Portfolio optimization

Question

I prepare Markowitz optimization model for 4 different local companies with Excel functions and Solver. And I get confusing result for me, maybe somebody will explain it. So, from historic information I calculated daily returns and from that daily returns I calculated arithmetic and geometric means. And for just one company I get a huge difference between AVERAGE and GEOMEAN means. Arithmetic mean: 0.145 and geometric mean: -0.00171. So, in result this company has the highest arithmetic mean and the lowest geometric mean of all 4 companies. I get that the model invests the lower percentage of cash to this company. Is everything fine with this result?

Answer 1

Given returns $r_1,r_2, \ldots , r_n$ the geometric mean $(G)$ and arithmetic mean $(A)$ returns are

$$G = \sqrt[n]{\prod_{j=1}^n (1+ r_j)} - 1, \quad A = \frac{1}{n}\sum_{j=1}^n r_j$$

The geometric mean return is smaller than the arithmetic mean return and the approximate difference is related to volatility $\sigma$ (return standard deviation) according to

$$G \approx A - \frac{\sigma^2}{2}$$

The geometric mean return is a more realistic measure when volatility is high, and the results you observe are plausible. For example given daily returns of $-40\%$ and $+65\%$, the artithmetic mean return is $12.5\%$ and the geometric mean return is

$$G = \sqrt{(1-0.4)(1+0.65)}- 1 \approx -0.5 \%$$

Given the large difference between the arithmetic mean and geometric mean returns of this security, the volatility must be very high and the mean-variance optimizer is likely to arrive at a very low allocation.