📘 14.1 Framework for Constructing Portfolios – Modern Portfolio Theory

What is Modern Portfolio Theory (MPT)?

Modern Portfolio Theory (MPT) is a framework for constructing portfolios that aims to maximize returns for a given level of risk. It emphasizes diversification to reduce risk and achieve optimal performance by combining different asset classes in a portfolio.

Key Concepts of MPT

📈 Risk and Return

MPT considers both risk (the possibility of loss) and return (the potential for profit) in portfolio construction. It seeks to find the balance between maximizing returns and minimizing risk.

⚖️ Diversification

Diversification is central to MPT. By combining assets that are not perfectly correlated, investors can reduce the overall risk of the portfolio. The goal is to find a mix of investments that perform well in different market conditions.

📊 Efficient Frontier

The efficient frontier represents a set of portfolios that offer the highest return for a given level of risk. MPT helps identify the optimal portfolio by plotting these portfolios on a risk-return graph.

📉 Correlation

In MPT, the correlation between asset returns is key. Assets with low or negative correlation help in risk reduction, as their price movements are less likely to be synchronized.

How to Optimize a Portfolio?

Portfolio optimization involves selecting the right mix of assets that will offer the highest return for the least amount of risk. MPT helps in finding this optimal combination by analyzing the expected returns, variances, and correlations of the assets.

📘 14.2 Assumptions of MPT

Assumptions of Modern Portfolio Theory

Modern Portfolio Theory (MPT) is built on certain assumptions that guide portfolio construction. These assumptions are critical to how portfolio managers approach risk, return, and diversification. Below are the key assumptions of MPT:

Risk-Averse Investors

MPT assumes that all investors are **risk-averse**, meaning they prefer lower risk for a given level of return. Investors aim to make decisions that minimize risk while maximizing expected returns.

Efficient Markets

The theory assumes that **all financial markets are efficient**, meaning all available information is quickly reflected in asset prices. As a result, investors cannot consistently outperform the market through individual stock picking.

Rational Decision-Making

Investors in MPT are assumed to make **rational decisions**. They evaluate investment options based on objective data, aiming to maximize utility (expected return) for a given level of risk.

One-Period Investment Horizon

MPT assumes that investors have a **one-period investment horizon**, meaning they make decisions based on a short-term perspective, typically over a single period such as a year.

No Transaction Costs

The theory assumes that there are **no transaction costs** or taxes involved in buying and selling assets, allowing for seamless portfolio rebalancing without additional costs.

Perfect Substitutes

MPT assumes that **assets are perfect substitutes** for each other, meaning investors can freely trade between assets without any changes in liquidity or market constraints.

📘 14.3 Definition of Risk-Averse, Risk-Seeking, and Risk-Neutral Investors

Understanding Investor Types

Investors can be categorized based on their willingness to take risks in pursuit of higher returns. The three main types of investors are **risk-averse**, **risk-seeking**, and **risk-neutral**. Each type has distinct characteristics and approaches to portfolio management.

Risk-Averse Investor

A **risk-averse investor** is one who prefers to avoid risk and seeks to minimize uncertainty in their investment decisions. These investors prioritize safety and are willing to accept lower returns in exchange for less risk. They tend to invest in stable, lower-risk assets such as bonds and blue-chip stocks.

Risk-Seeking Investor

A **risk-seeking investor** actively looks for investments with higher potential returns, even if it means taking on more risk. These investors are willing to accept significant volatility and uncertainty in the hopes of achieving high returns. They typically invest in speculative assets, such as stocks of start-ups or commodities.

Risk-Neutral Investor

A **risk-neutral investor** is indifferent to risk. These investors are primarily focused on maximizing their expected return without caring about the level of risk involved. They tend to make decisions based purely on potential rewards, rather than risk mitigation, and may invest in a mix of high-risk and low-risk assets depending on the expected return.

📘 14.4 Calculation of Expected Rate of Return

What is Expected Rate of Return?

The **expected rate of return** of an individual investment is the weighted average of all possible returns, with each return being weighted by its corresponding probability. This gives an investor an estimate of the potential returns they can expect from a particular investment, accounting for different market scenarios.

1. Example of Calculating Expected Rate of Return

Stock A

The expected return of Stock A (RA) is calculated using the probabilities of different states (Boom, Normal, and Recession) and their corresponding returns. The formula is as follows:

RA = 0.3(15%) + 0.5(10%) + 0.2(2%) = 9.9%

In Excel, this calculation would be:

=SUMPRODUCT(A2:A4, B2:B4)

Where A2:A4 are the probabilities, and B2:B4 are the returns for Stock A in the respective states.

Stock B

The expected return of Stock B (RB) is similarly calculated as:

RB = 0.3(25%) + 0.5(20%) + 0.2(1%) = 17.7%

In Excel, use the formula:

=SUMPRODUCT(A2:A4, C2:C4)

Where A2:A4 contains the probabilities, and C2:C4 contains the returns for Stock B in each scenario.

2. Calculating Variance of Return

Variance Formula

The **variance** measures the dispersion of returns around the expected return. It is calculated as:

Variance = SUM((Return - Expected Return)^2 * Probability)

Variance gives us an idea of how much an investment’s return is likely to fluctuate. A higher variance indicates higher risk.

Steps to Calculate Variance in Excel

For Stock A

To calculate the variance for Stock A, you need to:

  1. Calculate the **deviation** of each return from the expected return.
  2. Square each deviation to get the **squared deviation**.
  3. Multiply the squared deviation by the **probability** for each state.
  4. Sum the results to obtain the variance.

In Excel, the formula would look like this:

=SUMPRODUCT((B2:B4 - expected_return_A)^2, A2:A4)

For Stock B

For Stock B, the variance can be calculated similarly:

  1. Calculate the **deviation** of each return from the expected return for Stock B.
  2. Square the deviations and multiply by the **probabilities**.
  3. Sum them up to get the variance for Stock B.

In Excel, you would use:

=SUMPRODUCT((C2:C4 - expected_return_B)^2, A2:A4)

3. Portfolio Expected Return

How to Calculate Portfolio Expected Return

The **expected return of a portfolio** is the weighted average of the expected returns of the individual securities in the portfolio. It is calculated as:

Portfolio Expected Return = SUM(Weight of Asset * Expected Return of Asset)

In Excel, the formula would be:

=SUMPRODUCT(weights, expected_returns)

For example, if you have 4 assets in your portfolio with different weights, you can use this formula to calculate the overall expected return of your portfolio.

4. Portfolio Risk (Variance)

Portfolio Variance Formula

The formula for the portfolio variance with two assets is:

Portfolio Variance = w1^2 * σ1^2 + w2^2 * σ2^2 + 2 * w1 * w2 * Covariance(1,2)

Where:

  • w1 and w2 are the weights of Asset 1 and Asset 2 in the portfolio.
  • σ1 and σ2 are the standard deviations of the returns of Asset 1 and Asset 2.
  • Covariance(1,2) is the covariance between the returns of Asset 1 and Asset 2.

In Excel, this can be calculated by plugging the values of weights, standard deviations, and covariance into the formula.

📘 14.5 Portfolio Risk/Return of Two Securities

Risk-Return Opportunity Set

If two securities are perfectly correlated, the risk-return opportunity set is represented by a **straight line** connecting the two securities. The line contains portfolios formed by changing the weight of each security in the portfolio. The portfolio return for any combination of these two securities is the weighted average return of the securities.

The standard deviation of any combination of these two securities is the weighted average standard deviation of the securities. There is no benefit of diversification when the assets are perfectly positively correlated.

Graphical Presentation

Perfectly Correlated Securities

When two assets have perfect positive correlation, both the expected return and standard deviation of expected return are linear combinations. A graph of the possible portfolio return and standard deviation is a **straight line** connecting the two securities, representing no diversification benefit.

📘 14.6 The Concept of Efficient Frontier

What is the Efficient Frontier?

The **efficient frontier** shows the optimal return for a given level of risk or the lowest risk for a given level of expected return. It is a set of optimal portfolios that offer the highest expected return for a given level of risk.

Portfolios lying below the efficient frontier are sub-optimal because they do not provide enough return for the level of risk. Portfolios to the right of the frontier are also sub-optimal because they have higher risk for the given rate of return.

Graphical Representation

Umbrella Shaped Curve

Many securities can be combined in various combinations, and the one that offers the highest return for a given level of risk or the lowest risk for a given level of return will form an umbrella-shaped curve known as the **Efficient Frontier**.

📘 14.7 Portfolio Optimization Process

Steps for Portfolio Optimization

To construct and select the optimal portfolio, the portfolio manager must estimate the following:

  • The **expected return** of every asset class, securities, and investment opportunities in the investment universe.
  • The **standard deviation** of each asset’s expected returns.
  • The **correlation coefficient** among the entire set of asset classes, securities, and investment opportunities.

In addition to these inputs, the investor’s **constraints** must also be specified. From the feasible combinations of the various portfolios, the one that meets the investor’s objectives can be selected.

📘 14.8 Estimation Issues

Estimating Portfolio Risk

For constructing and selecting a portfolio, the portfolio manager has to estimate the **returns, risk**, and **correlations** among the securities in the investment universe. The accuracy of these estimations is crucial, as the output of the portfolio allocation depends on the quality of the statistical inputs.

The **number of correlation estimates** can be significant. For example, for a portfolio of 50 securities, the number of correlation estimates is \( \frac{50(50-1)}{2} = 1225 \). These potential sources of error are referred to as **estimation risk**.

Variance-Covariance Matrix

For portfolios with more than two securities, portfolio risk can be calculated by using the **variance-covariance matrix**. The variance-covariance matrix includes the variances of individual securities and the covariances between them.

The **variance-covariance matrix** can be easily calculated using spreadsheet functions or Visual Basic applications, as discussed in financial modeling textbooks.