Key Concepts in Asset Pricing Models

Why This Matters

Asset pricing models are how we answer a fundamental question in finance: what should this asset be worth? Whether you're valuing a stock, pricing an option, or constructing a portfolio, you're applying one of these frameworks. Exam questions will test your understanding of risk-return relationships, no-arbitrage principles, factor exposures, and present value mechanics.

Knowing the formulas matters, but knowing when each model applies and what assumptions drive it matters more. If you can explain why CAPM uses one factor while Fama-French uses three, or why binomial trees handle American options better than Black-Scholes, you're thinking like a financial mathematician. Focus on recognizing which model fits which scenario and comparing their underlying assumptions.

---

Risk-Return Equilibrium Models

These models establish the theoretical relationship between risk and expected return in efficient markets. The core principle: investors demand compensation for bearing systematic risk, and equilibrium prices reflect this trade-off.

Capital Asset Pricing Model (CAPM)

The CAPM says expected return is a linear function of beta:

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

Here, $\beta_i$ measures how sensitive asset $i$ is to movements in the overall market. $R_f$ is the risk-free rate, and $E(R_m) - R_f$ is the market risk premium.

• Only systematic risk is priced. Diversification eliminates idiosyncratic (firm-specific) risk, so investors aren't compensated for holding it.
• Single-factor simplicity makes CAPM a foundational benchmark, but empirical tests reveal anomalies it can't explain (e.g., the size effect and value premium).
• Key assumptions include mean-variance optimization by all investors, homogeneous expectations, and frictionless markets.

Intertemporal Capital Asset Pricing Model (ICAPM)

The ICAPM extends CAPM to a multi-period setting. Investors don't just care about next period's return; they also care about how investment opportunities might shift over time.

• State variables matter. Investors hedge against shifts in interest rates, volatility, or other economic conditions that affect future wealth. Each hedgeable state variable can generate its own risk premium.
• Dynamic risk premiums emerge because the price of risk itself fluctuates with economic states, unlike CAPM where the risk premium is static.

Arbitrage Pricing Theory (APT)

APT allows multiple systematic factors to drive returns, not just market risk. These could include inflation surprises, GDP growth, credit spreads, term structure changes, and more.

• No-arbitrage foundation. If prices deviate from factor-implied values, arbitrageurs exploit the mispricing until equilibrium restores. This is a weaker assumption than CAPM's full equilibrium.
• Factor identification is empirical. APT doesn't tell you which factors matter; you have to find them through statistical analysis. This gives APT flexibility but makes implementation harder.

---

Factor-Based Equity Models

These models refine equilibrium pricing by identifying specific characteristics that explain cross-sectional return differences. The insight: certain stock attributes (size, value, momentum) carry persistent risk premiums beyond market beta.

Fama-French Three-Factor Model

This model adds two factors to CAPM:

$E(R_i) = R_f + \beta_m(R_m - R_f) + \beta_s \cdot SMB + \beta_v \cdot HML$

• SMB (Small Minus Big) captures the size premium: small-cap stocks have historically outperformed large-cap stocks.
• HML (High Minus Low) captures the value premium: stocks with high book-to-market ratios have historically outperformed growth stocks.
• Portfolio construction tool. Managers use factor exposures (the betas) to target specific risk premiums or neutralize unwanted tilts.

These two anomalies were well-documented patterns that CAPM's single beta couldn't explain, which motivated the model's development.

Multifactor Models

Multifactor models extend beyond three factors. Common additions include:

• Momentum (winners keep winning over intermediate horizons)
• Profitability (more profitable firms earn higher returns)
• Investment intensity (firms that invest aggressively tend to earn lower returns)
• Liquidity (less liquid assets demand a premium)

A central debate: are factor premiums compensation for bearing genuine risk, or do they reflect behavioral anomalies like investor overreaction? Quantitative strategies rely heavily on these models for systematic stock selection and risk decomposition.

---

Option Pricing Models

These models value derivative contracts by constructing replicating portfolios or computing risk-neutral expectations. The unifying principle: in a no-arbitrage world, the price of an option equals the cost of hedging it perfectly.

Black-Scholes Option Pricing Model

The Black-Scholes formula gives a closed-form solution for European options:

$C = S_0 N(d_1) - Ke^{-rT}N(d_2)$

where:

• $S_0$ is the current stock price, $K$ is the strike price, $r$ is the risk-free rate, $T$ is time to expiration
• $N(\cdot)$ is the cumulative standard normal distribution function
• $d_1$ and $d_2$ depend on volatility ($\sigma$), time, the stock price, strike, and the risk-free rate

Key features:

• Risk-neutral valuation. You price the option as if all investors are indifferent to risk, then discount at the risk-free rate. This works because the replicating portfolio argument eliminates risk preferences from the equation.
• Assumes constant volatility and continuous trading, which real markets violate. This mismatch produces the volatility smile, where implied volatility varies with strike price rather than staying flat as the model predicts.

Binomial Option Pricing Model

The binomial model uses a discrete-time tree structure: at each time step, the asset price moves up by factor $u$ or down by factor $d$, building a lattice of possible outcomes.

• Handles American options. At each node, you compare the immediate exercise value to the continuation value (the discounted expected value of holding). This makes early exercise modeling straightforward.
• Flexible assumptions. You can incorporate changing volatility, discrete dividends, and varying interest rates across the tree.
• As you increase the number of time steps, the binomial model converges to the Black-Scholes price for European options.

---

Present Value and Cash Flow Models

These models value assets by discounting future cash flows to today. The foundation: a dollar today is worth more than a dollar tomorrow, so future payments must be adjusted for the time value of money.

Discounted Cash Flow (DCF) Model

The general DCF framework values any cash-flow-generating asset:

$V_0 = \sum_{t=1}^{n} \frac{CF_t}{(1+r)^t}$

• Discount rate reflects risk. Higher uncertainty demands a higher rate $r$, which reduces present value. For equities, $r$ is often estimated using CAPM or a multifactor model.
• Versatile application across stocks, bonds, real estate, and capital budgeting projects. The challenge is always in forecasting the cash flows $CF_t$ and choosing the right discount rate.

Dividend Discount Model (DDM)

The DDM is an equity-specific version of DCF that values a stock as the present value of all future dividends:

$P_0 = \sum_{t=1}^{\infty} \frac{D_t}{(1+r)^t}$

• Best for mature, stable firms with predictable dividend policies. It's inappropriate for growth companies that reinvest most earnings rather than paying dividends.
• The model explicitly ties stock value to cash returned to shareholders, making it a natural fit for income-focused analysis.

Gordon Growth Model

The Gordon Growth Model simplifies the DDM by assuming dividends grow at a constant rate $g$ forever:

$P_0 = \frac{D_1}{r - g}$

where $D_1$ is next year's expected dividend.

• Requires $r > g$. If the growth rate equals or exceeds the discount rate, the formula produces an infinite or negative value, which is meaningless.
• Useful as a quick intrinsic value estimate for blue-chip stocks with stable dividend histories. For example, if a stock pays $D_1 = \$2.00$, $r = 0.10$, and $g = 0.04$, then $P_0 = \frac{2.00}{0.10 - 0.04} = \$33.33$.

---

Quick Reference Table

---

Self-Check Questions

1. Which two models both rely on no-arbitrage arguments but apply to different asset classes (equities vs. derivatives)?

2. If you're valuing an American call option on a dividend-paying stock, which model should you use and why can't you use Black-Scholes directly?

3. Compare and contrast CAPM and the Fama-French Three-Factor Model. What anomalies motivated the addition of SMB and HML factors?

4. A company has unpredictable, rapidly changing dividends. Why would the Gordon Growth Model be inappropriate, and what alternative approach would you use?

5. An FRQ asks you to explain why APT is more flexible than CAPM but harder to implement. What are the key trade-offs you would discuss?