The starting point of the multifactor holy grail: CAPM

A Formula That Changed the Financial World#

In 1964, a scholar named William Sharpe proposed a seemingly simple formula. At that time, most people believed that investing in the stock market was akin to gambling, making it hard to imagine that a mathematical model could be used for analysis.

Sharpe spent two years developing this formula:

$E(R_i) = R_f + \beta_i \times [E(R_m) - R_f]$

This is the Capital Asset Pricing Model (CAPM). Although the formula appears concise, it later became a cornerstone of modern finance, earning Sharpe the Nobel Prize in Economic Sciences in 1990.

What is CAPM?#

The core question CAPM seeks to address is: What kind of relationship exists between risk and reward?

Core Concept#

The reasoning is quite intuitive: To achieve higher returns, one must take on higher risks.

This aligns with the fundamental principle of investing; investments with higher risks should theoretically offer higher expected returns, otherwise, why would investors take on those risks?

Detailed Explanation of the CAPM Formula#

Standard Formula#

$E(R_i) = R_f + \beta_i \times [E(R_m) - R_f]$

Parameter Explanation#

Parameter	Name	Description
E(R_i)	Expected Return	The expected return of the asset
R_f	Risk-Free Rate	The rate of risk-free assets like government bonds
β_i	Beta Coefficient	The asset's volatility relative to the market
E(R_m)	Market Expected Return	The expected return of the overall market
[E(R_m) - R_f]	Market Risk Premium	The additional return from investing in the stock market over risk-free assets

Beta Coefficient: An Important Indicator of Risk#

The beta coefficient measures the volatility of an individual asset relative to the overall market:

β = 1: Moves in sync with the market
β > 1: More volatile than the market
β < 1: Less volatile than the market
β = 0: Unrelated to market volatility

Interesting Example#

During the 2008 financial crisis, Netflix's beta coefficient was about 1.2, meaning that theoretically, if the market dropped by 10%, it should drop by 12%. However, Netflix actually rose during that period, as more people chose to watch movies at home during the economic downturn.

This example reminds us that the beta coefficient is more effective in "normal" market conditions but may fail during extreme events.

Practical Application: TSMC Valuation Example#

Let's use TSMC (2330) to demonstrate the application of CAPM:

# Assumed data (estimated for 2024)
risk_free_rate = 0.01    # Risk-free rate 1% (Taiwan 10-year government bond)
market_return = 0.08     # Market expected return 8% (Taiwan Weighted Index)
tsmc_beta = 1.2         # TSMC Beta coefficient

# CAPM calculation
expected_return = risk_free_rate + tsmc_beta * (market_return - risk_free_rate)
print(f"TSMC Expected Return: {expected_return:.2%}")
# Result: TSMC Expected Return: 9.40%

Investment Judgment Reference#

If TSMC's current expected return is below 9.40%, it may be overvalued according to CAPM; if it is above 9.40%, there may be an investment opportunity.

Limitations of CAPM#

Although CAPM is an important theoretical foundation, it has some obvious limitations:

Major Limitations#

Idealized Assumptions: Assumes a perfectly efficient market and completely rational investors
- In reality, investors often exhibit irrational behavior, and various frictions exist in the market.
Single Risk Factor: Considers only market risk, ignoring other influencing factors
- In practice, company size, value characteristics, profitability, and other factors can affect returns.
Dependence on Historical Data: The beta coefficient is calculated based on past data
- However, company characteristics and market conditions may change.

Challenges Faced by CAPM#

In the 1970s, researchers found that small-cap stocks consistently outperformed CAPM predictions. This discovery of the "size effect" had a significant impact on academia and directly contributed to the later development of the Fama-French Three-Factor Model.

Evolution from Single-Factor to Multi-Factor#

Although CAPM has its limitations, it laid an important foundation for factor investing:

Development History of Factor Investing#

1960s: CAPM (Market Factor)
1990s: Fama-French Three-Factor (Market + Size + Value)
2000s: Carhart Four-Factor (Adding Momentum Factor)
2010s: Fama-French Five-Factor (Adding Profitability and Investment Factors)
Present: Multi-factor models with hundreds of factors

Just like biological evolution, investment models have gradually evolved from the single factor of CAPM to today's complex multi-factor systems.

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