the lognormal distribution
DESCRIPTION
The Lognormal Distribution. The lognormal distribution is an asymmetric distribution with interesting applications for modeling the probability distributions of stock and other asset prices - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: The Lognormal Distribution](https://reader035.vdocument.in/reader035/viewer/2022062305/56814c24550346895db9293c/html5/thumbnails/1.jpg)
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
![Page 2: The Lognormal Distribution](https://reader035.vdocument.in/reader035/viewer/2022062305/56814c24550346895db9293c/html5/thumbnails/2.jpg)
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
![Page 3: The Lognormal Distribution](https://reader035.vdocument.in/reader035/viewer/2022062305/56814c24550346895db9293c/html5/thumbnails/3.jpg)
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
where and 2 are the mean and variance of the normal distribution of the ln(X) variable where e 2.718
250.0 e
![Page 4: The Lognormal Distribution](https://reader035.vdocument.in/reader035/viewer/2022062305/56814c24550346895db9293c/html5/thumbnails/4.jpg)
The Lognormal Distribution
• Digression
• Recall that the exponential and logarithmic functions mirror each other
• This implies the following result
• E.g. ln(1) = 0 since e0 =1
xeyx y )ln(
![Page 5: The Lognormal Distribution](https://reader035.vdocument.in/reader035/viewer/2022062305/56814c24550346895db9293c/html5/thumbnails/5.jpg)
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
![Page 6: The Lognormal Distribution](https://reader035.vdocument.in/reader035/viewer/2022062305/56814c24550346895db9293c/html5/thumbnails/6.jpg)
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
![Page 7: The Lognormal Distribution](https://reader035.vdocument.in/reader035/viewer/2022062305/56814c24550346895db9293c/html5/thumbnails/7.jpg)
The Lognormal Distribution
• The variance of a lognormal distribution is
12
22 ee
![Page 8: The Lognormal Distribution](https://reader035.vdocument.in/reader035/viewer/2022062305/56814c24550346895db9293c/html5/thumbnails/8.jpg)
Example: Relative Asset Prices andthe 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
1,11, 1ln/ln tttttt RSSr
![Page 9: The Lognormal Distribution](https://reader035.vdocument.in/reader035/viewer/2022062305/56814c24550346895db9293c/html5/thumbnails/9.jpg)
Example: Relative Asset Prices andthe 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
• Note that
TrT eSS ,0
0
01110 //// SSSSSSSS TTTTT
![Page 10: The Lognormal Distribution](https://reader035.vdocument.in/reader035/viewer/2022062305/56814c24550346895db9293c/html5/thumbnails/10.jpg)
Example: Relative Asset Prices andthe Lognormal Distribution
• Digression
• Recall that– ln(XY) = ln(X) + ln(Y)– ln(eX) = X
• Following these rules,
01110 /ln/ln/ln)/ln( SSSSSSSS TTTTT
1,01,2,1,0 rrrr TTTTT
![Page 11: The Lognormal Distribution](https://reader035.vdocument.in/reader035/viewer/2022062305/56814c24550346895db9293c/html5/thumbnails/11.jpg)
Example: Relative Asset Prices andthe 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
![Page 12: The Lognormal Distribution](https://reader035.vdocument.in/reader035/viewer/2022062305/56814c24550346895db9293c/html5/thumbnails/12.jpg)
Example: Relative Asset Prices andthe 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
![Page 13: The Lognormal Distribution](https://reader035.vdocument.in/reader035/viewer/2022062305/56814c24550346895db9293c/html5/thumbnails/13.jpg)
Sampling and Estimation
![Page 14: The Lognormal Distribution](https://reader035.vdocument.in/reader035/viewer/2022062305/56814c24550346895db9293c/html5/thumbnails/14.jpg)
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?
![Page 15: The Lognormal Distribution](https://reader035.vdocument.in/reader035/viewer/2022062305/56814c24550346895db9293c/html5/thumbnails/15.jpg)
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
![Page 16: The Lognormal Distribution](https://reader035.vdocument.in/reader035/viewer/2022062305/56814c24550346895db9293c/html5/thumbnails/16.jpg)
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
![Page 17: The Lognormal Distribution](https://reader035.vdocument.in/reader035/viewer/2022062305/56814c24550346895db9293c/html5/thumbnails/17.jpg)
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
![Page 18: The Lognormal Distribution](https://reader035.vdocument.in/reader035/viewer/2022062305/56814c24550346895db9293c/html5/thumbnails/18.jpg)
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
n
iiXn
X1
1
![Page 19: The Lognormal Distribution](https://reader035.vdocument.in/reader035/viewer/2022062305/56814c24550346895db9293c/html5/thumbnails/19.jpg)
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
![Page 20: The Lognormal Distribution](https://reader035.vdocument.in/reader035/viewer/2022062305/56814c24550346895db9293c/html5/thumbnails/20.jpg)
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
nX /
![Page 21: The Lognormal Distribution](https://reader035.vdocument.in/reader035/viewer/2022062305/56814c24550346895db9293c/html5/thumbnails/21.jpg)
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
1,0~
/N
nXZX
![Page 22: The Lognormal Distribution](https://reader035.vdocument.in/reader035/viewer/2022062305/56814c24550346895db9293c/html5/thumbnails/22.jpg)
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
10.XP
![Page 23: The Lognormal Distribution](https://reader035.vdocument.in/reader035/viewer/2022062305/56814c24550346895db9293c/html5/thumbnails/23.jpg)
Example of Sampling from a Normal Distribution
• Transforming the sample mean into a standard normal variable
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%
83.19/036.
12.10.10.
ZPZPXP
![Page 24: The Lognormal Distribution](https://reader035.vdocument.in/reader035/viewer/2022062305/56814c24550346895db9293c/html5/thumbnails/24.jpg)
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
nXp /ˆ
![Page 25: The Lognormal Distribution](https://reader035.vdocument.in/reader035/viewer/2022062305/56814c24550346895db9293c/html5/thumbnails/25.jpg)
Sampling Distribution of a Sample Proportion
• The mean and variance of the sampling distribution of the sample proportion are
• The standard error is obtained accordingly and the standardized variable Z follows the standard normal distribution
npppV
ppE
1ˆ
ˆ
![Page 26: The Lognormal Distribution](https://reader035.vdocument.in/reader035/viewer/2022062305/56814c24550346895db9293c/html5/thumbnails/26.jpg)
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
• The mean of the sampling distribution of the sample variance is equal to the population variance
n
ii XX
ns
1
221
1
![Page 27: The Lognormal Distribution](https://reader035.vdocument.in/reader035/viewer/2022062305/56814c24550346895db9293c/html5/thumbnails/27.jpg)
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
follows the chi-square distribution with (n – 1) degrees of freedom
2
21
sn
212
2~1
nsn
![Page 28: The Lognormal Distribution](https://reader035.vdocument.in/reader035/viewer/2022062305/56814c24550346895db9293c/html5/thumbnails/28.jpg)
Sampling Distribution of the Sample Variance
• The variance of the sampling distribution of the sample variance is
1
2 42
n
sV