8007 Exam II: Mathematical Foundations of Risk Measurement - 2015 Edition Questions and Answers

Your stockbroker randomly recommends stocks to his clients from a tip sheet he is given each day. Today, his tip sheet has 3 common stocks and 5 preferred stocks from Asian companies and 3 common stocks and 5 preferred stocks from European companies. What is the probability that he will recommend a common stock AND/OR a European stock to you when you call and ask for one stock to buy today?

I have a portfolio of two stocks. The weights are 60% and 40% respectively, the volatilities are both 20%, while the correlation of returns is 100%. The volatility of my portfolio is

Consider two securities X and Y with the following 5 annual returns:

X: +10%, +3%, -2%, +3%, +5%

Y: +7%, -2%, +3%, -5%, +10%

In this case the sample covariance between the two time series can be calculated as:

Find the roots, if they exist in the real numbers, of the quadratic equation

Maximum likelihood estimation is a method for:

The natural logarithm of x is:

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

Which of the following is consistent with the definition of a Type I error?

What is the sum of the first 20 terms of this sequence: 3, 5, 9, 17, 33, 65,…?

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 ?

An asset price S is lognormally distributed if:

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.

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

Find the first-order Taylor approximation p(x) for the function: at the point .

What is the indefinite integral of the function f(x) = ln(x), where ln(x) denotes the natural logarithmic function?

Let E(X ) = 1, E(Y ) = 3, Corr(X, Y ) = -0.2, E(X2 ) = 10 and E(Y2 ) = 13. Find the covariance between X and Y

If a time series has to be differenced twice in order to be transformed into a stationary series, the original series is said to be:

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

A 95% confidence interval for a parameter estimate can be interpreted as follows:

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

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?

The determinant of a matrix X is equal 2. Which of the following statements is true?

Let N(.) denote the cumulative distribution function of the standard normal probability distribution, and N ' its derivative. Which of the following is false?

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

A simple linear regression is based on 100 data points. The total sum of squares is 1.5 and the correlation between the dependent and explanatory variables is 0.5. What is the explained sum of squares?

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?

Let X be a random variable normally distributed with zero mean and let . Then the correlation between X and Y is:

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

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?

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

Which statement regarding the matrix below is true?

Two vectors are orthogonal when:

Let A be a square matrix and denote its determinant by x. Then the determinant of A transposed is:

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:

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?

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