Overview
Diversification is one of the primary goals of portfolio construction, if not the primary goal. From an expected average return perspectives, adding assets into a portfolio will give you a portfolio with an exptected return which the weighted average of the expected returns of its consituents. In particular, the portfolio's expected return will be less than the expected return of the asset with the highest expected return. That is, one could get a higher expected return just by investing in that single asset.
From a growth perspective, diversification can increase expected growth.
Diversification and Standard Deviation
Using return variance (or equivalently standard deviation) as a the measure of portfolio risk.
{% variance(r) = \mathbb{E}[\sum \frac{1}{n}(r_i - r)]^2 %}
{% variance(r) = \frac{1}{n^2} \sum \sigma_{ij} %}
{% variance(r) = \frac{1}{n^2} [\sum_{i} \sigma_{ii} + \sum_{i \neq j} \sigma_{ij}] %}
Uncorrelated Assets
When the assets are uncorrelated, we have
{% \sigma_{ij} = 0 %}
for i not equal to j.
Assuming that each asset has a variance of {% \sigma^2 %}, then the total variance is
{% variance = \frac{n}{n^2} \sigma^2 %}
{% variance = \frac{1}{n} \sigma^2 %}
That is, the variance decreases linearly with the number of assets and approaches zero asymptotically.
Correlated Assets
{% \sigma_{ij} = c %}
Then
{% \sum_{i \neq j} \sigma_{ij} = \frac{1}{n^2} (n^2 - n) \times c %}
which approaches c asymptotically. That is, when the assets are Correlated, c is the lowest achievable variance, showing
that some of the risk is diversifiable, and some is not.
Trend Diversification
The standard mathematical treatment of diversification given above ignores the trends developing within each asset or asset class. Theoretically, it is possible for two assets to trend in the direction while the day to day returns are uncorrelated, or even anticorrelated. From this perspective, the most effective way to diversify portfolio is to find assets that do not form down trends at the same time that the portfolio is trending down.
Note that we do not seek assets that trend in oppositive directions. We dont necessarily want our diversifying assets to trend down while our portfolio is trending up.
As an example, consider a manager trying to diversify an equity portfolio. Equities trend up most of the time (say 90%). If the manager found an asset that trended in the opposite direction to equities, they would get an asset that trends down 90% of the time. (A true anti equity trending asset would be short equities, but then you would end up with the risk free rate).
Trend Diversification