8002 PRM Certification - Exam II: Mathematical Foundations of Risk Measurement Questions and Answers

Which of the provided answers solves this system of equations?

2y – 3x = 3y +x

y2 + x2 = 68

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?

Suppose we perform a principle component analysis of the correlation matrix of the returns of 13 yields along the yield curve. The largest eigenvalue of the correlation matrix is 9.8. What percentage of return volatility is explained by the first component? (You may use the fact that the sum of the diagonal elements of a square matrix is always equal to the sum of its eigenvalues.)

Maximum likelihood estimation is a method for:

Consider the linear regression model for the returns of stock A and the returns of stock B. Stock A is 50% more volatile than stock B. Which of the following statements is TRUE?

An asset price S is lognormally distributed if:

Let a, b and c be real numbers. Which of the following statements is true?

What can be said about observations of random variables that are i.i.d. a normally distributed?

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?

What is the simplest form of this expression: log2(165/2)

You are to perform a simple linear regression using the dependent variable Y and the independent variable X (Y = a + bX). Suppose that cov(X,Y)=10, var(X)= 5, and that the mean of X is 1 and the mean of Y is 2. What are the values for the regression parameters a and b?

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

The Lagrangian of a constrained optimisation problem is given by L(x,y,λ) = 16x+8x2+4y-λ(4x+y-20), where λ is the Lagrange multiplier. What is the solution for x and y?

For each of the following functions, indicate whether its graph is concave or convex:

Y = 7x2 + 3x + 9

Y = 6 ln(3x)

Y = exp(-4x)

Which of the following is not a direct cause of autocorrelation or heteroskedasticity in the residuals of a regression model?

Which of the following properties is exhibited by multiplication, but not by addition?

A linear regression gives the following output:

Figures in square brackets are estimated standard errors of the coefficient estimates. What is the value of the test statistic for the hypothesis that the coefficient of is zero against the alternative that is less than zero?

Suppose that f(x) and g(x,y) are functions. What is the partial derivative of f(g(x,y)) with respect to y?

Which of the following can induce a 'multicollinearity' problem in a regression model?

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

A quadratic form is

Evaluate the derivative of exp(x2 + 2x + 1) at the point x = -1

The gradient of a smooth function is

An option has value 10 when the underlying price is 99 and value 9.5 when the underlying price is 101. Approximate the value of the option delta using a first order central finite difference.

Which of the following statements is true?

The quarterly compounded rate of return is 6% per annum. What is the corresponding effective annual return?

You want to test the hypothesis that a population parameter β of a regression model is zero. Your alternative hypothesis is that β≠0. Denote by SD(β) the estimated standard deviation of β, and by MEAN(β) the estimated mean of β. Which test statistic is appropriate, and what is its distribution?

What is a Hessian?

What is the probability of tossing a coin and getting exactly 2 heads out of 5 throws?

Which of the following is not a sequence?

What is the 40th term in the following series: 4, 14, 30, 52, …?

The fundamental theorem of analysis establishes a relation between

Suppose 60% of capital is invested in asset 1, with volatility 40% and the rest is invested in asset 2, with volatility 30%. If the two asset returns have a correlation of -0.5, what is the volatility of the portfolio?

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?

An indefinite integral of a polynomial function is

A 2-step binomial tree is used to value an American put option with strike 104, given that the underlying price is currently 100. At each step the underlying price can move up by 20% or down by 20% and the risk-neutral probability of an up move is 0.55. There are no dividends paid on the underlying and the discretely compounded risk free interest rate over each time step is 2%. What is the value of the option in this model?