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  • Unbiased parameter is one for which the expected value of the estimator is equal to the parameter you are trying to estimate
  • An unbiased parameter is also efficient if its sampling distribution has minimum variance
  • An estimator is also consistent if the accuracy of the parameter estimate increases as the sample size increases
  • If this estimator is linear, we call it the Best Linear Unbiased Estimator (BLUE)




ABC Annual stock prices
2001 2002 2003 2004 2005 2006
12% 5% -7% 11% 2% 11%

Assuming that the distribution of ABC stock returns is a population, what is the population variance and standard deviation?

  1. 05.00
  2. 06.80
  3. 45.22
  4. 80.20
The population variance is given by taking the mean of all squared deviations from the mean.

σ2 = [(12-5.67)2 + (5-5.67)2 + (-7-5.67)2 + (11-5.67)2 + (2-5.67)2 + (11-5.67)2] / 6 = 45.22 (%2)

The standard deviation is the square root of the variance:



The random variables X and Y have variances of 2 and 3 respectively, and covariance of 0.5.
The variance of 2X + 3Y is:


Var(X + Y) = Var(X) + Var(Y) +2*Cov(x,y)
Var(X - Y) = Var(X) + Var(Y) -2*Cov(x,y)
Var(cX) = c^2 * Var(X)
Cov (ax,by)  =  abCov(x,y)
So, Var(2X + 3Y) = 22 Var(X) + 32 Var(Y) +2*2*3*Cov(x,y)
Var(2X + 3Y) = 4*2 + 9*3 + 12*0.5 = 41



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?

  1. 0.50%
  2. 5.00%
  3. 7.40%
  4. 9.70%



Example- (FRM Exam 2009)

Which type of distribution produces the lowest probability for a variable to exceed a specified extreme value ‘X’ which is greater than the mean assuming the distributions all have the same mean and variance?

  1. A leptokurtic distribution with a kurtosis of 4
  2. A leptokurtic distribution with a kurtosis of 8
  3. A normal distribution
  4. A platykurtic distribution

By definition, a platykurtic distribution has thinner tails than both the normal distribution and any leptokurtic distribution. Therefore, for an extreme value X, the lowest probability of exceeding it will be found in the distribution with the thinner tails

  1. Incorrect.  A leptokurtic distribution has fatter tails than the normal distribution. The kurtosis indicates the level of fatness in the tails, the higher the kurtosis, the fatter the tails. Therefore, the probability of exceeding a specified extreme value will be higher
  2. Incorrect.  Since answer A. has a lower kurtosis, a distribution with a kurtosis of 8 will necessarily produce a larger probability in the tails
  3. Incorrect.  By definition, a normal distribution has thinner tails than a leptokurtic distribution and larger tails than a platykurtic distribution.


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