EMT Practice Test

1. Question Content...

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Question1: You want to test the hypothesis that a population parameter of a regression model is zero. Your alternative hypothesis is that 0. Denote by SD() the estimated standard deviation of , and by MEAN() the estimated mean of . Which test statistic is appropriate, and what is its distribution?

Question2: The bisection method can be used for solving f(x)=0 for a unique solution of x, when

Question3: When the errors in a linear regression show signs of positive autocorrelation, which of the statements below is true?

Question4: 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?

Question5: What is the probability of tossing a coin and getting exactly 2 heads out of 5 throws?

Question6: A quadratic form is

Question7: 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?

Question8: A 95% confidence interval for a parameter estimate can be interpreted as follows:

Question9: What is the simplest form of this expression: log2(165/2)

Question10: 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)?

Question11: An indefinite integral of a polynomial function is

Question12: Every covariance matrix must be positive semi-definite. If it were not then:

Question13: What is a Hessian?

Question14: Which of the following statements about variance and standard deviation are correct?
1. When calculated based on a sample of the population data, one has to correct for any bias in the result by using the number of degrees of freedom in the calculation
2. Variance is in square root units of the underlying data, whereas standard deviation is in units of the underlying data
3. When considering independent variables, variance is additive, while standard deviation is not

Question15: On average, one trade fails every 10 days. What is the probability that no trade will fail tomorrow?

Question16: What is the angle between the following two three dimensional vectors: a=(1,2,3), b=(-4,2,0)?

Question17: 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%?

Question18: 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%?

Question19: You are given the following values of a quadratic function f(x): f(0)=0, f(1)=-2, f(2)=-5. On the basis of these data, the derivative f'(0) is ...

Question20: 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

Question21: 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)?

Question22: 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?

Question23: Let X be a random variable distributed normally with mean 0 and standard deviation 1. What is the expected value of exp(X)?

Question24: Newton-Raphson iteration is used to find a solution of x5 - x3 + x = 1. If xn = 2, what is xn+1?

Question25: Which of the following can induce a 'multicollinearity' problem in a regression model?

Question26: What is the total derivative of the function f(x,y) = ln(x+y), where ln() denotes the natural logarithmic function?

Question27: 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:

Question28: You intend to invest $100 000 for five years. Four different interest payment options are available. Choose the interest option that yields the highest return over the five year period.

Question29: Consider two functions f(x) and g(x) with indefinite integrals F(x) and G(x), respectively. The indefinite integral of the product f(x)g(x) is given by

Question30: Let N(.) denote the cumulative distribution function and suppose that X and Y are standard normally distributed and uncorrelated. Using the fact that N(1.96)=0.975, the probability that X 0 and Y 1.96 is approximately

Question31: If A and B are two events with P(A) = 1/4, P(B) = 1/3 and P(A intersection B) =1/5, what is P(Bc | Ac) i.e. the probability of the complement of B when the complement of A is given?

Question32: Which of the following statements is not correct?

Question33: Let f(x) = c for x in [0,4] and 0 for other values of x.
What is the value of the constant c that makes f(x) a probability density function; and what if f(x) = cx for x in
[0,4]?

Question34: Over four consecutive years fund X returns 1%, 5%, -3%, 8%. What is the average growth rate of fund X over this period?

Question35: Which of the following is consistent with the definition of a Type I error?

Question36: 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.

Question37: 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.

Question38: Concerning a standard normal distribution and a Student's t distribution (with more than four degrees of freedom), which of the following is true?

Question39: Which of the provided answers solves this system of equations?
2y - 3x = 3y +x
y2 + x2 = 68

Question40: 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

Question41: 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?

Question42: 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)?

Question43: 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

Question44: I have $5m to invest in two stocks: 75% of my capital is invested in stock 1 which has price 100 and the rest is invested in stock 2, which has price 125. If the price of stock 1 falls to 90 and the price of stock 2 rises to 150, what is the return on my portfolio?

Question45: Simple linear regression involves one dependent variable, one independent variable and one error variable. In contrast, multiple linear regression uses...

Question46: You work for a brokerage firm that charges its client x per share. The volume of trade of a client of type A depends on the per share commission in the following manner. If the commission is x, the client of type A will trade e-ax shares on average each week. What is the optimal commission x that maximizes the income from client A, noting that a is greater than zero?