Capital Market Theory: Final Exam Guide (Topics 5, 6, and 7)

Vocabulary practice flashcards covering Capital Market Theory Topics 5, 6, and 7: the equity premium and risk-free rate puzzles, intertemporal consumption and Campbell-Shiller present-value relations, and complete markets with Arrow-Debreu pricing.

Lognormal Variable

A variable $x$ whose natural logarithm is normally distributed, written as $\log(x) \sim N(\mu, \sigma^2)$, satisfying the expectation formula $E[x^a] = e^{a\mu + \frac{1}{2}a^2\sigma^2}$ for any constant $a$.

Jensen's Inequality

A mathematical property stating that for any convex function $g$ ($g'' \ge 0$), the expectation of the function is at least the function of the expectation, $E[g(X)] \ge g(E[X])$, with the inequality reversing for concave functions.

Equity (Market) Risk Premium

The average extra return stocks earn over the risk-free asset, $E[R_m] - R_f$, which equals approximately $7.7\%$ per year in postwar data.

Sharpe Ratio

The ratio of expected excess return to return volatility, defined as $\frac{E[R_i] - R_f}{\sigma(R_i - R_f)}$, which equals approximately $0.43$ for the aggregate stock market.

Hansen-Jagannathan (HJ) Bound

The mathematical upper limit on the Sharpe ratio of any asset in an economy, stating that $\frac{|E[R_i] - R_f|}{\sigma(R_i - R_f)} \le \frac{\sigma_m}{E[m]}$, which requires SDF volatility to be at least $43\%$ of its mean to match market data.

Equity Premium Puzzle

The inability of the consumption-based asset pricing model with CRRA utility to explain the observed $7.7\%$ equity premium without requiring an implausibly high relative risk aversion ($\gamma \approx 55$, or at least $\gamma \approx 21.6$ via the HJ bound), caused by the smooth nature and low stock-market covariance of aggregate consumption.

Risk-Free Rate Puzzle

The failure that occurs when raising relative risk aversion to $\gamma \approx 55$ in the CRRA consumption model: the model predicts an excessively high risk-free rate that requires an implausible time-discount factor $\delta > 1$ ($\approx 1.16$) to reconcile with real data, while predicting counterfactually extreme fluctuations in the interest rate whenever consumption growth shifts.

Lifecycle Hypothesis

Modigliani's (1966) demographic framework positing that young individuals save while older individuals dissave, predicting that population and per-capita income growth raise aggregate savings and push the equilibrium risk-free rate down.

Surplus Consumption Ratio

In Campbell and Cochrane's (1999) habit-formation model, the proportion by which consumption exceeds the habit level, defined as $S_t = \frac{C_t - X_t}{C_t}$, which causes effective relative risk aversion ($\frac{\gamma}{S_t}$) to rise sharply in recessions.

Epstein-Zin Preferences

A recursive utility formulation that breaks the rigid constraint of CRRA utility by separating relative risk aversion ($\gamma$) from the elasticity of intertemporal substitution ($\frac{1}{\rho}$).

Preference for Early Resolution of Uncertainty

An economic preference in recursive Epstein-Zin utility where an agent with $\gamma > \rho$ strictly prefers to learn future consumption outcomes sooner rather than later, even if the information cannot be used to take any action.

Prospect Theory

Kahneman and Tversky's behavioral model of decision under risk featuring reference dependence, loss aversion, diminishing sensitivity, and nonlinear probability weighting.

Loss Aversion

The psychological phenomenon where losses hurt roughly twice as much as equivalent monetary gains, characterized by a value function that is steeper in the loss domain than in the gain domain.

Precautionary Saving

Additional saving undertaken by an agent to build a financial buffer against future consumption uncertainty.

Prudence

The condition of having a strictly positive third derivative of the utility function ($u'''(c) > 0$), which makes marginal utility convex so that uncertainty raises expected marginal utility and induces precautionary saving.

Relative Prudence

The dimensionless ratio $P(c) = -\frac{c\,u'''(c)}{u''(c)}$ (which equals $\gamma + 1$ under CRRA utility); saving increases with return risk if and only if $P(c) \ge 2$.

Certainty Equivalence

The property of quadratic utility where $u'''(c) = 0$, meaning marginal utility is linear and future consumption risk has zero influence on the optimal level of saving.

Mean-Preserving Spread

A transformation that adds mean-zero random noise to a payoff or return distribution, leaving the expected value unchanged while increasing its variance and risk.

Dividend-Discount Model (DDM)

A multi-period valuation framework derived from no-arbitrage equilibrium stating that the price of an asset equals the expected discounted sum of its future dividends: $p_t = E_t\left[\sum_{i=1}^{\infty} m_{t,t+i}\,d_{t+i}\right]$.

Transversality Condition

The mathematical boundary condition $\lim_{j \to \infty} E_t[m_{t,t+j}\,p_{t+j}] = 0$ that excludes asset price bubbles by ensuring the discounted price in the infinite future approaches zero.

Gordon Growth Model

A constant-growth formulation of the dividend-discount model yielding $p_t = \frac{(1 + g)\,d_t}{r - g} = \frac{d_{t+1}}{r - g}$, requiring the discount rate to exceed the dividend growth rate ($r > g$).

Excess Volatility

The empirical finding that stock prices fluctuate substantially more than can be justified by changes in the fundamental present value of expected future dividends under constant discount rates.

Shiller's Variance Bound

The inequality $\text{Var}(P^*) \ge \text{Var}(P)$ derived under constant expected returns, showing that the rational forecast price $P$ must be less volatile than the perfect-foresight realized present value of dividends $P^*$.

Campbell-Shiller (CS) Decomposition

A log-linear present-value identity approximating log prices into discounted cash-flow expectations and discounted discount-rate expectations: $p_t \approx \text{const} + p_{CF,t} - p_{DR,t}$.

Return Predictability

The empirical fact that the price-dividend ratio forecasts future multi-period excess returns rather than future dividend growth, confirming that stock price variations are driven predominantly by shifting discount rates.

Discount-Rate News ($N_{DR}$)

Revisions in investor expectations regarding future discount rates, defined as $N_{DR,t+1} = (E_{t+1} - E_t)\sum_{j=1}^{\infty} \rho^j\,r_{t+j+1}$, which enter unexpectedly with a negative sign into contemporaneous returns.

Complete Markets

A financial market setting where the asset span equals $\mathbb{R}^S$ (the payoff matrix $X$ has rank equal to the number of states $|S|$), enabling every state-contingent payoff profile to be synthesized.

Arrow-Debreu (A-D) Security

A primitive contingent claim that pays exactly 1 unit of consumption if a designated state of the world occurs tomorrow and 0 units in all other states.

State Price

The current price $q(s)$ of an Arrow-Debreu security for state $s$, which equals the physical probability times the stochastic discount factor: $q(s) = \pi(s)\,m(s)$.

Arbitrage

An investment strategy requiring zero upfront capital that produces non-negative payoffs across all states and a strictly positive payoff in at least one state.

Law of One Price (LOOP)

The economic principle that two assets or portfolios delivering identical state-contingent future payoffs must trade at the same price today to avoid instantaneous arbitrage.

Fundamental Theorem of Finance

The core theorem stating that market security prices rule out arbitrage if and only if there exist strictly positive state prices ($q(s) > 0$ for all $s$), or equivalently, strictly positive risk-neutral probabilities ($\pi^*(s) > 0$ for all $s$).

Risk-Neutral Probability

The normalized state price measure $\pi^*(s) = \frac{q(s)}{\sum_s q(s)} = \frac{m(s)}{E[m]}\,\pi(s)$, which reweights real probabilities upward for bad economic states where marginal utility is high.

State Price Density

The price per unit of physical probability, defined as $\frac{q(s)}{\pi(s)}$, which equals the agent's discounted marginal rate of substitution and governs the state-contingent consumption path.

Mutuality Principle

The complete-markets equilibrium theorem stating that all idiosyncratic individual risks are fully pooled and insured, leaving consumption exposed solely to non-diversifiable aggregate (systematic) risk.

Representative Agent

A theoretical aggregate consumer whose consumption equals total macroeconomic endowment and whose marginal rate of substitution defines the economy-wide stochastic discount factor.

Incomplete Markets

A market environment where the number of linearly independent assets is smaller than the number of states, preventing full risk sharing, eliminating the unique state-price vector, and causing the equilibrium to be only constrained Pareto efficient.