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Home/Financial Mathematics/Course 4

Course 4

Core Theory

7-8 hours

Modern Portfolio Theory

Markowitz mean-variance optimization, efficient frontier, and optimal portfolio construction

Learning Objectives

•Derive portfolio return and risk formulas for multi-asset portfolios
•Understand and construct the efficient frontier using Lagrangian optimization
•Master the two-fund separation theorem and Capital Market Line derivation
•Calculate minimum variance and tangency portfolios
•Quantify diversification benefits and understand systematic vs. idiosyncratic risk

1. Two-Asset Portfolio Analysis

Definition 1.1: Portfolio Return

For a portfolio with weights $w_1, w_2$ (where $w_1 + w_2 = 1$ ) in assets with returns $r_1, r_2$ , the portfolio return is:

r_p = w_1 r_1 + w_2 r_2

Expected return:

\mu_p = w_1 \mu_1 + w_2 \mu_2

Theorem 1.1: Two-Asset Portfolio Variance

The portfolio variance is:

\sigma_p^2 = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2w_1 w_2 \text{Cov}(r_1, r_2)

Using correlation $\rho_{12} = \text{Cov}(r_1, r_2)/(\sigma_1 \sigma_2)$ :

\sigma_p^2 = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2w_1 w_2 \rho_{12} \sigma_1 \sigma_2

Proof of Portfolio Variance Formula:

Starting from the definition:

\sigma_p^2 = \text{Var}(w_1 r_1 + w_2 r_2)

Using variance properties:

= w_1^2 \text{Var}(r_1) + w_2^2 \text{Var}(r_2) + 2w_1 w_2 \text{Cov}(r_1, r_2)

= w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2w_1 w_2 \rho_{12} \sigma_1 \sigma_2

∎

Example 1.1: Portfolio Risk with Different Correlations

Consider equal weights ( $w_1 = w_2 = 0.5$ ) in two assets with $\sigma_1 = 20\%$ , $\sigma_2 = 30\%$ .

Case 1: Perfect positive correlation ( $\rho = 1$ )

\sigma_p^2 = (0.5)^2(0.04) + (0.5)^2(0.09) + 2(0.5)(0.5)(1)(0.2)(0.3) = 0.01 + 0.0225 + 0.03 = 0.0625

\sigma_p = 25\% = \frac{20\% + 30\%}{2}

Case 2: Zero correlation ( $\rho = 0$ )

\sigma_p^2 = 0.01 + 0.0225 = 0.0325, \quad \sigma_p = 18.03\%

Case 3: Perfect negative correlation ( $\rho = -1$ )

\sigma_p^2 = 0.01 + 0.0225 - 0.03 = 0.0025, \quad \sigma_p = 5\%

Notice how diversification benefit increases as correlation decreases!

Theorem 1.2: Minimum Variance Portfolio Weights

The portfolio with minimum variance has weights:

w_1^{MV} = \frac{\sigma_2^2 - \rho_{12}\sigma_1\sigma_2}{\sigma_1^2 + \sigma_2^2 - 2\rho_{12}\sigma_1\sigma_2}

w_2^{MV} = 1 - w_1^{MV}

Proof of Minimum Variance Weights:

Minimize $\sigma_p^2$ with respect to $w_1$ (using $w_2 = 1 - w_1$ ):

\frac{d\sigma_p^2}{dw_1} = 2w_1\sigma_1^2 + 2(1-w_1)\sigma_2^2(-1) + 2\rho_{12}\sigma_1\sigma_2(1-2w_1) = 0

2w_1\sigma_1^2 - 2\sigma_2^2 + 2w_1\sigma_2^2 + 2\rho_{12}\sigma_1\sigma_2 - 4w_1\rho_{12}\sigma_1\sigma_2 = 0

w_1(\sigma_1^2 + \sigma_2^2 - 2\rho_{12}\sigma_1\sigma_2) = \sigma_2^2 - \rho_{12}\sigma_1\sigma_2

Therefore:

w_1^{MV} = \frac{\sigma_2^2 - \rho_{12}\sigma_1\sigma_2}{\sigma_1^2 + \sigma_2^2 - 2\rho_{12}\sigma_1\sigma_2}

∎

Remark 1.1: Special Cases

If $\sigma_1 = \sigma_2$ : $w_1^{MV} = w_2^{MV} = 0.5$ regardless of $\rho$
If $\rho = 1$ : $w_1^{MV} = \sigma_2/(\sigma_1 + \sigma_2)$ (inverse variance weighting)
If $\rho = -1$ : Can achieve zero variance with $w_1 = \sigma_2/(\sigma_1 + \sigma_2)$

2. Markowitz Mean-Variance Optimization

Definition 2.1: N-Asset Portfolio

For $n$ assets with weights $\mathbf{w} = (w_1, \ldots, w_n)^T$ where $\sum_{i=1}^n w_i = 1$ :

Expected return:

\mu_p = \mathbf{w}^T \boldsymbol{\mu} = \sum_{i=1}^n w_i \mu_i

Portfolio variance:

\sigma_p^2 = \mathbf{w}^T \Sigma \mathbf{w} = \sum_{i=1}^n \sum_{j=1}^n w_i w_j \sigma_{ij}

where $\Sigma$ is the covariance matrix with elements $\sigma_{ij} = \text{Cov}(r_i, r_j)$ .

Theorem 2.1: Markowitz Optimization Problem

The efficient frontier solves:

\min_{\mathbf{w}} \frac{1}{2}\mathbf{w}^T \Sigma \mathbf{w}

subject to:

\mathbf{w}^T \boldsymbol{\mu} = \mu_p \quad (\text{target return})

\mathbf{w}^T \mathbf{1} = 1 \quad (\text{budget constraint})

Proof of Efficient Frontier via Lagrangian:

Form the Lagrangian:

\mathcal{L} = \frac{1}{2}\mathbf{w}^T \Sigma \mathbf{w} - \lambda_1(\mathbf{w}^T \boldsymbol{\mu} - \mu_p) - \lambda_2(\mathbf{w}^T \mathbf{1} - 1)

First-order condition:

\frac{\partial \mathcal{L}}{\partial \mathbf{w}} = \Sigma \mathbf{w} - \lambda_1 \boldsymbol{\mu} - \lambda_2 \mathbf{1} = 0

Therefore:

\mathbf{w} = \Sigma^{-1}(\lambda_1 \boldsymbol{\mu} + \lambda_2 \mathbf{1})

The Lagrange multipliers $\lambda_1, \lambda_2$ are determined by the constraints. The efficient frontier is a hyperbola in $(\sigma_p, \mu_p)$ space.

∎

Proposition 2.1: Two-Fund Theorem (Risky Assets Only)

Any portfolio on the efficient frontier can be expressed as a combination of two frontier portfolios. Specifically, if $P$ and $Q$ are any two efficient portfolios, then any other efficient portfolio $R$ has weights:

\mathbf{w}_R = \alpha \mathbf{w}_P + (1-\alpha)\mathbf{w}_Q

This dramatically simplifies portfolio choice - investors only need to identify two efficient portfolios.

Example 2.1: Three-Asset Portfolio Optimization

Consider three assets with expected returns $\boldsymbol{\mu} = (8\%, 10\%, 12\%)^T$ and covariance matrix:

\Sigma = \begin{pmatrix} 0.04 & 0.01 & 0.015 \\ 0.01 & 0.06 & 0.02 \\ 0.015 & 0.02 & 0.09 \end{pmatrix}

To find the minimum variance portfolio, solve:

\mathbf{w}^{MV} = \frac{\Sigma^{-1}\mathbf{1}}{\mathbf{1}^T \Sigma^{-1}\mathbf{1}}

This requires matrix inversion and can be computed numerically.

3. Risk-Free Asset and the Capital Market Line

Definition 3.1: Capital Market Line

When a risk-free asset with return $r_f$ is available, the new efficient frontier is a straight line:

\mu_p = r_f + \frac{\mu_T - r_f}{\sigma_T} \sigma_p

where $T$ denotes the tangency portfolio (the risky portfolio with the highest Sharpe ratio).

Theorem 3.1: Tangency Portfolio Derivation

The tangency portfolio maximizes the Sharpe ratio:

SR = \frac{\mu_p - r_f}{\sigma_p}

The optimal weights are:

\mathbf{w}_T = \frac{\Sigma^{-1}(\boldsymbol{\mu} - r_f \mathbf{1})}{\mathbf{1}^T \Sigma^{-1}(\boldsymbol{\mu} - r_f \mathbf{1})}

Proof of Tangency Portfolio Formula:

For portfolio combining risk-free asset (weight $1-w$ ) and risky portfolio ( $\mathbf{w}$ ):

\mu_p = (1-\mathbf{w}^T \mathbf{1})r_f + \mathbf{w}^T \boldsymbol{\mu} = r_f + \mathbf{w}^T(\boldsymbol{\mu} - r_f\mathbf{1})

\sigma_p^2 = \mathbf{w}^T \Sigma \mathbf{w}

Sharpe ratio:

SR = \frac{\mathbf{w}^T(\boldsymbol{\mu} - r_f\mathbf{1})}{\sqrt{\mathbf{w}^T \Sigma \mathbf{w}}}

Maximize by differentiating (using homogeneity, scale doesn't matter):

\Sigma \mathbf{w} = \lambda(\boldsymbol{\mu} - r_f\mathbf{1})

\mathbf{w} = \lambda \Sigma^{-1}(\boldsymbol{\mu} - r_f\mathbf{1})

Normalize so weights sum to 1:

\mathbf{w}_T = \frac{\Sigma^{-1}(\boldsymbol{\mu} - r_f \mathbf{1})}{\mathbf{1}^T \Sigma^{-1}(\boldsymbol{\mu} - r_f \mathbf{1})}

∎

Theorem 3.2: Two-Fund Separation Theorem

Every investor holds a combination of:

The tangency portfolio $T$
The risk-free asset

The allocation between them depends on risk aversion, but the composition of $T$ is the same for all investors.

Corollary 3.1: Mutual Fund Theorem

Investors can achieve any point on the CML by holding just two mutual funds: a risk-free money market fund and the tangency portfolio (market portfolio). This provides the theoretical foundation for index funds.

Example 3.1: Optimal Allocation with Risk-Free Asset

Suppose $r_f = 3\%$ , tangency portfolio has $\mu_T = 11\%$ , $\sigma_T = 20\%$ .

An investor wants expected return $\mu_p = 9\%$ . What allocation?

\mu_p = (1-\alpha)r_f + \alpha \mu_T

0.09 = (1-\alpha)(0.03) + \alpha(0.11)

0.09 = 0.03 + 0.08\alpha \implies \alpha = 0.75

Portfolio: 75% in tangency, 25% in risk-free asset.

Portfolio risk:

\sigma_p = \alpha \sigma_T = 0.75 \times 20\% = 15\%

Sharpe ratio: $(9-3)/15 = 0.4$ (same as tangency portfolio!)

4. Diversification and Risk Decomposition

Theorem 4.1: Diversification with Many Assets

For $n$ equally-weighted assets with identical variance $\sigma^2$ and identical pairwise correlation $\rho$ :

\sigma_p^2 = \frac{\sigma^2}{n} + \left(1 - \frac{1}{n}\right)\rho\sigma^2

As $n \to \infty$ :

\lim_{n \to \infty} \sigma_p^2 = \rho\sigma^2

Proof of Diversification Limit:

Equal weights: $w_i = 1/n$ . Portfolio variance:

\sigma_p^2 = \sum_{i=1}^n \left(\frac{1}{n}\right)^2 \sigma^2 + \sum_{i=1}^n \sum_{j \neq i} \left(\frac{1}{n}\right)^2 \rho\sigma^2

= n \cdot \frac{\sigma^2}{n^2} + n(n-1) \cdot \frac{\rho\sigma^2}{n^2}

= \frac{\sigma^2}{n} + \frac{n-1}{n}\rho\sigma^2 = \frac{\sigma^2}{n} + \left(1 - \frac{1}{n}\right)\rho\sigma^2

Taking the limit:

\lim_{n \to \infty} \sigma_p^2 = 0 + \rho\sigma^2

∎

Definition 4.1: Systematic vs. Idiosyncratic Risk

From the diversification formula:

Idiosyncratic risk: $\sigma^2/n$ - diversifiable, disappears as $n \to \infty$
Systematic risk: $\rho\sigma^2$ - undiversifiable, remains even with infinite assets

Investors are only rewarded for bearing systematic risk (leads to CAPM in next course).

Remark 4.1: Practical Implications

Key insights from modern portfolio theory:

Diversification reduces risk without sacrificing expected return
Most benefits achieved with 20-30 well-chosen assets
International diversification offers additional benefits (lower correlations)
Cannot eliminate systematic (market) risk through diversification alone
Correlation matters more than individual asset volatilities

Key Takeaways

1.Portfolio risk depends on variances AND covariances - diversification exploits imperfect correlation
2.Efficient frontier represents optimal risk-return tradeoffs using Lagrangian optimization
3.With risk-free asset, CML becomes the new efficient frontier through tangency portfolio
4.Two-fund separation: all investors hold same risky portfolio, differing only in leverage
5.Diversification eliminates idiosyncratic risk but not systematic (market) risk
6.Sharpe ratio maximization leads to tangency portfolio - foundation for CAPM

Frequently Asked Questions

What is the key insight of Markowitz portfolio theory?

Don't evaluate securities in isolation - consider how they interact through correlations. By combining assets with less-than-perfect correlation, you can reduce portfolio risk below the weighted average of individual risks. This is the power of diversification.

What does the efficient frontier represent?

The efficient frontier is the set of portfolios that offer the highest expected return for each level of risk, or equivalently, the lowest risk for each level of return. Portfolios below the frontier are inefficient - you can get better risk-return tradeoffs.

Why is the minimum variance portfolio special?

It's the portfolio with the absolute lowest risk, found at the leftmost point of the efficient frontier. While it minimizes variance, it may not maximize Sharpe ratio or expected return. It's particularly important when short sales are constrained.

How does adding a risk-free asset change the efficient frontier?

With a risk-free asset, the efficient frontier becomes a straight line (the Capital Market Line) from the risk-free rate tangent to the risky efficient frontier. All investors hold the same risky portfolio (the tangency portfolio) combined with varying amounts of the risk-free asset.

What is the separation theorem?

Portfolio selection can be separated into two independent decisions: (1) finding the optimal risky portfolio (same for all investors), and (2) allocating between this risky portfolio and the risk-free asset based on individual risk preferences. This is also called the two-fund separation theorem.

Practice Quiz

Modern Portfolio Theory

Questions

Correct

Accuracy

A portfolio has 60% in asset A (return 10%, std 15%) and 40% in asset B (return 8%, std 10%). If correlation is 0.3, what is the portfolio return?

Easy

Not attempted

For the same portfolio, what is the portfolio variance?

Medium

Not attempted

What happens to portfolio risk as the number of assets increases with equal weights and identical pairwise correlation ρ?

Hard

Not attempted

The minimum variance portfolio with two assets has weights determined by:

Medium

Not attempted

The tangency portfolio maximizes:

Easy

Not attempted

If rf = 3%, tangency portfolio has μ = 11% and σ = 20%, an investor with 50% in risk-free asset has portfolio Sharpe ratio:

Medium

Not attempted

For uncorrelated assets with equal weights, how does portfolio standard deviation scale with n?

Hard

Not attempted

The global minimum variance portfolio:

Medium

Not attempted

An investor allocates 120% to the tangency portfolio and -20% to risk-free asset. This investor is:

Easy

Not attempted

The slope of the Capital Market Line equals:

Medium

Not attempted

Which assumption is NOT required for mean-variance optimization?

Hard

Not attempted

Diversification benefits are greatest when correlation between assets is:

Easy

Not attempted