Understand mean-variance portfolio theory and its application

Welcome to Mean-Variance Portfolio Theory!

Hello there! Today we are diving into one of the most famous areas of financial economics: Mean-Variance Portfolio Theory (MVPT). This chapter is a cornerstone of the Asset Valuations section of CM2.
Don't be intimidated by the name! At its heart, MVPT is simply about finding the "sweet spot" between making as much money as possible (Mean) and keeping the chance of things going wrong as low as possible (Variance). Think of it as learning how to bake the perfect cake: you want it to be as delicious as possible without it falling flat in the oven!

1. The Foundation: What is Mean-Variance Portfolio Theory?

Mean-Variance Portfolio Theory, introduced by Harry Markowitz, suggests that investors aren't just looking at how much an individual stock might grow. Instead, they look at how a portfolio (a collection of assets) performs as a whole.

Key Assumptions (The "Rules of the Game")

For the math to work simply, we make a few assumptions. Don't worry if these seem a bit unrealistic—they are just the "starting blocks" for the theory:

1. Risk Aversion: We assume investors are "risk-averse." This means if two investments have the same return, an investor will always choose the one with the lowest risk.
2. Single Period: We assume people invest for one fixed holding period (e.g., one year).
3. Decision Criteria: Investors make investment decisions based solely on the expected return and the variance (or standard deviation) of return.

When Does MVPT Lead to an Optimum Portfolio?

Under Expected Utility Theory, rational investors seek to maximize expected utility \( E[U(W)] \). Mean-variance analysis is strictly consistent with expected utility maximization under either of two conditions:
1. Quadratic Utility: The investor has a quadratic utility function, \( U(w) = w - b w^2 \) (with \( b > 0 \)), where expected utility depends purely on the mean and variance of wealth regardless of how asset returns are distributed.
2. Multivariate Normally Distributed Returns: Asset returns follow a joint normal distribution. Because a normal distribution is entirely determined by its means, variances, and covariances, any risk-averse expected utility maximizer will make decisions purely based on mean and variance.

2. Calculating Portfolio Return and Risk

Before we can pick the best portfolio, we need to know how to measure it.

Two-Asset Portfolio

If you hold Asset A and Asset B with weights \( w_A \) and \( w_B \) such that \( w_A + w_B = 1 \):
Expected Return:
\( E[R_p] = w_A E[R_A] + w_B E[R_B] \)
Variance:
\( \sigma_p^2 = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2w_A w_B \text{Cov}(R_A, R_B) = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2w_A w_B \rho_{AB} \sigma_A \sigma_B \)
where \( \rho_{AB} \) is the correlation coefficient between the two assets.

Portfolio of Many Risky Assets (\( n \) Assets)

In practice, actuaries and fund managers invest across many assets. For \( n \) risky assets with weights \( w_i \) satisfying \( \sum_{i=1}^n w_i = 1 \):

Portfolio Expected Return:
\( E[R_p] = \sum_{i=1}^n w_i E[R_i] = \mathbf{w}^T \mathbf{e} \)
where \( \mathbf{w} \) is the column vector of weights and \( \mathbf{e} \) is the column vector of expected returns.

Portfolio Variance:
\( \sigma_p^2 = \sum_{i=1}^n \sum_{j=1}^n w_i w_j \text{Cov}(R_i, R_j) = \sum_{i=1}^n w_i^2 \sigma_i^2 + \sum_{i=1}^n \sum_{j \neq i} w_i w_j C_{ij} \)
where \( C_{ij} = \text{Cov}(R_i, R_j) = \rho_{ij} \sigma_i \sigma_j \).
In matrix notation, this is:
\( \sigma_p^2 = \mathbf{w}^T \mathbf{V} \mathbf{w} \)
where \( \mathbf{V} \) is the \( n \times n \) covariance matrix of asset returns.

3. Diversification: Don't Put All Your Eggs in One Basket

Diversification is the classic "free lunch" in finance. By combining assets that don't move perfectly together, you can reduce portfolio risk without necessarily sacrificing expected return.

The Role of Correlation (\( \rho \)):

1. If \( \rho = +1 \): The assets move in perfect lockstep. There is no risk reduction—standard deviation is simply the weighted average of individual standard deviations.
2. If \( \rho < +1 \): Portfolio standard deviation is strictly less than the weighted average of the individual standard deviations.
3. If \( \rho = -1 \): Perfect negative correlation. It is theoretically possible to combine the two assets to construct a completely risk-free portfolio (\( \sigma_p = 0 \)).

Real-World Analogy: Imagine owning an umbrella shop and an ice cream shop. When it rains, the umbrella shop does great and the ice cream shop struggles. When it's sunny, the opposite happens. Together, your total income stays steady regardless of the weather!

Mathematical Benefits of Diversification Across \( n \) Assets

To see the power of diversification mathematically, consider an equally weighted portfolio of \( n \) assets, so each weight is \( w_i = \frac{1}{n} \). The variance formula becomes:
\( \sigma_p^2 = \sum_{i=1}^n \left(\frac{1}{n}\right)^2 \sigma_i^2 + \sum_{i=1}^n \sum_{j \neq i} \left(\frac{1}{n}\right)^2 C_{ij} \)
Let \( \bar{\sigma}^2 = \frac{1}{n} \sum_{i=1}^n \sigma_i^2 \) be the average variance of the individual assets, and \( \bar{C} = \frac{1}{n(n-1)} \sum_{i=1}^n \sum_{j \neq i} C_{ij} \) be the average covariance across all distinct pairs. Then:
\( \sigma_p^2 = \frac{1}{n} \bar{\sigma}^2 + \frac{n-1}{n} \bar{C} \)
As the number of assets \( n \to \infty \):
\( \frac{1}{n} \bar{\sigma}^2 \to 0 \quad \text{and} \quad \frac{n-1}{n} \bar{C} \to \bar{C} \)
Key Result: As a portfolio becomes well diversified, the individual asset variances (the asset-specific, diversifiable risk) wash out to zero. The portfolio variance converges to the average covariance (the non-diversifiable, systematic market risk)!

4. The Opportunity Set and the Efficient Frontier

If we plot every possible combination of risky assets on a graph (with Risk \( \sigma_p \) on the horizontal axis and Return \( E[R_p] \) on the vertical axis), we get the Opportunity Set (often shaped like a bullet or curve).

The Efficient Frontier

We do not want to be anywhere inside the bullet; rational investors only choose portfolios on the upper edge.
Definition: The Efficient Frontier is the set of portfolios that provide the maximum expected return for a given level of risk, or equivalently, the minimum risk for a given level of expected return.

Common Mistake: Students often think the entire boundary is efficient. Only the top half (portfolios lying on or above the Global Minimum Variance Portfolio) belongs to the Efficient Frontier. Points on the lower boundary are inefficient because an investor could achieve a higher expected return for the exact same risk on the upper half.

5. Choosing the Optimal Portfolio (Indifference Curves)

How does an individual investor select their specific optimum portfolio on the Efficient Frontier? It depends on their Utility Function.

We use Indifference Curves to represent an investor's preferences between expected return and variance:
- For a Risk-Averse investor, indifference curves slope upwards (taking more risk requires a higher expected return to maintain the same expected utility), and they are convex.
- Moving upwards and to the left represents higher expected utility.
- The Optimal Portfolio for an investor is the point where an indifference curve is tangent to the Efficient Frontier (the highest reachable utility curve).

6. Introducing the Risk-Free Asset

What if investors can also lend or borrow at a risk-free rate of return \( r_f \) (with zero standard deviation and zero covariance with risky assets)?
The Efficient Frontier of risky assets is transformed into a straight tangent line: the Capital Market Line (CML).

The Equation of the CML:

\( E[R_p] = r_f + \frac{E[R_M] - r_f}{\sigma_M} \sigma_p \)
where \( M \) is the "Market Portfolio" (the unique tangency portfolio on the risky efficient frontier) and \( \frac{E[R_M] - r_f}{\sigma_M} \) is the Sharpe ratio of the market portfolio.

Two-Fund Separation Theorem:

1. Investment Decision (Technical): Find the optimal combination of risky assets (Point \( M \)). Under homogeneous expectations, this risky portfolio is identical for all investors!
2. Financing Decision (Personal): Decide how much wealth to allocate between the risk-free asset and the tangency portfolio \( M \), depending solely on the individual's level of risk aversion.

Summary Checklist

Before you move on, make sure you can answer these:
- When does MVPT equate to expected utility maximization? (When returns are multivariate normal, or when the investor has quadratic utility).
- How is portfolio variance calculated for \( n \) assets? (\( \sigma_p^2 = \sum_i w_i^2 \sigma_i^2 + \sum_i \sum_{j \neq i} w_i w_j C_{ij} = \mathbf{w}^T \mathbf{V} \mathbf{w} \)).
- What happens to risk in an equally weighted portfolio as \( n \to \infty \)? (Individual variance terms tend to zero, leaving only the average covariance).
- What is the "Efficient Frontier"? (The upper boundary of the opportunity set offering maximum return for a given variance).
- What is the Separation Theorem? (Separating the choice of optimal risky portfolio from the individual financing allocation with cash).