- CFA Exams
- 2023 Level I
- Topic 9. Portfolio Management
- Learning Module 62. Portfolio Risk and Return: Part I
- Subject 3. Variance and Covariance of Returns

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##### Subject 3. Variance and Covariance of Returns PDF Download

Investment is all about reward versus variability (risk). The return measures the reward of an investment and dispersion is a measure of investment risk.

The

**variance**is a measure of how spread out a distribution is. It is computed as the average squared deviation of each number from its mean. The formula for the variance in a population is:

where μ is the mean and N is the number of scores.

To compute variance in a sample:

where m is the sample mean.

The formula for the

**standard deviation**is very simple: it is the square root of the variance. It is the most commonly used measure of spread.

The

**standard deviation of a portfolio**is a function of:

- The weighted average of the individual variances, plus
- The weighted covariances between all the assets in the portfolio.

In a two-asset portfolio:

The maximum amount of risk reduction is predetermined by the correlation coefficient.

__Thus, the correlation coefficient is the engine that drives the whole theory of portfolio diversification.__

*Example with perfect positive correlation (assume equal weights):*

What is the standard deviation of a portfolio (E), assuming the following data?

σ

_{1}= 0.1, w

_{1}= 0.5, σ

_{2}= 0.1, w

_{2}= 0.5, ρ

_{12}= 1

Solution:

Cov

_{12}= σ

_{1}x σ

_{2}x ρ

_{12}= 0.1 x 0.1 x 1 = 0.01

Standard Deviation of Portfolio [0.5

^{2}x 0.1

^{2}+ 0.5

^{2}x 0.1

^{2}+ 2 x 0.5 x 0.5 x 0.01]

^{1/2}= 0.10 (perfect correlation)

If there are three securities in the portfolio, its standard deviation is:

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