8007 Practice Exam Questions with Answers Exam II: Mathematical Foundations of Risk Measurement - 2015 Edition Certification

Concerning a standard normal distribution and a Student's t distribution (with more than four degrees of freedom), which of the following is true?

What is a Hessian?

The correlation between two asset returns is 1. What is the smallest eigenvalue of their correlation matrix?

Kurtosis(X) is defined as the fourth centred moment of X, divided by the square of the variance of X. Assuming X is a normally distributed variable, what is Kurtosis(X)?

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 less than 1?

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?

Suppose a discrete random variable can take on the values -1, 0 and 1 each with a probability of 1/3. Then the mean and variance of the variable is

In a 2-step binomial tree, at each step the underlying price can move up by a factor of u = 1.1 or down by a factor of d = 1/u. The continuously compounded risk free interest rate over each time step is 1% and there are no dividends paid on the underlying. Use the Cox, Ross, Rubinstein parameterization to find the risk neutral probability and hence find the value of a European put option with strike 102, given that the underlying price is currently 100.

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

The correlation between two asset returns is 0.5. What is the largest eigenvalue of their correlation matrix?

An underlying asset price is at 100, its annual volatility is 25% and the risk free interest rate is 5%. A European call option has a strike of 85 and a maturity of 40 days. Its Black-Scholes price is 15.52. The options sensitivities are: delta = 0.98; gamma = 0.006 and vega = 1.55. What is the delta-gamma-vega approximation to the new option price when the underlying asset price changes to 105 and the volatility changes to 28%?

Assume that 40% of all financial organizations investigated by authorities turn out to be fraudulent.

What is the probability of randomly investigating 2 different organizations and finding that neither is fraudulent; and what is the probability of finding exactly one being fraudulent?

A 2-year bond has a yield of 5% and an annual coupon of 5%. What is the Macaulay Duration of the bond?

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

When the errors in a linear regression show signs of positive autocorrelation, which of the statements below is true?

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

You are investigating the relationship between weather and stock market performance. To do this, you pick 100 stock market locations all over the world. For each location, you collect yesterday's mean temperature and humidity and yesterday's local index return. Performing a regression analysis on this data is an example of…

The fundamental theorem of analysis establishes a relation between

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

Identify the type and common element (that is, common ratio or common difference) of the following sequence: 6, 12, 24

What is the maximum value for f(x)= 8-(x+3)(x-3)?

Calculate the determinant of the following matrix:

In statistical hypothesis tests, 'Type I error' refers to the situation in which…

Evaluate the derivative of ln(1+ x2) at the point x = 1

In a binomial tree lattice, at each step the underlying price can move up by a factor of u = 1.1 or down by a factor of . The continuously compounded risk free interest rate over each time step is 1% and there are no dividends paid on the underlying. The risk neutral probability for an up move is:

The sum of the infinite series 1+1/2+1/3+1/4+1/5+.... equals:

Newton-Raphson iteration is used to find a solution of x5 - x3 + x = 1. If xn = 2, what is xn+1?

A 2-year bond has a yield of 5% and an annual coupon of 5%. What is the Modified Duration of the bond?

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?

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?

Simple linear regression involves one dependent variable, one independent variable and one error variable. In contrast, multiple linear regression uses…

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)

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?

A quadratic form is