2.11

# How does diversification work?

This step explains how we can use diversification to reduce the risk of our investment, by looking at a simple example.

Consider two assets, $AB$ and $XY$, with the same expected return, 5%:

and with the same risk measured by their standard deviation, 4%:

The two assets therefore have the same variability, and on average also yield the same return.

Would there be an advantage in investing in a portfolio which contains both assets, instead of investing everything in either of the two assets only?

If we form a portfolio, $P$, where assets $AB$ and $XY$ are held in equal proportion, the return on the portfolio will be an average of the returns on each one of the assets:

The expected return on $P$ is:

Therefore, the expected return on the portfolio will be the same as the expected return on each one of the assets.

But what about the variability of the returns on the portfolio? This will depend on how the two assets vary together, that is, on their covariance. The formula that gives the variance of the portfolio when the two assets are held in equal proportions is:

where ${{{\sigma}_A}_B},{_X}{_Y}$ is the covariance between the assets.

The covariance between $AB$ and $XY$ can be computed as the product of the correlation coefficients between the two assets, ${{{\rho}_A}_B},{_X}{_Y}$, and of their standard deviations, ${{{\sigma}_A}_B}$ and ${{{\sigma}_X}_Y}$:

The variance of the portfolio, ${{\sigma}^2_P}$, is thus given by:

If we carry out all the algebraic calculations and simplify, the above formula becomes:

Let us consider some values for the correlation coefficient between -1 and 1, which are the smallest and the largest possible values that it can take. The variance and the standard deviations of the portfolio $P$ will depend on the value of the correlation coefficient. The calculations give the values reproduced in the following table.

${{{\rho}_A}_B},{_X}{_Y}$ ${{\sigma}^2_P}$ ${{\sigma}_P}$
1 0.0016 0.040
0.5 0.0012 0.035
0 0.0008 0.028
- 0.5 0.0004 0.020
- 1 0 0.000

The last column of the table shows the standard deviation of the portfolio, for the different values of the correlation coefficient.

When the correlation coefficient is 1, there is no gain from diversification: the standard deviation of the portfolio is 0.04 = 4%, just like the standard deviation of the individual assets.

In all the other cases, however, the standard deviation of the portfolio is lower than that of the assets. When the two assets are less than perfectly positively correlated, diversification can therefore be effective in reducing the risk of the portfolio. If the assets are perfectly negatively correlated, that is when the correlation coefficient is equal to - 1, the portfolio standard deviation becomes zero: we have completely eliminated uncertainty, and the portfolio is now riskless.

Think about that result. Are you convinced by it? If it helps, think back to the umbrella firm and the ice-cream firm. What does it mean if the returns on their shares are perfectly negatively correlated?