Mean-Variance Portfolio Selection and the CAPM

Imagine you have money to invest. You care about two things: the return you hope to get, and the risk that your actual return might be a lot lower — or even a loss. This chapter shows a clear, step-by-step way to think about that trade-off. We will build the classic model that finds the portfolio with the highest expected return at each level of risk. Then we will see how this idea leads to one of the most famous equations in finance: the Capital Asset Pricing Model, or CAPM.

The Big Picture#

The whole chapter answers one core question: if investors care only about the average return and the variance (the spread, or risk) of their future wealth, what portfolios should they hold? And what does that imply about how every asset is priced? By working through the logic, we discover that in a world where everyone follows this mean-variance rule, a single “market” portfolio becomes the universal risky investment. Every asset’s expected return ends up proportional to its sensitivity to that market portfolio. Understanding this reasoning is the foundation for almost everything that follows in asset pricing.

Mean-variance preferences and the return distribution#

When we say an investor has mean-variance preferences, we mean that their happiness (their utility) depends only on two numbers describing the portfolio’s future return: the expected return and the variance (or equivalently the standard deviation , which we call volatility).

In the simplest case, an investor likes higher expected return and dislikes higher variance. So we can write their goal as choosing a portfolio that maximises something like

Here, is a personal number that shows how much the investor hates risk. A larger means a stronger dislike for variance, so the investor will accept a lower expected return in exchange for a smoother ride.

Of course, not every return distribution can be described fully by just its mean and variance. But this preference makes perfect sense if the returns follow a shape where mean and variance really do tell the whole story. The key family that works this way is the set of elliptical distributions. The most famous member is the normal distribution, but the family also includes the Student-t (with the same degrees of freedom across assets) and the Cauchy distribution. In an elliptical world, any mix of asset returns (a portfolio) is also elliptical, and its risk can be fully captured by its variance or standard deviation. So if you believe returns are roughly normal, or if you are comfortable ignoring extreme events and focusing only on those first two numbers, then mean-variance analysis is not just convenient — it is actually consistent with expected utility maximisation.

Even when the real world is not exactly elliptical, mean-variance analysis is still a very useful approximation. It gives us a clear language to talk about diversification, trade-offs, and equilibrium pricing. Throughout this chapter, we take the mean-variance objective as given and explore its consequences.

Mean-variance preferences: An investor likes higher expected return and dislikes higher return variance; their utility depends only on these two moments.

Elliptical distributions: A family of symmetric distributions where mean and covariance fully capture the distribution’s shape. Portfolios of elliptical assets remain elliptical, so variance is a complete risk measure.

📝 Section Recap: If investors care only about a portfolio’s expected return and variance — a sensible goal when returns are at least roughly elliptical — we can simplify the investment problem to picking the best possible combination of risk and reward.

The opportunity set and the efficient frontier without a riskless asset#

Let us start in a world with risky assets — stocks, bonds, commodities, etc. — but no risk-free bank account. Each asset has an expected return and a variance , and each pair of assets has a covariance . A portfolio is just a set of weights that sum to one: . The portfolio’s expected return and variance are

For any target expected return, there are many possible portfolios. Among them, the one with the lowest variance is called the minimum-variance portfolio (MVP) for that return. If we plot all possible portfolios in a risk-return space (standard deviation on the horizontal axis, expected return on the vertical), the set of all possible combinations forms a solid, curved region. The left boundary of this region is the minimum-variance frontier — the collection of portfolios that give the lowest possible volatility for each expected return.

The upper half of this frontier, from the global minimum-variance portfolio upwards, is called the efficient frontier. Any portfolio on the efficient frontier is mean-variance efficient: you cannot get a higher expected return without accepting more risk, and you cannot reduce risk without giving up expected return. The lower half contains portfolios that are also on the frontier but are obviously inefficient because you could get a higher return for the same risk.

Picture the global minimum-variance portfolio as the “nose” of the bullet-shaped frontier. Starting from that point, moving upward along the curve gives you more return and more risk. This trade-off is intuitive and forms the core of portfolio choice.

We can find the efficient frontier mathematically by solving a quadratic optimisation problem: minimise subject to the target return and the budget constraint . The solution involves the covariance matrix and the vector of expected returns, but the important idea is that every efficient portfolio is a weighted average of only two distinct “fund” portfolios. This property is called two-fund separation.

Two-fund separation: In a world with only risky assets, any mean-variance efficient portfolio can be built by blending just two distinct efficient portfolios (e.g., the global minimum-variance portfolio and another frontier portfolio). Every investor, regardless of risk aversion, can in theory hold the same two funds and just choose the mix.

Two-fund separation is a big simplification. Instead of analysing thousands of individual stocks, you can reduce the whole investment universe to just two building blocks. We will see that adding a riskless asset makes separation even sharper.

📝 Section Recap: Without a riskless asset, all efficient portfolios lie on the upper half of a bullet-shaped frontier. The powerful idea of two-fund separation says every efficient portfolio is a simple mix of just two frontier portfolios.

Adding a riskless asset and the tangent portfolio#

Now let us introduce a riskless asset that pays a guaranteed return (the risk-free rate). You can think of this as a government bond or a bank deposit with no default risk. You can invest a fraction in this asset and the rest in risky assets, so the total weights sum to one: . The portfolio’s expected return and volatility become

where and are the expected return and volatility of the risky component (the tangency portfolio that combines the risky assets). We can write the equation linking return and risk directly:

This is a straight line in risk-return space, starting at the point and passing through the risky portfolio . The slope, , is the Sharpe ratio — the extra return per unit of volatility.

The set of all such lines that touch the risky-asset frontier is infinite, but one of them is special: the line with the highest possible slope. It is tangent to the minimum-variance frontier of risky assets. This line is called the capital market line (CML), and the risky portfolio at the tangency point is the tangent portfolio .

Why is the tangent portfolio so important? A mean-variance investor will always want the steepest possible line — combining the riskless asset and the tangency portfolio gives the highest expected return at any risk level. Said another way, every investor, no matter how risk-averse, will hold a mix of the riskless asset and the exact same tangency portfolio of risky assets. A very cautious investor will lend most of her wealth at the risk-free rate and put only a little in ; an aggressive investor may even borrow at the risk-free rate to invest more than 100% in . Either way, the risky part is always the same .

This is a stronger version of separation: Tobin separation says the investment decision splits into two independent steps. First, figure out the tangency portfolio from market data (expected returns, variances, covariances). Second, decide how much to borrow or lend based on your own risk tolerance. The mix of risky assets is the same for everyone.

Capital market line (CML): The straight line from the risk-free rate to the tangency portfolio on the risky-asset frontier. It represents the best possible risk-return trade-off when a riskless asset exists.

Tobin separation: The choice of the optimal risky portfolio (the tangent portfolio) is separate from the choice of how much to invest in it versus the riskless asset. All investors hold the same risky mix.

📝 Section Recap: With a riskless asset, the efficient set becomes a straight line — the capital market line. The unique tangent portfolio of risky assets becomes the universal building block for every investor, independent of their risk aversion.

Market portfolio efficiency and the Capital Asset Pricing Model#

If we assume that all investors have the same beliefs about expected returns, variances, and covariances, and they all follow mean-variance optimisation, then every investor will hold a mix of the riskless asset and the same tangency portfolio. So in total, all the money invested in risky assets ends up held in exactly the same proportions as that tangency portfolio. That aggregate portfolio is, by definition, the market portfolio — a value-weighted portfolio of all risky assets in the economy.

Thus, in equilibrium, the tangency portfolio is the market portfolio. And because the tangency portfolio lies on the efficient frontier, we conclude that the market portfolio is mean-variance efficient.

That single insight leads directly to the Capital Asset Pricing Model (CAPM). If the market portfolio is efficient, then the expected return of any individual asset must line up in a specific linear relation with its covariance with the market. We can write this as

where is the expected return on the market portfolio, is the risk-free rate, and

The beta of an asset measures how much the asset’s return tends to move when the market moves. A stock with moves, on average, with the market. A defensive utility stock might have , while a high-growth tech stock might have .

The equation tells us that the only risk that earns a higher expected return is systematic risk (also called non-diversifiable risk) — the part that cannot be eliminated by holding the market portfolio. Idiosyncratic, firm-specific risk (the part uncorrelated with the market) does not get a risk premium because investors can diversify it away by holding a slice of the market.

The term is the market risk premium — the extra return investors demand for bearing one unit of market risk. So an asset’s expected excess return is times this premium.

Capital Asset Pricing Model (CAPM): An equilibrium model stating that the expected return on any asset or portfolio is a linear function of its market beta. Only systematic risk commands a risk premium.

Market portfolio: The value-weighted portfolio of all risky assets in the economy. Under homogeneous mean-variance preferences, it coincides with the tangency portfolio and is mean-variance efficient.

We can test the CAPM in a simple graph called the security market line (SML). It plots expected return on the vertical axis against beta on the horizontal axis. The model predicts that all assets and portfolios should lie exactly on the straight line from the risk-free rate (beta = 0, return = ) to the market portfolio (beta = 1, return = ). Any deviation would suggest an asset is mispriced relative to its systematic risk.

📝 Section Recap: When all investors use mean-variance analysis with identical views, the market portfolio must be efficient, giving us the CAPM relation: expected excess return equals beta times the market risk premium.

The stochastic discount factor behind the CAPM#

Modern asset pricing often uses a different language — a stochastic discount factor (SDF), also called the pricing kernel. The SDF, usually written as , is a random variable that tells you how much a payoff in each possible future state is worth today. The fundamental pricing equation is

where is today’s price and is the random future payoff. For returns , this becomes .

The CAPM also gives a specific form for the SDF. When the market portfolio is efficient, the SDF that prices all assets can be written as a linear function of the market return:

with constants and that work for the risk-free asset and the market portfolio. Because goes down when the market does well, it puts higher value on payoffs in bad times and lower value on payoffs in good times. This matches the idea that risk premiums exist because investors care more about wealth in bad states.

This SDF view also gives a prediction: the highest possible Sharpe ratio in the economy is bounded by the volatility of divided by its mean. In the CAPM world, that maximum Sharpe ratio is exactly the market’s Sharpe ratio, showing that the market portfolio offers the highest reward per unit of risk.

Stochastic discount factor (SDF): A random variable such that today’s price of any payoff is . It encodes how the market values payoffs in different states of the world. Under the CAPM, is linear in the market return.

📝 Section Recap: The CAPM can be restated in SDF language, where a linear function of the market return serves as the single factor that prices all assets. This shows the close link between the market portfolio’s efficiency and the shape of the discount factor.

The equity premium puzzle: a reality check#

The CAPM is elegant, but when we check it against data, it faces a big challenge. The most famous challenge is the equity premium puzzle. Historical data from many developed countries show that the average excess return on a broad stock market index over a short-term risk-free rate has been surprisingly high — for example, around 6–8% per year in the United States. Meanwhile, the volatility of consumption growth has been quite low. Standard economic models with reasonable levels of risk aversion would need investors to be extremely risk-averse to justify such a large equity premium, far beyond what other evidence suggests is likely.

In the CAPM, the market risk premium depends on average risk aversion and market variance. But when we try to match the observed premium using low consumption volatility and moderate risk aversion, the numbers don’t line up. Simply put, stocks seem to have given too high a return compared to bonds given the fairly mild risk to overall consumption.

This puzzle doesn’t mean the CAPM is useless — it still gives us the essential structure for thinking about systematic risk. But it does push us to explore richer models: ones with time-varying risk aversion, rare disasters, long-run risks, or habit formation. These advanced frameworks try to explain why investors demand such a large premium for holding stocks. For now, the puzzle is a healthy reminder that the CAPM is a powerful benchmark, not the final word.

📝 Section Recap: The CAPM’s difficulty in explaining the historically large spread between stock and bond returns — the equity premium puzzle — pushes us toward richer models that can accommodate higher and time-varying risk premiums.

Summary#

We started with the basic trade-off between expected return and variance, and built up to the powerful CAPM. By assuming investors care only about mean and variance — a reasonable simplification when returns are roughly normal — we discovered the efficient frontier and, with a riskless asset, the capital market line. The logic that every investor should hold the same tangency portfolio forced that portfolio to be the market portfolio, which then gave us the beta relationship: an asset’s expected excess return is proportional to its market beta. We even restated this result in the modern language of stochastic discount factors. Finally, we recognised that while the CAPM is foundational, it faces a serious empirical challenge from the equity premium puzzle, which sets the stage for more advanced models.

Here is a quick reference of the key ideas and why they matter: