PM 2 Portfolio Risk and Return: Part II

The goal of this reading is to identify the single optimal risky portfolio for every investor and, from there, to price any asset by its systematic risk alone. The building blocks are familiar from Part I: the expected return of a portfolio is a weighted average of its holdings, while portfolio risk depends on the individual risks and on how the assets move together. For a portfolio of N risky assets those relationships are:

The covariance between two assets ties the correlation to the two standard deviations, and an asset covaries with itself as its own variance:

Because assets that are not perfectly correlated partly offset one another, a portfolio can carry less risk than the assets inside it. To see the point concretely, let Asset 1 return 10 percent with a 20 percent standard deviation, and Asset 2 return 5 percent with a 10 percent standard deviation, the two being uncorrelated. Three mixes produce sharply different risk and return.

Correlation between Asset 1 and Asset 2 is 0.0.

Portfolio X returns 6.25 percent at only 9.01 percent risk, while Portfolio Z carries a standard deviation more than two-thirds higher (15.21 percent) for a return only about one-third higher (8.75 percent). A risk-free asset added to a risky portfolio sharpens the trade-off again, and the reason is its lack of any correlation with the other holdings. The set of all mixes of the risk-free asset with a chosen risky portfolio forms the capital allocation line (CAL). Where three advisers offer three different risky portfolios A, B, and C, the portfolio that yields the greatest expected return at every level of risk (say C) traces the highest CAL, and rational investors gravitate to it. The exact point an investor selects on that line depends only on how risk-averse the investor is: the most risk-averse hold mostly the risk-free asset, and the less risk-averse tilt toward the risky portfolio.

Does one optimal risky portfolio exist?

Whether a single best risky portfolio exists hinges on whether investors agree. The value of any asset is the present value of its future cash flows:

Estimating those cash flows and the required returns is subjective. Siemens AG closed at EUR111.84 on Xetra on 31 August 2018, a price that reflects the view of the marginal investor. Analyst B might judge fair value at EUR95 and conclude the stock is overvalued, assigning it a weight of zero, while Analyst C might see EUR125 and overweight it heavily. Because valuations differ, each investor can arrive at a different optimal risky portfolio. If instead we assume homogeneity of expectations, meaning everyone shares the same forecasts, the calculations converge on one optimal risky portfolio. Even without that assumption, market prices proxy for the marginal informed investor, so the market portfolio serves as the benchmark against which all other portfolios are judged. In it, each asset is weighted by its own market value as a share of the combined value of everything held.

The capital market line (CML) is the special case of the capital allocation line in which the risky portfolio is the market portfolio. Before defining it, it helps to separate two styles of investing. If markets are informationally efficient, the traded price already reflects all public information, so an investor cannot reliably beat the required return, and there is little reward for independent analysis. Portfolios built on that premise are passive portfolios; they usually replicate a market index such as the S&P 500, the Nikkei 300, or the CAC 40, and carry low costs. Active investors, by contrast, back their own estimates of cash flows and discount rates, overweighting assets they judge undervalued, underweighting or short selling those they judge overvalued.

What counts as the market

In theory the market holds every risky asset that has value, spanning equities, bonds, property, and even human capital. In practice many assets are neither tradable nor investable (the Taj Mahal and human capital are not tradable; some listed shares are closed to certain investors), so a broad local index stands in as a proxy. This reading uses the S&P 500. As of mid-2018 its constituents represented roughly 80 percent of US equity capitalization, and because US equities were about 40 percent of world markets, the index captured close to 32 percent of worldwide publicly traded equity. Under this convention, both the market return and the risk premium of the market describe US equities alone.

Deriving the line

The market portfolio sits where a line drawn from the risk-free asset just touches the Markowitz efficient frontier, and every point inside that frontier is inefficient. A portfolio that splits wealth between the risk-free asset (weight w1) and the market (weight 1 minus w1) has return and risk given by the two-asset formulas. Because the risk-free asset has zero standard deviation and zero covariance with the market, two of the three variance terms vanish, and the risk expression collapses to a straight line:

Substituting for the weight rewrites expected return directly in terms of portfolio risk, in the classic form of a line:

Its intercept equals the risk-free rate, while the slope, known as the market price of risk, equals the market risk premium scaled by market risk. The slope is positive because the market rewards risk, so moving up the line raises both risk and expected return together.

Everything between the risk-free point and the market portfolio M is a lending portfolio: the investor lends part of their wealth at the risk-free rate and holds the rest in the market. To move to the right of M, an investor borrows at the risk-free rate and invests more than 100 percent in the market, creating a borrowing (leveraged) portfolio in which the weight on the risk-free asset turns negative. Leverage lifts expected return, but it lifts risk in the same proportion.

When lending and borrowing rates differ, the CML is no longer a single straight line but bends downward at M. Its slope is the market price of risk using Rf up to M, and a smaller slope using the higher borrowing rate Rb beyond M:

Because the risk of an asset depends on the company it keeps, the key question is which portion of risk actually earns a return. Total risk splits into two parts. Systematic risk, also called non-diversifiable or market risk, affects the whole economy: interest rates, inflation, business cycles, political uncertainty, and widespread disasters. It cannot be diversified away. Nonsystematic risk, known interchangeably as diversifiable, idiosyncratic, company-specific, or industry-specific risk, stays local to one firm or industry: a failed drug trial or an airliner crash hurts the affected companies but leaves distant assets untouched. Investors erase nonsystematic risk by holding assets that are not highly correlated. The two combine at the level of variance:

Statements that call total risk the sum of systematic and nonsystematic risk are, strictly, statements about variance rather than standard deviation.

Why only systematic risk is priced

Suppose, for the sake of argument, that both kinds of risk earned a return. Investors would rush to buy assets loaded with nonsystematic risk, then diversify it away, collecting a reward for risk they no longer bore. That demand would bid such assets up until the reward for diversifiable risk fell to zero. So the only internally consistent conclusion is that in an efficient market diversifiable risk earns nothing, and investors are compensated only for systematic risk, which cannot be shed. The practical lesson is that risk-averse investors should hold well-diversified portfolios and accept only priced, systematic risk.

Building the full market portfolio is daunting: a 1,000-asset portfolio needs 1,000 return estimates, 1,000 standard deviations, and 499,500 correlations. A return-generating model sidesteps much of this by estimating an asset’s expected return from a small set of risk factors. Because only systematic risk is rewarded, the model needs only the asset’s exposure to systematic factors. The most general form is a multi-factor model:

The left side is the expected excess return over the risk-free rate; the betas are factor weights (factor loadings). Every model separates out the market factor, E(Rm). Factors come in three families. Macroeconomic models use variables such as growth, interest rates, inflation, productivity, employment, and consumer confidence. Fundamental models use company traits such as earnings, earnings growth, cash flow, research spending, and patents. Statistical models mine historical returns for factors that explain variance, though these may lack any economic meaning; a spurious example is which conference the Super Bowl winner belongs to. Because data mining invents meaningless factors, analysts prefer macroeconomic and fundamental models.

Three-factor and four-factor models

Eugene Fama and Kenneth French argued that returns are explained better by adding relative company size and relative book-to-market value to beta. Mark Carhart later added a momentum factor based on relative past returns. These sit at the multi-factor end of the spectrum and reappear in the section on extensions to the CAPM.

The single-index model and the market model

The simplest model uses one factor, the market, giving the single-index model:

This is consistent with the CML. Rearranging the CML for any security i expresses its excess return as total security risk relative to total market risk, multiplied by the market premium:

To connect the factor loading sigma_i divided by sigma_m to beta, decompose total risk. Replacing expected with realized returns introduces an error term e_i that captures non-market surprises:

Since returns unrelated to the market carry no correlation with it, the covariance term vanishes and total variance splits cleanly, so total risk is the square root of the two variances:

For a well-diversified portfolio, nonsystematic variance is zero, so total risk is just beta times market risk. Substituting beta_i sigma_m for sigma_i in the rearranged CML reduces the factor loading to beta and recovers the single-index model, confirming that for diversified portfolios the CML and the single-index model are one and the same.

The most common single-index implementation is the market model, which uses realized market return as the single index and is used to estimate beta and abnormal returns:

To be consistent with the earlier expression, the intercept equals Rf times (1 minus beta). The intercept alpha_i and slope beta_i are estimated from historical returns. Suppose a regression of Wal-Mart’s daily returns on the S&P 500 gives an alpha of 0.0001 and a beta of 0.9, so the expected daily return is 0.0001 + 0.90 × Rm. If, on a day the market rises 1 percent, Wal-Mart rises 2 percent, its company-specific return is 0.02 minus (0.0001 + 0.90 × 0.01) = 0.0109, an abnormal return of 1.09 percent.

Beta gauges the degree to which an asset responds to movements in the market. Starting from the single-index model in realized returns and taking the covariance of the asset with the market, the constant risk-free term drops out and the error term bears no correlation to the market, which leaves beta as covariance over market variance, which also equals correlation times the ratio of standard deviations:

For example, if an asset correlates 0.70 with the market and the asset and market standard deviations are 0.25 and 0.15, then beta is (0.70 × 0.25) divided by 0.15 = 1.17. If instead the covariance with the market is 0.026250 and market variance is 0.02250, then beta is 0.026250 divided by 0.02250 = 1.17. A positive beta means the asset moves with the market; a negative beta means it moves against it. A risk-free asset has a beta of zero because it does not covary with anything. Substituting sigma_m for sigma_i, and using that any asset correlates perfectly with itself, shows the market’s beta is 1:

Since the market beta is 1, the average beta of all stocks is 1. Most stocks in developed markets correlate above 0.70 with the market, and broad US indexes such as the S&P 500, the Dow Jones 30, and the NASDAQ 100 exceed 0.90, so consistently negative-beta assets are rare.

Estimating and applying beta

Beta can also be estimated directly by regressing historical security returns on market returns; the fitted line is the security characteristic line, its slope the beta. Shorter estimation windows (say 12 months) track current systematic risk more closely but are noisier; windows of three to five years are steadier but may misstate the future if the company has changed. Once beta is known, the CAPM, arranged with the risk-free rate isolated on the right side, converts it into an expected return:

Assets with beta above 1 are expected to out-return the market; those below 1 to under-return it. A negative-beta asset can even require a return below the risk-free rate, because it lowers overall portfolio risk. Insurance is the classic case: it pays out when the insured suffers a loss and costs a premium otherwise, so it carries a beta below zero and an expected return below zero, yet stays valuable for cutting risk.

The capital asset pricing model (CAPM), developed independently by William Sharpe, John Lintner, Jack Treynor, and Jan Mossin on Harry Markowitz’s foundation, states that expected returns differ only by systematic risk, beta. Two assets with the same beta earn the same expected return, whatever else they are. The relationship is the now-familiar line E(Ri) = Rf + beta_i[E(Rm) minus Rf].

Assumptions

The model rests on six simplifying assumptions:

1. Investors are risk-averse, utility-maximizing, and rational. They need not share the same degree of risk aversion, only be averse to risk and prefer more wealth to less.
2. Markets are frictionless: no transaction costs, no taxes, and no restrictions on short selling, with borrowing and lending possible at the risk-free rate.
3. Investors share a single holding period, which keeps the model to one period for tractability.
4. Investors have homogeneous expectations, so they agree on distributions and inputs and reach the same market portfolio.
5. All investments are infinitely divisible, allowing continuous functions.
6. Investors are price takers, none large enough to move prices.

Together these assumptions construct a marginal investor who, acting rationally, selects a portfolio that is efficient in mean-variance terms. Relaxing most of them changes the conclusions only slightly, though costs or restrictions on short selling can bias prices upward. Even with its limitations, the CAPM provides a benchmark and a first estimate of returns.

The security market line

The security market line (SML) plots the CAPM with beta on the horizontal axis and expected return on the vertical. It cuts the vertical axis at Rf, and its slope is the market risk premium, E(Rm) minus Rf. Unlike the CML, which applies only to efficient portfolios and uses total risk, the SML prices any security or portfolio, efficient or not, using systematic risk. Total risk and systematic risk coincide only for efficient portfolios, which carry no diversifiable risk.

Portfolio beta

The SML also prices combinations of securities. Writing each security through the CAPM and taking a weighted average shows that a portfolio’s beta is the weighted average of its component betas, and its expected return follows the same CAPM line:

The CAPM and the SML say what a return should be for a given level of risk; the actual return may differ. That gap makes the model useful in three main ways: estimating expected returns for capital budgeting, comparing a manager’s realized return with the CAPM benchmark for performance appraisal, and analyzing whether an asset is mispriced for security selection.

Estimating expected return

Investors typically treat the CAPM required return as their opening estimate when valuing assets and when discounting project cash flows in capital budgeting. The same figure sets the cost of capital for regulated firms and fair insurance premiums. In net present value analysis, cash flows are discounted at the CAPM required return matched to the project’s risk.

The CAPM is important but not the only return-generating model, and its assumptions create both theoretical and practical weaknesses.

Limitations

The theoretical limitations are structural. The CAPM is a single-factor model, pricing only beta risk and ignoring every other characteristic, which makes it rigid. It is also a single-period model, blind to multi-period objectives, so it can encourage myopic decisions. The practical limitations arise on implementation. The true market portfolio holds all assets, including non-investable ones such as human capital, so, as Richard Roll noted, it is unobservable and the model is not strictly testable. Analysts substitute proxies that vary by analyst and country and give different return estimates for the same asset. Estimating beta needs three to five years of history that may misrepresent the current company, and betas differ with the estimation window and return frequency. Empirically the CAPM predicts returns poorly, and it assumes homogeneous expectations that, if relaxed, would produce many optimal portfolios and many security market lines.

Extensions

Two families of models address these gaps. The theoretical example is arbitrage pricing theory (APT), developed by Stephen Ross, which keeps a linear return-risk relationship but allows many factors:

Here each lambda is the risk premium for a factor and each beta the portfolio’s sensitivity to it. APT allows any number of relevant factors and does not require them to be common across assets; a no-arbitrage condition pins down the factors and betas. Elegant as it is, APT does not specify the factors, so it is hard to apply, and in practice the CAPM is preferred to it.

The practical example is the Fama-French-Carhart four-factor model, which replaces the single beta with market, size, value, and momentum factors:

MKT is the excess market return, SMB the return of small over large stocks (size), HML the excess return of high book-to-market names over low ones, capturing value against growth, and UMD the return of past winners over past losers (momentum). In historical tests the loading on MKT comes out statistically indistinguishable from zero, so size, book-to-market, and momentum, rather than the market alone, explain returns. The model predicts US stock returns far better than the CAPM, but it is not grounded in an equilibrium theory and offers no guarantee of working in the future.

Performance evaluation covers measurement (return and risk), attribution (the sources of performance), and appraisal (skill versus luck). Four CAPM-based ratios are used for appraisal, each pairing reward with a measure of risk.

The Sharpe ratio

The Sharpe ratio, or reward-to-variability ratio, sets the portfolio risk premium against its total risk, and it equals the slope of the capital allocation line:

It is easy to compute and interpret and works ex ante or ex post, but it has two drawbacks: it uses total risk when only systematic risk is priced, and the raw number carries no meaning until compared with another portfolio’s Sharpe ratio. Rankings hold only when the numerator is positive; a negative premium reverses the ranking, since a riskier portfolio then looks less negative.

The Treynor ratio

The Treynor ratio fixes the first drawback by dividing the risk premium by beta rather than total risk:

Both the numerator and the beta must be positive for a meaningful result, so the measure fails for assets whose beta is negative. Like the Sharpe ratio, it ranks but does not measure the economic size of any performance difference.

M-squared (risk-adjusted performance)

M-squared, developed by Franco and Leah Modigliani, expresses risk-adjusted performance as a percentage return, assuming the portfolio is levered or de-levered until its volatility matches the market’s. It ranks portfolios identically to the Sharpe ratio because it is a rescaling of it:

Jensen’s alpha

Jensen’s alpha, like the Treynor ratio, uses systematic risk. It is the realized portfolio return minus the CAPM return for the portfolio’s beta, that is, the vertical distance from the SML:

A positive alpha means outperformance, a negative alpha underperformance, and the market’s own alpha is zero. Alpha ranks managers and quantifies the size of over- or underperformance, so it is widely used to appraise pension funds and mutual funds. It is often described as the maximum fee an investor should pay a manager.

These measures assume the market portfolio is the right benchmark, so a wrong benchmark (say judging a real estate fund against the S&P 500) or errors in measuring risk and return can mislead. Since many inputs come from historical data, projections presume that past performance persists.

Relaxing the assumption of identical expectations lets the CAPM guide active portfolio building. The tool is the security characteristic line (SCL), which plots a security’s excess return against the market’s excess return; rearranging Jensen’s alpha gives its equation, with alpha the intercept and beta the slope:

Security selection

If investors disagree about cash flows or systematic risk, an investor’s own estimate of value can differ from the CAPM price, which reflects the market consensus. When the investor’s estimated price exceeds the market price, the asset is undervalued; when it is lower, the asset is overvalued. Jensen’s alpha makes the best selection tool, since it rests on systematic risk and carries meaning even in absolute terms: a positive alpha flags a superior security, a negative alpha one likely to underperform once adjusted for risk. The SML gives the same signal graphically. Assets on the line are fairly priced; a point above the line offers more return than its risk warrants (a buy), while a point below the line is overvalued and, if permitted, a short-sale candidate.

Building the portfolio

Owning every asset is impractical, but much nonsystematic risk disappears within about 30 randomly selected securities drawn from different asset classes. A workable method starts from an index such as the S&P 500, treated as the market. Any security outside the index is judged by its alpha, computed against that index; a security with a positive alpha (for example, a new market with an alpha near 3 percent and a beta of 0.60) is added, and one with a significantly negative alpha may be short sold. Even a correctly priced security can carry a positive alpha simply because it correlates weakly with the index while still rewarding its systematic risk. Within the index, overvalued names (negative alpha) are dropped and undervalued names (positive alpha) are increased. Weights track the ratio of alpha to nonsystematic variance, and the information ratio, alpha over nonsystematic risk, captures abnormal return for each unit of extra risk, so a higher value marks a more valuable security.