Capital Market Theory

Capital Market Theory

This chapter develops the quantitative frameworks the Series 66 tests directly: Modern Portfolio Theory (diversification math, the efficient frontier), the Capital Asset Pricing Model (the risk-return equation, SML, alpha), the four risk-adjusted return ratios (Sharpe, Treynor, Jensen, Information — when to use which), the three forms of the Efficient Market Hypothesis, and the alternatives/critiques (Arbitrage Pricing Theory, behavioral finance). Two interactives anchor the math: a CAPM calculator (Section 2) lets you adjust risk-free rate, market return, and beta to see expected return and alpha; an efficient-frontier two-asset mixer (Section 1) lets you adjust portfolio weights and correlation to see how diversification reduces risk.

Diversification math — why correlation matters

The intuition behind diversification is that combining assets whose returns don't move together reduces portfolio volatility. The math says this precisely. For a two-asset portfolio with weights w_A and w_B:

The key insight is in the LAST term — the cross-correlation term. When ρ = +1 (perfectly correlated), the formula collapses to a weighted average of standard deviations — NO diversification benefit. When ρ < +1, the cross term shrinks, and total portfolio risk drops BELOW the weighted average. When ρ = −1 (perfectly negatively correlated), the cross term goes fully negative and a specific weighting drives portfolio variance to zero.

• ρ = +1.0. Assets move in lockstep. Combining them just averages their volatility — no diversification benefit.
• ρ = +0.5. Some shared movement, some independent — modest diversification. Typical for stocks within the same sector.
• ρ = 0.0. Movements are independent. Meaningful diversification benefit; combining reduces total risk substantially.
• ρ = −0.5. Counter-cyclical — one tends to rise when the other falls. Strong diversification benefit; common between stocks and high-quality bonds historically.
• ρ = −1.0. Perfect hedge. A specific weight combination eliminates portfolio variance entirely (theoretical limit).

The Series 66 doesn't require computing portfolio variance by hand, but DOES test recognition that correlation BELOW +1 produces diversification benefit, and that lower correlation (closer to −1) produces MORE benefit. The interactive below lets you adjust weights and correlation to feel the effect directly.

The efficient frontier — what it means and what it doesn't

Markowitz's efficient frontier is the SET of portfolios that maximize expected return for each level of risk (standard deviation). Plotted with return on the y-axis and standard deviation on the x-axis, the frontier forms a curved upper boundary — portfolios on the curve are EFFICIENT; portfolios below are inefficient (you could get more return for the same risk, or the same return for less risk).

• Minimum variance portfolio (MVP). The single point on the frontier with the LOWEST possible standard deviation. The frontier's leftmost point.
• Tangent portfolio. The point on the frontier where a line from the risk-free rate (y-intercept) is tangent to the curve. Has the highest Sharpe ratio of any feasible portfolio.
• Capital Market Line (CML). The line from the risk-free rate through the tangent portfolio. Investors who hold the tangent portfolio + risk-free asset (in any proportion, including leverage) achieve combinations along the CML — which DOMINATES the efficient frontier above the tangent point.
• Two-fund separation. MPT's elegant conclusion: ALL investors with mean-variance preferences should hold combinations of (1) the tangent portfolio of risky assets and (2) the risk-free asset. Differences in risk tolerance just shift the mix, not the composition of the risky piece.

What the efficient frontier ASSUMES (and where it breaks): inputs (expected returns, std devs, correlations) are KNOWN and STABLE. In reality, they're estimated with error and shift over time. The frontier is also highly sensitive to input estimates — small changes in assumed returns produce big shifts in the “optimal” weights. Practitioners use the frontier as a CONCEPTUAL ANCHOR, not a literal optimization output.

The market risk premium — the most contested input

The CAPM equation E(R) = R_f + β × (R_m − R_f) hinges on the MARKET RISK PREMIUM (R_m − R_f) — the extra return investors expect for bearing systematic risk. It's also the input practitioners disagree about most:

• Historical estimates. US stocks earned approximately 5-7% above T-bills over long horizons (1926-present, depending on the dataset). Used as a backward-looking benchmark.
• Forward-looking estimates. Implied from current dividend yields, growth expectations, and bond yields. Typically run LOWER (3-5%) in periods of high valuations and low bond yields.
• Survey-based estimates. Aggregated forecasts from professional investors. Tend to fall between historical and forward-looking estimates.
• Country and currency adjustments. Emerging-market risk premiums add a country-risk component; foreign-currency exposure adds another adjustment.

The Series 66 doesn't require choosing the “right” premium — it tests recognition that the MARKET RISK PREMIUM is the (R_m − R_f) term, that beta MULTIPLIES it to produce the security-specific risk premium, and that the result added to R_f gives the expected return. The next block is an interactive CAPM calculator that lets you adjust each input.

Beta — types, limitations, and what it doesn't capture

Beta is a single number, but the way it's estimated and the limitations it carries matter on the exam:

• Equity beta (levered beta). The most commonly cited number — sensitivity of an equity's returns to a benchmark's returns, including the effect of the company's capital structure (debt amplifies equity beta).
• Asset beta (unlevered beta). Removes the effect of debt — isolates the operating risk of the business. Useful for comparing companies with different capital structures or projecting beta for a new project.
• Portfolio beta. The WEIGHTED AVERAGE of constituent betas. A portfolio with 40% in a beta-0.8 stock and 60% in a beta-1.4 stock has beta = 0.4(0.8) + 0.6(1.4) = 1.16.
• Beta = 1.0. Same systematic risk as the market — moves about as much as the benchmark.
• Beta > 1.0. Higher systematic risk than market — amplified moves. Cyclical sectors (tech, financials, discretionary).
• Beta < 1.0 (positive). Lower systematic risk — defensive sectors (utilities, staples, healthcare).
• Beta < 0. Inverse relationship — rare; gold or short-volatility products may have negative beta against equity benchmarks in some regimes.

Limitations: (1) beta is ESTIMATED from historical data and may not predict the future; (2) it captures only SYSTEMATIC risk — unsystematic risk is invisible to beta; (3) beta is BENCHMARK-DEPENDENT (the same stock has different beta against the S&P 500, Russell 1000, and MSCI World); (4) beta assumes a linear relationship that breaks down in tail events.

CAPM — worked example

Sharpe ratio — excess return per unit of total risk

The Sharpe ratio is the canonical risk-adjusted return measure for total-volatility comparisons:

Interpretation: how much return PER UNIT of total volatility the portfolio delivered above the risk-free rate. Higher is better; the ratio rewards both higher excess return AND lower volatility.

• Sharpe of 0.5. 0.5 percentage points of excess return per 1 percentage point of std dev. Typical for diversified balanced portfolios.
• Sharpe of 1.0. Excess return equals std dev. Strong risk-adjusted performance.
• Sharpe above 1.5. Excellent; uncommon over long horizons; often signals either skill or short-term lucky path.
• Negative Sharpe. Portfolio underperformed the risk-free rate — investors would have done better in T-bills.

Use case: comparing portfolios or managers when TOTAL risk (not just systematic) is the relevant measure. Best when the investor holds the portfolio as a stand-alone allocation rather than adding it to a broader portfolio. Limitation: assumes returns are normally distributed; portfolios with skewed return distributions (selling tail risk for income) can show artificially high Sharpe ratios that don't reflect tail-risk exposure.

Treynor and Jensen — systematic-risk versions

Sharpe uses TOTAL risk (std dev). Treynor and Jensen use SYSTEMATIC risk (beta). They're the right measures when the portfolio is being added to an already-diversified mix — only systematic risk is added by the new piece.

Jensen's alpha is the DOLLAR measure of value added on a systematic-risk-adjusted basis; Treynor is the RATIO measure. Both compress the same information differently — alpha tells you HOW MUCH value, Treynor tells you the rate of value-add per unit of beta.

Information Ratio — consistency of active return

The Information Ratio measures how CONSISTENTLY an active manager outperforms a benchmark:

The numerator is the manager's excess return ABOVE the benchmark (not the risk-free rate). The denominator is the TRACKING ERROR — the standard deviation of (portfolio return − benchmark return). A high information ratio means the manager beat the benchmark by a lot AND consistently. A low ratio means either small outperformance or lots of volatility in the active return.

• IR of 0.50. Good. Suggests genuine skill if sustained over 5+ years.
• IR of 1.0. Exceptional. Rare over long horizons; often unsustainable.
• IR of 0 or below. No value added vs. the benchmark — investors should consider switching to passive at lower cost.

Use case: evaluating active managers against their stated benchmarks. The IR is more demanding than alpha because it penalizes inconsistent active returns even if average alpha is positive.

Active vs. passive — what EMH implies in practice

EMH's practical implication is that consistently beating the market through analysis is HARD — harder the stronger the EMH form that holds. The investor decisions that follow:

• If weak-form EMH holds. Past prices and volume contain no exploitable information. TECHNICAL ANALYSIS is useless; fundamental analysis and insider info may still work.
• If semi-strong-form EMH holds. All public information is in prices. Both technical AND fundamental analysis are useless. Only material non-public information could generate excess returns — which is illegal to trade on. Implication: ACCEPT MARKET RETURNS via low-cost index funds.
• If strong-form EMH holds. All information — public AND private — is reflected. Even insiders can't profit. Generally NOT considered to hold empirically (insider trading studies show abnormal returns exist).

The case for INDEX INVESTING flows from semi-strong EMH: if active managers can't consistently beat the market after fees, the rational choice is to OWN the market at the lowest possible cost. Empirical SPIVA studies (S&P Indices Versus Active) consistently show that majority of active large-cap US managers underperform their benchmarks over 5-10 year horizons after fees — consistent with semi-strong EMH at least in the most-analyzed markets.

The case for ACTIVE MANAGEMENT requires either (1) market inefficiencies that the manager can exploit, (2) less-efficient market segments (small-cap, emerging markets, distressed credit), (3) genuine information advantages, or (4) non-return objectives (downside protection, ESG screens, factor tilts).

Arbitrage Pricing Theory (APT) — the multi-factor alternative

APT, developed by Stephen Ross, generalizes CAPM by allowing MULTIPLE systematic risk factors instead of just one (the market). The expected return on an asset is the risk-free rate plus the asset's sensitivities (factor betas) to each of several factors, each multiplied by that factor's risk premium:

Where λ_i is the risk premium for factor i and β_i is the asset's sensitivity to that factor. Common factors include:

• Market factor — same as CAPM's market beta.
• Size factor (SMB) — small-minus-big; captures the size effect.
• Value factor (HML) — high book-to-market minus low; captures the value effect.
• Momentum factor — winners minus losers; captures the momentum effect.
• Quality factor — high-profitability minus low; more recently added.
• Macro factors — inflation, term-structure changes, industrial production, oil prices (in some formulations).

APT differs from CAPM in several important ways: (1) APT doesn't require a market portfolio; (2) APT doesn't require investors to hold mean-variance-optimal portfolios; (3) APT doesn't require returns to be normally distributed; (4) APT relies on a NO-ARBITRAGE argument (if APT were violated, riskless profit opportunities would exist and arbitrageurs would close them). Practical use: factor-tilted ETFs (small-cap value, momentum, quality) and multi-factor smart-beta strategies operationalize APT thinking.

Behavioral finance — the principled challenge to EMH

EMH assumes RATIONAL investors who process information without systematic errors. Behavioral finance (Kahneman, Tversky, Thaler, Shiller) documents that real investors exhibit predictable biases — the same biases catalogued in M3.3 (loss aversion, overconfidence, anchoring, herd behavior, recency, etc.). If biases are SYSTEMATIC and PERSISTENT, prices can deviate from fundamental value in predictable ways — violating EMH.

• Overreaction. Markets overshoot in response to dramatic news — followed by partial reversal as the initial overreaction is corrected. Underlies the contrarian-investing case.
• Underreaction. Markets fail to fully price gradual news — underlying the momentum anomaly. Gradual revelation of earnings strength produces sustained price drift.
• Bubbles. Asset prices detach from fundamentals when herd behavior + overconfidence + recency drive successive bidders to pay more than the previous one. Pop when the herd reverses.
• Limits to arbitrage. Even if some investors recognize the mispricing, ARBITRAGE may not eliminate it: shorting is costly, capital is limited, and arbitrageurs face career/funding risk if mispricings persist. So mispricings can endure.

The reconciliation most practitioners adopt: markets are MOSTLY efficient at most times, but biases produce persistent anomalies in specific market segments and during specific periods. Active management can add value in inefficient pockets; passive investing makes sense for the broadly efficient segments. Most academic finance now treats markets as partially efficient rather than fully efficient or fully inefficient.

Modern Portfolio Theory — baseline framework

Developed by Harry Markowitz in 1952 (Nobel Prize 1990), Modern Portfolio Theory established that portfolio risk can be REDUCED through diversification without necessarily sacrificing return. The core ideas:

• Focus on the portfolio as a whole, not individual securities. Individual security risk matters less than how the security combines with the rest of the portfolio.
• Efficient frontier. The set of portfolios offering the HIGHEST return for each level of risk — or equivalently, the LOWEST risk for each level of return.
• Diversification works because assets with LOW or NEGATIVE correlation partially offset each other's movements, reducing overall portfolio volatility.
• Risk decomposition. Total risk = systematic (market-wide, undiversifiable) + unsystematic (security-specific, diversifiable). Diversification eliminates unsystematic risk; only systematic risk should be priced.

Capital Asset Pricing Model — the basic equation

CAPM (Sharpe, Lintner, Mossin, 1960s) builds on MPT to describe the relationship between systematic risk and expected return:

• Investors are only compensated for SYSTEMATIC risk (beta), not unsystematic risk — consistent with MPT's conclusion that diversifiable risk shouldn't be priced.
• The Security Market Line (SML) plots expected return vs. beta.
• Securities ABOVE the SML are undervalued (provide excess return for their beta).
• Securities BELOW the SML are overvalued (don't provide enough return for their beta).

Efficient Market Hypothesis — three forms

EMH (Eugene Fama, 1970) suggests that securities prices reflect available information, making it difficult to CONSISTENTLY outperform the market. The three nested forms:

• Weak form. Prices reflect all PAST TRADING DATA (prices, volume). Technical analysis cannot generate excess returns. Fundamental analysis and insider information may still work.
• Semi-strong form. Prices reflect all PUBLICLY AVAILABLE INFORMATION (past data + earnings releases + news + analyst reports). Neither technical nor fundamental analysis can generate excess returns. Insider information may still work — but it's illegal to trade on.
• Strong form. Prices reflect ALL INFORMATION — public AND private. Even insiders cannot generate excess returns. Generally NOT considered to hold empirically.

Each form is NESTED in the next: if semi-strong holds, weak must hold too; if strong holds, semi-strong must hold. Most academic research supports weak form and partially supports semi-strong form for major liquid markets.

EMH forms — what each allows and prohibits

Practical implication: if you believe in the semi-strong form, ACTIVE STOCK SELECTION based on public information cannot reliably produce excess returns — the rational choice is low-cost passive index investing. The empirical case for semi-strong EMH is strongest in large-cap US equities and weakest in less-analyzed segments.