Normal Additive Returns and Lognormal Compounding for Terminal Wealth

# Squaring lognormal compounding with linear addition of normal returns


# Squaring lognormal compounding with linear addition of normal returns












Let’s say we start with $100 and invest it for 20 years in stocks and want to predict its terminal value as a random variable (RV). And let’s assume average yearly returns are 10% and volatility is 20%.

One way to do this is to assume returns are normally distributed (we only have two moments so it’s also parsimonious). Thus, the amount we expect to add next year is 10, giving us 110, and the volatility is 20 dollars numerically. The next year we do the same, adding now 11 dollars to get us to 121, and (if uncorrelated) after adding the standard deviations we will have a volatility of something like 28 dollars as our RV. Continue on for 20 years and we will have as our result a normal distribution with mean equal to 100 dollars times $1.1^{20}$, and a standard deviation that is whatever 20 compoundings gets us.

The other way to do this is to start with a certain 100 dollars and, each period, multiply the current distribution times a lognormal distribution. If you do the arithmetic right, the $\mu$ you use needs to be adjusted to account for the $\sigma^2/2$ term, as well as to deal with non-continuous compounding (I.e., you want $e^\mu$ to be equal to 1.1 if 10% is still to be the expected yearly market return.). If you work at it, you can get the right parameters to have the same approach happen; but every year you get the new RV by multiplying the old RV by this log-normal multiplier.

The RV from approach two will, at the end, still have an expected value that is the same (100 dollars times $1.1^{20}$). But, it will be asymmetrical, with a heavier tail to the right. In particular, the mode will be less than the expected value. One implication is there would be a larger confidence interval for outsized returns to the upside compared to the normal distribution approach.

I am trying to interpret and justify this. Log-normal returns get used, in large part, because it squares with continuous time stochastic calculus. But if you think about it as I am doing - discretizing carefully - I am trying to justify which approach gives a more correct answer, and/or the pluses or minuses of the two approaches. One thing log-normal returns does is eliminate prices dropping below zero, but it also introduces this asymmetry with a skew to the upside. If anything it was always my belief that empirical stock market returns are skewed a bit to the downside. So, is this just a matter of needing a hammer to drive a nail and living with the collateral damage (I.e. getting the stochastic calculus to work right) or is there some logical intuition about how these two approaches lead to different answers, or empirical data on market returns that justifies approach 2?

Simply put, if market returns are, on average 10% (with some known volatility), does an investor who views that as an annual compounding of normally distributed returns with the right parameters end up with a different view as to terminal wealth as someone who uses GBM/a log-normal multiplicative approach - adjusting the parameters accordingly to eliminate arithmetic divergence alone - and, if so, what is the logic? It seems one can argue that log-normal distributions, by their very nature, bias upwards annual returns (since returns that are normally distributed per an RV X are skewed upwards by using the log-norm RV $Y=e^X$, and is this just an illusion or is there an empirical or theoretical reason to accept on as more correct than the other, ceteris paribus?)