Lesson 11.2: Risk, Return, and Portfolio Theory

Introduction

In this lesson, students will learn about the critical concepts of risk and return in portfolio management. This lesson is designed to provide a comprehensive understanding of how to measure portfolio risk and return, the importance of diversification, and how these elements relate to modern portfolio theory. By the end of this lesson, you will be able to compute portfolio risk and return, explain how diversification reduces risk, and describe the efficient frontier and the optimal portfolio.

Learning Objectives

• Measure portfolio risk and return and understand the effect of diversification.
• Describe the efficient frontier and the foundations of modern portfolio theory.
• Compute portfolio risk and return for different investment scenarios.
• Explain how diversification aids in reducing risk within a portfolio.
• Understand the concept of the efficient frontier and identify the optimal portfolio within it.

Section 1: Measuring Portfolio Risk and Return

Understanding Risk and Return

Risk and return are the cornerstones of investment. Generally, the higher the potential return of an investment, the higher the risk associated with it. In finance, return refers to the gain or loss made on an investment relative to the amount invested, while risk is the uncertainty related to the investment returns.

Definitions

• Return: The profit or loss generated from an investment.
• Risk: The probability that the actual return on an investment will differ from the expected return.

Calculating Expected Return

The expected return of an investment can be calculated as follows:

$$\text{Expected Return} = \sum (p_i \cdot r_i)$$

where:

• $p_i$ is the probability of each possible return, and
• $r_i$ is the return associated with that probability.

Example: Expected Return Calculation

Suppose we have the following potential returns for a stock and their associated probabilities:

• 20% return with a probability of 0.5
• 10% return with a probability of 0.3
• -5% return with a probability of 0.2

The expected return would be:

$$\text{Expected Return} = (0.5 \cdot 0.20) + (0.3 \cdot 0.10) + (0.2 \cdot -0.05)$$

Calculating this gives:

$$\text{Expected Return} = 0.10 + 0.03 - 0.01 = 0.12 \text{ or } 12\%$$

Section 2: Portfolio Risk

Variance and Standard Deviation

Risk can often be quantified using variance and standard deviation. Variance measures the dispersion of returns from the expected return, indicating how much the returns of an asset deviate from their average.

• Variance can be expressed mathematically as:

$$\text{Variance} = E[(r - \mu)^2]$$

where:

• $r$ denotes the return, and
• $\mu$ denotes the expected return.

• Standard Deviation is the square root of the variance and offers a measure of risk in the same units as the returns themselves:

$$\text{Standard Deviation} = \sqrt{\text{Variance}}$$

Example: Calculating Portfolio Variance

Consider a portfolio consisting of two assets, A and B, with the following expected returns and variances:

• Asset A: Expected return = 10%, Variance = 0.04
• Asset B: Expected return = 20%, Variance = 0.09
• Correlation coefficient between A and B = 0.5

The formula for portfolio variance is:

$$\text{Portfolio Variance} = w_A^2 \cdot \sigma_A^2 + w_B^2 \cdot \sigma_B^2 + 2 \cdot w_A \cdot w_B \cdot \sigma_A \cdot \sigma_B \cdot

ho_{AB}$$

where:

• $w_A$ and $w_B$ are the weights of the assets in the portfolio (e.g., 0.6 and 0.4),
• $\sigma_A^2$ and $\sigma_B^2$ are the variances,

ho_{AB} is the correlation coefficient.

Plugging in the numbers:

$$\text{Portfolio Variance} = (0.6^2 \cdot 0.04) + (0.4^2 \cdot 0.09) + 2 \cdot 0.6 \cdot 0.4 \cdot \sqrt{0.04} \cdot \sqrt{0.09} \cdot 0.5$$

Calculating this gives:

1. $0.6^2 \cdot 0.04 = 0.0144$
2. $0.4^2 \cdot 0.09 = 0.0144$
3. $2 \cdot 0.6 \cdot 0.4 \cdot 0.2 \cdot 0.3 \cdot 0.5 = 0.0072$

Thus, the Portfolio Variance equals:

$$\text{Portfolio Variance} = 0.0144 + 0.0144 + 0.0072 = 0.036$$

Conclusion on Risk and Return

Investors must weigh the potential returns against the risks associated with different investment choices. Understanding how to calculate the expected return and variance equips investors with the knowledge to make insightful decisions about their portfolios.

Section 3: The Effect of Diversification

What is Diversification?

Diversification is the practice of spreading investments across a variety of assets to reduce risk. By holding a variety of assets, a portfolio's overall risk can be minimized because poor performance in one asset may be offset by better performance in another.

Benefits of Diversification

• Reduced Risk: As mentioned, diversification helps in reducing the specific risk associated with single securities.
• More Stable Returns: A well-diversified portfolio will generally show more stable returns over time.

Example: Diversification in Action

Imagine investing in two assets:

• Asset C: 15% expected return, variance of 0.01.
• Asset D: 8% expected return, variance of 0.04.
• The weights are 0.5 (C) and 0.5 (D).

Using the variance formula mentioned earlier:

1. We calculate their portfolio variance with correlation of 0.0 (independent assets).
2. Plugging into the formula:

$$ \text{Portfolio Variance} = 0.5^2 \cdot 0.01 + 0.5^2 \cdot 0.04 + 2 \cdot 0.5 \cdot 0.5 \cdot \sqrt{0.01} \cdot \sqrt{0.04} \cdot 0 $$

This results in:

$$\text{Portfolio Variance} = 0.0025 + 0.01 + 0 = 0.0125$$

The independent correlation significantly reduces the overall risk compared to investing in a single asset.

Section 4: The Efficient Frontier

Understanding the Efficient Frontier

The efficient frontier is a key concept in modern portfolio theory, which illustrates the set of optimal portfolios that offer the highest expected return for a defined level of risk.

Constructing the Efficient Frontier

To construct the efficient frontier, one must calculate the expected returns and risks for numerous portfolios, then plot these against each other on a graph.

Example: Creating the Frontier

1. Calculate returns and standard deviations for different combinations of assets (say, C and D) with varying weights.
2. Plot the results in a risk-return graph, with expected return on the y-axis and standard deviation on the x-axis.
3. The efficient frontier consists of the upward-sloping section of the graph that represents the optimal trade-off between risk and return.

Identifying the Optimal Portfolio

The optimal portfolio lies on the efficient frontier where the return is maximized for a given risk level. Investors will choose their position on the frontier based on their risk tolerance, with conservative investors selecting portfolios on the left side and aggressive investors on the right.

Conclusion

In this lesson, students has explored the critical relationship between risk and return in portfolio management. By measuring portfolio risk and return, understanding diversification effects, and identifying the efficient frontier, it's possible to construct portfolios that align individual risk preferences with the potential for returns. Mastering these concepts is integral to effective portfolio management and investment strategies.

Study Notes

• Return is the gain or loss made on an investment relative to the invested amount.
• Risk is the uncertainty related to the investment returns.
• Expected return is calculated as a weighted average of possible returns.
• Portfolio risk can be calculated using variance and standard deviation.
• Diversification reduces risk by spreading investments across various assets.
• The efficient frontier illustrates the best possible risk-return combinations of portfolios.
• Optimal portfolios exist on the efficient frontier, determined by an investor's risk tolerance.