The Lognormal Distribution The lognormal distribution is an asymmetric distribution with interesting...

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


• 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 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


• 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(


• 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


• 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 variance of a lognormal distribution is

12

22 ee

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


• 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


• 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


• 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


• 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?


• 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


• 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


• 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

n

iiXn

X1

1


• 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


• 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 /


• 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

n

XZ

X

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

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

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

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

n

pppV

ppE

1

ˆ

ˆ

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


• 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

n

sn


• The variance of the sampling distribution of the sample variance is

1

2 42

n

sV

The Lognormal Distribution The lognormal distribution is an asymmetric distribution with interesting...

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