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1. Let Z be a standard normal random variable. An event X is defined to happen if either z takes a value between ?1 and 1 or z takes any value greater than 1.5. What is the probability of event X happening if N(1) = 0.8413, N(0.5) = 0.6915 and N(-1.5) = 0.0668, where N() is the cumulative distribution function of a standard normal variable?
A. 0.083
B. 0.2166
C. 0.6826
D. 0.7494
Correct answer is D
Let A be the event that z takes a value between 1 and ?1 and B be the event that z takes a value greater than 11/2 . The probability of z being between 1 and ?1 is the area under the standard normal curve between 1 and -1. From the properties of a standard normal distribution, we know that:ffice ffice" />
N(-1) = 1.0 - N(1) = 1.0 ? 0.8413 = 0.1587
Therefore, the probability of z being between 1 and ?1 = P(A) = N(1) - N(-1) = 0.6826
The probability of z being greater than 11/2 = P(B) = 1 - N(11/2) = N(-11/2) = 0.0668
Event X = A U B and P(X) = P(A) + P(B) since A and B are mutually exclusive.
Hence, P(X) = 0.7494
2. Let X and Y be two random variables representing the annual returns of two different portfolios. If E[X] = 3, E[Y] = 4 and E[XY] = 11, then what is Cov[X, Y]?
A. -1
B. 0
C. 11
D. 12
Correct answer is A
Cov[X,Y] = E[XY] - E[X] * E[Y] = 11 - 3 * 4 = -1
3. You are given the following information about the returns of stock P and stock Q:
Variance of return of stock P = 100.0
Variance of return of stock Q = 225.0
Covariance between the return of stock P and the return of stock Q = 53.2
At the end of 1999, you are holding USD 4 million in stock P. You are considering a strategy of shifting USD 1 million into stock Q and keeping USD 3 million in stock P. What percentage of risk, as measured by standard deviation of return, can be reduced by this strategy?
ffice:smarttags" />A. 0.50%
B. 5.00%
C. 7.40%
D. 9.70%
Correct answer is B
The portfolio with 100% Stock P has a variance of return equal to 100 and a standard deviation of return equal to 10.
Moving the portfolio to 75% Stock P and 25% Stock Q changes the variance to:
Variance = (0.75)2 (100) + (0.25)2 (225) + 2(0.75)(0.25)(53.2) = 56.25 + 14.06 + 19.95 = 90.26
Standard deviation = (90.26)0.5 = 9.50
Hence, the percentage of risk reduced = (10 ? 9.5)/10 = 5.0%.
4. A relative value hedge fund manager holds a long position in Asset A and a short position in Asset B of roughly equal principal amounts. Asset A currently has a correlation with Asset B of .97. The risk manager decides to overwrite this correlation assumption in the variance-covariance based VAR model to a level of 0.30. What effect will this change in correlation from 0.97 to 0.30 have on the resulting VAR measure?
A. It increases VAR.
B. It decreases VAR.
C. It has no effect on VAR, but changes profit or loss of strategy.
D. Do not have enough information to answer.
Correct answer is A
A is correct. With a correlation of 0.97, any increase (decrease) in value in the short position in Asset B will be almost completely offset by a decrease (increase) in the value of the long position in Asset A, which implies a very low VaR. Reducing the correlation to 0.30 reduces the effectiveness of this hedge, which implies a higher VaR.
Reference: Philippe Jorion, Value at Risk, 3rd ed. Chapter 17.
B is incorrect. With a correlation of 0.97, any increase (decrease) in value in the short position in Asset B will be almost completely offset by a decrease (increase) in the value of the long position in Asset A, which implies a very low VaR.
C is incorrect. Changing the correlation between Asset A and Asset B will change VaR.volatility.
D is incorrect. Reducing the correlation between Asset A and Asset B will change VaR and the direction of that change can be determined from the information provided.
5. Which of the following statements about the linear regression of the return of a portfolio over the return of its benchmark presented below are correct?
I. The correlation is 0.71
II. 34% of the variation in the portfolio return is explained by variation in the benchmark return
III. The portfolio is the dependent variable
IV. For an estimated portfolio return of 12%, the confidence interval at 95% is [7.16%;16.84%]
A. II and IV
B. III and IV
C. I, II and III
D. II, III and IV
Correct answer is B
The portfolio return is the dependent variable and for an estimated portfolio return of 12%, the 95% confidence interval is [12% - 2 * 2.42%, 12% + 2 * 2.42%] or [7.16%, 16.84%].
However, the correlation is the square root of the coefficient of determination and is therefore equal to 0.81, and 66% of the variation in the portfolio returns is explained by variation in the benchmark return.
[此贴子已经被作者于2009-6-13 13:58:15编辑过] | 
发表于 2009-6-13 13:57
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