> For the complete documentation index, see [llms.txt](https://laurence-wilse-samson.gitbook.io/textbooks/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://laurence-wilse-samson.gitbook.io/textbooks/financial-economics-claims-prices-holders/part-ii-asset-pricing/chapter_04_portfolio_choice_capm.md). # Chapter 4: Portfolio Choice and the CAPM *Part II: Asset Pricing — Financial Economics: Claims, Prices, and Holders* *** ## Opening Episode: The Inventor's Own Portfolio Harry Markowitz published "Portfolio Selection" in the *Journal of Finance* in March 1952. It runs to fourteen pages, contains no data, and changed what the word "risk" means in finance. Before it, a good portfolio was a collection of good securities, chosen one at a time, and diversification was a folk maxim about eggs and baskets. After it, a portfolio was a single object with two numbers attached — an expected return and a variance — and the contribution of any security to the portfolio's risk depended not on that security's own variance but on how it covaried with everything else the investor held. The apparatus that follows from those two numbers is the subject of this chapter, and it earned Markowitz a share of the 1990 Nobel Prize. Some years later, a journalist asked him how he had invested his own retirement money. Markowitz was, at the time of the decision, an employee of the RAND Corporation with a TIAA-CREF account and a choice between a stock fund and a bond fund. As he described it afterwards to Jason Zweig, who reported the exchange, he did not run the optimization. He thought instead about how he would feel in the two outcomes he most wanted to avoid: being out of the stock market while it rose sharply, and being in it while it fell sharply. Splitting his contributions evenly between the two funds made both regrets equally survivable. He put half in stocks and half in bonds. The story is usually told as an embarrassment, and it is not quite one. Markowitz has since given a fuller account: the allocation was made early in his career, from a short menu, at a time when the computational apparatus his own paper required did not exist in any usable form; he revised it later. He has also made the defensive point that matters most here, which is that a 50/50 split is not a stupid portfolio. For an investor with a moderate tolerance for risk, it sits close to the part of the frontier the theory would have recommended anyway. The optimizer's answer and the regret-minimizer's answer were, in this instance, nearly the same answer. Keep both halves of that. The first half is why this chapter exists: the mean-variance apparatus is the most useful single piece of machinery in finance, it generates the first complete equilibrium theory of asset prices, and it is still what a pension trustee, an index provider, and a corporate treasurer are implicitly using when they argue about a discount rate. The second half is why the book does not stop there. The theory says what an unconstrained investor with well-defined beliefs should hold. Actual holders have menus, mandates, regrets, borrowing limits, and committees. Sections 4.1 through 4.8 develop the theory in full. Section 4.9 asks what happens to prices when a large fraction of the money in the market belongs to holders who *cannot* do what the theory tells them to do — and finds that the answer is visible in the data, in exactly the place the theory makes its sharpest prediction. *** ## 4.1 Mean, Variance, and the Arithmetic of Diversification An investor allocates wealth across $$N$$ risky assets. Write $$w = (w\_1, \dots, w\_N)'$$ for the vector of portfolio weights, with $$\sum\_i w\_i = 1$$; a negative weight is a short position. Write $$\mu$$ for the vector of expected net returns and $$\Sigma$$ for the covariance matrix, with entries $$\sigma\_{ij} = \mathrm{Cov}(r\_i, r\_j) = \rho\_{ij}\sigma\_i\sigma\_j$$ and $$\sigma\_{ii} = \sigma\_i^2$$. (Mean-variance algebra is written in net returns throughout, per the notation registry; §4.6 translates back into the gross-return language of Chapter 3.) The portfolio's return is $$r\_p = \sum\_i w\_i r\_i$$, so its expected return and variance are $$ \mu\_p = w'\mu = \sum\_i w\_i \mu\_i $$ $$ \sigma\_p^2 = w'\Sigma w = \sum\_i \sum\_j w\_i w\_j \sigma\_{ij} $$ The first line is unremarkable: expected return is linear in weights, so a portfolio's expected return is the weighted average of its holdings' expected returns and there is nothing to optimize. The second line is where the subject is. Variance is a *quadratic* form, and it is not the weighted average of the components' variances. Cross terms appear, and there are $$N(N-1)$$ of them against $$N$$ own-variance terms. Once $$N$$ is large, the portfolio's risk is almost entirely made of covariances. Make that precise. Suppose all $$N$$ assets have the same variance $$\bar\sigma^2$$ and every pair has the same covariance $$\bar c$$, and hold them in equal weights $$w\_i = 1/N$$. Then $$ \sigma\_p^2 = \frac{1}{N}\bar\sigma^2 + \left(1 - \frac{1}{N}\right)\bar c $$ The first term is the own-variance contribution, and it vanishes as $$N$$ grows. The second term does not: it converges to $$\bar c$$, the average covariance. As $$N \to \infty$$, $$ \sigma\_p^2 \longrightarrow \bar c = \bar\rho\bar\sigma^2 $$ **Diversification eliminates own variance and leaves covariance.** That sentence is the whole of Chapter 4's economics, and everything after it — the frontier, the CAPM, the beta — is bookkeeping around it. It is also the portfolio-choice statement of the result Chapter 3 §3.5 obtained from the pricing equation, that only $$\mathrm{Cov}(m, R)$$ is priced. An investor who can diversify will not pay to avoid a risk that diversification removes, so no risk premium can attach to it in equilibrium. Table 4.1 puts numbers on the limit. Take individual stocks with an annual standard deviation of 30 percent ($$\bar\sigma^2 = 0.09$$) and an average pairwise correlation of 0.20, so $$\bar c = 0.018$$. **Table 4.1: Portfolio risk as the number of holdings grows** | Number of stocks $$N$$ | Portfolio variance | Portfolio standard deviation | | ---------------------- | ------------------ | ---------------------------- | | 1 | 0.0900 | 30.0% | | 2 | 0.0540 | 23.2% | | 5 | 0.0324 | 18.0% | | 10 | 0.0252 | 15.9% | | 20 | 0.0216 | 14.7% | | 50 | 0.0194 | 13.9% | | 100 | 0.0187 | 13.7% | | $$\infty$$ | 0.0180 | 13.4% | *Source: Author's calculation from the diversification formula above, with average volatility 0.30 and average pairwise correlation 0.20.* Two readings. First, four-fifths of the variance of a single stock is diversifiable: 0.09 falls to 0.018. Second, the fall is fast and then stops. Twenty holdings capture most of the available reduction, and the last eighty of a hundred holdings buy almost nothing. This is why the practical advice "own a few dozen names" survives, and it is also why the advice is dangerous when stated without the correlation assumption. Push $$\bar\rho$$ to 0.5 and the limiting standard deviation is 21 percent rather than 13.4; in a crisis, when correlations rise toward one, the diversification a portfolio thought it owned is exactly what it discovers it does not have. Chapter 16 §16.5 is about what happens next. Figure 4.1 puts the table on a curve and adds the two cases the paragraph above only names. ![Figure 4.1: Diversification and its floor](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-7cff4064cb69b000b108f04c17da82b42f5f2757%2Ffig_04_01_diversification_floor.png?alt=media) **Figure 4.1: Diversification and its floor.** Portfolio standard deviation against the number of equally weighted holdings, on a logarithmic horizontal axis, at three average pairwise correlations. The middle curve is Table 4.1, and the markers are the table's own rows; the dashed line under it is the floor at 13.4 percent, which is the square root of the average covariance and which no number of holdings goes below. Two things the table cannot show are visible here. First, the shape: the fall is steep to about twenty names and then nearly flat, so the marker at twenty is already 92 percent of the whole reduction available, and the last eighty of a hundred holdings buy the remaining eight percent. Second, the correlation assumption is doing all the work. At an average correlation of 0.5 the floor is 21.2 percent rather than 13.4, and the curve reaches it just as quickly — the diversification is not slower to arrive, there is simply less of it. The bottom curve is the case in which the assets are uncorrelated, where the floor is zero and risk falls forever at the rate of one over the square root of N. That curve is the one the phrase "diversify away your risk" describes, and it is the only one of the three that is not a description of any equity market. The decomposition also names the two kinds of risk that the rest of Part II uses. **Idiosyncratic** (diversifiable, unsystematic) risk is the part that the $$1/N$$ term removes. **Systematic** (undiversifiable, market) risk is what is left. Nothing yet says that systematic risk earns a premium — that requires an equilibrium argument, and §4.5 supplies it — but the possibility is now visible. *** ## 4.2 The Efficient Frontier An investor who cares only about the mean and variance of terminal wealth will never hold a portfolio when another portfolio offers the same expected return at lower variance. The set of portfolios that survive this test is the **efficient frontier**: for each attainable $$\mu\_p$$, the minimum-variance portfolio delivering it, restricted to the upper half where higher risk buys higher return. Formally, the frontier solves $$ \min\_w w'\Sigma w \quad \text{subject to} \quad w'\mu = \mu\_p, \quad w'\mathbf{1} = 1 $$ Two facts about the solution matter more than the algebra of it. First, the frontier in $$(\sigma\_p, \mu\_p)$$ space is a hyperbola, opening rightward from a leftmost point — the **global minimum-variance portfolio**, the lowest-variance combination available at any expected return. Second, the frontier is generated by any two portfolios on it: if $$w^{(1)}$$ and $$w^{(2)}$$ are both frontier portfolios, so is $$a w^{(1)} + (1-a) w^{(2)}$$ for any scalar $$a$$. This is **two-fund separation** in its version without a riskless asset, and it is the reason the theory has a chance of describing an economy rather than an individual. Investors with different risk tolerances hold different portfolios, but all of them can be built from the same two. Why is the frontier curved rather than straight? Because of the covariance term. Combine two assets with correlation $$\rho\_{12} < 1$$ and the portfolio's standard deviation is strictly less than the weighted average of theirs: $$ \sigma\_p^2 = w\_1^2\sigma\_1^2 + w\_2^2\sigma\_2^2 + 2w\_1w\_2\rho\_{12}\sigma\_1\sigma\_2 $$ At $$\rho\_{12} = 1$$ this collapses to $$\sigma\_p = w\_1\sigma\_1 + w\_2\sigma\_2$$, a straight line, and there is no diversification benefit at all. Every unit of curvature in the frontier is a unit of correlation below one. Diversification, in this picture, *is* the bulge. One consequence deserves emphasis because students consistently get it backwards. A high-variance asset can improve a portfolio. Adding an asset with a large $$\sigma\_i$$ but a low correlation with what you already hold reduces $$\sigma\_p$$, even though it raises the average variance of the holdings. The question a mean-variance investor asks of a candidate security is never "how risky is it?" but "how does it covary with what I own?". Commodities, emerging-market equity, and volatility strategies are all sold on this argument, and the argument is correct in the theory; whether the correlations that justify it survive the moments when they are needed is an empirical matter, and §4.9 and Chapter 16 are unkind about it. *** ## 4.3 A Riskless Asset, Two-Fund Separation, and the Tangency Portfolio Now add a riskless asset with net return $$r\_f$$, in unlimited supply, available for both lending and borrowing at the same rate. This assumption does more work than any other in the chapter, and §4.9 is what happens when it fails. Put a fraction $$\theta$$ of wealth in a risky portfolio $$w$$ and $$1-\theta$$ in the riskless asset. The combination has $$ \mu\_p = r\_f + \theta(\mu\_w - r\_f), \qquad \sigma\_p = \theta\sigma\_w $$ Eliminate $$\theta$$: $$ \mu\_p = r\_f + \frac{\mu\_w - r\_f}{\sigma\_w}\sigma\_p $$ This is a straight line in $$(\sigma\_p, \mu\_p)$$ space, starting at $$r\_f$$ on the vertical axis and running through the risky portfolio $$w$$. Its slope is the **Sharpe ratio** of $$w$$, $$ \mathrm{SR}\_w = \frac{\mu\_w - r\_f}{\sigma\_w} $$ the risk premium per unit of standard deviation. Every investor prefers a steeper line to a flatter one, because a steeper line offers more expected return at every level of risk. So every investor wants the risky portfolio with the highest Sharpe ratio — the one whose line is tangent to the efficient frontier. That portfolio is the **tangency portfolio**, and its weights are $$ w^T \propto \Sigma^{-1}(\mu - r\_f\mathbf{1}), \qquad \text{normalized so that } \sum\_i w^T\_i = 1 $$ The formula is worth reading rather than memorizing. $$\mu - r\_f\mathbf{1}$$ is the vector of risk premia — the reward. $$\Sigma^{-1}$$ is the inverse covariance matrix — the risk, inverted. Optimal weights are rewards divided by risks, in the matrix sense, which means an asset earns weight for a high premium, for low variance, and for *low covariance with everything else*. It is the third channel that distinguishes portfolio theory from stock-picking. The line from $$r\_f$$ through $$w^T$$ is the **capital allocation line** (or, once §4.5 makes $$w^T$$ the market, the capital market line). It dominates the entire risky frontier except at the single point of tangency. This is the sharp form of **two-fund separation**: > **Every mean-variance investor holds only two things: the riskless asset and the tangency portfolio. Risk tolerance determines the split between them and nothing else.** A conservative investor lends and holds a small position in $$w^T$$; an aggressive one borrows and holds a levered position in $$w^T$$. Neither changes the *composition* of the risky part. The separation of the risk-tolerance decision from the portfolio-composition decision is the theoretical foundation of the entire index-fund industry: if the theorem is right, one risky product serves everybody, and the only thing that needs customizing is how much of it you own. Chapter 17 tells that story as an institutional history; it began as this line of algebra. Figure 4.2 draws the whole construction — the hyperbola of §4.2, the line from $$r\_f$$, and the single point where they touch — in the economy Table 4.2 sets out next. ![Figure 4.2: The efficient frontier, the capital allocation line, and the tangency portfolio](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-de624582716e3320343c01df1949b2bb680aebde%2Ffig_04_02_efficient_frontier.png?alt=media) **Figure 4.2: The efficient frontier, the capital allocation line, and the tangency portfolio.** Sections 4.2-4.3's central picture, which the text currently describes in words ("a hyperbola, opening rightward"; "a straight line, starting at $$r\_f$$"): the frontier, the global minimum-variance point, the capital allocation line, and the tangency point, all labelled. *Source: Author's construction from the worked example of Table 4.2.* *** ## 4.4 A Fully Worked Example Two risky assets and a riskless asset. All numbers are annual net returns. **Table 4.2: The example economy** | Asset | Expected return $$\mu\_i$$ | Standard deviation $$\sigma\_i$$ | | ---------- | -------------------------- | -------------------------------- | | A (equity) | 10% | 20% | | B (bonds) | 5% | 10% | | Riskless | 2% | 0 | *Correlation between A and B: zero.* *Source: Author's construction.* The zero correlation is a simplification chosen to keep the arithmetic transparent; nothing in the argument depends on it, and Problem 1 redoes the whole example with $$\rho = 0.25$$. **Step 1: the covariance matrix and the risk premia.** $$ \Sigma = \begin{pmatrix} 0.04 & 0 \cr 0 & 0.01 \end{pmatrix}, \qquad \mu - r\_f\mathbf{1} = \begin{pmatrix} 0.08 \cr 0.03 \end{pmatrix} $$ **Step 2: the tangency portfolio.** Because $$\Sigma$$ is diagonal, $$\Sigma^{-1}$$ is diagonal with entries $$1/0.04 = 25$$ and $$1/0.01 = 100$$. So $$ \Sigma^{-1}(\mu - r\_f\mathbf{1}) = \begin{pmatrix} 25 \times 0.08 \cr 100 \times 0.03 \end{pmatrix} = \begin{pmatrix} 2 \cr 3 \end{pmatrix} $$ These sum to 5. Normalizing, $$ w^T = (0.40, 0.60) $$ Forty percent in equity, sixty percent in bonds. Note what produced that split. Equity has the larger risk premium by far — 8 points against 3 — and still gets the smaller weight, because its variance is four times as large. In the diagonal case the tangency weight of each asset is proportional to $$(\mu\_i - r\_f)/\sigma\_i^2$$, premium per unit of variance, and bonds win that contest 3 to 2. **Step 3: the tangency portfolio's mean, variance, and Sharpe ratio.** $$ \mu\_T = 0.40(0.10) + 0.60(0.05) = 0.07 $$ $$ \sigma\_T^2 = 0.40^2(0.04) + 0.60^2(0.01) = 0.0064 + 0.0036 = 0.01, \qquad \sigma\_T = 0.10 $$ $$ \mathrm{SR}\_T = \frac{0.07 - 0.02}{0.10} = 0.50 $$ A portfolio with a 7 percent expected return and a 10 percent standard deviation, built out of one asset with a 20 percent standard deviation and one with a 10 percent standard deviation. The combination is *less* volatile than the safer of its two ingredients. That is the bulge of §4.2 in one number, and it is the payoff to $$\rho\_{AB} = 0$$. **Step 4: compare to the alternatives.** Table 4.3 walks along the risky frontier. **Table 4.3: Combinations of A and B, no riskless asset** | Weight in A | $$\mu\_p$$ | $$\sigma\_p$$ | Sharpe ratio | | ----------- | ---------- | ------------- | ------------ | | 0.00 | 5.00% | 10.00% | 0.300 | | 0.20 | 6.00% | 8.94% | 0.447 | | **0.40** | **7.00%** | **10.00%** | **0.500** | | 0.50 | 7.50% | 11.18% | 0.492 | | 0.60 | 8.00% | 12.65% | 0.474 | | 0.80 | 9.00% | 16.12% | 0.434 | | 1.00 | 10.00% | 20.00% | 0.400 | *Source: Author's calculation from Table 4.2.* Three things to read off it. The global minimum-variance portfolio is at $$w\_A = 0.20$$, with $$\sigma\_p = 8.94$$ percent — and it is *not* the tangency portfolio, because it gives up too much expected return for the variance it saves; its Sharpe ratio is 0.447 against the tangency portfolio's 0.500. Minimum variance and maximum Sharpe are different objectives and they pick different portfolios. Second, the 40/60 row has exactly the same standard deviation as holding bonds alone and two extra points of expected return: bonds alone is inefficient, dominated by a portfolio containing the riskier asset. Third, the Sharpe ratio is flat near its maximum — 0.492 at 50/50 against 0.500 at 40/60 — which is the mathematical fact behind the opening episode. Being somewhat wrong about the optimal weights costs very little. Being wrong about $$\mu$$ and $$\Sigma$$, which are estimated with enormous error, can cost a great deal more, and Chapter 6 returns to that. **Step 5: two-fund separation.** Any investor in this economy holds the 40/60 portfolio and adjusts the riskless share $$1-\theta$$. **Table 4.4: Points on the capital allocation line** | Share $$\theta$$ in tangency portfolio | Riskless share | $$\mu\_p$$ | $$\sigma\_p$$ | | -------------------------------------- | ---------------- | ---------- | ------------- | | 0.5 | 0.5 (lending) | 4.5% | 5% | | 1.0 | 0 | 7.0% | 10% | | 1.5 | −0.5 (borrowing) | 9.5% | 15% | | 2.0 | −1.0 (borrowing) | 12.0% | 20% | *Source: Author's calculation. Every row has Sharpe ratio 0.50.* Compare the last row of Table 4.4 with the last row of Table 4.3. Both have a 20 percent standard deviation. Holding equity alone delivers 10 percent expected return; borrowing an amount equal to your wealth and putting twice your wealth into the 40/60 portfolio delivers 12 percent. The two extra percentage points are the price of the leverage constraint, for anyone who faces one. Hold that number — §4.9 is about the investors who cannot execute the $$\theta = 2$$ row and must therefore settle for something like the first. > **Box 4.1 — The frontier you can estimate is not the frontier you drew** > > Every input above was handed to you. In practice $$\mu$$ and $$\Sigma$$ are estimated from a sample, and they are not estimated equally badly. With $$N$$ assets there are $$N$$ means and $$N(N+1)/2$$ covariances to fit, but the means are the hard ones: the precision of an estimated mean depends on the *span* of the sample and not on its frequency, so sampling daily rather than monthly buys twenty times the observations and essentially nothing about $$\mu$$. > > The optimizer then does the worst possible thing with that error. It reads a high estimated mean as an opportunity and a low estimated covariance as a diversification gain, so it loads on exactly the assets whose inputs are most overstated; the machinery maximizes estimation error as efficiently as it maximizes the Sharpe ratio. The symptom is familiar to anyone who has run one — enormous offsetting long and short positions that reverse when the sample is extended by a year. > > The standard responses amount to trusting the data less: shrink the inputs toward a prior, such as a common mean or a single-factor covariance matrix, or constrain the output, since a no-short-sale constraint is itself a form of shrinkage. Table 4.3 supplies the consolation. Because the Sharpe ratio is nearly flat near its maximum, a deliberately crude portfolio gives up very little of what a perfectly estimated one would deliver — and much less than a precisely optimized portfolio built on imprecise inputs. Chapter 6 §6.5 meets the same problem in the cross-section and arrives at the same answer. *** ## 4.5 From Portfolio Choice to Equilibrium: The CAPM Everything so far is a theory of one investor's decision, taking $$\mu$$, $$\Sigma$$, and $$r\_f$$ as given. The Capital Asset Pricing Model, developed independently by Sharpe (1964), Lintner (1965), and Mossin (1966), turns it into a theory of prices by adding two assumptions and imposing market clearing. **Assumption 1: everyone is a mean-variance investor** facing the same $$r\_f$$ for both borrowing and lending, and free of constraints on short sales or position sizes. **Assumption 2: everyone agrees** on $$\mu$$ and $$\Sigma$$ — homogeneous expectations. Now run the argument. By §4.3, every investor holds the riskless asset plus the tangency portfolio. By Assumption 2 they all compute the *same* tangency portfolio $$w^T$$. So every investor's risky holdings have identical proportions. But the risky assets in existence must be held by somebody, and if every holder's risky portfolio has the same composition, that composition must be the composition of the total supply. Aggregate risky supply, weighted by market value, is the **market portfolio**. Therefore $$ w^T = w^M $$ **The tangency portfolio is the market portfolio.** Everything else is a consequence. The riskless asset is in zero net supply (every dollar borrowed is a dollar lent), so in aggregate investors hold exactly the market portfolio and the capital allocation line becomes the **capital market line**. Now derive the pricing relation. Consider an investor holding the market portfolio who contemplates shifting a marginal weight $$\varepsilon$$ into asset $$i$$, financed by selling the market. The resulting portfolio has return $$r\_p = \varepsilon r\_i + (1-\varepsilon)r\_M$$, and $$ \mu\_p = \varepsilon\mu\_i + (1-\varepsilon)\mu\_M, \qquad \sigma\_p^2 = \varepsilon^2\sigma\_i^2 + (1-\varepsilon)^2\sigma\_M^2 + 2\varepsilon(1-\varepsilon)\sigma\_{iM} $$ Differentiate both at $$\varepsilon = 0$$ — the point where the investor is already holding the market and is deciding whether to tilt: $$ \left.\frac{d\mu\_p}{d\varepsilon}\right|\_{0} = \mu\_i - \mu\_M, \qquad \left.\frac{d\sigma\_p}{d\varepsilon}\right| \_{0} = \frac{\sigma \_{iM} - \sigma\_M^2}{\sigma\_M} $$ The ratio of these is the marginal rate at which the tilt trades risk for return. In equilibrium the market portfolio is the tangency portfolio, so no tilt improves the Sharpe ratio, and this marginal rate must equal the slope of the capital market line, $$(\mu\_M - r\_f)/\sigma\_M$$: $$ \frac{\mu\_i - \mu\_M}{(\sigma\_{iM} - \sigma\_M^2)/\sigma\_M} = \frac{\mu\_M - r\_f}{\sigma\_M} $$ Cross-multiply, cancel $$\sigma\_M$$, and collect terms in $$\mu\_i$$. The result is the **security market line**: $$ \boxed{\mu\_i - r\_f = \beta\_i(\mu\_M - r\_f), \qquad \beta\_i = \frac{\mathrm{Cov}(r\_i, r\_M)}{\mathrm{Var}(r\_M)}} $$ Read it slowly, because four separate claims are packed into it. **Only covariance with the market is priced.** An asset's own variance appears nowhere. A wildly volatile asset with $$\beta\_i = 0$$ has an expected return of exactly $$r\_f$$, no matter how frightening its price chart. This is the equilibrium payoff to §4.1's arithmetic: diversifiable risk is free to bear, so nobody is paid to bear it. It is also, precisely, Chapter 3 §3.5's conclusion that $$E\[R] - R\_f = -R\_f\mathrm{Cov}(m, R)$$, with the market return standing in for $$m$$. **The relation is linear in** $$\beta$$**, with a common price of risk.** The slope $$\lambda\_M = \mu\_M - r\_f$$ is the same for every asset. Assets differ only in quantity of risk, never in its price. This is what makes the model usable: estimate one number for the market and one $$\beta$$ per asset. **Beta is a regression coefficient, and betas aggregate.** $$\beta\_i$$ is exactly the slope from regressing $$r\_i$$ on $$r\_M$$, so it is estimable from returns data. And because covariance is linear, a portfolio's beta is the weighted average of its holdings' betas, $$\beta\_p = \sum\_i w\_i\beta\_i$$, with the market's own beta equal to one. **Everything is measured relative to the market.** The model has no free-standing notion of a risky asset. Risk is a *relation* between an asset and the aggregate portfolio, which is why it can be a theory of equilibrium at all. Return to §4.4's numbers, where the CAPM's arithmetic is exact by construction. With $$w^M = w^T = (0.40, 0.60)$$: $$ \mathrm{Cov}(r\_A, r\_M) = 0.40 \times 0.04 = 0.016, \qquad \mathrm{Cov}(r\_B, r\_M) = 0.60 \times 0.01 = 0.006 $$ $$ \beta\_A = \frac{0.016}{0.01} = 1.6, \qquad \beta\_B = \frac{0.006}{0.01} = 0.6 $$ Check that they aggregate: $$0.40(1.6) + 0.60(0.6) = 0.64 + 0.36 = 1.00$$. Now price them off the security market line, with $$\mu\_M - r\_f = 0.05$$: $$ \mu\_A = 0.02 + 1.6(0.05) = 0.10, \qquad \mu\_B = 0.02 + 0.6(0.05) = 0.05 $$ Exactly the expected returns we started from in Table 4.2. That is not a coincidence and it is not a test of anything: the tangency portfolio was constructed from those means, so recovering them is an internal consistency check on the algebra. It does, however, make one point worth making explicitly. Asset A has twice the standard deviation of asset B but only 2.67 times the beta and 2.67 times the risk premium. In a two-asset world with zero correlation, beta and volatility happen to rank assets the same way. In a world with many assets they do not, and the whole content of the CAPM is that the ranking that matters is the beta one. *** ## 4.6 ★ The CAPM as an SDF Restriction *Starred. This section restates §4.5 in the language of Chapter 3 §3.5 and shows that the two derivations are the same result. A reader who skips it loses a connection, not an argument.* Chapter 3 §3.6 previewed the claim: the CAPM is the restriction that the stochastic discount factor is **linear in the market return**. Here it is, worked. Switch to gross returns, per the notation registry's convention 1. Posit $$ m = a - bR\_M, \qquad b > 0 $$ and impose the two conditions that Chapter 3 §3.5 requires of any SDF: it must price the riskless asset, $$E\[m] = 1/R\_f$$, and it must price the market, $$E\[m R\_M] = 1$$. Two equations, two unknowns. Solving them gives $$ b = \frac{E\[R\_M] - R\_f}{R\_f\mathrm{Var}(R\_M)}, \qquad a = \frac{1}{R\_f} + bE\[R\_M] $$ Now take any asset $$i$$ and apply Chapter 3 §3.5's beta representation, $$E\[R\_i] - R\_f = -R\_f\mathrm{Cov}(m, R\_i)$$. Since $$\mathrm{Cov}(m, R\_i) = -b\mathrm{Cov}(R\_M, R\_i)$$, $$ E\[R\_i] - R\_f = R\_fb\mathrm{Cov}(R\_M, R\_i) = \frac{\mathrm{Cov}(R\_M, R\_i)}{\mathrm{Var}(R\_M)}\big(E\[R\_M] - R\_f\big) = \beta\_i\big(E\[R\_M] - R\_f\big) $$ The security market line, from the pricing equation, with no mention of portfolios, frontiers, or investors' preferences. The mean-variance derivation of §4.5 and the SDF derivation here are the same theorem in two notations, exactly as Chapter 3 §3.4's three routes to the option price were one argument in three notations. Check the constants against §4.4. There $$E\[R\_M] = 1.07$$, $$R\_f = 1.02$$, and $$\mathrm{Var}(R\_M) = 0.01$$, so $$ b = \frac{1.07/1.02 - 1}{0.01} = 4.902, \qquad a = 0.9804 + 4.902(1.07) = 6.225 $$ Verify asset A, with $$\mathrm{Cov}(R\_A, R\_M) = 0.016$$ and $$E\[R\_A] = 1.10$$: $$ E\[mR\_A] = E\[m]E\[R\_A] + \mathrm{Cov}(m, R\_A) = 0.9804(1.10) - 4.902(0.016) = 1.0784 - 0.0784 = 1.000 $$ It prices. So does asset B, and so does the market. Three payoffs to seeing it this way. **The economic content becomes visible as a claim about marginal utility.** Chapter 3 §3.5 established that $$m$$ is marginal utility growth: high in bad states, low in good ones. The CAPM asserts that the only thing making a state bad is a low market return. Every recession, credit event, liquidity freeze, and idiosyncratic personal disaster enters the pricing kernel only through its effect on the aggregate stock market. Stated that baldly, the assumption is obviously too strong, and every model in Chapters 5 and 6 relaxes it by adding arguments to $$m$$. **The CAPM's** $$m$$ **is not a legitimate SDF everywhere.** Chapter 3 §3.5 noted that no arbitrage requires a strictly *positive* $$m$$. But $$m = a - bR\_M$$ is a downward-sloping line: it crosses zero at $$R\_M = a/b = 1.27$$ and is negative above. In §4.4's economy, the CAPM's discount factor puts a negative price on a dollar delivered in states where the market returns more than 27 percent. That is an arbitrage, if such states exist. The CAPM is best read as a linear approximation to a positive kernel, accurate near the middle of the distribution and unreliable in the tails — which is one reason it does so badly on options, whose payoffs live entirely in the tails (Chapter 8). **The generalization writes itself.** Nothing in this section used the fact that the single factor was the market. Replace $$R\_M$$ with a vector of factors, $$m = a - \sum\_k b\_k f\_k$$, and the same three lines deliver a multi-beta security market line. That is the arbitrage pricing theory and the entire factor-model literature, and it is Chapter 6. *** ## 4.7 Roll's Critique and What the CAPM Does Not Deliver The CAPM was tested almost immediately and has been failing tests for fifty years. Understanding *what* has failed requires first understanding a logical objection that Roll (1977) raised, which is prior to any evidence. **The market portfolio is not observable.** The theory's $$w^M$$ is the value-weighted portfolio of every risky asset in existence: listed equity everywhere in the world, private firms, corporate and government bonds, real estate, commodities, collectibles, and — most awkwardly — human capital, which is plausibly the largest single asset in the economy and has no price. What empirical work uses instead is a proxy: the CRSP value-weighted index, or the S\&P 500, both of which are US listed equity and therefore a small and unrepresentative corner of the object the theory names. From this Roll drew two conclusions, and they bite differently. The first is that **the CAPM has essentially one testable implication, and it is not the security market line**. The claim $$\mu\_i - r\_f = \beta\_i(\mu\_M - r\_f)$$ holding exactly for all $$i$$ is *mathematically equivalent* to the statement that $$w^M$$ is mean-variance efficient. It is not an additional economic prediction; it is the same statement rearranged. So a test of the SML is a test of the efficiency of whatever portfolio you used to compute the betas. The second, sharper, conclusion follows. If your proxy happens to be mean-variance efficient in your sample, the SML will fit *perfectly and tautologically*, whatever the true market portfolio looks like. If your proxy is inefficient, the SML will fail, again whatever the truth is. Every test of the CAPM is therefore a **joint test** of the model and of the proposition that the proxy is the market. A rejection can always be attributed to the proxy, and a confirmation is never conclusive. Roll's own summary was that the theory had never been tested and probably could not be. This is not a license to ignore the evidence, and Chapter 6 takes the anomalies seriously as evidence about something. But it disciplines what the evidence can mean. When Chapter 6 reports that size, value, and momentum generate returns the CAPM cannot explain, the honest statement is that they generate returns *unexplained by covariance with a US listed-equity index* — a proposition about a proxy, not about the aggregate consumption risk of the economy. So what survives? **The frontier and separation survive intact.** They are results in decision theory and they hold whether or not the equilibrium argument does. Two-fund separation is why the index fund exists, and the fact that most investors would be better off with fewer, more diversified holdings is one of the most robust findings in household finance (Chapter 14). **Beta survives as a description of risk exposure**, and this is most of its practical use. Whatever it does or does not do for expected returns, beta measures how a position's value moves with the aggregate market, which is what a risk manager and a margin desk both need to know. Chapter 26 uses it in that role. **The cost-of-capital application survives, with a health warning.** The CAPM remains the dominant method for setting discount rates in corporate valuation — Chapter 22 treats it as the workhorse it is in practice. Its usefulness there does not depend on the SML being exactly right; it depends on the SML being a better disciplined guess than the alternative, which is a number typed into a spreadsheet. The health warning is that the empirical SML is *flatter* than the CAPM's, so the CAPM systematically overstates the cost of capital for high-beta projects and understates it for low-beta ones, biasing firms against safe, stable investments. **Performance benchmarking survives and is arguably the model's most consequential legacy.** Jensen's alpha — the intercept $$\alpha\_i$$ in $$r\_i - r\_f = \alpha\_i + \beta\_i(r\_M - r\_f) + \varepsilon\_i$$ — became the standard by which delegated managers are judged. That measurement convention shaped the asset-management industry more than any pricing result did, and Chapter 17 §17.3 traces the consequences. What does not survive is the model's central quantitative prediction. Beginning with Black, Jensen and Scholes (1972) and Fama and MacBeth (1973), and confirmed in every subsequent sample and market, the **empirical security market line is too flat**: high-beta assets earn less than the CAPM predicts and low-beta assets earn more, and the fitted intercept is above $$r\_f$$ rather than at it. The relation between beta and average return is positive but weak, and in some long samples statistically indistinguishable from zero. That flatness is not a small residual to be swept into measurement error. It is a large, stable, international regularity, and it is the subject of §4.9. *** ## 4.8 International Diversification and Home Bias Two-fund separation, applied globally, says the risky fund is the *world* market portfolio: a US investor should hold US equity in proportion to the US share of world market capitalization, and everything else abroad. Investors do not do this. French and Poterba (1991) documented that domestic equities made up roughly 94 percent of US investors' equity portfolios, about 98 percent in Japan and roughly 82 percent in the UK, at a time when the corresponding home markets were far smaller shares of world capitalization — an overweight so large that no plausible estimate of transaction costs or currency risk accounts for it. The bias has fallen substantially since, with the growth of cheap international index products, and it has not disappeared in any major market. Candidate explanations — hedging domestic non-tradable income and inflation, informational advantage in familiar names, institutional and regulatory constraints on foreign holdings, and plain familiarity — are examined in **Chapter 14 §14.2**, where home bias sits alongside the other departures of household portfolios from the theory. The cross-border *positions* themselves, and what they imply for capital flows and exchange rates, belong to *International Finance*. *** ## 4.9 Who Holds the Market Portfolio, and What Their Constraints Do to Its Price Section 4.7 recorded the CAPM's most durable empirical failure: the security market line is too flat. High-beta stocks have delivered lower Sharpe ratios than low-beta stocks, the fitted line's intercept sits above the riskless rate, and the pattern holds across US stocks, international stocks, Treasury bonds, corporate bonds, credit indices, futures, and currencies. Frazzini and Pedersen (2014) document the flatness in all of those asset classes, which is what makes it hard to dismiss as a US equity data-mining artifact. Figure 4.5 shows the flatness in the one asset class every reader can rebuild for free. ![Figure 4.5: How flat is the SML, in the data](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-d5e6811f3c9f87ddad77efba605db751ca6da3b2%2Ffig_04_05_how_flat_is_the_sml.png?alt=media) **Figure 4.5: How flat is the SML, in the data.** Average annualized excess return against estimated market beta, for French's ten beta-sorted deciles and his 49 industry portfolios, monthly since July 1963. The orange line is the CAPM's prediction — through the origin, with slope equal to the realized market premium of 7.2 percent. The blue line is what the cross-section actually delivers: an intercept of 6.3 percent and a slope of 1.7 percent. The two cross near the market portfolio, which they must, because the market is on both lines by construction; everywhere else they diverge, and the divergence is the anomaly. Read the red markers alone and the size of it is plain. Going from the lowest beta decile to the highest nearly triples market exposure, from 0.59 to 1.60, and buys about two extra points of average return — where the CAPM says it should buy seven. Two things the figure does not show are worth stating. It does not show that low beta is a free lunch: the low-beta deciles have lower average returns, just less lower than their betas predict, and capturing the difference requires the leverage that §4.9's constrained investors cannot use. And it is not evidence of irrationality anywhere. Every point on it is a portfolio somebody chose to hold. *Source: Kenneth R. French data library, Portfolios Formed on Beta, 49 Industry Portfolios and the three-factor file, monthly from July 1963; betas and means estimated over the full sample by the author.* The explanation this book takes seriously is about holders — and about which of them *cannot* hold what the theory assumes they hold — rather than about beliefs. It requires dropping exactly one of §4.3's assumptions. **The ancestor: Black (1972).** Fischer Black asked what happens when investors cannot borrow at $$r\_f$$, or at all. Section 4.3 used the riskless asset in exactly one place: to establish that every investor holds the *same* risky portfolio and adjusts leverage — the step §4.5 then turned into the claim that the tangency portfolio is the market. Remove unlimited borrowing and that step fails. Investors who want more risk than the tangency portfolio provides can no longer buy it with leverage; they must buy it by tilting the risky portfolio toward high-beta assets. In the resulting equilibrium the riskless rate is replaced by the return on a **zero-beta portfolio** $$R\_Z$$: $$ E\[r\_i] = E\[r\_Z] + \beta\_i\big(E\[r\_M] - E\[r\_Z]\big) $$ with $$E\[r\_Z] > r\_f$$: a higher intercept and a shallower slope, which is precisely the shape the data show — produced by a constraint on holders, not a mistake in their beliefs. Table 4.4 showed the mechanism in miniature: the investor who cannot execute the $$\theta = 2$$ row must reach a 20 percent standard deviation by holding equity outright, at 10 percent expected return instead of 12. **Frazzini and Pedersen (2014) make it a demand story with a testable factor.** Their model has some investors constrained — they face binding leverage or margin limits — and others not. Constrained investors with return targets above what an unlevered tangency portfolio delivers substitute *beta for leverage*: they overweight high-beta securities, because beta is the only source of amplified market exposure they are permitted to buy. That demand bids up the prices of high-beta assets and bids down their expected returns; the unconstrained investors who take the other side must be paid to hold the low-beta assets everyone else is shunning, which raises low-beta expected returns. Both movements compress the slope of the SML and lift its intercept. The constraint is the price mechanism. Who is constrained? The list is the one Chapter 16 §16.3 compiles, and it is long. Mutual funds registered under the Investment Company Act of 1940 are, with narrow exceptions, prohibited from borrowing for investment purposes; the same Act's diversification and illiquidity limits bind their other margins. Defined-benefit pension plans and most endowments operate under investment policy statements that cap or forbid leverage. Insurers face risk-based capital charges that price levered balance-sheet risk in a different currency. Individual investors face Regulation T margin limits and, more binding still, a reluctance to borrow against securities at all. Between them these holders own the great majority of the equity market. The unconstrained pool — hedge funds, proprietary desks, some sovereign funds — is real but small relative to the market it is asked to arbitrage, and its own capital is procyclical in exactly the way Chapter 16 §16.5 describes. Leverage is the sharpest of these constraints but not the only one. An endowment bound to a spending rule must deliver a fixed real draw every year, which converts the problem into liability matching, and a manager judged against a benchmark under a tracking-error budget optimizes in deviations from that benchmark, so his origin is the index rather than cash; in both cases a flawless mean-variance optimizer ends up holding something other than the tangency portfolio, for the same structural reason Black's borrowing-constrained investor does. Their test isolates the effect with a portfolio. Rank securities by estimated beta; go long the low-beta portfolio *levered up* to a beta of one, and short the high-beta portfolio *scaled down* to a beta of one. The result is market-neutral by construction and is called **betting against beta** (BAB). If the SML were correctly sloped it would earn nothing; if the SML is too flat, it earns the flatness. The arithmetic is exact and worth doing. Suppose the realized relation is $$ E\[r\_i] - r\_f = \hat\alpha + \hat\lambda\beta\_i $$ with $$\hat\alpha > 0$$ and $$\hat\lambda < \mu\_M - r\_f$$ — a fitted intercept and a fitted price of risk, the objects Chapter 6's Fama-MacBeth regressions estimate. A position in asset $$i$$ scaled to unit beta has excess return $$(\hat\alpha + \hat\lambda\beta\_i)/\beta\_i = \hat\alpha/\beta\_i + \hat\lambda$$. Long the low-beta leg and short the high-beta leg, the $$\hat\lambda$$ terms cancel and $$ E\[r\_{\mathrm{BAB}}] = \hat\alpha\left(\frac{1}{\beta\_L} - \frac{1}{\beta\_H}\right) $$ **The BAB return is the SML's intercept times the spread in reciprocal betas.** It is positive whenever the empirical line is too flat, and it is zero exactly when the CAPM holds. Put §4.4's economy through it. Keep the covariance structure and the market weights, so $$\beta\_A = 1.6$$ and $$\beta\_B = 0.6$$ and the market premium is still 5 percent, but suppose constrained demand has flattened the line to $$\hat\alpha = 3$$ percent and $$\hat\lambda = 2$$ percent. **Table 4.5: A flattened security market line in the example economy** | | $$\beta\_i$$ | CAPM $$E\[r\_i]$$ | Flattened $$E\[r\_i]$$ | Excess over $$r\_f$$ | Excess per unit beta | | ------------------- | ------------ | ----------------- | ---------------------- | -------------------- | -------------------- | | Asset A (high beta) | 1.6 | 10.0% | 8.2% | 6.2% | 3.875% | | Asset B (low beta) | 0.6 | 5.0% | 6.2% | 4.2% | 7.000% | | Market | 1.0 | 7.0% | 7.0% | 5.0% | 5.000% | *Source: Author's calculation. Both lines are drawn through the same market portfolio and the same riskless rate of 2 percent; the flattened line has an intercept of 3 percent and a slope of 2 percent.* The BAB portfolio is long $$1/0.6 = 1.67$$ units of B and short $$1/1.6 = 0.625$$ units of A: $$ E\[r\_{\mathrm{BAB}}] = \frac{0.042}{0.6} - \frac{0.062}{1.6} = 0.07000 - 0.03875 = 0.03125 $$ which is 3.125 percent, and matches the formula: $$0.03 \times (1/0.6 - 1/1.6) = 0.03 \times 1.0417 = 0.03125$$. A market-neutral position earning better than three percent a year, generated by nothing but the fact that some holders cannot borrow. Figure 4.4 draws both lines through the same market portfolio, with the two legs marked at $$\beta\_L$$ and $$\beta\_H$$. ![Figure 4.4: A flattened security market line](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-e279495dfa23e025f14c0f2878b3bfedb8687db1%2Ffig_04_04_flattened_sml.png?alt=media) **Figure 4.4: A flattened security market line.** The CAPM line and the empirical line drawn through the same market portfolio (Table 4.5). Constrained demand for beta lifts the intercept to $$r\_f + \hat\alpha$$ and compresses the slope to $$\hat\lambda$$, so low-beta assets are priced above the CAPM line and high-beta assets below it. The two betting-against-beta legs are marked at $$\beta\_L = 0.6$$ and $$\beta\_H = 1.6$$; scaled to unit beta they earn $$\hat\alpha/\beta\_i + \hat\lambda$$, and the difference between them is $$E\[r\_{\mathrm{BAB}}] = \hat\alpha(1/\beta\_L - 1/\beta\_H)$$. *Source: Author's calculation from the worked example of Table 4.2; the flattened line is Section 4.9's illustrative calibration, not an estimate.* Two closing observations, because this section is the book's argument in its first fully classical setting. **Nobody in the story is irrational.** Every investor is a mean-variance optimizer with correct beliefs; the constrained ones solve the same problem with one additional inequality, and the mispricing is that inequality's shadow price — the Lagrange multiplier Chapter 3 §3.7 said would enter the pricing relation whenever an investor sits at a corner. The CAPM fails not because its investors are wrong but because its budget set is. **The story has a supply side, and that is where the trouble is.** The BAB return is payment for supplying leverage to a market that wants it, so whoever collects it must themselves be levered — and is therefore exposed to the funding shocks that make leverage scarce. Frazzini and Pedersen's model predicts BAB does badly when funding tightens: the strategy loses money in the moments its economic function is most needed. That is what compensation for funding risk means, and it is why the constraint generating the premium will not be arbitraged away by the people it enriches. Chapters 16 and 19 develop that mechanism; Chapter 20 puts holder demand curves at the center of pricing outright. The chapter therefore ends where Chapter 3 §3.7 left off. The CAPM is a theory about a marginal investor who is unconstrained in everything. Ask who the marginal investor in high-beta stocks actually is, and the answer is a mutual fund that cannot lever. The price shows it. In Chapter 1 §1.2's terms, the flatness of the security market line is a constraint binding rather than news: nothing about the covariance structure changed, only what the holders of beta are permitted to borrow against it. *** ## Elsewhere in the Series * **Cross-border portfolio positions, capital flows, and the currency dimension of international investing** — *International Finance*. Section 4.8 states the home-bias fact and hands the positions data and the macroeconomic consequences to that volume. * Within this book, household portfolio behavior — participation, under-diversification, and the explanations for home bias — is **Chapter 14**. Everything else in this chapter is this book's own. *** ## Summary 1. **Variance is a quadratic form, and covariance is what survives diversification.** For $$N$$ equally weighted assets with common variance $$\bar\sigma^2$$ and common covariance $$\bar c$$, $$\sigma\_p^2 = \bar\sigma^2/N + (1-1/N)\bar c \to \bar c$$. With 30 percent individual volatility and an average correlation of 0.20, four-fifths of a single stock's variance is diversifiable and twenty holdings capture most of the reduction (Table 4.1). 2. **The efficient frontier is a hyperbola, and its curvature is correlation below one.** A high-variance asset can lower portfolio risk if its correlation is low enough; the question a portfolio investor asks of a security is how it covaries with what she owns, not how volatile it is. 3. **With a riskless asset, all mean-variance investors hold the same risky portfolio.** The tangency portfolio maximizes the Sharpe ratio and has weights $$w^T \propto \Sigma^{-1}(\mu - r\_f\mathbf{1})$$ — rewards divided by risks. Risk tolerance determines only the split between $$w^T$$ and the riskless asset. This is two-fund separation, and it is the theoretical charter of the index fund. 4. **The worked example.** With $$\mu = (10, 5)$$, $$\sigma = (20, 10)$$, $$\rho = 0$$, and $$r\_f = 2$$, all in percent: $$w^T = (0.40, 0.60)$$, $$\mu\_T = 7$$, $$\sigma\_T = 10$$, Sharpe 0.50. The tangency portfolio is less volatile than the safer of its two ingredients. The minimum-variance portfolio (20/80) is a different and inferior portfolio, with Sharpe 0.447. 5. **Add market clearing and the tangency portfolio becomes the market portfolio.** If all investors hold the same risky composition, that composition is the aggregate supply. This is the CAPM's one move. 6. **The security market line:** $$\mu\_i - r\_f = \beta\_i(\mu\_M - r\_f)$$**.** Only covariance with the market is priced; own variance earns nothing; the price of risk is common to all assets and the quantity of risk is $$\beta\_i$$. In the example, $$\beta\_A = 1.6$$ and $$\beta\_B = 0.6$$, and the SML reproduces the assumed means exactly. 7. **★ The CAPM is the restriction that** $$m$$ **is linear in the market return**, $$m = a - bR\_M$$ with $$b = (E\[R\_M]-R\_f)/(R\_f\mathrm{Var}(R\_M))$$. Substituted into Chapter 3 §3.5's beta representation it yields the SML directly. Economically it asserts that a state is bad only if the market is down. The implied $$m$$ turns negative for large $$R\_M$$, so the CAPM is a linear approximation to a positive pricing kernel, not a kernel. 8. **Roll's critique.** The market portfolio is unobservable, and the SML's exact validity is mathematically equivalent to the efficiency of the portfolio used to compute betas. Every test is therefore a joint test of the model and the proxy: an efficient proxy fits tautologically, an inefficient one fails regardless of the truth. 9. **What survives.** The frontier, two-fund separation, beta as a risk-exposure measure, the CAPM cost of capital as a disciplined guess, and Jensen's alpha as the benchmarking convention that shaped the asset-management industry. What does not survive is the model's quantitative prediction: the empirical SML is too flat, with an intercept above $$r\_f$$, in every market and asset class examined since Black, Jensen and Scholes (1972). 10. **The flatness is a holder story.** Black (1972) showed that restricted borrowing replaces $$r\_f$$ with a higher zero-beta return and flattens the line. Frazzini and Pedersen (2014) made it a demand mechanism: leverage-constrained holders — 1940 Act mutual funds above all, plus pensions, insurers, and margin-limited households (Chapter 16 §16.3) — substitute beta for leverage, bidding up high-beta assets. The betting-against-beta portfolio earns $$\hat\alpha(1/\beta\_L - 1/\beta\_H)$$, which is 3.125 percent in the chapter's example economy. Nobody is irrational; the budget set, not the belief, is what the CAPM gets wrong. *** ## Key Terms * **Portfolio weight** $$w\_i$$: The share of a portfolio's value held in asset $$i$$; weights sum to one and may be negative (a short position) * **Diversification**: The reduction in portfolio variance obtained by spreading holdings across imperfectly correlated assets; it removes own variance and leaves average covariance * **Idiosyncratic (diversifiable) risk**: The component of an asset's variance that vanishes from a large portfolio, and therefore earns no premium in equilibrium * **Systematic risk**: The component that does not vanish; the only kind for which any investor is compensated * **Efficient frontier**: The set of portfolios with minimum variance for each attainable expected return, restricted to the upward-sloping branch * **Global minimum-variance portfolio**: The lowest-variance portfolio available; distinct from, and generally worse than, the tangency portfolio * **Sharpe ratio** $$\mathrm{SR}$$: $$(\mu\_p - r\_f)/\sigma\_p$$; the slope of the line from the riskless asset through a portfolio * **Tangency portfolio** $$w^T$$: The risky portfolio with the highest Sharpe ratio, $$w^T \propto \Sigma^{-1}(\mu - r\_f\mathbf{1})$$ * **Two-fund separation**: The result that every mean-variance investor holds only the riskless asset and the tangency portfolio, differing solely in the proportions * **Capital market line**: The line from $$r\_f$$ through the market portfolio in mean-standard-deviation space; the efficient frontier once a riskless asset exists * **Market portfolio** $$w^M$$: The value-weighted portfolio of all risky assets; in CAPM equilibrium, identical to the tangency portfolio * **Beta** $$\beta\_i$$: $$\mathrm{Cov}(r\_i, r\_M)/\mathrm{Var}(r\_M)$$; the quantity of systematic risk, equal to the regression slope of the asset's return on the market's * **Security market line (SML)**: The equilibrium relation $$\mu\_i - r\_f = \beta\_i(\mu\_M - r\_f)$$, linear in beta with a common price of risk * **Jensen's alpha** $$\alpha\_i$$: The intercept in a regression of an asset's excess return on the market's; zero for every asset under the CAPM, and the standard measure of delegated-manager performance * **Roll's critique**: The argument that the market portfolio is unobservable and that the SML is equivalent to the efficiency of the proxy, so every CAPM test is a joint test * **Zero-beta portfolio** $$R\_Z$$: The minimum-variance portfolio uncorrelated with the market; it replaces the riskless rate in Black's (1972) restricted-borrowing CAPM * **Betting against beta (BAB)**: A market-neutral portfolio long the low-beta assets levered to unit beta and short the high-beta assets scaled to unit beta; its return is $$\hat\alpha(1/\beta\_L - 1/\beta\_H)$$ * **Home bias**: The tendency of investors to hold far more of their domestic market than world market weights imply *** ## Readings ### Required * Markowitz, H. (1952). "Portfolio Selection." *Journal of Finance* 7(1): 77-91. *Fourteen pages that replace "pick good stocks" with "choose a point on a frontier"; read it for the argument that a security's risk is a property of the portfolio, not of the security.* * Sharpe, W. (1964). "Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk." *Journal of Finance* 19(3): 425-442. *The step from Markowitz's decision problem to an equilibrium, and the origin of the security market line — the derivation in §4.5 is Sharpe's argument in modern notation.* ### Recommended * Roll, R. (1977). "A Critique of the Asset Pricing Theory's Tests: Part I." *Journal of Financial Economics* 4(2): 129-176. *Shows that the SML's validity and the market proxy's mean-variance efficiency are the same statement, so no test of the CAPM can separate the model from the proxy.* * Black, F. (1972). "Capital Market Equilibrium with Restricted Borrowing." *Journal of Business* 45(3): 444-455. *Derives the CAPM without a riskless asset and gets a flatter line with a zero-beta intercept — the theoretical ancestor of everything in §4.9.* * Frazzini, A. and L. Pedersen (2014). "Betting Against Beta." *Journal of Financial Economics* 111(1): 1-25. *Turns Black's constraint into a demand mechanism and a tradable factor, and documents the flat SML across equities, bonds, credit, futures, and currencies.* * Roll, R. (1992). "A Mean/Variance Analysis of Tracking Error." *Journal of Portfolio Management* 18(4): 13-22. *Benchmark-relative optimization worked out: a manager minimizing tracking error is efficient in deviations from an index, which is a different frontier from §4.2's; it is the tracking-error constraint §4.9 names alongside leverage.* * Dybvig, P. (1999). "Using Asset Allocation to Protect Spending." *Financial Analysts Journal* 55(1): 49-62. *An endowment's version of the problem, and the most concrete instance of a holder who cannot hold the market: the spending obligation, not risk aversion, determines the portfolio.* * Pástor, Ľ. (2000). "Portfolio Selection and Asset Pricing Models." *Journal of Finance* 55(1): 179-223. *The estimation-error answer to §4.4 and to Box 4.1: portfolio weights are derived from a prior over pricing models rather than from sample moments, which makes explicit how much of an optimized portfolio is data and how much is belief.* * Dimson, E., P. Marsh and M. Staunton (2014). *Credit Suisse Global Investment Returns Yearbook 2014*. Credit Suisse Research Institute. *Over a century of returns and correlations for some twenty countries; the source that puts real numbers behind §4.8's diversification-gain arithmetic, and the same series Chapter 5 uses for the premium.* * Cochrane, J. Lecture notes on portfolio theory, University of Chicago. *Restates §§4.1-4.3 in the discount-factor language of §4.6, which is the cleanest available bridge between this chapter's two halves.* * Carroll, C. D. Graduate lecture notes on portfolio choice under CARA utility with normally distributed returns, Johns Hopkins University. *Works §4.1's arithmetic in the preference form Chapter 7 §7.2 then requires, where demand is linear in the expected excess return and independent of wealth.* *** ## Discussion Questions 1. **Markowitz's fifty-fifty.** The opening episode reports that the inventor of mean-variance optimization allocated his own retirement contributions by minimizing anticipated regret rather than by solving his own model. Table 4.3 shows that the Sharpe ratio is nearly flat near its maximum. Does that arithmetic exonerate him, or does it indict the model — if being roughly right costs almost nothing, what exactly is the optimization buying? Distinguish the case where the inputs $$\mu$$ and $$\Sigma$$ are known from the case where they are estimated. 2. **Who supplies the leverage?** Section 4.9 argues that the BAB premium is compensation for supplying leverage to holders who cannot borrow. Name the institutions that could plausibly collect it, and for each one say where its own borrowing comes from and what would happen to its position if that funding were withdrawn. If every potential supplier of leverage is itself levered, in what sense is the premium a free lunch, and in what sense is it not? 3. **A constraint or a mistake?** Suppose you observe that low-beta stocks earn higher Sharpe ratios than high-beta stocks. Sketch three explanations: a leverage constraint (§4.9), a behavioral preference for lottery-like payoffs, and a mismeasured market portfolio (§4.7). What observable would distinguish them? Which of the three would predict that the effect is stronger among stocks held mainly by mutual funds? 4. **Beta for the cost of capital.** A firm evaluating a low-risk regulated-utility project uses the CAPM to set its discount rate. Given the empirical flatness of the SML, in which direction is the resulting hurdle rate biased, and what does that do to the firm's investment in safe versus risky projects? (Chapter 22 gives the corporate treatment; answer it here from §4.7.) 5. **What the market portfolio would have to include.** Roll's objection is that the market portfolio contains human capital, private firms, and real estate. Pick one of the three and describe how including it would change the estimated beta of a listed technology company relative to a listed utility. Is the direction of the change predictable enough to be useful? *** ## Problems **Problem 1 — Frontier and tangency with correlated assets.** Two risky assets: C has $$\mu\_C = 10$$ percent and $$\sigma\_C = 20$$ percent; D has $$\mu\_D = 6$$ percent and $$\sigma\_D = 10$$ percent. Their correlation is $$\rho\_{CD} = 0.25$$. The riskless rate is $$r\_f = 2$$ percent. (a) Write down $$\Sigma$$. (b) Compute the mean and standard deviation of an equally weighted portfolio of C and D. (c) Find the global minimum-variance portfolio and its mean and standard deviation. Verify that its standard deviation is below that of either asset. (d) Find the tangency portfolio weights and its Sharpe ratio. (e) Compare the Sharpe ratio of the tangency portfolio with that of the minimum-variance portfolio and with that of holding C alone. **Problem 2 — Betas and the security market line.** Continue with the economy of Problem 1, and suppose it is in CAPM equilibrium, so the tangency portfolio is the market portfolio. (a) Compute $$\mathrm{Cov}(r\_C, r\_M)$$ and $$\mathrm{Cov}(r\_D, r\_M)$$, and hence $$\beta\_C$$ and $$\beta\_D$$. (b) Verify that the weighted average of the two betas is one. (c) Verify that the security market line reproduces $$\mu\_C$$ and $$\mu\_D$$. (d) A third asset, E, has $$\beta\_E = 1.2$$ and an expected return of 8.5%. Compute its Jensen's alpha. Is it cheap or dear? What trade would a mean-variance investor put on, and what happens to the alpha as others do the same? **Problem 3 — Two-fund separation and the leverage constraint.** Return to the chapter's example economy, all figures in percent: $$\mu = (10, 5)$$, $$\sigma = (20, 10)$$, $$\rho = 0$$, $$r\_f = 2$$, so $$w^T = (0.40, 0.60)$$ with $$\mu\_T = 7$$ and $$\sigma\_T = 10$$. (a) An investor targets an expected return of 9 percent. What share $$\theta$$ of the tangency portfolio achieves it, what is the standard deviation, and what is the Sharpe ratio? (b) A second investor targets the same 9 percent but may not borrow and must be fully invested in the two risky assets. Find her weights, her standard deviation, and her Sharpe ratio. (c) Compute the beta of each investor's portfolio with respect to the market. Comment on what is and is not the same across the two. (d) The constrained investor's weight in asset A exceeds the market's. If constrained investors are a large share of the market, what must happen to the expected returns on A and B for the market to clear? Relate your answer to Table 4.5. **Problem 4 — Diversification arithmetic.** Individual stocks have annual standard deviation 40 percent and average pairwise correlation 0.15. (a) Compute the standard deviation of an equally weighted portfolio of 5, 25, and 100 stocks, and the limit as $$N \to \infty$$. (b) How many stocks are needed to bring the portfolio's standard deviation within 5 percent (relative) of the limit? (c) Now suppose a crisis raises the average pairwise correlation to 0.60 with individual volatilities unchanged. Recompute the limit. By how much does the standard deviation of an already-diversified 100-stock portfolio rise? (d) Explain in two sentences why (c) is a problem for a levered holder specifically, rather than for any holder. **Problem 5 — The empirical SML and betting against beta.** In a market with $$r\_f = 3$$ percent, average excess returns line up as $$E\[r\_i] - r\_f = \hat\alpha + \hat\lambda\beta\_i$$ with $$\hat\alpha = 2.5$$ percent and $$\hat\lambda = 3$$ percent. The market portfolio has beta one. (a) What is the market risk premium implied by this line? What would the CAPM require of $$\hat\alpha$$? (b) A low-beta portfolio has $$\beta\_L = 0.7$$ and a high-beta portfolio has $$\beta\_H = 1.5$$. Compute each one's expected excess return and its excess return per unit of beta. (c) Construct the BAB portfolio: state the leverage applied to each leg and verify that the combined beta is zero. Compute its expected return, and check it against the formula $$\hat\alpha(1/\beta\_L - 1/\beta\_H)$$. (d) Suppose funding costs rise so that the BAB investor must borrow at $$r\_f$$ plus two percentage points rather than at $$r\_f$$ to lever the long leg. Recompute the strategy's return. What does this tell you about when BAB loses money? **Problem 6 ★ — The CAPM's stochastic discount factor.** Use the chapter's example economy, in gross returns: $$R\_f = 1.02$$, $$E\[R\_M] = 1.07$$, $$\mathrm{Var}(R\_M) = 0.01$$. (a) Solve $$E\[m] = 1/R\_f$$ and $$E\[mR\_M] = 1$$ for the coefficients in $$m = a - bR\_M$$. (b) Verify that this $$m$$ prices asset A, using $$E\[R\_A] = 1.10$$ and $$\mathrm{Cov}(R\_A, R\_M) = 0.016$$. (c) At what value of $$R\_M$$ does $$m$$ turn negative? Explain why a negative $$m$$ is inconsistent with the absence of arbitrage (Chapter 3 §3.5), and what that implies about using the CAPM to price a deep out-of-the-money index call. (d) Show that a mean-preserving increase in $$\mathrm{Var}(R\_M)$$, holding $$E\[R\_M]$$ and $$R\_f$$ fixed, lowers $$b$$. Interpret the comparative static: what happens to the price of market risk, and why? *** ## Selected Solutions *Solutions to Problems 1 and 2 follow. Solutions to the remainder are in the instructor materials.* **Problem 1.** (a) $$\sigma\_{CD} = 0.25 \times 0.20 \times 0.10 = 0.005$$, so $$ \Sigma = \begin{pmatrix} 0.04 & 0.005 \cr 0.005 & 0.01\end{pmatrix} $$ (b) $$\mu\_p = 0.5(0.10) + 0.5(0.06) = 8.00$$ percent. And $$\sigma\_p^2 = 0.25(0.04) + 0.25(0.01) + 2(0.25)(0.005) = 0.01 + 0.0025 + 0.0025 = 0.015$$, so $$\sigma\_p = 12.25$$ percent. (c) For two assets, $$w\_C = (\sigma\_D^2 - \sigma\_{CD})/(\sigma\_C^2 + \sigma\_D^2 - 2\sigma\_{CD}) = (0.01 - 0.005)/(0.04 + 0.01 - 0.01) = 0.005/0.04 = 0.125$$. So the minimum-variance portfolio is **12.5% C, 87.5% D**, with $$\mu\_p = 0.125(0.10) + 0.875(0.06) = 6.50$$ percent and $$ \sigma\_p^2 = 0.125^2(0.04) + 0.875^2(0.01) + 2(0.125)(0.875)(0.005) = 0.009375 $$ so $$\sigma\_p = 9.68$$ percent, below D's 10 percent. Even the safer asset is improved by a small holding of the riskier one, because $$\rho\_{CD} < \sigma\_D/\sigma\_C$$. (d) $$\mu - r\_f\mathbf{1} = (0.08, 0.04)'$$. With $$\det\Sigma = 0.04(0.01) - 0.005^2 = 0.000375$$, $$ \Sigma^{-1}(\mu - r\_f\mathbf{1}) = \frac{1}{0.000375}\begin{pmatrix} 0.01 & -0.005 \cr -0.005 & 0.04\end{pmatrix}\begin{pmatrix}0.08 \cr 0.04\end{pmatrix} = \frac{1}{0.000375}\begin{pmatrix}0.0006 \cr 0.0012\end{pmatrix} = \begin{pmatrix}1.6 \cr 3.2\end{pmatrix} $$ These sum to 4.8, so $$w^T = (1/3, 2/3)$$. Then $$\mu\_T = (1/3)(0.10) + (2/3)(0.06) = 7.333$$ percent and $$ \sigma\_T^2 = (1/9)(0.04) + (4/9)(0.01) + 2(1/3)(2/3)(0.005) = 0.011111 $$ so $$\sigma\_T = 10.54$$ percent and $$\mathrm{SR}\_T = (0.07333 - 0.02)/0.10541 = 0.506$$. (e) Minimum variance: $$(0.065 - 0.02)/0.09682 = 0.465$$. Asset C alone: $$(0.10 - 0.02)/0.20 = 0.400$$. The tangency portfolio's 0.506 beats both, as it must — it maximizes exactly this ratio. Note again that the minimum-variance portfolio is not the tangency portfolio. **Problem 2.** (a) $$\mathrm{Cov}(r\_C, r\_M) = w^T\_C\sigma\_C^2 + w^T\_D\sigma\_{CD} = (1/3)(0.04) + (2/3)(0.005) = 0.016667$$. And $$\mathrm{Cov}(r\_D, r\_M) = (1/3)(0.005) + (2/3)(0.01) = 0.008333$$. Dividing each by $$\sigma\_M^2 = 0.011111$$: $$ \beta\_C = 1.50, \qquad \beta\_D = 0.75 $$ (b) $$(1/3)(1.50) + (2/3)(0.75) = 0.50 + 0.50 = 1.00$$. Betas are covariances scaled by a constant, so they aggregate exactly as weights do, and the market's beta with itself is one by definition. (c) The market premium is $$\mu\_M - r\_f = 0.073333 - 0.02 = 5.333$$ percent. Then $$ \mu\_C = 0.02 + 1.50(0.053333) = 0.10, \qquad \mu\_D = 0.02 + 0.75(0.053333) = 0.06 $$ which are the assumed means. This is a consistency check, not a test: the tangency weights were derived from these means, so the SML must return them. (d) The SML says asset E should earn $$0.02 + 1.2(0.053333) = 8.40$$ percent. It earns 8.50 percent, so $$\alpha\_E = +0.10$$ percentage points — cheap, though barely. A mean-variance investor buys E and finances it by shorting a portfolio with the same beta: 1.2 units of the market, or equivalently 1.2 units of the tangency portfolio funded at $$r\_f$$. The position has zero market beta and a positive expected return. As investors take it, E's price rises, its expected return falls toward 8.40 percent, and $$\alpha\_E$$ goes to zero. That last step is the part that requires someone to be unconstrained — which is precisely what §4.9 denies for a large share of the market. *** ## Data Exercise: How Flat Is the Security Market Line? **Part A — Betas and average returns (free data: Ken French's data library).** From Kenneth French's data library at Dartmouth, download three monthly files: **Portfolios Formed on Beta** (the decile portfolios, value-weighted returns), **49 Industry Portfolios** (value-weighted returns), and **Fama/French 3 Factors**, which supplies the market excess return `Mkt-RF` and the riskless rate `RF`. Use the longest common sample. 1. For each of the ten beta-sorted decile portfolios, compute the full-sample average monthly excess return (portfolio return minus `RF`) and annualize it. Estimate each portfolio's market beta by regressing its monthly excess return on `Mkt-RF`. 2. Plot average annualized excess return on the vertical axis against estimated beta on the horizontal. On the same axes draw the CAPM's prediction: a line through the origin with slope equal to the annualized average of `Mkt-RF`. 3. Fit the empirical line by regressing the ten average excess returns on the ten betas. Report the fitted intercept $$\hat\alpha$$ and slope $$\hat\lambda$$, with standard errors. Compare $$\hat\lambda$$ with the market premium and $$\hat\alpha$$ with zero. Which of the ten deciles lies furthest above the CAPM line, and which furthest below? 4. Repeat steps 1-3 with the 49 industry portfolios. Industries are not sorted on beta, so their betas span a narrower range; report how that affects the precision of the fitted slope and say what it implies about why sorting on beta is the right experimental design. **Part B — Build the BAB portfolio and stress it.** 5. Using the decile portfolios, form the BAB return each month: go long the bottom three deciles scaled by $$1/\beta\_L$$ and short the top three scaled by $$1/\beta\_H$$, using betas estimated on a rolling prior window (60 months) so the construction is implementable in real time. Report the average return, standard deviation, Sharpe ratio, and the realized market beta (which should be near zero). 6. Regress your BAB return on `Mkt-RF` and report the alpha. Then check the formula of §4.9: does $$\hat\alpha(1/\beta\_L - 1/\beta\_H)$$, using your Part A estimates, approximate the average BAB return you computed? 7. Split the sample by funding conditions — a simple proxy is the TED spread or, over a longer sample, changes in the three-month Treasury-bill yield, both free from FRED. Report BAB's average return in months when funding conditions tighten versus months when they ease. Frazzini and Pedersen predict the strategy does worse when funding tightens; does your sample agree? 8. Write one paragraph on what your results do and do not establish, given §4.7. You have shown that a US-listed-equity proxy is not mean-variance efficient. Say what would have to be true of the unobserved market portfolio for the CAPM still to hold, and whether you find that plausible. **Part C ★ (if you have WRDS).** Repeat Part A at the individual-security level using CRSP monthly returns: estimate pre-formation betas on rolling 60-month windows, sort into deciles each year, and compute post-formation average returns and betas. This distinguishes the flatness of the SML from the attenuation caused by estimating betas on the same data used to form the portfolios. Then merge Thomson Reuters 13F institutional holdings and split the cross-section by the share of each stock held by mutual funds. Section 4.9 predicts the SML is flattest among the stocks most heavily held by leverage-constrained institutions; test it, and report the sign and significance of the interaction.