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The Lognormal Distribution
• The lognormal distribution is an asymmetric distribution with
interesting applications for modeling the probability distributions
of stock and other asset prices
• A continuous random variable X follows a lognormal distribution
if its natural logarithm, ln(X), follows a normal distribution
• We can also say that if the natural log of a random variable,
ln(X), follows a normal distribution, the random variable, X,
follows a lognormal distribution
The Lognormal Distribution
• Interesting observations about the lognormal distribution
– The lognormal distribution is asymmetric (skewed to the right)
– The lognormal distribution is bounded below by 0 (lowest possible
value)
– The lognormal distribution fits well data on asset prices (note that
prices are bounded below by 0)
• Note also that the normal distribution fits well data on asset returns
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 is
e  0.50 2 
where  and 2 are the mean and variance of the normal
distribution of the ln(X) variable where e  2.718
The Lognormal Distribution
• Digression
• Recall that the exponential and logarithmic functions mirror each
other
• This implies the following result
ln(x)  y  e y  x
• E.g. ln(1) = 0 since e0 =1
The Lognormal Distribution
• Therefore, if X is lognormal, we can write
ln (X) = ln (eY) = Y
where Y is normal
• The expected value of X is equal to the expected value of eY
• But, this is not equal to e, but to the expression for the mean
shown above
The Lognormal Distribution
• Intuitive explanation
– As the variance of the associated distribution increases, the
lognormal distribution spreads out
– The distribution can spread out upwards, but is bounded below by 0
– Thus, the mean of the lognormal distribution increases
The Lognormal Distribution
• The variance of a lognormal distribution is
2   e
2   2
e

1

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
• E.g., if S0 = $30 and S1 = $34.5, then the relative price is
$34.5/$30 = 1.15, meaning that the holding period return 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  lnSt 1 / St   ln1  Rt ,t 1 
Example: Relative Asset Prices and
the Lognormal Distribution
• For the above example, the continuously compounded return is
r0,1 = ln($34.5/$30) = ln(1.15) = 0.1397 or 13.98%, 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
ST  S0e
r0,T
• Note that
ST / S0  ST / ST 1 ST 1 / ST S1 / S0 
Example: Relative Asset Prices and
the Lognormal Distribution
• Digression
• Recall that
– ln(XY) = ln(X) + ln(Y)
– ln(eX) = X
• Following these rules,
ln(ST / S0 )  lnST / ST 1   lnST 1 / ST     lnS1 / S0 
r0,T  rT 1,T  rT 2,T 1    r0,1
Example: Relative Asset Prices and
the Lognormal Distribution
• It is commonly assumed in investments that returns are
represented by random variables that are independently and
identically distributed (IID)
• This means that investors cannot predict future returns based on
past returns (weak-form market efficiency) and the distribution of
returns is stationary
• Following the previous results, the mean continuously
compounded return between periods 0 and T is the sum of the
continuously compounded returns of the interim one-period
returns
Example: Relative Asset Prices and
the Lognormal Distribution
• If the one-period continuously compounded returns are normally
distributed, their sum will also be normal
• Even if they are not, by the CLT, their sum will be normal
• So, we can model the relative stock price as a lognormal
variable whose natural log, given by the continuously
compounded return is distributed normally
– Application: option pricing models like Black-Scholes include the
volatility of continuously compounded returns on the underlying
asset obtained through historical data
Sampling and Estimation
Random Sampling from a Population
• In inferential statistics, we are interested in making an inference
about the characteristics of a population through information
obtained in a subset called sample
• Examples
– What is the mean annual return of all stocks in the NYSE?
– What is the mean value of all residential property in the area of
Chicago?
– What is the variance of P/E ratios of all firms in Nasdaq?
Random Sampling from a Population
• To make an inference about a population parameter
(characteristic), we draw a random sample from the population
• Suppose we select a sample of size n from a population of size
N
• A random sampling procedure is one in which every possible
sample of n observations from the population is equally likely to
occur
Random Sampling from a Population
• Example: We want to estimate the mean ROE of all 8,000+
banks in the US
– Draw a random sample of 300 banks
– Analyze the sample information
– Use that information to make an inference about the population
mean
• To make an inference about a population parameter, we use
sample statistics, which are quantities obtained from sample
information
• E.g., To make an inference about the population mean, we
calculate the statistic of the sample mean
Random Sampling from a Population
• Note: Drawing several samples from a population will result in
several values of a sample statistic, such as the sample mean
• A sample statistic is a random variable that follows a distribution
called sampling distribution
• Note: We say that the sample mean will be our estimate of the
“true” population parameter, the population mean
• The difference between the sample mean and the “true”
population mean is called the sampling error
Sampling Distribution of the Sample Mean
• Suppose we attempt to make an inference about the population
mean by drawing a sample from the population and calculating
the sample mean
• The sample mean of a random sample of size n from a
population is given by
1 n
X   Xi
n i 1
Sampling Distribution of the Sample Mean
• Digression
• Central Limit Theorem
– Suppose X1, X2, …, Xn are n independent random variables from a
population with mean  and variance 2. Then the sum or average
of those variables will be approximately normal with mean  and
variance 2/n as the sample size becomes large
• Implication:
– If we view each member of a random sample as an independent
random variable, then the mean of those random variables,
meaning the sample mean, will be normally distributed as the
sample size gets large
Sampling Distribution of the Sample Mean
• The CLT applies when sample size is greater or equal than 30
– Note: In most applications with financial data, sample size will be
significantly greater than 30
• Using the results of the CLT, the sampling distribution of the
sample mean will have a mean equal to  and a variance equal
to 2/n
• The corresponding standard deviation of the sample mean,
called the standard error of the sample mean, will be
X / n
Sampling Distribution of the Sample Mean
•
Implication: The variance of the sampling distribution of the sample
mean decreases as the sample size n increases
•
The larger is the sample drawn from a population, the more certain is
the inference made about the population mean based on sample
information, such as the sample mean
•
Example: Suppose we draw a random sample from a normal
population distribution
– The sample mean will also follow a normal distribution
– The variable Z follows the standard normal distribution
Z
X 
~ N 0,1
X / n
Example of Sampling from a
Normal Distribution
• Suppose that, based on historical data, annual percentage
salary increases for CEOs of mid-size firms are normally
distributed with mean 12.2% and st. deviation of 3.6%
• What is the probability that the sample mean in a random
sample of 9 will be less than 10%
• We are looking for
P X  .10
Example of Sampling from a
Normal Distribution
• Transforming the sample mean into a standard normal variable
.10  .12 

P X  .10  P Z 
  PZ  1.83
.036 / 9 

which is equal to FZ(-1.83) = 1 - FZ(1.83) = .0336, which is the
probability that the sample mean will be less than 10%
Sampling Distribution of a
Sample Proportion
• If X follows a binomial distribution, then to find the probability of
a certain number of successes in n trials, we need to know the
probability of a success p
• To make inferences about the population proportion p (the
probability of a success as described above), we use the
sample proportion
• The sample proportion is the ratio of the number of successes
(X) in a sample of size n
pˆ  X / n
Sampling Distribution of a
Sample Proportion
• The mean and variance of the sampling distribution of the
sample proportion are
E  pˆ   p
p1  p 
V  pˆ  
n
• The standard error is obtained accordingly and the standardized
variable Z follows the standard normal distribution
Sampling Distribution of the
Sample Variance
• Suppose we draw a random sample n from a population and
want to make an inference about the population variance
• This inference can be based on the sample variance defined as
follows
1
s2 
n
 X i  X 2

n 1
i 1
• The mean of the sampling distribution of the sample variance is
equal to the population variance
Sampling Distribution of the
Sample Variance
• In many applications, the population distribution of the random
variable of interest will be normal
• It can be shown that, in this case
n  1s 2
2
follows the chi-square distribution with (n – 1) degrees of
freedom
n  1s 2
~  n21
2
Sampling Distribution of the
Sample Variance
• The variance of the sampling distribution of the sample variance
is
 
2
2
Vs 
4
n 1