Portfolio optimization is the process of choosing asset weights to balance expected return against risk. The classic framework is to maximize return for a given level of risk, or equivalently minimize risk for a target return.

1. The mean-variance problem

Let $w$ be the vector of portfolio weights, $\mu$ the vector of expected returns, and $\Sigma$ the covariance matrix of asset returns. The portfolio expected return is

\[\mu_p = w^T \mu\]

and the portfolio variance is

\[\sigma_p^2 = w^T \Sigma w.\]

The Markowitz problem is:

\[\min_w \; \frac{1}{2} w^T \Sigma w\]

subject to

\[w^T \mu = \mu_{target}, \qquad \sum_i w_i = 1.\]

This is the foundation of modern portfolio theory.

2. Why diversification matters

Diversification reduces risk because asset returns are not perfectly correlated. If the covariance matrix has off-diagonal terms smaller than variance terms, the portfolio variance can be reduced relative to holding a single asset.

In a two-asset case,

\[\sigma_p^2 = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \sigma_{12},\]

where $\sigma_{12}$ is the covariance between the two assets. This shows that mixing assets can lower the overall variance if the correlation is less than 1.

3. The efficient frontier

By sweeping through different target returns, we trace out the efficient frontier. Each point on this frontier is a portfolio with the minimum risk for a given expected return.

The risk-return tradeoff is often summarized as:

\[\text{maximize } w^T \mu - \frac{\gamma}{2} w^T \Sigma w,\]

where $\gamma$ controls the investor’s risk aversion. A higher $\gamma$ places more weight on minimizing variance.

4. A practical example

Suppose an investor holds three assets with expected returns and covariance matrix:

\[\mu = \begin{bmatrix} 0.08 \\ 0.10 \\ 0.12 \end{bmatrix}, \qquad \Sigma = \begin{bmatrix} 0.04 & 0.01 & 0.02 \\ 0.01 & 0.09 & 0.03 \\ 0.02 & 0.03 & 0.07 \end{bmatrix}.\]

The optimal portfolio depends on the target return or the chosen risk penalty. In practice, the problem is solved numerically by quadratic programming or by using a convex optimization routine.

5. Implementation idea

A simple optimization routine looks like this conceptually:

  1. Define expected returns $\mu$.
  2. Define the covariance matrix $\Sigma$.
  3. Impose budget and target-return constraints.
  4. Solve the quadratic program.
  5. Inspect the resulting weights and efficient frontier.

The general optimization problem can be written as:

\[\min_w \; \frac{1}{2} w^T \Sigma w - \lambda w^T \mu\]

subject to $\sum_i w_i = 1$.

This form highlights the tradeoff between expected return and risk.

6. Takeaways

Portfolio optimization is most useful when you want to:

  • control total portfolio risk,
  • choose a reasonable balance between return and volatility,
  • diversify across assets with imperfect correlation,
  • build a disciplined quantitative investment framework.

In practice, the optimal portfolio depends on assumptions about expected returns, covariances, and the investor’s risk preferences. Even a simple mean-variance model can provide a strong starting point.

This tutorial is intentionally compact, but it illustrates the central intuition behind portfolio optimization: choose weights that efficiently trade off expected return and risk.