How Diversification Reduces Portfolio Volatility Through Covariance

# Relationship between diversification and standard deviation


# Relationship between diversification and standard deviation












Explain the relationship between diversification and standard deviation:

There are two general principles that should govern investment behaviors in a world of efficient markets, where one has the same information as other market participants have:

The first principle is the principle of diversification---of "not putting all one's eggs in one basket".

The second is the principle that one can obtain a higher returns over the very long run (though not necessarily in the short run) by investing in riskier assets. To put the second point differently, market participants require a higher return from an asset, and will correspondingly pay a lower price for the income stream from it, the greater the risk. This topic deals with the first of these principles.

## Answer by markowitz (score 1)

https://quant.stackexchange.com/a/44128

The material linked above by Emma are useful. However a short answer to your question can be the next: any equally weighted ptf, with $N$ assets, have his standard deviations ($\sigma_N$). Essentially diversification says that if we add another asset, always in equally weighted scheme, the new standard deviation become $\sigma_{N+1} <= \sigma_N$. Perfect correlation case apart the disequality is strict.

## Answer by jthg (score 1)

https://quant.stackexchange.com/a/44130

Here are two examples on how diversification reduces standard deviation.

Diversification in a 2-asset portfolio.

We have that the variance of a 2-asset portfolio is given by

$$ \sigma_p^2 = \omega_a^2 Var[r_a]+(1-\omega_a)^2 Var[r_b]+2\omega(1-\omega)Std[r_a]Std[r_b]\rho_{ab}$$

Where $\omega_a$ is the weight in asset $a$, $Var[r_a],Var[r_b]$ are the assets variances, $Std[r_a], Std[r_b]$ are the standard deviations of the assets, and $\rho_{ab}$ is the correlation between the them.

If the two assets were the same, e.g. the same stock, the correlation would be perfect, i.e. $\rho_{ab}=1$, and portfolio variance would just be

$$ \sigma_p^2 = \omega_a^2 Var[r_a]+(1-\omega_a)^2 Var[r_b]+2\omega(1-\omega)Std[r_a]Std[r_b]$$

The correlation between any assets is always between 1 and -1, so for any two assets

$$ \sigma_p^2 \leq \omega_a^2 Var[r_a]+(1-\omega_a)^2 Var[r_b]+2\omega(1-\omega)Std[r_a]Std[r_b]$$

Which means that the portfolio variance of two assets will always be less than or equal to the weighted variance-contribution from each individual asset.

Portfolio of equal weights.

For a portfolio of $N$ correlated and equally weighted assets ($ w_i = \frac{1}{N}$) we have that $$ \sigma_p^2 = \frac{1}{N^2} \sum\limits_{i=1}^N Var[r_i] + \frac{1}{N^2} \sum\limits_{i=1}^N \sum\limits_{j\neq i,\,j=1}^N Cov[r_i,r_j] $$

The average values these individual assets are

$$ \bar{Var} = \frac{1}{N} \sum\limits_{i=1}^N Var[r_i]$$ $$ \bar{Cov} = \frac{1}{N(N-1)} \sum\limits_{i=1}^N\sum\limits_{j\neq i,j=1}^N Cov[r_i,r_j]$$

From which it follows that $$ \sigma_p^2 = \frac{1}{N^2} N \bar{Var} + \frac{1}{N^2}N(N-1)\bar{Cov} = \underbrace{\frac{1}{N} \bar{Var}}_{\rightarrow 0}+\underbrace{(1-\frac{1}{N})\bar{Cov}}_{\rightarrow \bar{Cov}}$$

From that we can conclude that when the number of assets $N$ goes to infinity, the variance of the portfolio goes to $\bar{Cov}$. So basically, diversification is the elimination of asset-specific (idiosyncratic) standard deviation (risk) from investing in multiple assets.

All of above is based on Financial Markets and Investments by Claus Munk (2018, Chapter 4.3). I do not know if this is available online, but if not i can also recommend Investments by Bodie, Kane & Marcus (2014).