Capital Market Theory - CAPM and the Efficient Frontier › Limitations and Empirical Challenges
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Despite its theoretical elegance, CAPM has faced significant empirical challenges since its development. You need to understand these limitations before applying the model.
The Market Portfolio Problem
CAPM's central prediction involves the market portfolio, but this portfolio is inherently unobservable. It should include all investable assets globally: stocks, bonds, real estate, commodities, art, human capital, and more. Richard Roll's critique, published in 1977, argued that CAPM is essentially untestable because any test uses a proxy (like the S&P 500) that differs from the true market portfolio. A rejection of CAPM using a proxy may simply indicate that the proxy is inefficient, not that the theory is wrong.
Empirical Anomalies
Decades of empirical research have identified systematic patterns in returns that CAPM fails to explain:
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Size effect: Small-cap stocks have historically earned higher returns than large-cap stocks, even after adjusting for their higher betas. This anomaly was documented by Rolf Banz in 1981 and has persisted across markets and time periods, though it has weakened in recent decades.
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Value effect: Stocks with high book-to-market ratios (value stocks) have outperformed stocks with low book-to-market ratios (growth stocks), controlling for beta. Eugene Fama and Kenneth French documented this extensively, leading to their famous three-factor model, which we'll explore in the next chapter on Arbitrage Pricing Theory and Multi-Factor Models.
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Momentum effect: Stocks that have performed well over the past 3-12 months tend to continue outperforming, while recent losers continue underperforming. This effect, documented by Jegadeesh and Titman (1993), contradicts the efficient market hypothesis implicit in CAPM.
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Low-volatility anomaly: Contrary to CAPM's prediction that higher risk should command higher returns, low-volatility stocks have historically outperformed high-volatility stocks on a risk-adjusted basis. This challenges the fundamental risk-return tradeoff at the heart of the model.
Beta Instability
Beta estimates are unstable over time. A stock's beta can change significantly as the company's business evolves, financial leverage changes, or market conditions shift. This instability creates challenges for forward-looking applications:
import numpy as np
from scipy import stats
# Demonstrate beta instability over rolling windows
np.random.seed(303)
n_periods_long = 120 # 10 years monthly
# True beta that changes over time
true_beta_path = np.concatenate(
[
np.ones(30) * 0.8, # First 2.5 years: defensive
np.ones(30) * 1.4, # Next 2.5 years: aggressive after acquisition
np.ones(30) * 1.1, # Next 2.5 years: normalizing
np.ones(30) * 0.9, # Final 2.5 years: mature company
]
)
market_long = np.random.normal(0.007, 0.04, n_periods_long)
stock_long = np.array(
[
true_beta_path[i] * market_long[i] + np.random.normal(0, 0.03)
for i in range(n_periods_long)
]
)
# Rolling beta estimation (36-month window)
window = 36
rolling_betas = []
rolling_dates = list(range(window, n_periods_long + 1))
for end in rolling_dates:
start = end - window
slope, _, _, _, _ = stats.linregress(
market_long[start:end], stock_long[start:end]
)
rolling_betas.append(slope)We plot the estimated beta over time against the true beta to observe the lag and instability.
The figure illustrates how rolling beta estimates lag changes in true beta due to the historical estimation window. When making forward-looking decisions, we must decide whether historical beta represents future risk exposure.
Model Extensions
The empirical failures of CAPM led to important extensions:
-
Multi-factor models: Rather than a single market factor, these models include additional factors (size, value, momentum, quality) that help explain cross-sectional return differences. The Fama-French three-factor and five-factor models are widely used alternatives, which we'll cover in the next chapter.
-
Conditional CAPM: Time-varying betas and risk premiums can capture changing market conditions. Beta might be higher during recessions when correlations increase.
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Consumption CAPM: Replacing market returns with consumption growth provides a stronger theoretical foundation but faces its own empirical challenges.
Despite these limitations, CAPM remains valuable as a benchmark model, a way to decompose returns into systematic and idiosyncratic components, and a framework for thinking about risk and return relationships. Its simplicity makes it a practical tool even when more sophisticated models are available.