In consumption-based asset pricing, the risk-free rate puzzle of Weil (1989) is the finding that with time-separable preferences the average real risk-free rate implied by observed consumption growth is too high relative to observed safe rates unless risk aversion is implausibly low, while the high risk aversion needed for the equity premium makes the mismatch worse. Choi (2014) showed that an external habit ratio model, calibrated on Korean annual data, provides only a partial resolution: it matches the mean observed rate for some safe-rate proxies (estimated CD yields, hybrid bond series) but fails unexplainedly for the time-deposit rate, assumes homoscedastic lognormal consumption growth, does not jointly match the equity premium, and leaves the habit-persistence parameter chosen from a grid rather than disciplined by data or theory. The open problem is to determine whether habit formation can resolve the risk-free rate puzzle in the Korean market robustly across safe-rate proxy choices, under realistic (possibly heteroscedastic or disaster-prone) consumption dynamics, jointly with the equity premium, and with the habit parameter estimated or derived from primitives.

Background

Consumption-based asset pricing prices assets by the Euler equation of a representative consumer: the expected intertemporal marginal rate of substitution equals the gross return of any traded asset. With time-separable constant-relative-risk-aversion (CRRA) preferences and jointly lognormal, homoscedastic consumption growth, the implied average log real risk-free rate is

rf≈−ln⁡δ+ρ μc−12ρ2σc2,r_f \approx -\ln\delta + \rho\,\mu_c - \tfrac{1}{2}\rho^2\sigma_c^2,

where δ\delta is the subjective discount factor, ρ\rho is relative risk aversion, and μc,σc2\mu_c,\sigma_c^2 are the mean and variance of log consumption growth. The equity premium puzzle of Mehra and Prescott (1985) is that matching the historical average excess return of equities over safe assets requires risk aversion far above conventional ranges; the risk-free rate puzzle of Weil (1989) is the companion failure that, with smooth consumption (small σc\sigma_c), the same equation implies an average safe rate well above the observed one unless the discount factor or risk aversion takes implausible values, so that fixing one puzzle aggravates the other.

Habit formation relaxes time separability: current utility depends not only on current consumption but on a habit level built from past consumption. In external (or "catching up with the Joneses", Abel 1990) specifications the habit is driven by aggregate past consumption, so a representative consumer's marginal utility today moves with yesterday's aggregate consumption. Constantinides (1990) introduced habit formation as a resolution of the equity premium puzzle; Campbell and Cochrane (1999) showed, for US data, that a surplus-consumption external habit model (in which utility depends on the surplus of consumption over a slow-moving habit level, rather than on their ratio) can jointly match a high equity premium and a low, nearly constant risk-free rate. In the simpler external habit ratio model used by Choi (2014), utility takes the form U(Ct,Xt)=(Ct/Xt)1−ρ/(1−ρ)U(C_t,X_t)=(C_t/X_t)^{1-\rho}/(1-\rho) with the habit Xt=Ct−1κX_t=C_{t-1}^{\kappa}, where κ∈(0,1)\kappa\in(0,1) measures the degree of time nonseparability; under joint lognormality and homoscedasticity the implied log risk-free rate acquires an additional habit term that mirrors past consumption and lowers the implied rate as κ\kappa rises, which is the channel by which the model can meet the data at moderate risk aversion.

Choi (2014) calibrated this model on Korean annual data (samples spanning 1965/1975–2008), where the choice of a proxy for the risk-free rate is genuinely unresolved: Korean Treasury Bills exist only from 2000, so researchers have used time-deposit rates, certificate-of-deposit (CD) yields, monetary stabilization bond yields (short-term Bank of Korea securities), and spliced hybrid series, whose average real returns differ by several percentage points per year. The paper's acceptance criterion is the field's standard calibration convention: the first-moment restriction, testing whether the mean implied risk-free rate differs statistically from the mean observed real return, within conventionally acceptable risk aversion (roughly 1–10). It finds the external habit ratio model satisfies the restriction for the CD-yield and hybrid-bond proxies at moderate risk aversion but fails for the time-deposit rate proxy, an anomaly it reports but cannot explain; its derivations assume homoscedastic, jointly lognormal consumption growth (excluding time-varying volatility or rare-disaster mechanisms of the type studied by Barro 2006); it does not jointly fit the equity premium; and κ\kappa is selected on a grid rather than estimated or derived. Heaton (1993) showed, for US data, that the interaction of time-nonseparable preferences with time aggregation can seriously bias inferences from annual data, a caveat that applies to annual Korean calibrations.

Problem Statement

Determine whether habit formation resolves the risk-free rate puzzle in the Korean market robustly and jointly. Concretely: specify a habit-based consumption asset-pricing model (external habit, driven by aggregate past consumption, or internal habit, driven by the consumer's own past consumption; in ratio or surplus-consumption form), estimated or calibrated on Korean data with a realistic consumption-growth process (allowing time-varying volatility or rare-disaster-type tails rather than imposing homoscedastic lognormality), and establish whether it can (i) match the first moment of the observed real return for each standard Korean safe-rate proxy — time-deposit rates, CD yields, monetary stabilization bond and Treasury Bill yields, and their spliced hybrids — within conventionally acceptable relative risk aversion, where the source study's model fails unexplainedly for the time-deposit rate; (ii) simultaneously match the equity premium, so that both puzzles are resolved by one preference structure; (iii) discipline the habit-persistence parameter by estimation or derivation from primitives rather than grid selection; and (iv) survive correction for the time-aggregation bias that annual data induces in time-nonseparable preferences. Clauses (i)–(iv) are joint conditions on a single habit-based specification under one parameterization that simultaneously matches the observed first moment for every standard safe-rate proxy — rather than fitting each proxy separately — matches the equity premium under the same mean-matching convention and conventionally acceptable risk aversion as (i), disciplines the habit-persistence parameter by estimation or derivation, and remains valid once the time-aggregation bias is corrected. Either such a model exists with the supporting moment evidence across proxies, or no such specification succeeds within the conventionally acceptable risk-aversion range, in which case the component responsible for the failure — proxy-specific premia, consumption dynamics, or the preference structure itself — is also to be determined.

References

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