Hi, As per introductory paragraph, if Assest A and B have same expectation and variance and if Assest A is positively skewed and Assest B is symmetrical, then Assest A might be preferred by some investors. Can someone please let me know the reason for this. Many thanks Sunil
Intuitively: Asset A is less likely to experience big negative returns than asset B, since it is positively skewed, even though the mean and variance are the same. Or, if you prefer a more theoretical argument: An investor wanting to maximise their expected log-wealth will want to maximise the odd moments and minimise the even moments of return. If the return is R then, without loss of generality, wealth is 1+R, and maximising log(1+R) is the goal. We can Taylor expand log(1+R): log(1+R) = R - R^2 / 2 + R^3 / 3 - R^4 / 4 + ... From this expression, we can see that to maximise log-wealth we need to maximise the odd moments of the return (mean, skew, ...) and minimise the even moments (volatility, kurtosis, ...). Maximising expected log-wealth is a common idea used in finance because it maximises the expected geometric return. You can also think of it as maximising expected utility with a logarithmic utility function. Through this lens, it's incorporating some risk aversion because log is a concave function.
Thanks. I don't know if my understanding is correct, but if Asset A is positively skewed then the probability of return more than mean is less than what is for Asset B. Or probability of higher returns for Asset A is less than Asset B. Does this make any sense.
What you described is a decent rule of thumb for most 'nice' probability distributions, in particular continuous ones, but I wouldn't be confident that the idea holds in general. E.g. see here: https://www.tandfonline.com/doi/full/10.1080/10691898.2005.11910556 which shows some examples of positively skewed distributions where the median is above the mean rather than below.
