Unit 3: Portfolio theory and asset pricing
How to combine assets into portfolios, why diversification works, and what return the market requires for bearing risk. Change any number on the left of a tool and every result, chart and explanation updates at once. Each tool starts from a worked example in the lecture notes, so you can check it against your notes and the video, then try your own numbers.
Chapter 18. Portfolios: one risky and one risk-free asset, two risky assets, many risky assets
The first portfolio decision is how much to put in something risky and how much in something safe. The second is how to combine two risky assets, where the answer depends on how their returns move together.
Capital allocation line
Put a share w of your money in a risky portfolio and the rest in T-bills, which pay the risk-free rate for sure. Every choice of w gives one complete portfolio, and together they form a straight line called the capital allocation line (CAL). Its slope, the Sharpe ratio, is the extra expected return you get for each extra point of risk (standard deviation).
Starts from Worked example 18.1 (video 18-1).
Two-asset portfolio
Split your money between two stocks, A and B. The expected return is simply the weighted average of the two, but the risk is not: it depends on the correlation, a number from −1 to +1 that measures how closely the two returns move together. The lower the correlation, the more the ups of one stock offset the downs of the other.
Starts from Worked examples 18.2 (video 20-1) and 18.3 (video 18-2).
Chapter 19. The minimum-variance portfolio and the optimal risky portfolio
Among all mixes of two risky assets, one has the lowest risk and one has the best reward per unit of risk. Once T-bills are available, every investor should hold the second one and choose only how much of it to hold.
Minimum-variance and optimal risky portfolio
The minimum-variance portfolio (MVP) is the mix of A and B with the smallest possible risk. The optimal risky portfolio O is the mix with the highest Sharpe ratio: the point where the steepest line from the risk-free rate just touches the curve of possible portfolios (the tangency portfolio). Your risk aversion A then decides how much of your money goes into O and how much stays in T-bills.
Starts from Worked examples 19.1 (video 19-1), 19.2 and 19.3 (video 19-2).
Chapter 20. Modern portfolio theory and diversification
Adding stocks to a portfolio removes the risk that comes from each firm's own news, but not the risk that all stocks share. That shared risk sets a floor that no amount of diversification can go below.
How many stocks are enough?
Hold N stocks in equal amounts. Each stock has the same risk, and every pair of stocks has the same correlation. Watch how the portfolio's standard deviation falls as N grows, and how it levels off at the systematic risk (the risk from economy-wide events that hit all firms).
Starts from Worked example 20.1 (video 20-1).
Chapter 21. The capital asset pricing model and the security market line
If only systematic risk is rewarded, how much return should each stock offer? The CAPM answers: the risk-free rate plus beta times the market risk premium. Beta measures how strongly a stock moves with the market.
Beta from correlation
Beta tells you how much a stock tends to move when the market moves 1%. It depends on the stock's own volatility and on how closely it follows the market (its correlation with the market). A very volatile stock can still have a low beta if most of its ups and downs come from its own news.
Starts from Worked example 21.1 (video 21-1).
CAPM and the security market line
The security market line (SML) plots the return the CAPM requires against beta. Compare it with what analysts forecast the stock will return. The difference is alpha: a stock above the line (positive alpha) offers too much for its risk and is underpriced; a stock below the line is overpriced.
Starts from Worked examples 21.2, 21.3 and 21.4 (video 21-3).
Chapter 22. The CAPM in the real world: tests and evidence
Testing the CAPM takes two steps. First, estimate each stock's beta from its past monthly returns (the first pass). Second, check whether stocks with higher betas really earned higher average returns (the second pass). Both tools below use simulated data, so you can see what the tests look for.
First-pass (index-model) regression
We simulate monthly excess returns (returns above the T-bill rate) for the market and for one stock, using a true alpha, a true beta and some firm-specific noise that you choose. Then we fit a straight line through the dots, exactly as you would with real data: the slope estimates beta, the intercept estimates alpha, and R-squared is the share of the stock's ups and downs explained by the market. The market's average excess return is fixed at 0.42% per month.
Built on the section "Index-model (first-pass) regression" in Chapter 22 (video 22-1); the starting values follow Try it 22.1. This chapter has no worked example.
Second pass: do high-beta portfolios earn more?
Sort stocks into portfolios by their beta and plot each portfolio's average excess return against its beta. The CAPM predicts the dots lie on the SML: a line starting at zero and rising by the market risk premium for each unit of beta. Pick what the simulated world rewards and see what the fitted line looks like. Fama and French (1992) found the line was essentially flat.
Built on the sections "Does the CAPM work?" and "Fama and French (1992): beta, size and book-to-market" in Chapter 22 (video 22-1). This chapter has no worked example.
Chapter 23. Arbitrage pricing theory, the single-index model and multifactor models
An arbitrage costs nothing, carries no risk and still makes a profit. Because traders jump on such chances, related prices stay consistent, and that idea alone gives a pricing model with one or several risk factors.
Arbitrage across markets: gasoline
If gasoline sells for more in the U.S. than it costs to buy it in Europe and ship it across, traders can buy in Europe, ship and sell in the U.S. for a profit. Compare the two shipping routes.
Starts from Worked example 23.1 (video 23-1).
Arbitrage between well-diversified portfolios
A well-diversified portfolio holds so many stocks that only factor risk (beta) is left. Two such portfolios should offer the same risk premium per unit of beta. If one offers more, you can copy the other's beta with it and T-bills, buy the copy, short (sell borrowed shares of) the expensive one, and lock in a profit whatever the factor does.
Starts from Worked example 23.2 (video 23-2).
Two-factor model
In a multifactor model, a stock's expected return is the risk-free rate plus a reward for each source of systematic risk: its sensitivity to the factor (the factor beta) times the risk premium the market pays for that factor. Here the two factors are surprises in GDP growth and in interest rates.
Starts from Worked example 23.3 (video 23-3).