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# Log normal distribution

## Where do you meet this distribution?

• Finance, Economics : Change of stock price

## Shape of Distribution

### Basic Properties

• Two parameters $M, S$ are required (How can you get these).
$S>0$
• Continuous distribution defined on semi-bounded range $x>0$
• This distribution is always asymmetric.

### Probability

• Cumulative distribution function
$F(x)=\Phi\left(\frac{\ln x-M}{S}\right)$

where
$\Phi(\cdot)$is cumulative distribution function of standard normal distribution.

• Probability density function
$f(x)=\frac{1}{Sx}\phi\left(\frac{\ln x-M}{S}\right)$

where
$\phi(\cdot)$is probability density function of standard normal distribution.

• How to compute these on Excel.

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A B
Data Description
0.5 Value for which you want the distribution
0.1 Value of parameter M
2 Value of parameter S
Formula Description (Result)
=NTLOGNORMDIST(A2,A3,A4,TRUE) Cumulative distribution function for the terms above
=NTLOGNORMDIST(A2,A3,A4,FALSE) Probability density function for the terms above

• Function reference : NTLOGNORMDIST

### Quantile

• Inverse function of cumulative distribution function
$F^{-1}(P)=\exp\left[S\Phi^{-1}(P)+M\right]$

where
$\Phi(\cdot)$is cumulative distribution function of standard normal distribution.

• How to compute this on Excel.

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A B
Data Description
0.7 Probability associated with the distribution
0.1 Value of parameter M
2 Value of parameter S
Formula Description (Result)
=NTLOGNORMINV(A2,A3,A4) Inverse of the cumulative distribution function for the terms above

## Characteristics

### Mean – Where is the “center” of the distribution? (Definition)

• Mean of the distribution is given as
$m\sqrt{\omega}$

where

$m=\exp(M),\;\omega=\exp(S^2)$
• How to compute this on Excel

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A B
Data Description
0.1 Value of parameter M
2 Value of parameter S
Formula Description (Result)
=NTLOGNORMMEAN(A2,A3) Mean of the distribution for the terms above
• Function reference : NTLOGNORMMEAN

### Standard Deviation – How wide does the distribution spread? (Definition)

• Variance of the distribution is given as
$m^2\omega(\omega-1)$

where

$m=\exp(M),\;\omega=\exp(S^2)$

Standard Deviation is a positive square root of Variance.

• How to compute this on Excel

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A B
Data Description
0.1 Value of parameter M
2 Value of parameter S
Formula Description (Result)
=NTLOGNORMSTDEV(A2,A3) Standard deviation of the distribution for the terms above
• Function reference : NTLOGNORMSTDEV

### Skewness – Which side is the distribution distorted into? (Definition)

• Skewness of the distribution is given as
$(\omega+2)\sqrt{\omega-1}$

where

$\omega=\exp(S^2)$
• How to compute this on Excel

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A B
Data Description
0.1 Value of parameter M
2 Value of parameter S
Formula Description (Result)
=NTLOGNORMSKEW(A2,A3) Skewness of the distribution for the terms above
• Function reference : NTLOGNORMSKEW

### Kurtosis – Sharp or Dull, consequently Fat Tail or Thin Tail (Definition)

• Kurtosis of the distribution is given as
$\omega^4+2\omega^3+3\omega^2-6$

where

$\omega=\exp(S^2)$
• How to compute this on Excel

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A B
Data Description
0.1 Value of parameter M
2 Value of parameter S
Formula Description (Result)
=NTLOGNORMKURT(A2,A3) Kurtosis of the distribution for the terms above
• Function reference : NTLOGNORMKURT

## Random Numbers

• Random number x is generated by inverse function method, which is for uniform random U,
$x=\exp\left[S\Phi^{-1}(U)+M\right]$

where
$\Phi(\cdot)$is cumulative distribution function of standard normal distribution.

• How to generate random numbers on Excel.

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Data Description
0.1 Value of parameter M
2 Value of parameter S
Formula Description (Result)
=NTRANDLOGNORM(100,A2,A3,0) 100 log normal deviates based on Mersenne-Twister algorithm for which the parameters above

Note The formula in the example must be entered as an array formula. After copying the example to a blank worksheet, select the range A7:A106 starting with the formula cell. Press F2, and then press CTRL+SHIFT+ENTER.

• Function reference : NTRANDLOGNORM