Key Concepts in Asset Pricing Models

Why This Matters

Asset pricing models form the backbone of modern financial theory—they're how we answer the fundamental question: 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. You need to know not just the formulas, but when each model applies and what assumptions drive it.

Don't just memorize equations—understand what problem each model solves and what limitations it carries. The real exam payoff comes from recognizing which model fits which scenario and being able to compare their underlying assumptions. 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.

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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)

• Expected return is linear in beta—the model states $E(R_i) = R_f + \beta_i(E(R_m) - R_f)$, where beta measures sensitivity to market movements
• Only systematic risk is priced—diversification eliminates idiosyncratic risk, so investors aren't compensated for holding it
• Single-factor simplicity makes CAPM a foundational benchmark, though empirical tests reveal anomalies it can't explain

Intertemporal Capital Asset Pricing Model (ICAPM)

• Multi-period extension of CAPM—accounts for how investment opportunities and hedging demands change over time
• State variables matter—investors hedge against shifts in interest rates, volatility, or other economic conditions that affect future wealth
• Dynamic risk premiums emerge because the price of risk itself fluctuates with economic states

Arbitrage Pricing Theory (APT)

• Multiple systematic factors drive returns—not just market risk, but potentially inflation, GDP growth, credit spreads, and more
• No-arbitrage foundation—if prices deviate from factor-implied values, arbitrageurs exploit the mispricing until equilibrium restores
• Factor identification is empirical, giving APT flexibility but requiring careful statistical analysis to implement

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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

• Adds SMB and HML factors—$E(R_i) = R_f + \beta_m(R_m - R_f) + \beta_s \cdot SMB + \beta_v \cdot HML$, capturing size and value premiums
• Small-cap and value stocks outperform historically, which CAPM's single beta cannot explain
• Portfolio construction tool—managers use factor exposures to target specific risk premiums or neutralize unwanted tilts

Multifactor Models

• Extend beyond three factors—common additions include momentum (winners keep winning), profitability, investment intensity, and liquidity
• Risk vs. mispricing debate—are factor premiums compensation for bearing risk, or do they reflect behavioral anomalies?
• Quantitative strategies rely heavily on these models for systematic stock selection and risk decomposition

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Option Pricing Models

These models value derivative contracts by constructing replicating portfolios or 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

• Closed-form solution for European options—$C = S_0 N(d_1) - Ke^{-rT}N(d_2)$, where $d_1$ and $d_2$ depend on volatility, time, and rates
• Risk-neutral valuation—price the option as if investors are indifferent to risk, then discount at the risk-free rate
• Assumes constant volatility and continuous trading, which real markets violate—hence the volatility smile phenomenon

Binomial Option Pricing Model

• Discrete-time tree structure—asset price moves up or down each period with known probabilities, building a lattice of possible outcomes
• Handles American options—at each node, compare immediate exercise value to continuation value, enabling early exercise modeling
• Flexible assumptions—can incorporate changing volatility, dividends, and interest rates across the tree

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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

• General valuation framework—$V_0 = \sum_{t=1}^{n} \frac{CF_t}{(1+r)^t}$, applicable to any asset generating predictable cash flows
• Discount rate reflects risk—higher uncertainty demands a higher rate, reducing present value
• Versatile application across stocks, bonds, real estate, and capital budgeting projects

Dividend Discount Model (DDM)

• Equity-specific DCF—values 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—requires predictable dividend policy; inappropriate for growth companies that reinvest earnings
• Income investor focus—explicitly ties stock value to cash returned to shareholders

Gordon Growth Model

• Constant-growth simplification—assumes dividends grow at rate $g$ forever, yielding $P_0 = \frac{D_1}{r - g}$
• Requires $r > g$—if growth exceeds the discount rate, the formula breaks down (infinite value)
• Quick intrinsic value estimate for blue-chip stocks with stable dividend histories

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Quick Reference Table

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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?