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.

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.

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).

• This formulation is correct but there are some imprecision. In the first example you write: “The correlation between any assets is always between 0 and 1, so for any two assets” the below formula is correct but the correlation move between $-1$ and $1$. – markowitz Feb 19 at 14:53
• After you write: “Which means that the portfolio variance of two assets will always be less than or equal to the variance-contribution from each individual asset.” This phrase is incorrect in general, and the example is so. This phrase is true only for equally weighted ptf. In the general case the contribution of any single additional asset is not trivial to show. In the second example you write: “So basically, diversification is the elimination of standard deviation (risk) from investing in multiple assets.” – markowitz Feb 19 at 14:53
• The correct word is not “elimination” but “reduction”. Probably you wanted to say that elimination is about idiosyncratic risk. – markowitz Feb 19 at 14:53
• Ok, now the explanation is correct. – markowitz Feb 19 at 14:58