. A function is a rule that assigns to every value x at least one value of y.