I think John Lim was addressing the concept that if there is a probability of some loss for one year, that “the growing improbability of a loss is offset by the increasing magnitude of potential losses” (Mark Ktrizman, Financial Analysts Journal, 2015:71 (1), referenced in John Clements’ comment above) with longer time horizons. Although I am not a statistician, my experience from taking a few statistical classes left me with a nagging feeling after reading John Lim’s article. After going back to my statistical book, I finally figured it out what concerned me.
The statement “If the standard deviation of returns over one year is 30%, or 0.3, the standard deviation of the return over n years equals 0.3 * √n.” is incorrect. N is the number of annual return data points used in the analysis of the mean and standard deviation of one-year returns. The analysis of yearly returns can only provide information on a probability of returns on any random year, that is, the standard deviation of returns in any given year is 30%. This analysis says nothing about the returns for a 30-year period. A completely different analysis is required to address the probability of returns for a 30-year period. A Monte Carlo analysis could be used to calculate the probability of total return for a longer periods of time using numerous (N observations) combination of 30 one-year intervals using the given probability of one-year returns stated above. This method was probably used to generate Figure A in Kritzman article that show risk decreases with time horizon for a their specific one-year return probability. The probability of returns for a 30-year period could also be calculated using the returns of random 30-year periods from the stock market data, in which each random 30-year period is one observation (N).
My analysis does not address the discussion of time diversification using the utility function presented later in the Kritzman article.
Comments
I think John Lim was addressing the concept that if there is a probability of some loss for one year, that “the growing improbability of a loss is offset by the increasing magnitude of potential losses” (Mark Ktrizman, Financial Analysts Journal, 2015:71 (1), referenced in John Clements’ comment above) with longer time horizons. Although I am not a statistician, my experience from taking a few statistical classes left me with a nagging feeling after reading John Lim’s article. After going back to my statistical book, I finally figured it out what concerned me. The statement “If the standard deviation of returns over one year is 30%, or 0.3, the standard deviation of the return over n years equals 0.3 * √n.” is incorrect. N is the number of annual return data points used in the analysis of the mean and standard deviation of one-year returns. The analysis of yearly returns can only provide information on a probability of returns on any random year, that is, the standard deviation of returns in any given year is 30%. This analysis says nothing about the returns for a 30-year period. A completely different analysis is required to address the probability of returns for a 30-year period. A Monte Carlo analysis could be used to calculate the probability of total return for a longer periods of time using numerous (N observations) combination of 30 one-year intervals using the given probability of one-year returns stated above. This method was probably used to generate Figure A in Kritzman article that show risk decreases with time horizon for a their specific one-year return probability. The probability of returns for a 30-year period could also be calculated using the returns of random 30-year periods from the stock market data, in which each random 30-year period is one observation (N). My analysis does not address the discussion of time diversification using the utility function presented later in the Kritzman article.
Post: Time Can Take a Toll
Link to comment from October 15, 2021