Stochastic Discount Factor (SDF)

The SDF, denoted $m_{t+1}$, bridges the gap between a future stochastic payoff $x_{t+1}$ and its current price $p_t$.

Core Identity: $p_t = E_t[m_{t+1} \cdot x_{t+1}]$

• No-Arbitrage Guarantee: The existence of a strictly positive SDF is mathematically equivalent to the absence of arbitrage opportunities in a market.
• State-Price Density: In discrete states, $m_{t+1}$ represents the price of a pure contingent claim (Arrow-Debreu security) divided by the physical probability of that state occurring.
• Unified Framework: This single equation prices everything from risk-free bonds to complex derivatives by adjusting for both the passage of time and the riskiness of the cash flow.

The SDF reveals how the risk-free rate is determined by two distinct economic forces.

Conditional Expectation: $R_f = \frac{1}{E_t[m_{t+1}]}$

• Time Preference: A higher impatience (lower expected SDF) implies a higher risk-free rate, as investors demand more compensation to defer consumption.
• Precautionary Savings: The volatility of the SDF matters. By Jensen's inequality, a more volatile SDF increases its expected value, which paradoxically lowers the risk-free rate as investors rush to save in safe assets.
• Log-Normal Framework: Under log-normality, $r_f = -\ln \beta + \gamma E[\Delta c] - \frac{1}{2}\gamma^2 \sigma^2(\Delta c)$, explicitly showing the tension between intertemporal substitution and precautionary motives.

The SDF translates covariance with consumption into a risk premium, formalizing the intuition that assets paying off in bad times are more valuable.

Covariance Pricing: $E[R_i] - R_f = -R_f \cdot Cov(m, R_i)$

• Consumption Beta: An asset's risk is not its standalone volatility, but its covariance with the SDF (marginal utility). Assets that covary positively with consumption (negatively with the SDF) require a high premium.
• Factor Models: The Capital Asset Pricing Model (CAPM) and Fama-French models are special cases where the SDF is a linear function of market returns or other factors: $m = a + b'f$.
• Sharpe Ratio Bound: The volatility of the SDF relative to its mean sets the maximum possible Sharpe ratio in the economy: $\frac{\sigma(m)}{E[m]} \geq \frac{|E[R_i] - R_f|}{\sigma(R_i)}$.

A valid SDF must be strictly positive in all future states of the world to prevent arbitrage.

Economic Rationale: $m_{t+1} > 0$ in every state

• Marginal Utility: In consumption-based models, $m_{t+1} = \beta \frac{u'(c_{t+1})}{u'(c_t)}$. Since marginal utility is always positive, the SDF inherits this strict positivity naturally.
• Arbitrage Detection: If a candidate SDF takes a negative value in any state, it implies a negative state price, creating an arbitrage opportunity where an investor can receive a positive payoff for a negative cost.
• Hansen-Jagannathan Bound: The minimum variance SDF that prices a set of assets correctly must satisfy a specific mean-variance frontier, and any violation of positivity implies the model is misspecified.

The SDF extends naturally to multiple periods through a recursive law of motion, enabling the pricing of long-dated claims.

Telescoping Product: $M_{t+k} = \prod_{j=1}^{k} m_{t+j}$

• Long-Term Pricing: The price of a payoff at time $t+k$ is $p_t = E_t[M_{t+k} \cdot x_{t+k}]$. The multi-period SDF $M_{t+k}$ compounds the short-term drivers of risk and time preference.
• Epstein-Zin Preferences: In recursive utility models, the SDF depends on both current consumption growth and the return on the aggregate wealth portfolio, breaking the tight link between the risk-free rate and risk aversion found in power utility.
• Term Structure of Risk: The multi-period SDF explains why risk premiums for long-horizon claims (like equity strips) can differ significantly from short-horizon premiums, depending on the persistence of shocks to the SDF.

The empirical volatility of the SDF required to explain observed risk premiums is a critical diagnostic of model validity.

Empirical Challenge: $\sigma(m) \gg$ predicted by standard models

• The Puzzle: To justify the historical 6-7% equity premium with standard power utility, the coefficient of relative risk aversion must be implausibly high (often > 30), implying an extremely volatile SDF.
• Long-Run Risk Resolution: Models that incorporate a small, highly persistent component in consumption growth generate a highly volatile SDF through the recursive utility channel, resolving the puzzle with moderate risk aversion.
• Rare Disasters: A small, time-varying probability of a catastrophic economic collapse drastically increases the unconditional volatility of the SDF, as marginal utility spikes in the disaster state, rationalizing high equity premiums.