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Normal Distribution
  • Continuous Distribution
  • Described by mean & variance
  • Symmetric about its mean
  • Standard Normal Distribution - Mean = 0; Variance =1



 

Sample Question

Example

If Z is a standard normal R.V. An event X is defined to happen if either
-1< Z < 1 or Z > 1.5. What is the prob. of event X happening if N(1) = 0.8413, N(0.5) = 0.6915 and N(-1.5) = 0.0668, where N is the CDF of a standard normal variable?

Solution

P(X)= P(-1< Z < 1) + P(Z > 1.5)
= N(1) - (1-N(1)) + N(-1.5)

= 2*0.8413-1 + 0.0668

= 0.7494


 

 
Example

Which of the following is likely to be a probability distribution function?
For X=[1,2,3,4,5],             Prob[Xi]= 49/(75-Xi2)
For X=[0,5,10,15],            Prob[Xi]= Xi/30
For X=[1,4,9,16,25],         Prob[Xi]= [(Xi)1/2 – 1]/5

Solution

The correct answer is For X=[0,5,10,15], Prob[Xi]= Xi/30
For all values of X, probability lies within [0,1] and sum of all the probabilities is equal to 1.

Z-Score

 

No. of σ a given observation is away from population mean.
 

Z = (x-µ)/σ


Sample Question

 

Example

At a particular time, the market value of assets of the firm is $100 Mn and the market value of debt is $80 Mn. The standard deviation of assets is $10 Mn. What is the distance to default?

Solution

z = (A-K)/σA
   = (100-80)/10
   = 2

Skewness and Kurtosis
  • Skewness
  • Kurtosis


Skewness:

  • Positively:

Mean > median > mode

  • Negatively:

Mean < median < mode

Skewness of Normal = 0


       


 

Kurtosis:
 

Leptokurtic: More peaked than normal (fat tails); excess kurtosis > 0
 

Platykurtic: Flatter than a normal; excess kurtosis <0

Kurtosis of Normal = 3


 

Sample Question

Example

If distributions of returns from financial instruments are leptokurtotic. How does it compare with a normal distribution of the same mean and variance?

Solution

Leptokurtic refers to a distribution with fatter tails than the normal, which implies greater kurtosis.

 

Roy's Safety First Criterion


For optimal portfolio, minimize SF Ratio,

SF Ratio = [E(RP) – RL] / σP

Shortfall Risk = Probability corresponding to SF Ratio
 




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