# Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

## [PGP-I FM] Diversifiable vs. Non-diversifiable Risk: The Math (Wonkish)

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The result that variance of the market portfolio is a weighted average of covariance of the underlying stocks with the market is a general result. This post fills in the mathematical blanks.

Diversifiable vs. Non-diversifiable Risk: The Math

Consider the variance of a Markowitz portfolio $M$ containing $N$ assets:

$\displaystyle \sigma^2_M = \sum_{i=1}^N \omega^2_i \sigma^2_i + \sum_{i=1}^N \sum_{\substack{j=1 \\ j \ne i}}^N \omega_i \omega_j \sigma_{ij}$

Now if we let the weight of each asset in the portfolio to be the same, i.e. $\omega_i = \omega = 1/N$ $\forall i$, and consider the “average variance” as:

$\displaystyle \overline{\sigma^2} =\frac{1}{N} \sum_{i=1}^N \sigma^2_i$

and “average covariance” as:

$\displaystyle \overline{\sigma_{ij}} = \frac{1}{N (N - 1)} \displaystyle \sum_{i=1}^N \sum_{\substack{j=1 \\ j \ne i}}^N \sigma_{ij}$

then the above portfolio variance simplifies to:

\begin{aligned} \displaystyle \sigma^2_M &= \sum_{i=1}^N \frac{1}{N^2} \sigma^2_i + \sum_{i=1}^N \sum_{\substack{j=1 \\ j \ne i}}^N \frac{1}{N^2} \sigma_{ij} \\ &= \frac{1}{N^2} \sum_{i=1}^N \sigma^2_i + \frac{1}{N^2} \sum_{i=1}^N \sum_{\substack{j=1 \\ j \ne i}}^N \sigma_{ij} \\ &= \frac{N \overline{\sigma^2}}{N^2} + \frac{N(N - 1)}{N^2} \overline{\sigma_{ij}} \\ &= \frac{ \overline{\sigma^2}}{N} + (1 - \frac{1)}{N} \overline{\sigma_{ij}} \end{aligned}

Then as $\displaystyle N \rightarrow \infty$, the portfolio variance $\sigma^2_M$ converges to:

$\displaystyle \sigma^2_M = \overline{\sigma_{ij}}$

That is, as the number of assets in the portfolio go up, the variance of individual assets become unimportant, and its the covariance terms that dominate. This is just our diversification. Graphically this can be represented as:

Diversification (Click on the figure to zoom; Source: Brealey-Myers, 9th Ed.)

Unique Risk (or alternatively, Diversifable Riskor Unsystematic Riskor Idiosyncratic Risk) is the “average variance” of the individual assets.

As number of assets in the portfolio increase, this “average variance” tends to zero. The only risk, then, that matters is the one that remains after diversification has done its work. And this is just the average covariance between all assets in the portfolio. This is called Market Risk (or alternatively, Systematic Risk, or Undiversifiable Risk). And accordingly, the covariance of an asset with the market portfolio is called its market risk.

The fact that portfolio variance after diversification is just the weighted average of covariance between assets can be seen by first noting that:

\displaystyle \begin{aligned} \sigma_{iM} &= E[r_i - \overline{r_i}][r_M - \overline{r_M}] \\ &= E[r_i - \overline{r_i}][\sum_{j=1}^N \omega_j r_j - \sum_{j=1}^N \omega_j \overline{r_j}] \end{aligned}

Since the expectations add up, we can take out the summation sign outside the expectation, and it follows that:

\displaystyle \begin{aligned} \sigma_{iM} &= \sum_{j=1}^N \omega_j E[r_i - \overline{r_i}][r_j - \overline{r_j}] \\ & \Rightarrow \boxed{\sigma_{iM} = \sum_{j=1}^N \omega_j \sigma_{ij}} \end{aligned}

That is the covariance of any asset with the market portfolio is nothing but the weighted average of its covariance with all other assets in the portfolio.

Next, note that we can write:

\displaystyle \begin{aligned} \sigma^2_M &= \sum_{i=1}^N \omega^2_i \sigma^2_i + \sum_{i=1}^N \sum_{\substack{j=1 \\ j \ne i}}^N \omega_i \omega_j \sigma_{ij} \\&= \sum_{i=1}^N \sum_{j=1}^N \omega_i \omega_j \sigma_{ij} \end{aligned}

(If you are looking for the variance terms, note the change in the limits in the summation operator, and recall that $\sigma_{ii} = \sigma^2_i$)

Then if we substitute our result that $\sigma_{iM} = \sum_{j=1}^N \omega_j \sigma_{ij}$, we see that:

\displaystyle \begin{aligned} \sigma^2_M &= \sum_{i=1}^N \sum_{j=1}^N \omega_i \omega_j \sigma_{ij} \\ &= \sum_{i=1}^N \omega_i \sigma_{iM} \end{aligned}

That is the variance of the market portfolio is just the weighted average of the covariance of all assets in the portfolio with itself. Again, this result is important enough to warrant a separate ‘box’:

$\boxed{\sigma^2_M = \sum_{i=1}^N \omega_i \sigma_{iM}}$

Combine this with our observation that in the limit individual variances (unique risks) disappear and we have our economic result that:

Moral of the Story 4: The risk of an individual asset is determined not by its individual variance, but by its covariance with the market portfoliobecause the diversifiable/unique/idiosyncratic risk can be diversified away.