FNCE30001 Investments
CAPM and APT
Idiosyncratic risk vanishes in a diversified portfolio, so the market refuses to pay for it; what remains — co-movement with the market, captured by beta — is the only risk that earns a premium. The CAPM turns that into a one-line pricing equation, E[r] = r_f + β(E[r_M] − r_f), and the Security Market Line (SML) draws it: expected return rises linearly with beta, not with σ. The gap between a stock's actual expected return and its SML-required return is its alpha — positive alpha plots above the line (under-priced, a buy), negative below (over-priced, a sell), and in equilibrium every alpha is zero. Because true beta is unobservable, you estimate it by regressing the stock's excess return on the market's: the fitted single-index model / Security Characteristic Line gives the slope (beta), an intercept (alpha) and an R² that splits total variance into a systematic and a firm-specific share. Finally, multifactor models (Fama-French size and value) and the APT (a no-arbitrage, multifactor alternative with far weaker assumptions) extend the one-factor picture, and Roll's critique warns that the true market portfolio is unobservable, so every CAPM test is a joint test of the model and its proxy.
What this chapter covers
- 016.1 The CAPM pricing equation and the SML
- 026.2 Reading beta: defensive, market, aggressive
- 036.3 Alpha and the buy/sell decision (above vs below the SML)
- 048.1 The single-index model / Security Characteristic Line
- 05R² as the systematic share and the covariance shortcut
- 068.2 Fama-French multifactor models (SMB, HML)
- 078.3 APT, no-arbitrage and Roll's critique / the proxy problem
Worked example: alpha and the buy/sell call
- +1Identify. CAPM equilibrium pricing — compare each stock's actual forecast return to its SML-required return for its beta.
- +1Woodside required return: r_f + β(E[r_M] − r_f) = 4 + 1.3(6) = 11.8%.
- +1Woodside alpha: 13 − 11.8 = +1.2% → above the SML → under-priced → BUY.
- +1Wesfarmers required return: 4 + 0.7(6) = 8.2%.
- +1Wesfarmers alpha: 7.5 − 8.2 = −0.7% → below the SML → over-priced → SELL.
- +1Interpret: despite Woodside's higher raw return, the call is about return relative to beta. Woodside earns more than its systematic risk warrants (+α); Wesfarmers earns less (−α).
Key terms
- CAPM / Security Market Line
- E[r_i] = r_f + β_i(E[r_M] − r_f). The SML starts at r_f (β = 0), passes through the market (β = 1), and has slope equal to the market risk premium. Beta is systematic risk: β_M = 1, β_rf = 0, β < 1 is defensive, β > 1 is aggressive. Portfolio beta is additive: β_p = Σw_iβ_i.
- Alpha (mispricing)
- α_i = E[r_i]^actual − [r_f + β_i(E[r_M] − r_f)]. Positive alpha plots above the SML (under-priced, buy); negative below (over-priced, sell). CAPM mispricing is closed by price pressure as investors shade toward +α — not by the riskless arbitrage of the APT. In equilibrium every alpha is zero.
- Single-index model / SCL
- Regress the stock's excess return on the market's: r_i − r_f = α + β(r_M − r_f) + e. The fitted Security Characteristic Line gives the index-model beta (= CAPM beta), and total variance splits into a systematic piece β²σ_M² and a firm-specific piece σ²(e).
- R² and the covariance shortcut
- R² = β²σ_M² / (β²σ_M² + σ²(e)) is the systematic share of a stock's variance; 1 − R² is the diversifiable share. The index model also gives cov(r_i, r_j) = β_iβ_jσ_M², collapsing the covariance-matrix dimensionality problem to one beta per asset plus the market variance.
- APT and Roll's critique
- The Arbitrage Pricing Theory derives E[r_p] = r_f + Σβ_k(E[r_Fk] − r_f) from a factor structure, diversification and no-arbitrage — far weaker assumptions than the CAPM, needing only a few arbitrageurs, but silent on what the factors are. Roll's critique notes the true market portfolio is unobservable, so any CAPM test using a proxy (e.g. the ASX 200) is a joint test of the model and the proxy's efficiency.
CAPM and APT FAQ
What is alpha, and how does it give a buy or sell signal?
Alpha is actual expected return minus the CAPM-required return for the stock's beta: α = E[r] − [r_f + β(E[r_M] − r_f)]. Positive alpha means the stock plots above the SML — it offers more than its systematic risk warrants, so it is under-priced and a buy. Negative alpha is over-priced, a sell. The signal is about return relative to beta, so a stock with a high raw return can still be a sell if its beta is high enough that its required return is higher still.
What is the difference between the CML and the SML?
The SML prices every asset off its systematic risk beta; total risk σ is irrelevant on it, and every correctly-priced asset (efficient or not) lies on it. The CML relates return to total risk σ and holds only for efficient complete portfolios. Plotting a single stock against σ on a CML, or expecting an inefficient stock to lie on the CML, is the classic mix-up. Use beta with the SML; use σ with the CML.
How do I read a beta regression (the single-index model)?
The slope is the estimated beta — below 1 is defensive, above 1 aggressive. The intercept is alpha; if it is statistically insignificant there is no evidence of mispricing or skill, consistent with efficiency. R² is the systematic share of the stock's variance, so 1 − R² is the firm-specific (diversifiable) share. A low-beta, low-R² stock is volatile held alone, but most of that volatility diversifies away in a portfolio, leaving little priced systematic risk.
How does the APT differ from the CAPM?
The APT rests on a factor structure, enough assets to diversify idiosyncratic risk, and no-arbitrage — it needs only a few informed arbitrageurs, not all investors holding the market portfolio, so its assumptions are far weaker. Two well-diversified portfolios with the same factor beta must have the same expected return, or a riskless long-short arbitrage exists. The cost of that generality is that the APT is silent on what the priced factors are, whereas the CAPM names the single market factor. Fama-French add empirical size (SMB) and value (HML) factors to bridge the gap.
Exam move
Anchor on the one-line CAPM and the SML picture, then run the alpha decision automatically: compute the SML-required return for the beta, subtract from the actual, and translate the sign into above/below the line and buy/sell. Keep the CML-vs-SML distinction sharp (total σ vs systematic β) — it is a recurring conceptual trap. For the index model, be able to read a regression output: slope is beta, intercept is alpha (and whether it is significant), R² is the systematic share, and the covariance shortcut β_iβ_jσ_M² explains why the index model is used. For multifactor and APT, name SMB (size) and HML (value), state the APT's weaker no-arbitrage assumptions, and be ready to explain Roll's critique — the market proxy is unobservable, so a CAPM rejection may just mean the proxy is wrong.
