CAPM & Factor Models

Two Fund Separation left us with a result about portfolio construction: every investor should hold two things, a risk-free asset and the risky portfolio with the highest Sharpe ratio. That answers what to hold, not what anything is worth. If every investor follows the advice, though, the requirement that markets clear turns portfolio construction into a statement about how expected returns are determined.

That statement is the Capital Asset Pricing Model (CAPM), which connects expected return to one measure of risk. That connection explains why some stocks should earn more than others, and it gives us a benchmark for judging whether an investment is cheap or expensive. The model’s empirical fit is not particularly good, and we’ll get to that. But the CAPM’s structure is what every model after it is built on, so it is worth learning even as a stepping stone.

From Tangent Portfolio to Market Portfolio

Suppose we grant that conclusion and let every investor hold the tangent portfolio as their risky allocation. Some hold more of it (aggressive investors who borrow), some hold less (conservative investors who lend), but the risky mix itself is the same for everyone.

If every investor demands the same risky portfolio, aggregate demand for risky assets is a scaled-up copy of the tangent portfolio. Market clearing requires demand to equal supply, and the supply is every risky asset that trades, at its value weight: the market portfolio. So the tangent portfolio we picked on Sharpe-ratio grounds must be the market portfolio.

This is an equilibrium result. We do not need to know anyone’s risk preferences or how much they invest, only two assumptions: that investors follow Two Fund Separation, and that markets clear. Under those assumptions we get one object under three names. The tangent portfolio, the market portfolio, and the optimal risky portfolio are all the same thing.

In practice the assumptions do not hold perfectly, and active managers make their living on the gaps. The result explains why beating a passive index is difficult all the same, since an active manager competes against a benchmark the theory says is already optimal.

Why Diversifiable Risk Earns No Premium

Total risk splits into two parts. Systematic risk (also called market risk) is driven by broad economic forces: recessions, interest rate shifts, geopolitical shocks. It affects every asset to some degree. Idiosyncratic risk (also called firm-specific risk) is driven by events unique to one company: a product recall, a management shake-up, an earnings surprise. It is independent across firms.

In a large diversified portfolio, idiosyncratic risks cancel out. If one company has a bad quarter, another has a good one, and the net effect on the portfolio is close to zero. This is the law of large numbers applied to investing.

Suppose we hold 200 stocks in equal weight, each with a firm-specific standard deviation of $σ_{i d i o} = 20%$ , and take those firm-specific risks to be independent across firms. Independence lets us divide that 20% by the square root of the number of holdings, and we get a portfolio idiosyncratic standard deviation of $20%/ 200 \approx 1.4%$ . The firm-specific noise nearly vanishes. The systematic component of each stock’s risk is correlated across all of them, so it does not cancel. That part stays.

Since investors can eliminate idiosyncratic risk for free by diversifying, the market will not pay a premium for bearing it. Why compensate anyone for a risk that costs nothing to remove? Compensation comes only for risk nobody can escape. Systematic risk survives diversification. It affects every asset, so no amount of portfolio mixing eliminates it. That is the risk investors are paid to hold.

Beta

If only systematic risk matters for pricing, we need a measure of an asset’s exposure to it. That measure is beta:

β_{i} = \frac{Cov ( r _{i} , r _{m} )}{Var ( r _{m} )}

where $r_{i}$ and $r_{m}$ are the returns on asset $i$ and on the market portfolio.

Beta measures the sensitivity of an asset’s return to the market return. A stock with $β = 1$ moves one-for-one with the market. For a stock with $β = 1.3$ , a 10% market rise takes the stock up roughly 13%, and a 5% market fall takes it down about 6.5%. A stock with $β = 0.6$ captures about 60% of the market’s moves, and a stock with $β = 0$ has no systematic exposure at all.

Beta is about sensitivity, not about total volatility. A stock can have high total $σ$ (lots of idiosyncratic noise) but low beta (little market exposure). A biotech startup waiting on FDA approval can be both volatile and low-beta, because its outcome barely depends on whether the economy is booming or contracting. Only the beta component of risk is priced.

The CAPM and the Security Market Line

Combining the equilibrium result (the tangent portfolio is the market portfolio) with the pricing logic (only systematic risk earns a premium) gives us the Capital Asset Pricing Model:

E [r_{i}] = r_{f} + β_{i} \times (E [r_{m}] - r_{f})

The term $E [r_{m}] - r_{f}$ is the market risk premium: the market’s expected return in excess of the risk-free rate. Beta scales it. Higher beta means more systematic exposure, which means higher expected return.

Take a stock with $β = 1.3$ , a market expected return of 10%, and a risk-free rate of 2%. We compute the market risk premium as the difference between the two, 8%, and scale it by the stock’s beta:

E [r] = 2% + 1.3 \times 8% = 2% + 10.4% = 12.4%

The CAPM predicts this stock should earn 12.4%. Running the same arithmetic at $β = 0.6$ , we get $2% + 0.6 \times 8% = 6.8%$ .

We plot $E [r]$ on the vertical axis against $β$ on the horizontal, and the CAPM traces a straight line from $(0, r_{f})$ through $(1, E [r_{m}])$ . That line is the Security Market Line (SML), and in equilibrium every asset sits on it.

Drag the asset’s beta below and watch the return the CAPM predicts for it move along the line.

E[r] MARKET: 10.0% RISK-FREE RATE: 2.0% ASSET β: 1.3 ASSET E[r]: 12.4% 0% 5% 10% 15% 20% 25% 0.0 0.5 1.0 1.5 2.0 2.5 r f M Asset β E[r] β 1.3 CAPM E[r] 12.4% ACTUAL E[r] 12.4% ALPHA +0.0% Assets above the SML have positive alpha. Assets below have negative alpha.

Raising the expected market return steepens the line, and raising the risk-free rate lifts its intercept while flattening its slope. Raising or lowering the asset’s own expected return moves it off the line, opening the vertical gap that is alpha.

The Capital Allocation Line from the last post plotted $E [r]$ against $σ$ , total risk. The SML plots $E [r]$ against $β$ , systematic risk alone. The CAL is about constructing portfolios, and the SML is about pricing individual assets. Total volatility includes idiosyncratic risk, which is free to diversify away. Only the systematic component, $β$ , earns compensation.

Alpha

Alpha is the difference between an asset’s actual expected return and what the CAPM predicts:

α_{i} = E [r_{i}] - [r_{f} + β_{i} (E [r_{m}] - r_{f})]

In equilibrium, alpha is zero for every asset. Positive alpha means the asset’s expected return is higher than its systematic risk warrants, which puts it above the SML. Negative alpha means the expected return is lower than the risk warrants, which puts it below.

Take the same $β = 1.3$ stock and give it an expected return of 14% instead. The CAPM says it should earn 12.4%, so we get an alpha of $14% - 12.4% = + 1.6%$ . A second stock with the same beta and an expected return of 11% leaves an alpha of $11% - 12.4% = - 1.4%$ .

Alpha is what active managers chase. An investor who spots positive alpha before the market corrects earns excess returns, and whether anyone can do that reliably after costs is a separate (and contentious) debate. The efficient market hypothesis holds that prices already reflect all available information, which makes sustained alpha hard to come by. But alpha as a performance benchmark is universal, whichever way that debate runs.

Multi-Factor Models

The CAPM uses one factor: the market. Empirically, it does not fully explain the cross-section of returns. Small stocks tend to outperform large stocks after adjusting for market beta. Value stocks (high book-to-market ratio) tend to outperform growth stocks (low book-to-market). These patterns persist across decades and geographies.

Fama and French (1993) added two factors. SMB (Small Minus Big) is the return spread between small-cap and large-cap portfolios, which captures the size effect, and HML (High Minus Low) is the return spread between value and growth portfolios, which captures the value effect. Their three-factor model (FF3) prices exposure to all three:

E [r_{i}] = r_{f} + β_{m} \times MKT + β_{s} \times SMB + β_{h} \times HML

where each $β$ measures the asset’s sensitivity to one factor, and MKT, SMB, and HML are the premia those factors carry. A stock can load heavily on the market, lightly on size, and negatively on value.

Take a stock with $β_{m} = 1.1$ , $β_{s} = 0.4$ , and $β_{h} = - 0.2$ , against premia of 8% for MKT, 3% for SMB, and 4% for HML, and a risk-free rate of 2%. We weight each premium by its beta and add the three products to the risk-free rate:

E [r] = 2 + 1.1 \times 8 + 0.4 \times 3 + (- 0.2) \times 4 = 2 + 8.8 + 1.2 - 0.8 = 11.2%

Our stock’s expected return comes to 11.2%. It loads positively on the market and on the size factor (behaves like a small-cap stock) and slightly negatively on the value factor (behaves like a growth stock). The FF3 model attributes that return to three distinct sources of systematic risk rather than one.

Alpha generalizes the same way. Controlling for more factors can make what looked like CAPM alpha disappear: a stock that appears to beat the market on a risk-adjusted basis may be earning nothing more than fair compensation for its size and value exposure. Multi-factor alpha is the return left over after accounting for all the factors in the model, not the market alone.

Toggle between CAPM and FF3 below, then move the betas and premia to see what each factor contributes.

CAPM FF3 RISK-FREE RATE: 2.0% MKT β MKT: 1.1 PREMIUM: 8.0% SMB β SMB: 0.4 PREMIUM: 3.0% HML β HML: -0.2 PREMIUM: 4.0% RF +2.0% MKT +8.8% SMB +1.2% HML -0.8% TOTAL +11.2% 0% 2% 4% 6% 8% 10% 12% E[r] 11.2% RF 2.0% MKT +8.8% SMB +1.2% HML -0.8% Expected return decomposes into a risk-free rate plus compensation for each source of systematic risk.

Each factor’s contribution is drawn as its own bar, and the total is the risk-free rate plus the contributions shown. Switching to CAPM leaves the market as the only factor on the chart.

Arbitrage Pricing Theory

Arbitrage Pricing Theory (APT) generalizes the multi-factor approach: if idiosyncratic risk can be diversified away, expected return must be a linear function of factor betas. Beyond that structure, the theory does not say which factors to use or how many. The CAPM is APT with one factor, the market, and FF3 is APT with three. Researchers have proposed momentum, profitability, investment quality, liquidity, and others.

The procedure is the same in every case: identify the systematic factors, estimate the betas, price the risk. The open question is which factors correspond to sources of risk that investors cannot diversify away and which are statistical artifacts that happen to show up in historical data. That debate continues. The pricing logic underneath it is settled: expected return compensates for exposure to systematic factors, and nothing else.

What’s Next

We now have pricing models that say what an asset should earn, and alpha as the gap between what it earns and what it should. What we still lack is a use for that gap. Nothing so far says how much to tilt a portfolio toward an asset with positive alpha, or how to compute the risk of a portfolio holding hundreds of assets rather than two. The next post connects pricing to portfolio construction.