**Normal Distribution**

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

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**Sample Question**

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?

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

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Which of the following is likely to be a probability distribution function?

For X=[1,2,3,4,5],Â Â Â Â Â Â Â Â Â Â Â Â Prob[X_{i}]= 49/(75-X_{i}^{2})

For X=[0,5,10,15],Â Â Â Â Â Â Â Â Â Â Â Prob[X_{i}]= X_{i}/30

For X=[1,4,9,16,25],Â Â Â Â Â Â Â Â Prob[X_{i}]= [(X_{i})^{1/2}Â â€“ 1]/5

The correct answer is For X=[0,5,10,15], Prob[X_{i}]= X_{i}/30

For all values of X, probability lies within [0,1] and sum of all the probabilities is equal to 1.

**Z-Score**

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

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Z = (x-Âµ)/Ïƒ

**Sample Question**

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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?

z = (A-K)/Ïƒ_{A}

Â Â = (100-80)/10

Â Â = 2

**Skewness and Kurtosis**

- Skewness
- Kurtosis

**Skewness:**

**Positively:**

Mean > median > mode

**Negatively:**

Mean < median < mode

Skewness of Normal = 0

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**Kurtosis:**

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

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**Platykurtic: **Flatter than a normal; excess kurtosis <0

Kurtosis of Normal = 3

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**Sample Question**

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

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

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**Roy's Safety First Criterion**

For optimal portfolio, minimize SF Ratio,

SF Ratio = [E(R

_{P}) â€“ R

_{L}] / Ïƒ

_{P}

Shortfall Risk = Probability corresponding to SF Ratio

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