Chapter 3: Introduction to Modern Portfolio Theory and Capital Market Theory
Modern Portfolio Theory (MPT) Principles
Developed by Harry Markowitz in 1952, Modern Portfolio Theory (MPT) revolutionized financial economics by demonstrating that an asset's risk should not be assessed in isolation, but by how it contributes to the overall risk and return of an aggregated portfolio.
Core Assumptions of MPT
- Investors are rational, risk-averse utility maximizers who prefer higher expected return for a given level of risk.
- Asset returns are normally distributed and fully described by their first two statistical moments: mean (expected return) and variance (risk).
- Investors make decisions based on a single-period investment horizon.
- Capital markets are frictionless: zero transaction costs, no taxes, and all assets are infinitely divisible.
Mathematical Formulations
1. Expected Rate of Return for an Individual Security
For a security with $N$ potential economic states, where state $i$ has probability $P_i$ and return $R_i$:
$$E(R) = \sum_{i=1}^{N} P_i R_i$$
2. Variance and Standard Deviation of Return
$$\sigma^2 = \sum_{i=1}^{N} P_i [R_i - E(R)]^2$$
$$\sigma = \sqrt{\sigma^2}$$
3. Expected Return of a Portfolio
For a portfolio of $k$ assets with weights $w_i$ such that $\sum w_i = 1$:
$$E(R_p) = \sum_{i=1}^{k} w_i E(R_i)$$
4. Two-Asset Portfolio Variance and the Covariance Term
$$\sigma_p^2 = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \text{Cov}(R_1, R_2)$$
Where Covariance is linked to the Correlation Coefficient ($\rho_{12}$):
$$\text{Cov}(R_1, R_2) = \rho_{12} \sigma_1 \sigma_2 \quad \text{where } -1 \le \rho_{12} \le +1$$
$$\sigma_p^2 = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \rho_{12} \sigma_1 \sigma_2$$
Impact of Correlation on Portfolio Risk:
• If rho = +1.0 : Perfect positive correlation; zero diversification benefit.
• If rho = 0.0 : Uncorrelated; substantial risk reduction achieved.
• If rho = -1.0 : Perfect negative correlation; portfolio risk can be reduced to ZERO!
The Geometry of Portfolio Optimization
When thousands of risky assets are combined across all possible weight combinations, they form the Opportunity Set of Risky Portfolios (Investment Feasible Set):
Expected Return E(R)
^
│ * * * * (Efficient Frontier)
│ * [Portfolio A - Higher Return]
│ *
│ GMVP ───> *
│ * * * * * *
│ * * *
│ * * Feasible *
│ * * Set *
│ * * * * * * * *
└─────────────────────────────────────────────────────────────>
Standard Deviation (Sigma)
1. The Global Minimum Variance Portfolio (GMVP)
The portfolio located at the leftmost tip of the feasible set represents the Global Minimum Variance Portfolio (GMVP)—the portfolio with the lowest possible variance achievable from the universe of risky assets.
2. The Markowitz Efficient Frontier
The upper boundary of the feasible set extending upwards and to the right from the GMVP constitutes the Efficient Frontier:
- Every portfolio lying on the Efficient Frontier represents an optimal asset combination that either:
- Maximizes expected return for a specified target level of risk ($\sigma$).
- Minimizes portfolio risk for a specified target level of expected return.
- Any portfolio lying below the efficient frontier is sub-optimal (inefficient) because an investor can achieve higher return for the same risk, or lower risk for the same return, by moving to the frontier.
Category III AIF Role on the Frontier
By introducing assets with negative or near-zero correlation ($\rho \le 0$) through market-neutral and long/short pairs, Category III AIFs expand the feasible set outwards, shifting the Efficient Frontier upwards and to the left, enabling higher returns at lower total portfolio standard deviation.
Integrating the Risk-Free Asset: Capital Market Theory
Modern Portfolio Theory focuses exclusively on combinations of risky assets. Capital Market Theory introduces a Risk-Free Asset ($R_f$) (such as sovereign Treasury Bills), having zero variance ($\sigma_{Rf} = 0$) and zero correlation with risky assets.
Expected Return E(R)
^
│ / Capital Market Line (CML)
│ /
│ /
│ Market (M) * (Tangency Portfolio)
│ / * * * (Efficient Frontier)
│ / *
│ / *
│ / *
│ / *
│ /
│ Risk-Free (Rf) *
└─────────────────────────────────────────────────────────────>
Standard Deviation (Sigma)
1. The Capital Allocation Line (CAL) and Capital Market Line (CML)
Combining the risk-free asset with a portfolio of risky assets generates a linear relationship. The line drawn from $R_f$ tangent to the Markowitz Efficient Frontier is the Capital Market Line (CML):
- The point of tangency is the Market Portfolio ($M$), containing all available risky assets weighted by their market capitalizations.
- The equation of the Capital Market Line is:
$$E(R_p) = R_f + \left[ \frac{E(R_m) - R_f}{\sigma_m} \right] \sigma_p$$
Where $\frac{E(R_m) - R_f}{\sigma_m}$ is the Sharpe Ratio of the Market Portfolio (the market price of risk).
2. Capital Asset Pricing Model (CAPM) and Beta ($\beta$)
While the CML evaluates total risk ($\sigma$) for fully diversified portfolios, the Capital Asset Pricing Model (CAPM) (developed by Sharpe, Lintner, and Mossin) evaluates individual securities based on their non-diversifiable systematic risk:
$$E(R_i) = R_f + \beta_i [E(R_m) - R_f]$$
Where Beta ($\beta_i$) measures the sensitivity of security $i$'s returns to the market portfolio:
$$\beta_i = \frac{\text{Cov}(R_i, R_m)}{\sigma_m^2} = \frac{\rho_{im} \sigma_i}{\sigma_m}$$
3. The Security Market Line (SML)
The graphical representation of CAPM is the Security Market Line (SML), plotting expected return against Beta:
- Assets plotting above the SML are undervalued (providing excess return or positive Jensen's Alpha ($\alpha > 0$)).
- Assets plotting below the SML are overvalued ($\alpha < 0$).
- Category III AIF managers actively seek positive alpha ($\alpha > 0$) by identifying mispriced securities while hedging out market beta ($\beta_m$).