Asset Premium Puzzles
Are They Really Puzzles?
1 Introduction
Since the 1980s, financial economists have grappled with two so-called “puzzles” in asset pricing theory: the Equity Premium Puzzle (EPP) and the Risk-Free Rate Puzzle (RFRP). These puzzles refer to the pronounced gap between theoretical predictions derived from standard rational-expectations models with representative-agent utility maximization and the empirical observations on asset returns and risk-free rates. In particular, Mehra and Prescott (1985) and Weil (1989) spotlighted the disparity between observed excess returns on equities and the relatively tame variability of aggregate consumption, labeling the phenomenon a “puzzle” under the canonical equilibrium framework.
Beneath both puzzles lies a single “master equation,” namely the Euler equation, which in its most general form states:
\[ \mathbb{E}_t \big[m_t R_{t+1}\big] = 1, \]
where \(m_t\) (the Stochastic Discount Factor) is closely tied to the marginal utility of consumption, and \(R_{t+1}\) represents the return on various assets. The equity premium and risk-free rate each follow from this broad formulation, yet empirical magnitudes diverge significantly from conventional theoretical predictions.
2 Theoretical Framework
2.2 Risk-Free Rate Puzzle
A closely related conundrum is the Risk-Free Rate Puzzle, initially highlighted by Weil (1989). Under the same CRRA framework and rational expectations, the Euler equation for the risk-free asset implies:
\[ 1 = \mathbb{E}_t \Big[\beta \Big(\frac{C_{t+1}}{C_t}\Big)^{-\gamma} R_{f,t+1}\Big]. \]
Approximating in logs,
\[ r_f \approx \delta + \gamma \,g_c - \frac{1}{2}\,\gamma\,(\gamma + 1)\,\sigma_c^2, \]
where:
- \(r_f = \ln(R_f)\): the log of the risk-free rate,
- \(\delta = -\ln(\beta)\): time preference rate,
- \(g_c = \mathbb{E}[\ln(C_{t+1}/C_t)]\): average consumption growth rate.
Empirically, long-run real risk-free rates are typically in the 1%–3% range, whereas the above equation might predict rates of 4%–8% given plausible values for \(\gamma\), \(\delta\), and \(g_c\). Again, the severe mismatch between model forecasts and observed data has led researchers to classify it as a “puzzle.”
2.3 Critical Assumptions
To preserve tractability, the standard model assumes:
- A representative agent — but who truly “represents” the market?
- Time-separability of the utility function — ensuring that period utilities add linearly over time.
- Global concavity of CRRA utility — guaranteeing diminishing marginal utility at all consumption levels.
While these assumptions yield elegant closed-form solutions, they may excessively simplify real-world heterogeneity. Crucially, when the economy scales up over time, CRRA utility remains well-defined, but in practice this might obscure the role of vastly different consumption paths across distinct wealth brackets.
3 Critical Perspective
3.1 Rethinking ‘Rational Expectations’
Traditionally, Rational Expectations is seen as a condition that agents use all available information efficiently, forming unbiased forecasts. However, from a purely mathematical standpoint, “expectations” and “covariances” are simply operators for dealing with means and correlations of random variables. Such operators—particularly bilinear forms and inner products—require specific algebraic properties (linearity, symmetry, Cauchy–Schwarz inequality, etc.). While these simplifications can be powerful in physics or engineering, in economics they might be overly restrictive when applied to highly heterogeneous populations and institutions.
3.2 The Euler Equation
By construction, the Euler condition \(\mathbb{E}_t [m_{t+1} R_{t+1}] = 1\) is mathematically akin to an inner product on a probability space. This yields a focus on second moments (variance, covariance) and often leads to elliptical distribution assumptions (e.g., normal, Student-\(t\)). Real-world wealth distributions and market participation, however, may be far from elliptical in their statistical properties—especially when only a small fraction of the population holds the majority of risky assets.
3.3 Implications of Globally Concave CRRA Utility
CRRA utility, with its global concavity, implies a declining marginal utility as consumption grows. If aggregate consumption (\(C_t\)) trends upward over time, the ratio of marginal utilities \(\bigl[u'(C_{t+1}) / u'(C_t)\bigr]\) naturally declines, ensuring an inverse relationship between the SDF (\(m_{t+1}\)) and any asset (or variable) with a long-term growth trend. In equity markets, returns \(R_{t+1}\) also tend to grow over time, so \(m_t\) and \(R_{t+1}\) end up negatively correlated by construction.
If one then chooses a suitably volatile variable (with sufficient high variance) to stand in for \(m_{t+1}\), one can reconcile observed excess returns with the theoretical predictions—effectively defusing the puzzle. In that sense, the puzzle may be an artifact of incomplete modeling of real-world heterogeneity.
5 Conclusion
The Equity Premium Puzzle and Risk-Free Rate Puzzle have dominated discussions in asset pricing for decades. However, labeling them as genuine “puzzles” may reflect an artifact of restrictive models that hinge on a single representative agent, uniform preferences, and high-level assumptions about consumption growth. By introducing heterogeneous market participation, particularly the reality that a small fraction of wealthy agents holds the lion’s share of risky assets, one finds that what appears to be a puzzle for the average consumer is, in fact, quite explicable among those who actually drive stock prices.
In short, when empirical ownership and wealth concentration data are properly accounted for, the puzzling gaps between theory and observation can diminish or disappear. The challenge remains to integrate heterogeneous agent frameworks with accurate micro-level data on wealth and consumption in order to provide a more comprehensive understanding of asset prices—an endeavor that holds promise for reconciling the so-called “puzzles” with empirical reality.