Lesson 2.4: Capstone 1 - The MVN in Action: Portfolio Theory

It's time for the payoff. This lesson applies the Multivariate Normal theory to solve the most fundamental problem in modern finance: calculating the expected return and risk (volatility) of a diversified portfolio. We will use the mean vector and covariance matrix to see, mathematically, why diversification works.

Part 1: Setting the Stage

An investor wants to build a portfolio with two stocks: a tech company (T) and a utility company (U). From historical data, we assume their annual returns are jointly Multivariate Normal.

The Parameters of Our System

The system is described by a mean vector $\bm{\mu}$ and a covariance matrix $\mathbf{\Sigma}$ .

Mean Vector ( $\bm{\mu}$ ): Expected Returns

\bm{\mu} = \begin{bmatrix} E[R_T] \\\\ E[R_U] \end{bmatrix} = \begin{bmatrix} 0.12 \\\\ 0.06 \end{bmatrix}

(Tech: 12% expected return, Utility: 6%)

Covariance Matrix ( $\mathbf{\Sigma}$ ): Risk & Interaction

\mathbf{\Sigma} = \begin{bmatrix} 0.20^2 & 0.006 \\\\ 0.006 & 0.10^2 \end{bmatrix} = \begin{bmatrix} 0.04 & 0.006 \\\\ 0.006 & 0.01 \end{bmatrix}

(Tech Vol: 20%, Utility Vol: 10%)

The investor decides on a simple 50/50 portfolio. This is our **weights vector**, $\mathbf{w}$ :

\mathbf{w} = \begin{bmatrix} 0.5 \\\\ 0.5 \end{bmatrix}

The portfolio return, $R_P$ , is a linear combination of the asset returns: $R_P = \mathbf{w}^T \mathbf{R}$ .

Part 2: Calculating Portfolio Return and Risk

Calculating Expected Portfolio Return:

E[R_P]

This is a direct application of the linear transformation property on the mean vector.

Formula: Expected Portfolio Return

E[R_P] = \mu_P = \mathbf{w}^T \bm{\mu}

Let's calculate it for our portfolio:

\mu_P = \begin{bmatrix} 0.5 & 0.5 \end{bmatrix} \begin{bmatrix} 0.12 \\\\ 0.06 \end{bmatrix} = (0.5 \cdot 0.12) + (0.5 \cdot 0.06)

\mu_P = 0.06 + 0.03 = 0.09

The expected return of our 50/50 portfolio is 9%, which is simply the weighted average of the individual expected returns.

Calculating Portfolio Variance:

\sigma_P^2

This is where the covariance matrix becomes essential. It captures the diversification effect.

Formula: Portfolio Variance

\text{Var}(R_P) = \sigma_P^2 = \mathbf{w}^T \mathbf{\Sigma} \mathbf{w}

Let's calculate the risk of our portfolio:

\sigma_P^2 = \begin{bmatrix} 0.5 & 0.5 \end{bmatrix} \begin{bmatrix} 0.04 & 0.006 \\\\ 0.006 & 0.01 \end{bmatrix} \begin{bmatrix} 0.5 \\\\ 0.5 \end{bmatrix}

First, multiply the first two matrices:

\begin{bmatrix} (0.5 \cdot 0.04 + 0.5 \cdot 0.006) & (0.5 \cdot 0.006 + 0.5 \cdot 0.01) \end{bmatrix} = \begin{bmatrix} 0.023 & 0.008 \end{bmatrix}

Now, multiply the result by the final vector:

\sigma_P^2 = \begin{bmatrix} 0.023 & 0.008 \end{bmatrix} \begin{bmatrix} 0.5 \\\\ 0.5 \end{bmatrix} = (0.023 \cdot 0.5) + (0.008 \cdot 0.5)

\sigma_P^2 = 0.0115 + 0.004 = 0.0155

The Magic of Diversification

Our portfolio variance is $\sigma_P^2 = 0.0155$ . To make it interpretable, we find the standard deviation (volatility):

\sigma_P = \sqrt{0.0155} \approx 0.1245

The volatility of our portfolio is 12.45%.

Let's compare this to the simple weighted average of the individual volatilities:

\text{Weighted Avg. Vol} = 0.5 \cdot (20\%) + 0.5 \cdot (10\%) = 15\%

Our portfolio's risk (12.45%) is LESS than the average risk of its components (15%)!

This "free lunch" is **diversification**. The positive but low covariance ( $0.006$ ) means the assets don't move perfectly together. When one goes down, the other might only go down a little, or even go up, smoothing out the ride. The covariance matrix, $\mathbf{\Sigma}$ , is what mathematically captures this effect.

What's Next? Deeper Theory & The 'Sampling' Family

We've successfully applied the MVN to a real-world problem. This completes our foundational study of the Normal distribution family in its purest form.

Now, we must turn to a new set of distributions that arise not from natural phenomena, but from the process of **sampling data**. To understand how we test hypotheses, we must first meet the distributions that govern our test statistics.

In the next lesson, we begin our study of the "sampling distributions" by meeting the parent of them all: the **Chi-Squared (χ²) Distribution**.

Lesson 2.3: Performing Surgery on the MVN

Lesson 2.5: The χ² (Chi-Squared) Distribution