Finance-Theory · Unit 19 · Video 1 · Interactive Practice

Testing the CAPM with Portfolios: Size, Beta and a Flatter Line

IKey Formulas

FormulaNameWhat it says
E[Ri]=Rf+βi(E[Rm]−Rf)\text{E}[R_i] = R_f + \beta_i(\text{E}[R_m] - R_f)CAPM: the security market lineThe prediction for every portfolio and security; testing it means comparing realised averages with this line
R2R^2Goodness of fitThe share of a return's variation explained by the market; low for single stocks (Biogen 17.5%17.5\%, Motorola 33.4%33.4\%), and not a test of the CAPM, which concerns expected returns
E[Ri]=E[Rz]+βi(E[Rm]−E[Rz])\text{E}[R_i] = \text{E}[R_z] + \beta_i(\text{E}[R_m] - \text{E}[R_z])Black's zero-beta CAPME[Rz]\text{E}[R_z], the expected return on a risky portfolio uncorrelated with the market, takes the place of RfR_f
E[Rz]>Rf  ⇒  E[Rm]−E[Rz]<E[Rm]−Rf\text{E}[R_z] > R_f \;\Rightarrow\; \text{E}[R_m] - \text{E}[R_z] < \text{E}[R_m] - R_fA flatter lineBoth lines pass through the market at β=1\beta = 1; the zero-beta line starts higher and rises more slowly

Key Insight: One stock is too noisy to judge the model, so test it on portfolios. Size deciles from 1960 to 2000 sit close to the SML, with the smallest decile above it; beta-sorted portfolios trace a line flatter than the SML, which the zero-beta version fits.

IINoise Averages Out in a Portfolio

A portfolio averages away stock-specific noise, so the market explains far more of its variation than of one stock's.

IIISize Deciles Against the SML

The CAPM puts all ten size deciles on one line, 0.47%+β×0.49%0.47\% + \beta \times 0.49\% per month.

💡 A fit this close is the video's finding for 1960–2000; it is not typical of the finance literature.

IVBlack's Zero-Beta Line

Black's line pivots through the market point MM; raising E[Rz]\text{E}[R_z] above RfR_f makes it flatter.

💡 The points and the standard line are read off the video's chart of beta-sorted portfolios (annual returns, Rf≈3.6%R_f \approx 3.6\%, MM at 11.2%11.2\%), so the best-fitting E[Rz]\text{E}[R_z] here illustrates the idea; it is not a published estimate.

VQuiz Questions

Problem 1 · The CAPM Prediction for a Portfolio

Given: over a test period the Treasury bill rate averages Rf=0.40%R_f = 0.40\% per month and the market risk premium is E[Rm]−Rf=0.55%\text{E}[R_m] - R_f = 0.55\% per month. A size-sorted portfolio has β=1.30\beta = 1.30. Find its CAPM-predicted expected return per month.

✅ Correct! 0.40%+1.30×0.55%=0.40%+0.715%=1.115%0.40\% + 1.30 \times 0.55\% = 0.40\% + 0.715\% = 1.115\% per month: the risk-free rate plus beta times the premium.
❌ Not quite. 1.30×0.55%=0.715%1.30 \times 0.55\% = 0.715\% is only the risk premium the portfolio earns for its beta. The prediction adds it to Rf=0.40%R_f = 0.40\%.
❌ Not quite. 1.30×(0.40%+0.55%)=1.235%1.30 \times (0.40\% + 0.55\%) = 1.235\% scales the whole market return by beta. Beta multiplies only the premium; the risk-free part is earned once.
❌ Not quite. 0.95%0.95\% is the market's expected return, the prediction for β=1\beta = 1. A portfolio with β=1.30\beta = 1.30 carries more market risk and is predicted to earn more.
Show solution

The security market line, in monthly units throughout:

E[Rp]=Rf+βp(E[Rm]−Rf)=0.40%+1.30×0.55%\text{E}[R_p] = R_f + \beta_p(\text{E}[R_m] - R_f) = 0.40\% + 1.30 \times 0.55\% =0.40%+0.715%=1.115% per month= 0.40\% + 0.715\% = 1.115\% \text{ per month}

Testing the CAPM on this portfolio means comparing its realised average monthly return with 1.115%1.115\%. A realised average above it puts the portfolio above the SML, as the smallest size decile was in the video.

Problem 2 · What a Low R-Squared Means

Given: stock SS's monthly CAPM regression on the market has R2=12%R^2 = 12\%.

What does R2=12%R^2 = 12\% tell you?

An equal-weighted portfolio holds 40 stocks like SS: similar betas, stock-specific noise unrelated across stocks. Which R2R^2 is most plausible for the portfolio?

✅ Correct! R2R^2 measures fit, not the model's validity. In a portfolio the stock-specific noise largely averages out, so the market explains most of what remains, but not all of it.
❌ Not quite. R2R^2 is the share of the return's variation that the market's movements explain.
❌ Not quite. Ask what happens to unrelated stock-specific noise when 40 stocks are averaged.
❌ Not quite. The CAPM predicts expected returns and makes no claim about R2R^2. A low R2R^2 is typical of single stocks (Biogen's was 17.5%17.5\%) and does not reject the model.
❌ Not quite. R2R^2 and beta are different numbers: Biogen and Motorola had almost the same beta, about 1.41.4, with R2R^2 of 17.5%17.5\% and 33.4%33.4\%.
❌ Not quite. R2R^2 says nothing about the level of expected returns; the SML sets that from beta. R2R^2 only measures how tightly returns follow the market.
❌ Not quite. R2R^2 is not fixed by the stocks: averaging 40 stocks shrinks the unrelated stock-specific noise while the common market movement stays, so the market explains far more.
❌ Not quite. Adding stocks with unrelated noise does not add noise to the portfolio; the noise partly cancels, so R2R^2 rises.
❌ Not quite. Forty stocks shrink the noise but do not remove it, and the market is not the only common factor. A portfolio's R2R^2 is high, never exactly 100%100\%.
Show solution

Part 1. R2R^2 is the fraction of the variation in SS's return that the market's movements explain: 12%12\%. The remaining 88%88\% is stock-specific (idiosyncratic) noise. The CAPM is a statement about expected returns, not about fit, so a low R2R^2 does not reject it; with returns this noisy, a regression on one stock cannot test it sharply either.

Part 2. Each stock's return is a common part (driven by the market) plus its own noise. Averaging 40 stocks keeps the common part, since every stock shares it, while the unrelated noise terms largely cancel. The market therefore explains most of the portfolio's variation: 85%85\% is plausible. It cannot stay at 12%12\% or fall, and it cannot reach 100%100\% because 40 stocks leave some noise and other common factors.

This is why the video tests the CAPM on portfolios rather than single stocks.

Problem 3 · Standard versus Zero-Beta Prediction

Given: monthly Rf=0.40%R_f = 0.40\% and market risk premium 0.55%0.55\%, so E[Rm]=0.95%\text{E}[R_m] = 0.95\%. Estimated from the data, a zero-beta portfolio's expected return is E[Rz]=0.70%\text{E}[R_z] = 0.70\% per month. A beta-sorted portfolio has β=1.60\beta = 1.60.

Standard CAPM prediction, per month

Zero-beta CAPM prediction, per month

✅ Correct! Standard: 0.40%+1.60×0.55%=1.28%0.40\% + 1.60 \times 0.55\% = 1.28\%. Zero-beta: 0.70%+1.60×0.25%=1.10%0.70\% + 1.60 \times 0.25\% = 1.10\%. For a high-beta portfolio the flatter line predicts less.
❌ Check the standard prediction. E[Rp]=Rf+βp(E[Rm]−Rf)\text{E}[R_p] = R_f + \beta_p(\text{E}[R_m] - R_f).
❌ Check the zero-beta prediction. E[Rp]=E[Rz]+βp(E[Rm]−E[Rz])\text{E}[R_p] = \text{E}[R_z] + \beta_p(\text{E}[R_m] - \text{E}[R_z]).
❌ Not quite. 1.60×0.55%=0.88%1.60 \times 0.55\% = 0.88\% is the premium for the portfolio's beta; add Rf=0.40%R_f = 0.40\%.
❌ Not quite. 1.60×0.95%=1.52%1.60 \times 0.95\% = 1.52\% scales the whole market return by beta. Beta multiplies only the premium over RfR_f.
❌ Not quite. 0.95%0.95\% is the market's expected return, the prediction for β=1\beta = 1 only.
❌ Not quite. 0.70%+1.60×0.55%0.70\% + 1.60 \times 0.55\% swaps the intercept but keeps the old premium. In Black's model the premium becomes E[Rm]−E[Rz]=0.95%−0.70%=0.25%\text{E}[R_m] - \text{E}[R_z] = 0.95\% - 0.70\% = 0.25\%.
❌ Not quite. 0.40%+1.60×0.25%0.40\% + 1.60 \times 0.25\% uses the new premium but the old intercept. In Black's model E[Rz]\text{E}[R_z] replaces RfR_f in both places.
❌ Not quite. 0.70%+0.25%0.70\% + 0.25\% adds the premium once, which is the answer for β=1\beta = 1. Scale the premium by β=1.60\beta = 1.60.
Show solution

Standard CAPM:

E[Rp]=0.40%+1.60×0.55%=0.40%+0.88%=1.28%\text{E}[R_p] = 0.40\% + 1.60 \times 0.55\% = 0.40\% + 0.88\% = 1.28\%

Zero-beta CAPM: the premium is measured over E[Rz]\text{E}[R_z], not RfR_f:

E[Rm]−E[Rz]=0.95%−0.70%=0.25%\text{E}[R_m] - \text{E}[R_z] = 0.95\% - 0.70\% = 0.25\% E[Rp]=0.70%+1.60×0.25%=0.70%+0.40%=1.10%\text{E}[R_p] = 0.70\% + 1.60 \times 0.25\% = 0.70\% + 0.40\% = 1.10\%

Because E[Rz]>Rf\text{E}[R_z] > R_f, the zero-beta line starts higher and has a smaller slope (0.25%0.25\% against 0.55%0.55\% per unit of beta). Both lines give 0.95%0.95\% at β=1\beta = 1; above β=1\beta = 1 the zero-beta line predicts less, which is how it fits high-beta portfolios that earned less than the standard SML.

Problem 4 · Alphas Along a Flatter Line

Given: in some sample, beta-sorted portfolios lie exactly on Black's zero-beta line with E[Rz]=0.70%\text{E}[R_z] = 0.70\% per month, while Rf=0.40%R_f = 0.40\% and E[Rm]=0.95%\text{E}[R_m] = 0.95\% per month. Find each portfolio's alpha measured against the standard SML, in percentage points per month.

Portfolio with β=0.75\beta = 0.75

Portfolio with β=1.10\beta = 1.10

✅ Correct! Alpha against the standard SML is (E[Rz]−Rf)(1−β)=0.30%×(1−β)(\text{E}[R_z] - R_f)(1 - \beta) = 0.30\% \times (1 - \beta): positive below β=1\beta = 1, negative above it. That is the flatter-line pattern: low-beta portfolios beat the SML, high-beta portfolios fall short.
❌ Check the β=0.75\beta = 0.75 alpha. Alpha = return on the zero-beta line minus the standard SML prediction at the same beta.
❌ Check the β=1.10\beta = 1.10 alpha. Alpha = return on the zero-beta line minus the standard SML prediction at the same beta.
❌ Sign error. The zero-beta line gives 0.8875%0.8875\% and the SML 0.8125%0.8125\%; the realised return is above the SML, so alpha is positive.
❌ Not quite. The two lines meet only at β=1\beta = 1. At β=0.75\beta = 0.75 the zero-beta line is above the SML.
❌ Not quite. 0.30×0.750.30 \times 0.75 scales the intercept gap by β\beta. The gap is 0.300.30 at β=0\beta = 0 and shrinks to 00 at β=1\beta = 1, so it scales by 1−β1 - \beta.
❌ Sign error. Above β=1\beta = 1 the flatter zero-beta line lies below the SML: 0.975%0.975\% against 1.005%1.005\%.
❌ Not quite. β=1.10\beta = 1.10 is close to 11 but not equal; the lines cross only at the market, β=1\beta = 1.
❌ Not quite. 0.30×1.100.30 \times 1.10 scales the intercept gap by β\beta and gets the sign wrong. The gap scales by 1−β=−0.101 - \beta = -0.10.
Show solution

Zero-beta premium: E[Rm]−E[Rz]=0.95%−0.70%=0.25%\text{E}[R_m] - \text{E}[R_z] = 0.95\% - 0.70\% = 0.25\%. Standard premium: 0.55%0.55\%.

β=0.75\beta = 0.75:

zero-beta line: 0.70%+0.75×0.25%=0.8875%\text{zero-beta line: } 0.70\% + 0.75 \times 0.25\% = 0.8875\% SML: 0.40%+0.75×0.55%=0.8125%\text{SML: } 0.40\% + 0.75 \times 0.55\% = 0.8125\% α=0.8875%−0.8125%=+0.075 points\alpha = 0.8875\% - 0.8125\% = +0.075 \text{ points}

β=1.10\beta = 1.10:

zero-beta line: 0.70%+1.10×0.25%=0.975%\text{zero-beta line: } 0.70\% + 1.10 \times 0.25\% = 0.975\% SML: 0.40%+1.10×0.55%=1.005%\text{SML: } 0.40\% + 1.10 \times 0.55\% = 1.005\% α=0.975%−1.005%=−0.03 points\alpha = 0.975\% - 1.005\% = -0.03 \text{ points}

In general, subtracting the two lines gives α=(E[Rz]−Rf)(1−β)=0.30%×(1−β)\alpha = (\text{E}[R_z] - R_f)(1 - \beta) = 0.30\% \times (1 - \beta). If the zero-beta model is right, a test against the standard SML finds positive alphas for low-beta portfolios and negative ones for high-beta portfolios, the flatter line the video found.

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