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1
Q

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. 11.82
B. 12.33
C. 12.49
D. 12.78

A

C. 12.49

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2
Q

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?

A. 0.32
B. 0.64
C. 0.96
D. 1.92

A

B. 0.64

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3
Q

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

A. 2
B. 1.95
C. 1.86
D. 1.75

A

B. 1.95

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4
Q

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

A. One dependent variable, many independent variables, one error variable
B. Many dependent variables, one independent variable, one error variable
C. One dependent variable, one independent variable, many error variables
D. Many dependent variables, many independent variables, many error variables

A

A. One dependent variable, many independent variables, one error variable

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5
Q

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

A. f’(g(x,y))
B. f(dg/dy)
C. f(g(x,y)) dg/dy
D. f’(g(x,y)) dg/dy

A

D. f’(g(x,y)) dg/dy

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6
Q

If the annual volatility of returns is 25% what is the variance of the quarterly returns?

A. 0.1250
B. 0.0156
C. 0.0625
D. None of the above

A

B. 0.0156

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7
Q

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

A. x = 1; y = square root of 67
B. x = 2; y = 8
C. x = 2; y = -8
D. x = -2; y = -8

A

C. x = 2; y = -8

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8
Q

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

A. The regression coefficient will be too high and the standard error of the regression coefficient will be understated
B. The regression coefficient will be too low and the standard error of the regression coefficient will be overstated
C. The regression coefficient will be unbiased, but the standard error of the regression coefficient will be understated
D. The regression coefficient will be unbiased, but the standard error of the regression coefficient will be overstated

A

D. The regression coefficient will be unbiased, but the standard error of the regression coefficient will be overstated

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9
Q

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 ?

A. 29
B. -25
C. 1
D. 17

A

A. 29

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10
Q

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

A. The probability that the real value of the parameter is within this interval is 95%.
B. The probability that the real value of the parameter is outside this interval is 95%.
C. The probability that the estimated value of the parameter is within this interval is 95%.
D. The probability that the estimated value of the parameter is outside this interval is 95%.

A

A. The probability that the real value of the parameter is within this interval is 95%.

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11
Q

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:

A. 0.40729
B. 0.00109
C. 0.00087
D. 0.32583

A

B. 0.00109

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12
Q

Which of the following is not a sequence?

A. , , , … , , …
B. , , , , …
C. , , , , , , …
D. 30

A

D. 30

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13
Q

Which of the following can be used to evaluate a regression model?

(i) Magnitude of R2
(ii) Magnitude of TSS (total sum of squares)
(iii) Tests for statistical significance
(iv) Sign and magnitude of each regression parameter

A. (i) and (iv)
B. (i), (ii), and (iii)
C. (i), (iii), and (iv)
D. (i), (ii), (iii), and (iv)

A

C. (i), (iii), and (iv)

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14
Q

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

A. Some portfolios could have a negative variance
B. One or more of its eigenvalues would be negative
C. There would be no Cholesky decomposition matrix
D. All the above statements are true

A

D. All the above statements are true

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15
Q

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?

A. 0.75
B. 1.125
C. 0.3333
D. 0.375

A

D. 0.375

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16
Q

Which statement regarding the matrix below is true?

A. It is not positive definite
B. It is positive semi-definite
C. It is positive definite
D. It is negative definite

A

A. It is not positive definite

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17
Q

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 ?

A. [0, 1.5]
B. [1, 2]
C. [0, 3]
D. None of the above

A

C. [0, 3]

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18
Q

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

A. 0.25%
B. 0.488%
C. 0.49%
D. 0.495%

A

B. 0.488%

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19
Q 
Let N(.) denote the cumulative distribution function of the standard normal probability
distribution, and N' its derivative. Which of the following is false?

A. N(0) = 0.5
B. N’(0) 0
C. N(x) 0 as x
D. N’(x) 0 as x

A

C. N(x) 0 as x

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20
Q

Suppose I trade an option and I wish to hedge that option for delta and vega. Another
option is available to trade. To complete the hedge I would

A. trade the underlying in such a way as to make the portfolio delta and vega neutral.
B. trade the other option in such a way as to make the portfolio delta and vega neutral.
C. trade the other option in such a way as to make the portfolio vega neutral, and then trade the underlying in such a way as to make the portfolio delta neutral.
D. trade the underlying in such a way as to make the portfolio delta neutral, and then trade the other option in such a way as to make the portfolio vega neutral.

A

C. trade the other option in such a way as to make the portfolio vega neutral, and then trade the underlying in such a way as to make the portfolio delta neutral.

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21
Q

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.

A. 5.19
B. 5.66
C. 6.31
D. 4.18

A

C. 6.31

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22
Q

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

A. 1 / (x+y)
B. (x + y) / (x+y)
C. -x/(x+y) - y/(x+y)
D. ln(x+y) x + ln(x+y) y

A

B. (x + y) / (x+y)

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23
Q

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

A. 0.5
B. 1
C. 1.5
D. None of the above

A

C. 1.5

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24
Q

A 2-step binomial tree is used to value an American put option with strike 105, given that
the underlying price is currently 100. At each step the underlying price can move up by 10
or down by 10 and the risk-neutral probability of an up move is 0.6. There are no dividends
paid on the underlying and the continuously compounded risk free interest rate over each
time step is 1%. What is the value of the option in this model?

A. 7.12
B. 6.59
C. 7.44
D. 7.29

A

A. 7.12

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25
Q

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?

A. Changing the volatilities to be greater than 100%
B. Changing all the correlations to be unity
C. Changing all the correlations to be zero
D. All of the above

A

B. Changing all the correlations to be unity

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26
Q

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

A. Normal distribution
B. Student’s t distribution
C. F-distribution
D. Binomial distribution

A

C. F-distribution

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27
Q

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

A. x = -1, y = 0
B. x = 0, y = 20
C. x = 5, y = 0
D. None of the above

A

B. x = 0, y = 20

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28
Q

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

A. E(exp(X)) = 1.6487
B. E(exp(X)) = 1
C. E(exp(X)) = 2.7183
D. E(exp(X)) = 0.6065

A

A. E(exp(X)) = 1.6487

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29
Q

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?

A. 5/7
B. 8/11
C. 6/11
D. None of these

A

D. None of these

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30
Q

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

A. Estimation
B. Regression
C. Hypothesis testing
D. Prediction

A

D. Prediction

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31
Q

I have a portfolio of two stocks. The weights are equal. The one volatility is 30% while the
other is 40%. The minimum and maximum possible values of the volatility of my portfolio
are:

A. 30% and 40%
B. 5% and 35%
C. 10% and 40%
D. 10% and 70%

A

B. 5% and 35%

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32
Q

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?

A. 0.02
B. 0.14
C. 0.25
D. 0.50

A

B. 0.14

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33
Q

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

A. 240
B. 4598
C. 4840
D. 4960

A

C. 4840

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34
Q

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

A. All three statements are correct
B. Statements 1 and 2 are correct
C. Statements 1 and 3 are correct
D. Statements 2 and 3 are correct

A

A. All three statements are correct

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35
Q

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)?

A. The commodity prices are stationary
B. The commodity returns are stationary
C. The commodity returns are integrated of order 1
D. None of the above

A

D. None of the above

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36
Q

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

A. 90 degrees
B. 180 degrees
C. 57 degrees
D. 45 degrees

A

A. 90 degrees

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37
Q

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?

A. $9420
B. $7350
C. $7230
D. $7100

A

B. $7350

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38
Q

Which of the following statements is true?

A. Discrete and continuous compounding produce the same results if the discount rate is positive.
B. Continuous compounding is the better method because it results in higher present values compared to discrete compounding.
C. Continuous compounding can be thought as making the compounding period infinitesimally small.
D. The constant plays an important role in the mathematical description of continuous compounding.

A

C. Continuous compounding can be thought as making the compounding period infinitesimally small.

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39
Q

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

A. 0.5
B. 0
C. 1
D. 2

A

B. 0

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40
Q

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

A. A structural break in the dependent variable
B. A high positive correlation between two explanatory variables
C. The omission of a relevant explanatory variable
D. Using an inappropriate functional form in the model

A

B. A high positive correlation between two explanatory variables

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41
Q

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

A. 1 048 574
B. 1 048 595
C. 2 097 170
D. 2 097 172

A

C. 2 097 170

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42
Q

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

A. x ln(x) - x
B. ln(x) - x
C. 1/x
D. exp(x)

A

A. x ln(x) - x

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43
Q

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. b=0.5, a=2.5
B. b=0.5, a=1.5
C. b=2, a=4
D. b=2, a=0

A

D. b=2, a=0

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44
Q

The fundamental theorem of analysis establishes a relation between

A. First and second derivative of a function
B. The derivative of a function and the slope of its graph
C. Integration and differentiation of functions
D. The derivative of a function and the derivative of its inverse function

A

C. Integration and differentiation of functions

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45
Q

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

A. 1.50%
B. 6%
C. 6.14%
D. None of the above

A

C. 6.14%

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46
Q

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. 0
B. 3
C. 2
D. 1

A

B. 3

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47
Q

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

A. Simple time-series regression
B. Multiple time-series regression
C. Simple cross-section regression
D. Multiple cross-section regression

A

D. Multiple cross-section regression

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48
Q

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

A. 8
B. -1
C. 17
D. None of these

A

C. 17

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49
Q

Bond convexity is closely related to …

A. The derivative of the bond’s present value with respect to yield
B. The second derivative of the bond’s present value with respect to yield
C. The integral of the bond’s present value with respect to yield
D. The sensitivity of the bond’s present value with respect to yield

A

B. The second derivative of the bond’s present value with respect to yield

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50
Q

The Newton-Raphson method

A. is based on finding a middle point between left and right end of the search interval
B. is based on Taylor series and uses the first derivative
C. can be used for continuous but not differentiable functions
D. does provide an error bound along with every iteration

A

B. is based on Taylor series and uses the first derivative

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51
Q

The natural logarithm of x is:

A. the inverse function of exp(x)
B. log(e)
C. always greater than x, for x>0
D. 46

A

A. the inverse function of exp(x)

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52
Q

Which of the following statements is not correct?

A. Every linear function is also a quadratic function.
B. A function is defined by its domain together with its action.
C. For finite and small domains, the action of a function may be specified by a list.
D. A function is a rule that assigns to every value x at least one value of y.

A

D. A function is a rule that assigns to every value x at least one value of y.

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53
Q

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

A. associativity
B. commutativity
C. distributivity
D. invertibility

A

C. distributivity

54
Q

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

A. 2.67%
B. 2.75%
C. 2.49%
D. None of the above

A

A. 2.67%

55
Q

An underlying asset price is at 100, its annual volatility is 25% and the risk free interest rate
is 5%. A European put option has a strike of 105 and a maturity of 90 days. Its Black-
Scholes price is 7.11. The options sensitivities are: delta = -0.59; gamma = 0.03; vega =
19.29. Find the delta-gamma approximation to the new option price when the underlying
asset price changes to 105

A. 6.49
B. 5.03
C. 4.59
D. 4.54

A

D. 4.54

56
Q

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.

A. -4
B. 0.25
C. -0.5
D. -0.25

A

D. -0.25

57
Q

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

A. All three statements are correct
B. Statements 1 and 2 are correct
C. Statements 1 and 3 are correct
D. Statements 2 and 3 are correct

A

C. Statements 1 and 3 are correct

58
Q

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
A. in the interval ]-2.5,-2[
B. equal to -2
C. in the interval ]-2,+[
D. in the interval ]-,-2.5]

A

C. in the interval ]-2,+[

59
Q

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

A. the PDF is non-negative.
B. the definite integral of the CDF from minus infinity to plus infinity is undefined.
C. the CDF approaches 1 as its argument approaches infinity.
D. the definite integral of the PDF from minus infinity to plus infinity is zero.

A

D. the definite integral of the PDF from minus infinity to plus infinity is zero.

60
Q

Variance reduction is:

A. A technique that is applied in regression models to improve the accuracy of the coefficient estimates
B. A numerical method for finding portfolio weights to minimize the variance of a portfolio that has a given expected return
C. A numerical method for finding the variance of the underlying that is implicit in a market price of an option
D. A method for reducing the number of simulations required in a Monte Carlo simulation

A

D. A method for reducing the number of simulations required in a Monte Carlo simulation

61
Q

A linear regression gives the following output:
Figures in square brackets are estimated standard errors of the coefficient estimates. What
is thevalue of the test statistic for the hypothesis that the coefficient of is zero against the
alternative that is less than zero?

A. 0.125
B. 2.5
C. -1.25
D. -2.5

A

D. -2.5

62
Q

When a number is written with a fraction as an exponent, such as , which of the following is
the correct computation?

A. Take the square-root of 75 and raise it to the 5th power
B. Divide 75 by 2, then raise it to the 5th power
C. Multiply 75 by 2.5
D. Square 75, then take the fifth root of it

A

A. Take the square-root of 75 and raise it to the 5th power

63
Q

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. 2/5 and 1/2
B. 2/5 and 3/5
C. 1/3 and 8/17
D. 9/25 and 12/25

A

D. 9/25 and 12/25

64
Q

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

A. 0.095
B. 0.905
C. 0.95
D. 0.100

A

B. 0.905

65
Q

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

A. 10
B. 32
C. 5/2 + log2(16)
D. log2 (5/2) + log2(16)

A

A. 10

66
Q

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

A. -x
B. -x+1
C. x-1
D. x+1

A

B. -x+1

67
Q

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

A. 4%
B. 14.4%
C. 20%
D. 24%

A

C. 20%

68
Q

The gradient of a smooth function is

A. a vector that shows the direction of fastest change of a function
B. matrix of second partial derivatives of a function
C. infinite at a maximum point
D. a matrix containing the function’s second partial derivatives

A

A. a vector that shows the direction of fastest change of a function

69
Q 
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

A. -2.8
B. 1.3
C. -1.2
D. None of the above

A

C. -1.2

70
Q

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.)

A. 64%
B. 75%
C. 98%
D. Cannot be determined without estimates of the volatilities of the individual returns

A

B. 75%

71
Q

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

A. 10 basis point rise
B. 24 basis fall
C. 24 basis point rise
D. 30 basis points fall.

A

D. 30 basis points fall.

72
Q

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:

A. 0.5290
B. 0.5292
C. 0.5286
D. 0.5288

A

D. 0.5288

73
Q

Which of the following statements concerning class intervals used for grouping of data is
correct?
When grouping data, attention must be paid to the following with regards to class intervals:
1. Class intervals should not overlap
2. Class intervals should be of equal size unless there is a specific need to highlight data
within a specific subgroup
3. The class intervals should be large enough so that they not obscure interesting variation
within the group

A. Statements 2 and 3 are correct
B. Statements 1 and 2 are correct
C. All three statements are correct
D. Statements 1 and 3 are correct

A

B. Statements 1 and 2 are correct

74
Q

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

A. The commutativity of multiplication is defined by
B. The existence of negatives is defined by
C. The distributivity of multiplication is defined by
D. The associativity of multiplication is defined by

A

C. The distributivity of multiplication is defined by

75
Q

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 50%. The volatility of my portfolio is

A. 16%
B. 17.4%
C. 20%
D. 24.4%

A

B. 17.4%

76
Q

What is a Hessian?

A. Correlation matrix of market indices
B. The vector of partial derivatives of a contingent claim
C. A matrix of second derivatives of a function
D. The point at which a minimum of a multidimensional function is achieved

A

C. A matrix of second derivatives of a function

77
Q

At what point x does the function f(x) = x3 - 4x2 + 1 have a local minimum?

A. -0.666666667
B. 0
C. 2.66667
D. 2

A

C. 2.66667

78
Q

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?

A. The stocks must be positively correlated ( )
B. Beta must be positive ( )
C. Beta must be greater in absolute value than the correlation of the stocks ( )
D. Alpha must be positive ( )

A

C. Beta must be greater in absolute value than the correlation of the stocks ( )

79
Q

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%?

A. 17.33
B. 18.75
C. 19.23
D. 20.54

A

D. 20.54

80
Q

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

A. The probability of a Type I error is 100% minus the significance level
B. A Type I error would have occurred if the performance of a stock was positively correlated with the performance of a hedge fund, but in a linear regression, the hypothesis of positive correlation was rejected
C. A Type I error would have occurred if the performance of a stock was positively correlated with the performance of a hedge fund, but in a linear regression, the hypothesis of no correlation was rejected
D. A Type I occurs whenever data series are serially correlated

A

B. A Type I error would have occurred if the performance of a stock was positively correlated with the performance of a hedge fund, but in a linear regression, the hypothesis of positive correlation was rejected

81
Q

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.

A. a lump-sum payment of $22 500 on maturity (in five years)
B. an annually compounded rate of 4.15%
C. a quarterly-compounded rate of 4.1%
D. a continuously-compounded rate of 4%

A

C. a quarterly-compounded rate of 4.1%

82
Q

For the function f(x) =3x-x3 which of the following is true?

A. x = 0 is a minimum
B. x = -3 is a maximum
C. x = 2 is a maximum
D. None of these

A

D. None of these

83
Q

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

A. The null hypothesis is accepted when in fact it should have been rejected
B. The null hypothesis is rejected when in fact it should have been accepted
C. Both null hypothesis and alternative hypothesis are rejected
D. Both null hypothesis and alternative hypothesis are accepted

A

B. The null hypothesis is rejected when in fact it should have been accepted

84
Q

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?

A. 11/16
B. 7/8
C. 9/16
D. None of these

A

A. 11/16

85
Q

Two vectors are orthogonal when:

A. one is a scalar multiple of the other
B. their components are linearly dependent
C. their determinant is zero
D. their scalar product (sum product) is zero

A

D. their scalar product (sum product) is zero

86
Q

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?

A. 1
B. a
C. 42
D. a2

A

C. 42

87
Q

An indefinite integral of a polynomial function is

A. always positive
B. always increasing
C. always less than the function itself
D. none of the above

A

D. none of the above

88
Q

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

A. The estimated mean divided by the estimated variance has a t-distribution
B. The estimated mean divided by the estimated variance has a Chi2-distribution
C. The estimated mean divided by the estimated standard deviation has a t-distribution
D. The estimated mean divided by the estimated standard deviation has a Chi2-distribution

A

C. The estimated mean divided by the estimated standard deviation has a t-distribution

89
Q

Consider the following distribution data for a random variable X: What is the mean and
variance of X?

A. 3.6 and 7.15
B. 3.4 and 3.84
C. 3.5 and 3.45
D. None of these

A

D. None of these

90
Q

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

A. 1
B. 0.5
C. 0
D. None of the above

A

C. 0

91
Q

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?

A. -2.50%
B. -5%
C. 2.50%
D. 5%

A

A. -2.50%

92
Q

An asset price S is lognormally distributed if:

A. the change in price (dS) is normally distributed
B. 1/S is normally distributed
C. ln(dS/S) is normally distributed
D. ln(1+dS/S) is normally distributed

A

D. ln(1+dS/S) is normally distributed

93
Q

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:

A. non-linear
B. integrated of order 2
C. differential
D. non-functional

A

B. integrated of order 2

94
Q

In a quadratic Taylor approximation, a function is approximated by:

A. a constant
B. a straight line
C. a parabola
D. a cubic polynomial

A

C. a parabola

95
Q

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

A. (0, 2, 1)
B. (0, 0, 0)
C. (1, 1, 1)
D. (0, 1, -1)

A

D. (0, 1, -1)

96
Q

A typical leptokurtotic distribution can be described as a distribution that is relative to a
normal distribution

A. peaked and thin at the center and with heavy (fat) tails
B. peaked and thin at the center and with thin tails
C. flat and thick at the center and with heavy (fat) tails
D. flat and thick at the center and with thin tails

A

A. peaked and thin at the center and with heavy (fat) tails

97
Q

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

A. Some portfolios could have a negative variance
B. It could not be used to simulate correlated asset paths
C. The associated correlation matrix would not be positive semi-definite
D. All the above statements are true

A

D. All the above statements are true

98
Q

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?

A. 66.60%
B. 90%
C. 95%
D. 99%

A

C. 95%

99
Q

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

A. mean is 0, variance is 2/3
B. mean is 0, variance is 1/3
C. mean is 0, variance is 1/2
D. mean is 1/3, variance is 1/3

A

A. mean is 0, variance is 2/3

100
Q

A quadratic form is

A. defined as a positive definite Hessian matrix.
B. an algebraic expression in two variables, x and y,involving , and terms.
C. a specific solution of the Black-Scholes pricing formula
D. an algebraic expression in two variables, x and y, involving , , and terms.

A

B. an algebraic expression in two variables, x and y,involving , and terms.

101
Q

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

A. 2
B. 1.95
C. 1.86
D. 1.75

A

C. 1.86

102
Q

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

A. If the discriminant is negative, there are no real solutions
B. If the discriminant is zero, there is only one solution
C. If the discriminant is negative there are two different real solutions
D. If the discriminant is positive there are two different real solutions

A

C. If the discriminant is negative there are two different real solutions

103
Q

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.

A. (i) and (ii)
B. (i) only
C. (i) and (iii)
D. (i), (ii), and (iii)

A

C. (i) and (iii)

104
Q

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?

A. 0.7367
B. 0.3483
C. 0.2646
D. None of these

A

B. 0.3483

105
Q

The first derivative of a function f(x) is zero at some point, the second derivative is also
zero at this point. This means that:

A. f has necessarily a minimum at this point
B. f has necessarily a maximum at this point
C. f has necessarily neither a minimum nor a maximum at this point
D. f might have either a minimum or a maximum or neither of them at this point

A

D. f might have either a minimum or a maximum or neither of them at this point

106
Q

Consider an investment fund with the following annual return rates over 8 years: +6%, -6%,
+12%, -12%, +3%, -3%, +9%, -9% .
What can you say about the annual geometric and arithmetic mean returns of this
investment fund?

A. The arithmetic mean return is zero and the geometric mean return is negative
B. The arithmetic mean return is negative and the geometric mean return is zero
C. The arithmetic mean return is equal to the geometric mean return
D. None of the above

A

A. The arithmetic mean return is zero and the geometric mean return is negative

107
Q

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

A. 0.5
B. 0
C. 1
D. 2

A

C. 1

108
Q

There are two portfolios with no overlapping of stocks or bonds. Portfolio 1 has 6 stocks
and 6 bonds. Portfolio 2 has 4 stocks and 8 bonds. If we randomly select one stock, what is
the probability that it came from Portfolio1?

A. 0.3
B. 0.5
C. 0.6
D. None of these

A

C. 0.6

109
Q

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. 36%
B. 36.33%
C. 26.33%
D. 20.78%

A

D. 20.78%

110
Q

Maximum likelihood estimation is a method for:

A. Finding parameter estimates of a given density function
B. Estimating the solution of a partial differential equation
C. Solving a portfolio optimization problem
D. Estimating the implied volatility of a simple European option

A

A. Finding parameter estimates of a given density function

111
Q

You invest $2m in a bank savings account with a constant interest rate of 5% p.a. What is
the value of the investment in 2 years time if interest is compounded quarterly?

A. $2,208,972
B. $2,210,342
C. $2.205,000
D. None of them

A

A. $2,208,972

112
Q

Calculate the determinant of the following matrix:

A. 4.25
B. -4.25
C. 4
D. 2

A

D. 2

113
Q

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

A. negative
B. zero
C. not defined
D. positive

A

B. zero

114
Q

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?

A. test statistic = SD()/MEAN(), normal distribution
B. test statistic = MEAN()/SD(), normal distribution
C. test statistic = SD()/MEAN(), t distribution
D. test statistic = MEAN()/SD(), t distribution

A

D. test statistic = MEAN()/SD(), t distribution

115
Q

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

A. F(x)G(x)
B. F(x)g(x) + f(x)G(x)
C. F(x)g(x) - F(x)g’(x)dx
D. f(x)G(x) - F(x)g’(x)dx

A

C. F(x)g(x) - F(x)g’(x)dx

116
Q

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. concave, concave, concave
B. concave, convex, convex
C. convex, concave, concave
D. convex, convex, concave

A

C. convex, concave, concave

117
Q

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

A. 4 and -2
B. -4 and 2
C. 1 and 0
D. No real roots

A

D. No real roots

118
Q

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%?

A. 10
B. 100
C. 25
D. 5

A

A. 10

119
Q

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

A. A large negative correlation between the dependent variable and one of the explanatory variables
B. A high positive correlation between the dependent variable and one of the explanatory variables
C. A high positive correlation between two explanatory variables
D. The omission of a relevant explanatory variable

A

C. A high positive correlation between two explanatory variables

120
Q

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

A. det(2X) =
B. det(2X) = 2 det(X)
C. det(2X) = det(X)2
D. det(2X) = 4 det(X)

A

D. det(2X) = 4 det(X)

121
Q

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

A. x -1
B. x
C. ln(x)
D. -x

A

B. x

122
Q

Solve the simultaneous linear equations: x + 2y - 2 = 0 and y - 3x = 8

A. x = 1, y = 0.5
B. x = -2, y = 2
C. x = 2, y = 0
D. None of the above

A

B. x = -2, y = 2

123
Q

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

A. arithmetic sequence, common difference 2
B. arithmetic sequence, common ratio 2
C. geometric sequence, common ratio 2
D. geometric sequence, common ratio 3

A

C. geometric sequence, common ratio 2

124
Q

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

A. The distributions have the same kurtosis.
B. The normal distribution has higher kurtosis than the t distribution.
C. The normal distribution has lower kurtosis than the t distribution.
D. Which has the higher kurtosis depends on the degrees of freedom of the t distribution.

A

C. The normal distribution has lower kurtosis than the t distribution.

125
Q

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

A. The function f(x) is continuous and monotonic
B. The function f(x) is differentiable
C. The function f(x) is differentiable and we have an explicit expression for the derivative
D. The function f(x) is continuous

A

A. The function f(x) is continuous and monotonic

126
Q

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

A. 2.362
B. 1.623
C. 1.638
D. 0.377

A

C. 1.638

127
Q

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

A. 12
B. Infinity
C. 128
D. 20

A

B. Infinity

128
Q

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?

A. 12/29
B. 37/45
C. 3/4
D. None of these

A

B. 37/45

129
Q 
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]?

A. 1/4 and 1/7
B. 1/7 and 1/9
C. 1/4 and 1/6
D. None of the above

A

D. None of the above

130
Q

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

A. 8/15
B. 9/23
C. 10/32
D. None of these

A

C. 10/32

131
Q

Which of the following statements is true for symmetric positive definite matrices?

A. Its eigenvalues are all positive
B. One of its eigenvalues equals 0
C. If a is its eigenvalue, then -a is also its eigenvalue
D. If a is its eigenvalue, then is also its eigenvalue

A

A. Its eigenvalues are all positive

132
Q

Consider a binomial lattice where a security price S moves up by a factor u with probability
p, or down by a factor d with probability 1 - p. If we set d > 1/u then which of the following
will be TRUE?

A. The lattice will not recombine
B. The probability of an up move will not be constant
C. There will always be a downward drift in the lattice
D. None of the above

A

D. None of the above

Finance (2 decks)

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