Lesson 2.1: The Superpower of the Normal Distribution

This lesson reveals the most elegant and powerful property of the Normal distribution: its 'closure' under addition. We'll prove that if you add independent Normal variables together, the result is always another Normal variable. This single theorem is the theoretical engine that powers both Modern Portfolio Theory and the statistical inference of the OLS estimator.

Part 1: The Practical Problem - Portfolio Returns

Let's consider a simple portfolio with two assets, A and B. We assume their daily returns are independent and Normally distributed:

Return of A: $R_A \sim \mathcal{N}(\mu_A, \sigma_A^2)$
Return of B: $R_B \sim \mathcal{N}(\mu_B, \sigma_B^2)$

We build a portfolio $P$ with 60% in Asset A and 40% in Asset B. The portfolio's return, $R_P$ , is a **linear combination** of the asset returns:

R_P = 0.6 R_A + 0.4 R_B

The crucial question for any risk manager is: What is the probability distribution of $R_P$ ? Is it Normal? And what are its mean and variance?

The Linear Combination Theorem

If $X_1, X_2, \dots, X_n$ are independent random variables and $X_i \sim \mathcal{N}(\mu_i, \sigma_i^2)$ , then any linear combination $Y = a_1 X_1 + a_2 X_2 + \dots + a_n X_n$ is also Normally distributed.

This "closure" property is unique and incredibly powerful. It guarantees that our portfolio return $R_P$ will have a predictable bell shape, allowing us to model its risk perfectly.

Part 2: Finding the Parameters of the New Distribution

Knowing that the result is Normal is half the battle. Now we need to find its mean and variance. Let $Y = aX_1 + bX_2$ .

Deriving the Mean (

\mu_Y

)

Using the linearity of expectation, the derivation is straightforward:

\mu_Y = E[Y] = E[aX_1 + bX_2] = aE[X_1] + bE[X_2] = a\mu_1 + b\mu_2

Deriving the Variance (

\sigma_Y^2

)

For the variance of a sum, we must account for the covariance. But because we assume $X_1$ and $X_2$ are independent, their covariance is zero.

\sigma_Y^2 = \text{Var}(aX_1 + bX_2) = a^2\text{Var}(X_1) + b^2\text{Var}(X_2) + 2ab \cdot \text{Cov}(X_1, X_2)

Since $\text{Cov}(X_1, X_2) = 0$ :

\sigma_Y^2 = a^2\sigma_1^2 + b^2\sigma_2^2

Solving our Portfolio Problem: The return of our portfolio $R_P = 0.6 R_A + 0.4 R_B$ is Normally distributed with:

Mean: $\mu_P = 0.6\mu_A + 0.4\mu_B$
Variance: $\sigma_P^2 = (0.6)^2\sigma_A^2 + (0.4)^2\sigma_B^2 = 0.36\sigma_A^2 + 0.16\sigma_B^2$

Part 3: The Rigorous Proof (Using MGFs)

Proof: Why is the sum of independent Normals also Normal?

This is a classic and elegant proof that uses the Moment Generating Functions we mastered in Module 1.

Step 1: Recall the MGF for a Normal variable. If $X \sim \mathcal{N}(\mu, \sigma^2)$ , its MGF is:

M_X(t) = e^{\mu t + \sigma^2 t^2 / 2}

Step 2: Use the key property of MGFs for sums of independent variables. If $Y = X_1 + X_2$ , where $X_1$ and $X_2$ are independent, then the MGF of Y is the product of the individual MGFs:

M_Y(t) = M_{X_1}(t) \cdot M_{X_2}(t)

Step 3: Substitute the Normal MGFs.

M_Y(t) = \left( e^{\mu_1 t + \sigma_1^2 t^2 / 2} \right) \cdot \left( e^{\mu_2 t + \sigma_2^2 t^2 / 2} \right)

Step 4: Combine the exponents by adding them.

M_Y(t) = e^{(\mu_1 + \mu_2)t + (\sigma_1^2 + \sigma_2^2)t^2 / 2}

Step 5: Recognize the result. Look at this final formula! It is the MGF of a new Normal distribution with a mean of $(\mu_1+\mu_2)$ and a variance of $(\sigma_1^2+\sigma_2^2)$ .

Because the resulting MGF is the MGF of a Normal distribution, the distribution of Y *must* be Normal. This completes the proof.

The Payoff: Why This Theorem is a Cornerstone

Quantitative Finance (Modern Portfolio Theory): This theorem is the mathematical foundation of MPT. It assumes individual asset returns are (Multivariate) Normal. The theorem then guarantees that the overall portfolio return—a weighted sum of asset returns—is also Normal. This allows managers to use the simple and well-understood properties of the Normal distribution to model the risk and return of their entire portfolio.
Econometrics (The OLS Estimator): The OLS estimator $\hat{\beta}$ is a linear combination of the response variable $Y$ (or the error terms $\epsilon$ ). If we assume that the error terms are independent and Normally distributed, this theorem guarantees that the sampling distribution of $\hat{\beta}$ will also be Normal. This is the critical assumption that allows us to perform t-tests and construct confidence intervals for our regression coefficients. Without it, the entire framework of inference in OLS would break down.

What's Next? Handling Dependencies

We've proven this powerful result under the convenient assumption of independence. But in the real world, asset returns are rarely independent—they have correlations.

To handle this, we must upgrade our tools to the matrix world. The next lesson introduces the **Multivariate Normal Distribution**, which replaces single variance numbers ( $\sigma^2$ ) with a full Covariance Matrix ( $\Sigma$ ) to model these complex dependencies.

Lesson 2.0: The King of Distributions: The Normal Curve

Lesson 2.2: The Multivariate Normal Distribution (MVN)