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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 lnSt 1 / St ln1 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 ) lnST / ST 1 lnST 1 / ST lnS1 / 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 PZ 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 p1 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 1s 2 2 follows the chi-square distribution with (n – 1) degrees of freedom n 1s 2 ~ n21 2 Sampling Distribution of the Sample Variance • The variance of the sampling distribution of the sample variance is 2 2 Vs 4 n 1