Essay on 4 Lognormal Distribution

McGill University
Advanced Business Statistics
MGSC-372

MGSC 372

The Lognormal Distribution

The Lognormal Distribution

A continuous random variable X follows a lognormal distribution if its natural logarithm, ln(X), follows a normal distribution. The lognormal distribution is an asymmetric distribution with interesting applications for modeling the probability distributions of stock and other asset prices

The Lognormal Distribution
• Properties of the lognormal distribution
– Skewed to the right
– Strictly positive (i.e. bounded below by 0)
– The lognormal distribution may be used to model data on asset prices (note that prices are bounded below by 0)

Histograms of lognormal variable X and normal distribution ln(X)

The Lognormal Distribution

• The lognormal distribution is described by two parameters, its mean and variance, as in the case of a normal distribution
• The mean of a lognormal distribution X is given by

E ( X ) e

 

1 2

2



• The variance of a lognormal distribution X is given by

Var (X )  e

(2    2 )

2

(e  1)

where  and 2 are the mean and variance of the normal distribution of the ln(X) variable and e  2.718 is the natural base for logarithms.
• The median of a lognormal distribution X is given by eµ .

The Lognormal Distribution

Recall that the exponential and logarithmic functions are inverse functions so that

ln( x)  y  e y  x
This relationship is used to switch between the lognormal variable X and the normal variable y = ln X

Lognormal Distribution - Example

The material failure mechanism for a specific metal alloy has determined that the ductile strength X of the material has a lognormal distribution with parameters  = 5 and  = 0.1
(a) Compute E(X) and Var(X)
(b) Compute P(X > 120)
(c) Compute P(110  X  130)
(d) What is the value of median ductile strength?
(e) If the smallest 5% of strength values were unacceptable, what would the minimum acceptable strength be?

Lognormal Distribution –Example Solution

(a)

2 u 2 e5.005 e5.005 149.16
E ( X ) e
2
2
2
u



Var ( X ) e
(e
 1) 223

(b)
P( X  120) 1  P( X 120) ln120  5.0
1  P( Z 
)
0 .1
1  F ( 2.13)
1  0.0166
0.9834

Lognormal Distribution –Example Solution

(c) P (110  X 130) P ( ln 110  5.0 Z  ln 130  5.0 )

0.1

P ( 2.99 Z  1.32)
F ( 1.32)  F ( 2.99)
0.0934  0.0014
0.092
(d)

Median eu e5 148.41

0.1

Lognormal Distribution –Example Solution

(e) The value of X, call it L,, for which P( X  L) 0.05 is determined as follows: ln L  5.0
P( Z 
) 0.05
0.1
P( Z   1.64) 0.05 ln L  5.0
 1.64
0.1
L 125.964

Example: Relative Asset Prices and the Lognormal Distribution
• Consider the relative price of an asset between periods 0 and 1, defined as S1/S0, which is equal to 1 + R0,1
• For example, if S0 = $30 and S1 = $34.5, then the relative price is
$34.5/$30 = 1.15 or 1+ 0.15, meaning that the holding period return R0,1 is 15%
• The continuously compounded return rt,t+1 associated with a holding period return of Rt,t+1 is given by the natural log of the relative price

rT ,T 1

 ST 1
 ln 
  ln  1  RT ,T 1 
 ST 

Example: Relative Asset Prices and the Lognormal Distribution
•

For the above example, the continuously compounded return is given by r0,1 = ln($34.5/$30) = ln(1.15) = 0.1397 or 13.97%, which is lower than the holding period return of 15%.

•

To generalize, note that between periods 0 and T, r0,T = ln(ST/S0) or we can write r ST  S0 e 0,T

•

Note that

ST
ST ST  1 ST  2
S2 S1

.
.
...... .
S0 ST  1 ST  2 ST  3
S1 S0

Example: Relative Asset Prices and the Lognormal Distribution
• Recall that ln(XY) = ln(X) + ln(Y) so that

S 
S 
 S 
S  ln T  ln T   ln T  1     ln 1 
 ST  1 
 ST  2 
 S0 
 S0  or, equivalently,

r0 ,T rT  1,T  rT  2 ,T  1    r0 ,1
We see that the mean continuously compounded return between periods 0 and T is the sum of the continuously compounded returns of

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