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8002 Sample Questions Answers

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?

Options:

A.

36%

B.

36.33%

C.

26.33%

D.

20.78%

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Questions 5

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

Options:

A.

arithmetic sequence, common difference 2

B.

arithmetic sequence, common ratio 2

C.

geometric sequence, common ratio 2

D.

geometric sequence, common ratio 3

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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?

Options:

A.

0.02

B.

0.14

C.

0.25

D.

0.50

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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%?

Options:

A.

10

B.

100

C.

25

D.

5

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Questions 8

Variance reduction is:

Options:

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

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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 ?

Options:

A.

[0, 1.5]

B.

[1, 2]

C.

[0, 3]

D.

None of the above

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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)?

Options:

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

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Questions 11

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

Options:

A.

0.5

B.

0

C.

1

D.

2

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Questions 12

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

Options:

A.

-x

B.

-x+1

C.

x-1

D.

x+1

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Questions 13

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

Options:

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.

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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.

Options:

A.

-4

B.

0.25

C.

-0.5

D.

-0.25

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Questions 15

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

Options:

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

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Questions 16

Which of the following statements is not correct?

Options:

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.

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Questions 17

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

Options:

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

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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?

Options:

A.

x = -1, y = 0

B.

x = 0, y = 20

C.

x = 5, y = 0

D.

None of the above

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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.

Options:

A.

(i) and (ii)

B.

(i) only

C.

(i) and (iii)

D.

(i), (ii), and (iii)

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Questions 20

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

Options:

A.

0.5

B.

1

C.

1.5

D.

None of the above

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Questions 21

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

Options:

A.

0.5

B.

0

C.

1

D.

2

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Questions 22

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

Options:

A.

(0, 2, 1)

B.

(0, 0, 0)

C.

(1, 1, 1)

D.

(0, 1, -1)

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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)?

Options:

A.

E(exp(X)) = 1.6487

B.

E(exp(X)) = 1

C.

E(exp(X)) = 2.7183

D.

E(exp(X)) = 0.6065

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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?

Options:

A.

0.7367

B.

0.3483

C.

0.2646

D.

None of these

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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?

Options:

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

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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 ?

Options:

A.

29

B.

-25

C.

1

D.

17

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Questions 27

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

Options:

A.

x = 0 is a minimum

B.

x = -3 is a maximum

C.

x = 2 is a maximum

D.

None of these

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Questions 28

What is a Hessian?

Options:

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

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Questions 29

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

Options:

A.

-0.666666667

B.

0

C.

2.66667

D.

2

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Questions 30

Calculate the determinant of the following matrix:

Options:

A.

4.25

B.

-4.25

C.

4

D.

2

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Questions 31

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

Options:

A.

x = 1, y = 0.5

B.

x = -2, y = 2

C.

x = 2, y = 0

D.

None of the above

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Questions 32

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

Options:

A.

x -1

B.

x

C.

ln(x)

D.

-x

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Questions 33

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

Options:

A.

Estimation

B.

Regression

C.

Hypothesis testing

D.

Prediction

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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?

Options:

A.

$9420

B.

$7350

C.

$7230

D.

$7100

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Questions 35

A quadratic form is

Options:

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.

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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

Options:

A.

-2.8

B.

1.3

C.

-1.2

D.

None of the above

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Questions 37

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

Options:

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%.

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Questions 38

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

Options:

A.

Normal distribution

B.

Student's t distribution

C.

F-distribution

D.

Binomial distribution

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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?

Options:

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.

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Exam Code: 8002
Exam Name: PRM Certification - Exam II: Mathematical Foundations of Risk Measurement
Last Update: Jul 10, 2024
Questions: 132
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