Modern Portfolio Theory

Modern Portfolio Theory (MPT) is the foundational framework for understanding how investors can construct optimal portfolios. Developed by Harry Markowitz in his 1952 paper "Portfolio Selection," MPT earned him the Nobel Prize in Economics in 1990. This theory fundamentally changed how we think about risk and return in investment management.

The Core Insight of MPT

MPT's revolutionary insight is simple but powerful: portfolio risk is not just the sum of individual security risks. Instead, portfolio risk depends critically on how securities move relative to each other--their correlation.

Key MPT principles:

• Investors are risk averse: They prefer less risk for a given return level
• Investment decisions should be based on expected return and risk (measured by standard deviation)
• Diversification reduces risk: Combining securities with low or negative correlations reduces portfolio volatility
• Correlation matters: Two stocks with identical individual risks can create a portfolio with much lower risk if they're not perfectly correlated

CFP Exam Tip: MPT focuses on portfolio-level risk, not individual security risk. The exam frequently tests your understanding that diversification benefits begin whenever correlation is less than +1.

---

Correlation and Diversification

Correlation measures how two securities move relative to each other, ranging from +1 to -1:

Key diversification insight: Diversification benefits occur whenever correlation is less than +1. You do NOT need negative correlation to reduce risk--any correlation below perfect positive (+1) provides some benefit.

Systematic vs. Unsystematic Risk

MPT divides total risk into two components:

Systematic Risk (Market Risk):

• Cannot be diversified away
• Affects all securities in the market
• Measured by beta
• Examples: Interest rate changes, inflation, recessions, geopolitical events

Unsystematic Risk (Company-Specific Risk):

• CAN be diversified away
• Affects individual companies or industries
• Eliminated by holding a well-diversified portfolio (typically 20-30 stocks)
• Examples: Management changes, product recalls, labor strikes, lawsuits

CFP Exam Tip: Standard deviation measures total risk (systematic + unsystematic). Beta measures only systematic risk. For a well-diversified portfolio, beta is the appropriate risk measure.

---

The Efficient Frontier

The efficient frontier is a curve representing the set of portfolios that offer:

• The highest expected return for each level of risk, OR
• The lowest risk for each level of expected return

Understanding the Efficient Frontier Graph

The efficient frontier is plotted with:

• X-axis: Risk (standard deviation)
• Y-axis: Expected return

Portfolio positions:

• On the efficient frontier: Optimal--cannot improve return without more risk
• Below the efficient frontier: Suboptimal--can get more return for the same risk
• Above the efficient frontier: Unattainable--no portfolio can achieve this

The Minimum Variance Portfolio

The Minimum Variance Portfolio (MVP) is the leftmost point on the efficient frontier--the portfolio with the lowest possible risk. The efficient frontier is the upper portion of the curve above the MVP.

Indifference Curves: Investors have different risk tolerances, represented by indifference curves showing combinations of risk and return that provide equal satisfaction. The optimal portfolio for any investor is where their indifference curve is tangent to the efficient frontier.

---

The Risk-Free Asset and Capital Market Line

Adding the Risk-Free Asset

When Markowitz developed MPT, he considered only risky assets. The addition of a risk-free asset (such as Treasury bills) creates new possibilities:

• Investors can combine the risk-free asset with risky portfolios
• This creates a straight line from the risk-free rate to the efficient frontier
• The tangent point where this line touches the efficient frontier is the market portfolio

The Capital Market Line (CML)

The Capital Market Line (CML) is a straight line representing the risk-return relationship for portfolios combining the risk-free asset with the market portfolio.

CML Formula:

E(Rp) = Rf + [(Rm - Rf) / sigma_m] x sigma_p

Where:

• E(Rp) = Expected return of the portfolio
• Rf = Risk-free rate of return
• Rm = Expected return of the market portfolio
• sigma_m = Standard deviation of the market portfolio
• sigma_p = Standard deviation of the portfolio

CML characteristics:

• Y-intercept: The risk-free rate (Rf)
• Slope: (Rm - Rf) / sigma_m = reward-to-risk ratio
• X-axis: Standard deviation (total risk)
• Only efficient portfolios lie on the CML

CML Example:

Given:

• Risk-free rate (Rf) = 3%
• Market return (Rm) = 10%
• Market standard deviation (sigma_m) = 15%
• Portfolio standard deviation (sigma_p) = 12%

E(Rp) = 3% + [(10% - 3%) / 15%] x 12%
E(Rp) = 3% + [0.467] x 12%
E(Rp) = 3% + 5.6%
E(Rp) = 8.6%

The Market Portfolio

The market portfolio is the theoretical portfolio containing all risky assets in proportion to their market values. In practice, broad market indices like the S&P 500 or Russell 3000 are used as proxies.

Key characteristics:

• Contains only systematic risk (all unsystematic risk is diversified away)
• Has a beta of 1.0 by definition
• Represents the tangent point on the efficient frontier

---

Capital Asset Pricing Model (CAPM)

The Capital Asset Pricing Model (CAPM), developed by William Sharpe, John Lintner, and Jan Mossin in the 1960s, extends MPT to determine the expected return for individual securities based on their systematic risk.

The CAPM Formula

E(R) = Rf + Beta x (Rm - Rf)

Where:

• E(R) = Expected (required) return of the security
• Rf = Risk-free rate of return
• Beta = Systematic risk of the security relative to the market
• Rm = Expected return of the market
• (Rm - Rf) = Market risk premium

CFP Exam Tip: The CAPM formula is on the CFP exam formula sheet. Memorize the components and be able to calculate expected return, beta, or the market risk premium when given the other values.

Understanding Beta

Beta measures a security's sensitivity to market movements:

Beta examples:

• Beta = 1.5: If market rises 10%, security expected to rise 15%
• Beta = 0.8: If market falls 10%, security expected to fall 8%

CAPM Calculation Examples

Example 1: Calculate Expected Return

Given:

• Risk-free rate (Rf) = 4%
• Market return (Rm) = 11%
• Security beta = 1.3

E(R) = 4% + 1.3 x (11% - 4%)
E(R) = 4% + 1.3 x 7%
E(R) = 4% + 9.1%
E(R) = 13.1%

Example 2: Calculate Beta (Dalton Review Style)

Given:

• Risk-free rate = 3%
• Market risk premium = 9%
• Security beta = 1.5

What is the expected return?

E(R) = 3% + 1.5 x 9%
E(R) = 3% + 13.5%
E(R) = 16.5%

CFP Exam Tip: Note that (Rm - Rf) is the market risk premium. If you're given the market risk premium directly (as in Example 2), use it directly in the formula without calculating it.

Example 3: Solve for Beta

Given:

• Expected return = 12%
• Risk-free rate = 3%
• Market return = 9%

12% = 3% + Beta x (9% - 3%)
12% = 3% + Beta x 6%
9% = Beta x 6%
Beta = 9% / 6% = 1.5

---

The Security Market Line (SML)

The Security Market Line (SML) is the graphical representation of the CAPM. While the CML shows the relationship for efficient portfolios, the SML applies to all securities and portfolios.

SML vs. CML

Using the SML to Identify Mispriced Securities

The SML can identify whether securities are fairly priced:

Example: A security has beta = 1.2 and expected return = 15%. If Rf = 4% and Rm = 10%:

Required return (per CAPM) = 4% + 1.2 x (10% - 4%) = 4% + 7.2% = 11.2%

Since expected return (15%) > required return (11.2%), the security is undervalued and plots above the SML. It offers more return than required for its level of risk.

---

MPT Assumptions and Limitations

Key Assumptions of MPT/CAPM

1. Investors are risk averse and make decisions based on expected return and standard deviation
2. Markets are efficient: All information is reflected in prices
3. Investors have homogeneous expectations: Same forecasts for returns, volatility, and correlations
4. Single-period investment horizon: All investors plan for the same time period
5. Unlimited borrowing and lending at the risk-free rate
6. No taxes or transaction costs
7. Assets are infinitely divisible: Can buy any fraction of a security

Limitations in Practice

• Returns are not normally distributed (fat tails, skewness)
• Correlations change over time, especially during market stress
• Historical data may not predict future behavior
• Beta is not stable over time
• The market portfolio is theoretical and cannot be precisely identified

CFP Exam Tip: The exam may ask about MPT assumptions or limitations. Remember that while MPT provides a valuable framework, real-world markets don't perfectly match its assumptions.

---

Key Formulas Summary

Portfolio Beta Calculation

The beta of a portfolio is the weighted average of the component betas:

Portfolio Beta = (w1 x beta1) + (w2 x beta2) + ... + (wn x beta_n)

Example: A portfolio has 60% in Stock A (beta = 1.2) and 40% in Stock B (beta = 0.8):

Portfolio Beta = (0.60 x 1.2) + (0.40 x 0.8)
Portfolio Beta = 0.72 + 0.32
Portfolio Beta = 1.04