How Diversification Affects Geometric Portfolio Returns

# Why is the performance of a portfolio based on geometric means boosted by positive correlation?


# Why is the performance of a portfolio based on geometric means boosted by positive correlation?












https://qoppac.blogspot.com/2017/02/can-you-eat-geometric-returns.html

The blog post above by Rob Carver discusses the use of geometric means to evaluate investments. The section "The consequences of using geometric returns" gives the following example.

Assuming correlation of 0.85:

- 1 asset: arithmetic mean 5%, geometric mean 1.3%

- 5 assets: arithmetic mean 5%, geometric mean 1.8%

Unfortunately the calculation is not given so could someone run me through how this was calculated?

I want to understand the intuition as to why the geometric mean is improved by the positive correlation what would happen if the correlation was zero or negative?

Thanks

Baz

## Answer by Enrico Schumann (score 3, accepted)

https://quant.stackexchange.com/a/71497

The geometric return is not improved by high correlation; it increases if the number of assets increase. More assets will decrease variance, and so the geometric mean will approach the arithmetic mean. See also this answer Simulating Correlated Stock Returns in Python (SciPy)

## Answer by Dr. Ferry L. J. Vos (score 2)

https://quant.stackexchange.com/a/75124

For a portfolio, the relation between geometric mean return (GM) and arithmetic mean return (AM) is determined by the volatility (V) of the returns of that asset can be calculated as:

$$GM = AM - 1/2 \times V^2$$

This shows that if you lower the volatility of your portfolio without changing the arithmetic return, you will improve the geometric mean. This answers your question on the intuition for what happens here: by having not-perfect correlation between the assets you lower the portfolio volatility (diversification) relative to the case where you hold only one asset.

Apparently, in this example, the volatility used for the one asset class is

$$V = \sqrt{( 2 \times (AM-GM) )} = \sqrt{( 2 \times (0.05-0.013) )} = 27\%$$

The volatility of an equally weighted portfolio of 5 assets - each with the same volatility but with an assumed uniform mutual return correlation of 0.85 - would be:

$$V = \sqrt{( 5 \times (0.20 \times 27\%)^2 + 10 \times 2 \times 0.85 \times 0.20 \times 27\% \times 0.20 \times 27\%)} = 25.3\%$$

Since the expected arithmetic return for the five asset portfolio is still 5% (because in this example each assets is expected to return 5%) the GM of the 5-asset portfolio calculates to

$$\begin{align}GM &= AM - 1/2 V^2 \\ &= 0.05 - 1/2 \times (0.0253)^2 \\ &= 1.8\%\end{align}$$

In general GM is lower than AM because volatility detracts from multiperiod average return. A simple example

an arithmetic return of 20% followed by an arithmetic return of -10% does not result in a return of 10%. It results in a return of $1.2 \times 0.9 - 1 = 8\%$.

So the arithmetic average one-period return over the two periods is $(20\% + (-10\%))/2 = 10\%$. The geometric average one-period return is $( (1+0.20) \times (1+0.90) )^{1/2} -1 = 3.9\%$.

## Answer by Newquant (score 1)

https://quant.stackexchange.com/a/75136

Like others mentioned, it's not correlation, but the number of assets in the portfolio that draws the geometric growth rate up to the arithmetic growth rate.

The annual geometric growth rate is given by:

$r - \frac{1}{2} \sigma^2$

In the portfolio sense, $r$ is the weighted average arithmetic returns of the underlying assets, and $\sigma$ is the net volatility of the constituents. The volatility section is more complicated to calculate. For $N$ equally weighted assets, the volatility of the portfolio is given by:

$σ_{average} * \sqrt{(ρ_{average} +1)/N}$

So you could expect that for large $N$, the volatility of the portfolio decreases proportionately to $\sqrt{1/N}$.

To better answer your question, backsolving from mean to geo-mean, the volatility of 1 asset looks to be ~ 35%. For the 5 asset portfolio the average volatility looks to be ~ 41.5%. If the correlation moved to 0 the geo-mean would be 3.28%, if the correlation turned to -0.85 the geo-mean would become 4.71%.