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Questions 4

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?

Questions 5

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

Questions 6

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?

Questions 7

When calculating the implied volatility from an option price we use the bisection method and know initially that the volatility is somewhere between 1% and 100%. How many iterations do we need in order to determine the implied volatility with accuracy of 0.1%?

Questions 9

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 ?

Questions 10

You are given the following regressions of the first difference of the log of a commodity price on the lagged price and of the first difference of the log return on the lagged log return. Each regression is based on 100 data points and figures in square brackets denote the estimated standard errors of the coefficient estimates:

Which of the following hypotheses can be accepted based on these regressions at the 5% confidence level (corresponding to a critical value of the Dickey Fuller test statistic of – 2.89)?

Questions 12

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

Questions 13

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

Questions 14

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.

Questions 15

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

Questions 18

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?

Questions 19

Which of the following statements are true about Maximum Likelihood Estimation?

(i) MLE can be applied even if the error terms are not i.i.d. normal.

(ii) MLE involves integrating a likelihood function or a log-likelihood function.

(iii) MLE yields parameter estimates that are consistent.

Questions 20

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

Questions 22

The gradient of a function f(x, y, z) = x + y2 - x y z at the point x = y = z = 1 is

Questions 23

Let X be a random variable distributed normally with mean 0 and standard deviation 1. What is the expected value of exp(X)?

Questions 24

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?

Questions 25

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?

Questions 26

What is the maximum value of the function F(x, y)=x2+y2 in the domain defined by inequalities x ≤ 1, y ≥ -2, y-x ≤ 3 ?

Questions 32

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

Questions 33

Exploring a regression model for values of the independent variable that have not been observed is most accurately described as…

Questions 34

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?

Questions 36

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

Questions 37

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

Questions 38

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

Questions 39

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