# How is Covariance and Correlation used in Portfolio Theory?

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The process of combining numerous securities to reduce risk is known as diversification. It is necessary to consider the impact of covariance or correlation on portfolio risk more closely to understand the mechanism and power of diversification.

Let’s study the issue category-wise −

• when security returns are perfectly positively correlated,

• when security returns are perfectly negatively correlated, and

• when security returns are not correlated.

## Security Returns Perfectly Positively Correlated

When net assets returns are perfectly and positively correlated, the given correlation coefficient between the two securities will be +1. Thus, the returns of the two securities will move up or down together. The portfolio variance for perfectly positive correlation is calculated using the formula −

$$\mathrm{(σ_{ρ})^{2} = (𝑥_{1})^{2}(σ_{1})^{2} + (𝑥_{2})^{2}(σ_{2})^{2} + 2 𝑟_{12}\:𝑥_{1}𝑥_{2}\:σ_{1}σ_{2}}$$

Since𝑟12 = 1, this may be rewritten as −

$$\mathrm{(σ_{ρ})^{2} = (𝑥_{1})^{2}(σ_{1})^{2} + (𝑥_{2})^{2}(σ_{2})^{2} + 2 \times 1 \times 2\:σ_{1}σ_{2}}$$

The right-hand side of the equation has the same form as the expansion of the identity (a + b)2 = a2 + 2ab + b2

Hence, it may be reduced as,

$$\mathrm{(σ_{ρ})^{2} = (𝑥_{1}σ_{1} + 𝑥_{2}σ_{2})^{2}}$$

The standard deviation (SD) then becomes $(σ_{ρ} = 𝑥_{1}σ_{1} + 𝑥_{2}σ_{2})$ which is simply the net average of the standard deviations of the individual securities.

## Example

Standard deviation of security P = 50

Standard deviation of security Q = 30

Proportion of investment in P = 0.4

Proportion of investment in Q = 0.6

Correlation coefficient = +1.0

Portfolio standard deviation may be calculated as −

$$\mathrm{σ_{ρ} = 𝑥_{1}σ_{1} + 𝑥_{2}σ_{2}=(0.4 \times 50)+ (0.6 \times 30) = 38}$$

Being the weighted average of the SD of individual securities, the portfolio SD will lie between the standard deviations of the two singular individual securities. In the given example, it will vary between 50 and 30.

For example, if the proportion of investment in P and Q are 0.75 and 0.25 respectively, portfolio standard deviation becomes −

$$\mathrm{σ_{ρ} = 𝑥_{1}σ_{1} + 𝑥_{2}σ_{2}=(0.75 \times 50) + (0.25 \times 30) = 45}$$

Thus, when the security returns of assets are perfectly positively correlated, then diversification provides only risk averaging and no risk reduction. This happens because the portfolio risk cannot be reduced below the risk of each individual asset. Hence, diversification is not productive when security returns are perfectly positively correlated.

## Security Returns Perfectly Negatively Correlated

In this case, the correlation coefficient between them becomes -1. The two returns from the portfolio will move in exactly opposite directions. The portfolio variance is given by −

$$\mathrm{(σ_{ρ})^{2} = (𝑥_{1})^{2}(σ_{1})^{2} + (𝑥_{2})^{2}(σ_{2})^{2} + 2 𝑟_{12}\:𝑥_{1}𝑥_{2}\:σ_{1}σ_{2}}$$

Since 𝑟12 = −1,this may be rewritten as −

$$\mathrm{(σ_{ρ})^{2} = (𝑥_{1})^{2}(σ_{1})^{2} + (𝑥_{2})^{2}(σ_{2})^{2} - 2 \times 1 \times 2 σ_{1}σ_{2}}$$

The right-hand side of the equation has the same form as the expansion of the identity (a - b)2 = a2 - 2ab + b2

Hence, it may be reduced as,

$$\mathrm{({σ_{ρ}})^{2} = (𝑥_{1}σ_{1} - 𝑥_{2}σ_{2})^{2}}$$

The standard deviation (SD) then becomes $(σ_{ρ} = 𝑥_{1}σ_{1} - 𝑥_{2}σ_{2})$

For the illustrative portfolio considered above, we can calculate the portfolio standard deviation when the correlation coefficient is −1.

$$\mathrm{σ_{ρ} = 𝑥_{1}σ_{1} - 𝑥_{2}σ_{2}=(0.40 \times 50) − (0.60 \times 30) = 2}$$

The portfolio risk may go as low as zero. For example, when P and Q are 0.375 and 0.625 respectively, portfolio standard deviation becomes −

$$\mathrm{σ_{ρ}=(0.375 × 50) − (0.625 × 30) = 0}$$

Here, although the portfolio has two risky assets, the portfolio overall has no risk. Thus, the portfolio risk may be zero when security returns are perfectly negatively correlated.

## Security Returns Uncorrelated

When the returns of two securities are entirely uncorrelated, the correlation coefficient would be zero. The formula for portfolio variance is −

$$\mathrm{(σ_{ρ})^{2} = (𝑥_{1})^{2}(σ_{1})^{2} + (𝑥_{2})^{2}(σ_{2})^{2} + 2\:𝑟_{12}\:𝑥_{1}𝑥_{2}\:σ_{1}σ_{2}}$$

Since 𝑟12 = 0, the last term in the equation becomes zero; the formula can be rewritten as −

$$\mathrm{(σ_{ρ})^{2} = (𝑥_{1})^{2}(σ_{1})^{2} + (𝑥_{2})^{2}(σ_{2})^{2}}$$

The standard deviation then becomes −

$$\mathrm{σ_{ρ}=\sqrt{𝑥_{1}σ_{1} + 𝑥_{2}σ_{2}}}$$

For our illustrative portfolio,

$$\mathrm{σ_{ρ}=\sqrt{(0.4)^{2}(50)^{2} + (0.6)^{2}(30)^{2}}=\sqrt{400 + 324}= 26.91}$$

The portfolio SD is less than the standard deviations of individual securities in the portfolio. Thus, when the security returns are completely uncorrelated, diversification diminishes the risk and becomes productive.

We may conclude that risk is always reduced except when the security returns of a two asset portfolio are perfectly positively correlated. With the correlation coefficient declining from +1 to -1, the portfolio SD also declines automatically. However, the risk reduction is the most palpable when the security returns are negatively correlated.