Appendix Two: Lognormal Distribution of Annual Returns
Stochastic analysis of investments requires an understanding of the statistical properties of returns. The assumption that annual returns follow a lognormal distribution is relatively robust. It is based on the Central Limit Theorem as follows.
Suppose that a year is made up of many (say x=200) trading days, with daily returns (dyear,day) that are serially independent and of finite variance, but the form of the distribution is unspecified. Daily returns compound into annual returns (ryear):
(A7)
Taking the log of each side:
(A8)
According to the Central Limit Theorem, the sum of n independent random variables with finite variance converges to a normal distribution when n is large. Since dt,I is an independent series with finite variance, so is log[1+dt,I]. And 200 is large. Thus, log[1+rt] is approximately normally distributed and so [1+rt] approximately follows the corresponding lognormal distribution.
(A9)
This result requires no assumption about the shape of the distribution of daily returns. If daily returns, themselves, are lognormally distributed, then the annual returns will be exactly lognormally distributed (being the product of independent lognormally distributed variables).
The variable, log[1+rt], is also known as the continuously compounded rate of return.
The mean and variance of rt can be expressed in terms of μ and σ2 using the moment generating function of a normally distributed (1+rt).[18]
(A10)
(A11)
Notes
- [18]See Aitchison and Brown (1957) for a detailed treatment on the lognormal distribution and its application in economics.
