IFT Notes for Level I CFA^® Program

LM02 Portfolio Risk and Return Part I

Part 3

12. Portfolio Risk & Portfolio of Two Risky Assets

Portfolio risk depends on:

Risk of individual assets.
Weight of each asset.
Covariance or correlation between the assets.

12.1 Portfolio of Two Risky Assets

Portfolio Return

A portfolio is usually composed of more than one asset in different proportions. Portfolio return is the weighted average of the returns of the individual investments.

Assume you have three different stocks A, B, and C in your portfolio with weights of 50%, 25%, and 25% respectively. The returns on A, B, and C are 20%, 10%, and 10% respectively. The portfolio return is the weighted average return of three stocks: = 15%.

Risk of a two-asset portfolio is given by:

σ_p= $\sqrt{w_1^2 \times \sigma_1^2 + w_2^2 \times \sigma_2^2 + 2 \times w_1 \times w_2 \times \rho_{1,2} \times \sigma_1 \times \sigma_2}$

σ_p= $\sqrt{w_1^2 \times \sigma_1^2 + w_2^2 \times \sigma_2^2 + 2 \times w_1 \times w_2 \times Cov_{1,2}}$

Covariance = $Cov_{1,2} = \rho_{1,2} \times \rho_1 \times \rho_2$

where: $\rho_{1,2}$ = correlation coefficient that gives the correlation between returns R₁ and R_2.

Impact of correlation on portfolio risk

As correlation decreases, the diversification benefit increases i.e. the risk of the portfolio decreases.

Example

Let us revisit our portfolio of X and Y. Earlier, we considered the impact of changing weights of the two assets. Now, we will also consider the impact of different correlations. The relevant data is reproduced here. X has an expected return of 12% and a standard deviation of 16%. Y has an expected return of 20% and a standard deviation of 30%. What is the expected return and risk of the portfolio for different weights assuming the following correlations: -1, -0.5, 0, 0.5, and 1?

Solution:

R_p= w_x R_x+ w_y R_y?_p² = $w_x^2 \sigma_x^2 + w_y^2 \sigma_y^2 + 2w_x w_y \rho_{xy} \sigma_x \sigma_y$
R_x= 12%, ?_x= 16%; R_y= 20%; ?_y= 30%
Weight of X (%)	Portfolio return	Portfolio risk when correlation between X and Y is:
		-1	-0.5	0	0.5	1
0	20	30	30	30	30	30
20	18.4	20.8	22.5	24.2	25.7	27.2
50	16	7	13	17	20.2	23
60	15.2	2.4	10.9	15.4	18.7	21.6
70	14.4	2.2	10.28	14.3	17.5	20.2
100	12	16	16	16	16	16

This chart is not exhaustive. It is plotted only for the weights used above.

Consider the line represented with a correlation of 1. As the weight of X is reduced and weight of Y is increased the risk and return both increase. There is no diversification benefit.
Consider a correlation of 0. As the weight of X is reduced and weight of Y is increased the risk initially decreases and the return increases. Hence there is clearly a diversification benefit.
The diversification benefit increases as correlation decreases.
With a correlation of -1, there is certain weight of X and Y when the risk is zero.

13. Portfolio of Many Risky Assets

In the previous example, we saw how a two-asset portfolio’s risk reduces as the correlation between the assets decreases. To be more generic, for a portfolio with more than two assets, the variance is given by:

$\sigma_P^2 = \frac{\sigma^2}{N} + \frac{N-1}{N} * \text{Cov}$

where:

$\sigma_P^2$ = portfolio variance

$\sigma^2$ = average variance

$\text{Cov}$ = average covariance

N = number of assets in the portfolio

Instructor’s tip:

The probability of being tested on this formula is low.

14. The Power of Diversification

When creating a diversified portfolio, the choice of assets and the correlation among assets is important. Adding perfectly positively correlated assets to a portfolio does little to reduce the portfolio’s risk. Similarly, assets that have lower correlations will reduce risk. When two assets with a correlation of -1.0 are combined to form a portfolio, risk can be reduced to zero.