Further learning

How a mix changes risk

A portfolio's future return is unknown. An expected return is an average of possible future returns, weighted by their probabilities. In practice, we estimate it from data or a model. It is not a promised return.

If all the money is invested in two stocks, the expected return is:

Here, μ means expected return, p means portfolio, and w is the share of money in each stock. The weights add to one. All returns must refer to the same period.

Variance measures the average squared distance from the expected return. Its square root, the standard deviation σ, has the same units as returns. Correlation ρ describes how two returns move together. It ranges from −1 to +1. It changes the risk of their mix:

Holding more names helps less when they move together. Correlation can change, and zero correlation does not imply independence. A lower standard deviation also does not remove every type of loss.

MIT: Portfolio Theory ↗ develops the mean, standard deviation, and correlation of portfolio returns.

What moved with the market?

A stock can rise because many stocks rose that day. It can also move because of company news. These effects are hard to separate perfectly, but a simple model gives us a starting language:

Here, r_i,t is stock i's return in period t. The market index return is r_m,t. Beta β measures how strongly the stock moves with that index in the fitted model. Here, “fitted” means estimated from past data. Alpha α is the constant term. The remaining part ε is the residual: what this model does not explain.

A risk factor is a common source of return changes. Models can include several factors. For example, researchers use market returns and returns of stock groups based on company size or value measures.

In this setting, exposure means how sensitive a portfolio is to a factor. High returns may partly reflect such exposure. A positive fitted alpha is not proof of skill: the sample may be short, costs may be missing, or the chosen model may leave out other drivers.

Kenneth French: Data Library ↗ provides factor and portfolio return series, with descriptions of how they are built.

Information, expectations, and learning

A price reflects trades made by people with different information and beliefs. Public news may already be widely known when a model reads it. The idea of market efficiency asks how much available information is already reflected in prices. It does not mean that prices are always correct.

Markets also change as policies, participants, and tools change. This connects finance to several learning ideas:

• Ensembles combine views. Their errors may still overlap if they use the same information.
• Online learning updates a method after feedback arrives. Fast changes can respond to new conditions but also to noise.
• LLMs can turn text into useful information. Understanding a sentence does not mean its price effect is known.
• Agents organize tools, memory, and decisions. More steps do not automatically produce better decisions.

Yale: Financial Markets ↗ includes lessons on efficient markets, risk, and human behavior.

Course StockBench starter ↗ brings prices, company data, and news into simulated trading decisions. The original StockBench project ↗ explains the research behind it.