Capital Market Theory — the ground floor

One new ingredient

Chapter 15 built portfolios out of risky assets only. Capital market theory builds on that and adds one thing: a risk-free asset. That single addition has big effects on portfolio choice.

It also answers a basic question of finance: how should the risk of an investment affect its expected return? The first coherent answer was the Capital Asset Pricing Model (CAPM). It is the dominant model for valuing risky assets and estimating the required rate of return.

CAPM was developed in the early 1960s, independently and at the same time, by four people:

Note: Chapter 17 of the same workbook says Sharpe and Lintner introduced CAPM in 1964. So Lintner appears as 1965 in Chapter 16 and 1964 in Chapter 17.

The assumptions

Every MPT assumption still holds. Capital market theory adds these:

1. All investors are Markowitz-efficient — they target points on the efficient frontier. Where exactly depends on each investor's risk-return utility function.
2. Investors can borrow or lend any amount at the risk-free rate. In practice you can lend at the risk-free rate by buying government T-bills, but you cannot borrow at it. A higher borrowing rate does not change the general results.
3. All investors have homogeneous expectations — identical probability distributions of future returns. Small differences have minor effects.
4. All investors share the same one-period horizon (a month, six months, a year). A different horizon would require matching risk measures and risk-free assets.
5. All investments are infinitely divisible. Changing this has little impact.
6. No taxes or transaction costs. Relaxing this modifies results but not the basic theory.
7. No inflation, or inflation is fully anticipated.
8. Capital markets are in equilibrium — every investment is correctly priced for its risk. This one is very important and cannot be relaxed.

What makes an asset risk-free

A risky asset has uncertain future returns. A risk-free asset has certain returns. So:

• its standard deviation is zero
• its correlation (and covariance) with every risky asset is zero
• it earns the risk-free rate (rf)
• it sits on the vertical axis of a risk–return graph, because it has no risk

Mixing risk-free with risky

Put some money in the risk-free asset and the rest in a risky portfolio.

• Return is a weighted average, as always.
• Risk collapses to just (weight in risky asset) × (risk of risky asset), because the risk-free asset has zero variance and zero covariance.

So risk and return move in straight-line proportion. On a graph, every mix sits on a straight line from rf to the risky portfolio.

The Capital Market Line

Draw straight lines from rf to different portfolios on the efficient frontier. The steeper the line, the more return per unit of risk. The steepest line just touches the frontier at one point, called M.

Every rational investor prefers mixes of rf and M over any other mix. So:

• M is the most desirable risky portfolio — the Market Portfolio.
• The line from rf through the tangent point M is the Capital Market Line (CML).
• With a risk-free asset available, the CML is the new efficient frontier. Risky portfolios below it are no longer efficient.

Investors differ only in how much they put in M. The rest goes into the risk-free asset.

Lending and borrowing along the CML

• Left of M: you lend at rf — part in the risk-free asset, part in M.
• At M: 100% in the market portfolio.
• Right of M: you borrow at rf and put your own money plus the loan into M. That is leverage.

The market portfolio

In theory, M contains every risky asset — domestic and foreign stocks and bonds, real estate, coins, stamps, art, antiques, and anything else marketable, from around the world. Each asset is held in proportion to its market value. If an asset's weight in M were too high for its market value, excess demand would push its price up until the two matched.

You cannot build this in practice, so proxies are used — usually broad-based equity market indices. There is no agreement on which proxy to use. And an incorrect proxy distorts both beta estimates and the position and slope of the SML.

Two kinds of risk

Because M holds everything, it is the most diversified portfolio possible. Complete diversification removes all unsystematic (unique, non-market) risk. What remains is systematic (market) risk, which cannot be diversified away.

Total risk = market risk + non-market risk.

As you add securities, total risk drops steeply at first, then flattens. It never reaches zero. The workbook says the point where it flattens varies by country, and that 20 to 30 securities is the evidence found globally. (Chapter 17 of the same workbook gives 30 to 35 — see that chapter.)

CAPM and beta

Here is the key logic. If an investor holds M, only systematic risk is left. Unsystematic risk was diversified away for free, so nobody pays you for bearing it. Investors are compensated only for systematic risk.

That systematic risk is measured by beta:

Beta = Cov(security, market) ÷ Variance(market)

And CAPM says:

E(Ri) = RFR + βi × [E(Rm) − RFR]

The Security Market Line

The SML is the graph of CAPM: expected return on the y-axis, beta on the x-axis. Under equilibrium, every asset and portfolio should plot on the SML.

• Plots above the SML → more return than its beta warrants → undervalued.
• Plots below the SML → overvalued.

Does CAPM hold up?

• Beta is not stable for individual stocks, but is stable for portfolios. The larger the portfolio (over 50 stocks) and the longer the period (over 26 weeks), the more stable the estimate.
• Betas regress toward the mean: high betas drift down toward 1.00, low betas drift up toward 1.00.
• Sharpe and Cooper found a positive risk–return link, though not completely linear.
• Black, Jensen and Scholes found a positive linear relationship between monthly excess return and portfolio beta.

Beyond CAPM: Arbitrage Pricing Theory

CAPM is a single-factor model — the market. Arbitrage Pricing Theory (APT), developed by Stephen Ross, uses several factors.

APT assumes: perfectly competitive markets; investors always prefer more wealth to less with certainty; and returns are a linear function of K factors.

APT does not assume a mean-variance-efficient market portfolio of all risky assets, or normally distributed returns.

Each asset gets several betas, one per factor — for example inflation, GNP growth, interest rate changes. Like CAPM, APT treats unique risk as diversifiable. Its main practical challenge is identifying the risk factors.

Beyond the book

The workbook says broad equity indices serve as the market-portfolio proxy. In India, analysts typically estimate a stock's beta against a broad index such as the Nifty 50 or the Sensex. Remember the workbook's warning: pick a different proxy and you get a different beta.