PRMIA 8007 Exam II: Mathematical Foundations of Risk Measurement - 2015 Edition Online Training

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1. The class intervals should be large enough so that they not obscure interesting variation within the group

Statements 2 and 3 are correct

Statements 1 and 2 are correct

All three statements are correct

Statements 1 and 3 are correct

2. 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%?

17.33

18.75

19.23

20.54

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

-1

None of these

4. What is a Hessian?

Correlation matrix of market indices

The vector of partial derivatives of a contingent claim

A matrix of second derivatives of a function

The point at which a minimum of a multidimensional function is achieved

5. 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…

Simple time-series regression

Multiple time-series regression

Simple cross-section regression

Multiple cross-section regression

6. The fundamental theorem of analysis establishes a relation between

First and second derivative of a function

The derivative of a function and the slope of its graph

Integration and differentiation of functions

The derivative of a function and the derivative of its inverse function

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

0.1250

0.0156

0.0625

None of the above

8. 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?

11.82

12.33

12.49

12.78

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

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

peaked and thin at the center and with thin tails

flat and thick at the center and with heavy (fat) tails

flat and thick at the center and with thin tails

10. 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?

5/7

8/11

6/11

None of these

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11. Let X be a random variable distributed normally with mean 0 and standard deviation 1 .

What is the expected value of exp(X)?

E(exp(X)) = 1.6487

E(exp(X)) = 1

E(exp(X)) = 2.7183

E(exp(X)) = 0.6065

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

1 / (x+y)

(x + y) / (x+y)

-x/(x+y) - y/(x+y)

ln(x+y) x + ln(x+y) y

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

0.5

14. 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?

0.32

0.64

0.96

1.92

15. 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?

$9420

$7350

$7230

$7100

16. 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 …

in the interval ]-2.5,-2[

equal to -2

in the interval ]-2,+[

in the interval ]-,-2.5]

17. 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?

-2.50%

-5%

2.50%

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

a constant

a straight line

a parabola

a cubic polynomial

19. The determinant of a matrix X is equal 2 .

Which of the following statements is true?

det(2X) =

det(2X) = 2 det(X)

det(2X) = det(X)2

det(2X) = 4 det(X)

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

x = 0 is a minimum

x = -3 is a maximum

x = 2 is a maximum

None of these

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21. 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 lump-sum payment of $22 500 on maturity (in five years)

an annually compounded rate of 4.15%

a quarterly-compounded rate of 4.1%

a continuously-compounded rate of 4%

22. 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.)

64%

75%

98%

Cannot be determined without estimates of the volatilities of the individual returns

23. Let N(.) denote the cumulative distribution function of the standard normal probability distribution, and N' its derivative .

Which of the following is false?

N(0) = 0.5

N'(0) 0

N(x) 0 as x

N'(x) 0 as x

24. A 2-year bond has a yield of 5% and an annual coupon of 5% .

What is the Modified Duration of the bond?

1.95

1.86

1.75

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

One dependent variable, many independent variables, one error variable

Many dependent variables, one independent variable, one error variable

One dependent variable, one independent variable, many error variables

Many dependent variables, many independent variables, many error variables

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

the PDF is non-negative.

the definite integral of the CDF from minus infinity to plus infinity is undefined.

the CDF approaches 1 as its argument approaches infinity.

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

27. Which of the provided answers solves this system of equations?

2y C 3x = 3y +x

y2 + x2 = 68

x = 1; y = square root of 67

x = 2; y = 8

x = 2; y = -8

x = -2; y = -8

28. 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:

0.5290

0.5292

0.5286

0.5288

29. 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?

0.75

1.125

0.3333

0.375

30. 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:

0.40729

0.00109

0.00087

0.32583

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

90 degrees

180 degrees

57 degrees

45 degrees

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

Its eigenvalues are all positive

One of its eigenvalues equals 0

If a is its eigenvalue, then -a is also its eigenvalue

If a is its eigenvalue, then is also its eigenvalue

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

x = 1, y = 0.5

x = -2, y = 2

x = 2, y = 0

None of the above

34. Calculate the determinant of the following matrix:

4.25

-4.25

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

-0.666666667

2.66667

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

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

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

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

The probability that the estimated value of the parameter is outside this interval is 95%.

37. 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

trade the underlying in such a way as to make the portfolio delta and vega neutral.

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

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.

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.

38. 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)

concave, concave, concave

concave, convex, convex

convex, concave, concave

convex, convex, concave

39. 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 C 2.89)?

The commodity prices are stationary

The commodity returns are stationary

The commodity returns are integrated of order 1

None of the above

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

Normal distribution

Student's t distribution

F-distribution

Binomial distribution

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41. 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?

12/29

37/45

3/4

None of these

42. 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 ?

-25

43. 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.

(i) and (ii)

(i) only

(i) and (iii)

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

44. 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.

-4

0.25

-0.5

-0.25

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

Some portfolios could have a negative variance

One or more of its eigenvalues would be negative

There would be no Cholesky decomposition matrix

All the above statements are true

46. 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

(i) and (iv)

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

(i), (iii), and (iv)

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

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

associativity

commutativity

distributivity

invertibility

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