Asset Pricing and Risk-Sharing under DB vs DC Pensions: Coimbra, Gomes, Michaelides & Shen (2026)

What this is. The paper’s core results, model equations, solution method, and datasets: enough to know what it found and how without reading all 46 pages. To replicate or extend it, read the full source at the original.

TL;DR

The paper builds a general equilibrium incomplete-markets model with an explicit defined-benefit (DB) pension fund. Calibrated to U.S. data, the model matches the historical equity premium (7.46% vs. 7.55% in data), the riskless rate (1.16% vs. 0.86%), and the Sharpe ratio (0.39 vs. 0.36) better than a standard pass-through model that ignores the fund’s endowment and asset demands. The DB fund’s relatively conservative portfolio lowers the riskless rate and raises the equity premium; stochastic contribution rates create a new risk channel that raises consumption volatility for workers and firms. A shift to a DC-only economy produces a higher riskless rate (3.34% vs. 1.16%), a lower equity premium (4.96% vs. 7.46%), a lower Sharpe ratio (0.27 vs. 0.39), higher consumption volatility for retirees, and lower consumption volatility for workers.

Core results

Magnitudes and significance are as reported. Locators point into the source PDF (page numbers match the journal pagination printed on each page).

#	Result	Locator	Magnitude
R1	Baseline DB model closely matches key asset pricing moments	Table II, p. 164	Equity premium 7.46% (data 7.55%); riskless rate 1.16% (data 0.86%); Sharpe ratio 0.39 (data 0.36); stock market participation 59.1% (data 51.1%)
R2	DB model dominates pure-pass-through (PPG) model at same calibration: much higher equity premium and Sharpe ratio	Table III, p. 167	Baseline: equity premium 7.46%, Sharpe 0.39 vs. PPG: 5.53%, 0.31; PPG riskless rate 4.04% vs. baseline 1.16%
R3	DB pension fund’s conservative bond demand is the first pricing channel: raising equity premium and lowering riskless rate	Table V, p. 171	Varying risky share from 42% to 72% moves equity premium from 10.60% to 5.48% and Sharpe ratio from 0.55 to 0.28
R4	Stochastic DB contribution rates create a new risk channel: higher cross-sectional consumption volatility for workers	Table IV, p. 170	SD consumption growth ages 20-35: 10.7% (baseline) vs. 10.0% (rPPG1); ages 36-65: 8.4% vs. 7.6%; retirees (66+): 2.2% vs. 2.6%
R5	Asset pricing results are robust across alternative DB fund portfolio allocation rules	Table II cols (1)-(3), p. 164	Fixed vs. elastic vs. reaching-for-yield: equity premium 7.46%, 7.58%, 7.39%; Sharpe ratio 0.39, 0.39, 0.38
R6	DC-only economy (DB phased out) has a substantially higher riskless rate and lower equity premium	Table IX col (2), p. 182	Riskless rate 3.34% vs. 1.16%; equity premium 4.96% vs. 7.46%; Sharpe ratio 0.27 vs. 0.39
R7	In the DC-only economy, retiree consumption volatility increases while worker consumption volatility decreases	Table IX col (2), p. 182	SD cons. growth ages 20-35: 10.1% vs. 10.7%; ages 36-65: 7.4% vs. 8.4%; retirees (66+): 2.8% vs. 2.2%
R8	Lower stock market participation costs in DC economy raise participation but have modest aggregate price effects	Table IX col (3), p. 182	Participation rises to 79.7% (vs. 57.2% at baseline DC costs); equity premium 4.60% and riskless rate 3.44%, both close to base DC scenario

Overall (paper’s conclusion). The endowment and asset demands of DB pension funds matter for asset pricing and risk sharing in ways that purely PPG models miss. The shift toward DC plans is characterized by a higher riskless rate, a lower equity premium and Sharpe ratio, and a redistribution of consumption risk from retirees to workers.

Theory / model

The model is an incomplete-markets overlapping-generations (OLG) production economy, using the framework of Storesletten, Telmer, and Yaron (2007) as a benchmark. Households live from age 20 (adult age 1) to 100 (adult age 81), working until age 65 and retiring thereafter. Two household types (A and B) have heterogeneous discount factors and EIS but the same risk aversion.

Production technology (eq. 1, p. 149).

$$Y_t = Z_t K_t^{\alpha} L_t^{1-\alpha}$$

where $Z_t = G_t U_t$ is aggregate productivity ($G_t = (1+g)^t$ is deterministic growth; $U_t$ is a two-state Markov business-cycle shock), $K$ is the beginning-of-period capital stock, and $L$ is labor supply (eq. 2, p. 149).

Stochastic depreciation (eq. 3, p. 149).

$$\delta_t = \bar{\delta}(U_t) + \sigma^{\delta}(U_t) \cdot \eta_t$$

where $\eta_t$ is i.i.d. standard normal and both the conditional mean and standard deviation of depreciation are correlated with $U_t$. This device avoids explicit adjustment costs while generating return volatility in the incomplete-markets setting, sharing the stochastic depreciation device for tractability with Favilukis, Ludvigson, and Van Nieuwerburgh (2017).

Household preferences: Epstein-Zin-Weil (eq. 9, p. 151).

$$V_{a,t} = \left\{ (1-\beta)(C^i_{a,t})^{1-1/\psi} + \beta\left( \mathbb{E}_t\!\left(p_a V^{1-\gamma}_{a+1,t+1}\right) \right)^{(1-1/\psi)/(1-\gamma)} \right\}^{1/(1-1/\psi)}$$

where $\beta$ is the discount factor, $\gamma$ is relative risk aversion, and $\psi$ is the elasticity of intertemporal substitution (EIS). Type-B households: $\beta^B = 0.965$, $\psi^B = 0.65$; type-A: $\beta^A = 0.83$, $\psi^A = 0.25$; both have $\gamma = 6$ (Table I, p. 160).

Labor income (eqs. 13-16, pp. 152-153). Individual labor income is $H^i_{a,t} = W_t L^i_{a,t}$, with individual productivity $L^i_{a,t} = P^i_{a,t} \epsilon^i_t$ (permanent component times transitory shock). The permanent component follows:

$$P^i_{a,t} = \exp(f(a))\, P^i_{a-1,t-1}\, \xi^i_t$$

where $f(a)$ is a deterministic age profile (hump-shaped). Following Guvenen, Ozkan, and Song (2014), $\ln \xi^i_t$ is a mixture of two normals conditional on the aggregate state $U_t$, capturing countercyclical earnings risk.

Retirement income (eq. 17, p. 153). Retired households receive:

$$H^i_{a,t} = (\lambda^{ss} + \lambda^{db})\, P^i_{a^R, t^R}\, W_t, \qquad a > a^R$$

where $\lambda^{ss}$ and $\lambda^{db}$ are the social security and DB pension replacement ratios, calibrated to 0.4596 and 0.2225 respectively (Table I).

Pension fund endowment and return (eq. 19, p. 155).

$$R^P_t = \alpha^P R^K_t + (1 - \alpha^P) R^B_t$$

where $\alpha^P$ is the risky (equity) share of the pension fund portfolio, calibrated to 52% to match the historical Flow of Funds average.

Pension fund budget constraint and contribution rates. The fund keeps endowment $\omega^P$ constant and adjusts contribution rates each year. In the general case (eq. 22-24, pp. 156-157), the shortfall before adjustments is:

$$\tilde{\omega}^P_t = (1 + R^P_t)\, \omega^P + \sum_{a=20}^{65} \int_{i \in I^a} \bar{\tau}^{db} L^i_{a,t}\, w_t\, di - \sum_{a=66}^{100} \int_{i \in I^a} \left[\lambda^{db} \exp(f(a^R))\, w_t\, P^i_{a^R, t^R}\right] di$$

The fraction $\theta^P$ of the shortfall is absorbed by employer contributions:

$$\tau^{kdb}_t = \theta^P \cdot \frac{\omega^P - \tilde{\omega}^P_t}{k_t - \alpha^P \omega^P}$$

and the remainder $(1 - \theta^P)$ by employee contributions:

$$\tau^{db}_t - \bar{\tau}^{db} = (1 - \theta^P) \cdot \frac{\omega^P - \tilde{\omega}^P_t}{\displaystyle\sum_{a=20}^{65} \int_{i \in I^a} L^i_{a,t}\, w_t\, di}$$

Baseline calibration: $\theta^P = 0.5$ (equal split). This stochastic adjustment is the new risk channel: return shocks feed into net wages and firm profits, raising cross-sectional consumption volatility. Building on Constantinides and Duffie (1996), stochastic contribution rates raise idiosyncratic income risk for workers and firms, which raises the equity premium.

Government budget constraint (eq. 8, p. 151).

$$C^G_t + (1 + R^B_t)\, B_t = B_{t+1} + T_t$$

Bond supply is calibrated to a debt-to-GDP ratio of 42% (average U.S. Treasury holdings by the public). Interest payments are financed by taxes on capital income (rate $\tau^K = 40\%$), bond interest ($\tau^B = 20\%$), wages ($\tau^W$), and bequests.

Equilibrium conditions (eqs. 34-36, p. 160). Markets clear in capital, bonds, and the consumption good:

$$k_{t+1} = \int \int P^i_{a,t}\, k^i_{a,t+1}\, da\, di$$ $$b_{t+1} = \int \int P^i_{a,t}\, b^i_{a,t+1}\, da\, di$$ $$U_t k_t^{\alpha} L_t^{1-\alpha} = \frac{C^G_t}{G_t^{1/(1-\alpha)}} + (1+g)^{1/(1-\alpha)} k_{t+1} - (1-\delta_t) k_t + \int \int P^i_{a,t}\, c^i_{a,t}\, da\, di$$

Method

The paper contributes a calibrated structural model, not a new econometric method. The solution follows the krusell-smith approximate-aggregation approach, building on overlapping-generations, epstein-zin-weil preferences, and incomplete-markets-olg techniques.

Household optimization (eq. 30, pp. 158-159). Households solve the Bellman equation:

$$V_a(x^i_{a,t}, E^i_a, k_t, U_t, \eta_t, P^B_t) = \max_{k^i_{a+1,t+1},\, b^i_{a+1,t+1}} \left\{ (1-\beta) (c^i_{a,t})^{1-1/\psi} + \beta \left( \mathbb{E}_t \left[ \left(\frac{P^i_{a+1,t+1}}{P^i_{a,t}} (1+g)\right)^{1-\rho} p_a V^{1-\rho}(x^i_{a+1,t+1}, E^i_{a+1}; k_{t+1}, U_{t+1}, \eta_{t+1}, P^B_{t+1}) \right] \right)^{(1-1/\psi)/(1-\rho)} \right\}^{1/(1-1/\psi)}$$

subject to: $k^i \geq 0$, $b^i \geq 0$, budget constraint $c + b' + k' = x$ (eq. 32), and the wealth transition (eqs. 26-27) that includes after-tax capital and bond income, net labor income (less social security and DB contribution taxes for workers), and retirement income (for retirees).

State variables: age $a$, normalized cash-on-hand $x^i_{a,t}$, entry-cost dummy $E^i_a$, plus four aggregate variables $(k_t, U_t, \eta_t, P^B_t)$.

Aggregate forecasting rules (eqs. 28-29, p. 158).

$$k_{t+1} = \Gamma_K(k_t, U_t, \eta_t)$$ $$P^B_{t+1} = \Gamma_P(P^B_t, k_t, U_t, \eta_t)$$

These log-linear rules are estimated on simulated data and iterated to convergence (following Krusell and Smith 1998, and Gomes and Michaelides 2008).

Calibration procedure. Aggregate parameters are calibrated to NBER business cycle frequencies (Markov chain $\pi_r = 16/37$), capital share $\alpha = 34\%$, depreciation $\text{mean}(\delta) = 10\%$, $\text{vol}(\delta) = 10\%$. Household parameters are chosen to jointly match: the standard deviation of consumption growth, the riskless rate level, and limited stock market participation. The pension fund risky share ($\alpha^P = 52\%$) is calibrated to the 1970-2023 Flow of Funds average (Table I, p. 160; Section I.H.3, p. 162).

Alternative DB pension fund portfolio rules (eqs. 37-38, p. 166).

The baseline uses a constant $\alpha^P$. Two alternatives are also studied: elastic allocation proportional to the equity premium:

$$\alpha^P_t = a^P + c^P \bigl(\mathbb{E}[R^K_t] - R^B_t\bigr), \qquad c^P = 0.25$$

and reaching-for-yield allocation varying with the riskless rate:

$$\alpha^P_t = a^P + b^P R^B_t, \qquad b^P = -2$$

Both deliver nearly identical asset pricing moments (Table II, cols 1-3), so the results are insensitive to the precise specification.

DC-only counterfactual (eq. 39, p. 178). Setting $\lambda^{db} = 0$ and $\omega^P = 0$ removes the DB pension fund entirely; households finance retirement from private savings and social security only. The DC economy incorporates tax benefits via a reduced capital gains tax rate scaled to the increase in private household wealth, and a 10% early-withdrawal penalty (Sections V.B.1-V.B.2, pp. 179-180). The numerical solution adds an outer loop to find the fixed point for the implied capital gains tax adjustment.

Empirical specifications

The paper does not estimate regression equations. All quantitative results are steady-state moments from the calibrated structural model, compared against empirical counterparts. The “specifications” are the alternative calibrated economies:

Baseline DB economy (R1, R5): $\alpha^P = 52\%$, $\theta^P = 0.5$, two household types (A and B), calibrated to match riskless rate SD, participation rate, and equity premium. Key moments computed at the stationary distribution of the model (Table II, p. 164). Asset pricing data from CRSP; real risk-free rate from Croce et al. (2012); participation from SCF; consumption/GDP from NIPA 1929-2023.

Comparison with PPG model (R2): same parameter values, but $\omega^P = 0$ so the fund is a pure pass-through with no endowment and constant contribution rates (Table III, p. 167). Two recalibrations (rPPG1, rPPG2) additionally match the riskless rate or consumption growth SD of the baseline.

Risk channel decomposition (R3): the pension fund risky share $\alpha^P$ is varied from 42% to 72% (Table V, p. 171) to isolate the bond-demand channel; all other parameters are held at the baseline.

Consumption risk sharing by age group (R4): cross-sectional standard deviation of consumption growth reported by cohort (ages 20-35, 36-65, 66+) for the baseline and rPPG1 economy (Table IV, p. 170).

Adjustment rule robustness (R5): $\theta^P$ is varied from 0.2 (mostly employee adjustment) to 0.8 (mostly employer adjustment) with the same aggregate parameters (Table VIII, p. 177).

DC-only counterfactual (R6-R8): the DB fund is shut down ($\lambda^{db} = 0$, $\omega^P = 0$) and the model is solved for the new stationary equilibrium. Three scenarios: (1) same participation costs, (2) lower participation costs ($F^0 = 3\%$, $F^1 = 0.1\%$), (3) higher debt-to-GDP (0.6). Asset pricing and macro moments from Table IX (p. 182) are compared to the baseline.

Datasets used

Dataset	Role in paper	Wiki page
CRSP (Center for Research in Security Prices)	Asset pricing moments: equity return mean and SD, riskless rate (via Croce et al. 2012 for real rate)	WRDS / CRSP (licensed)
NIPA tables (BEA / Federal Reserve Bank of St. Louis)	Real consumption growth mean and SD; capital-output ratio (tables 1.1.3 and 1.1.5, 1929-2023)	FRED, free
Flow of Funds (Federal Reserve)	DB pension fund total financial assets and endowment-to-GDP ratio (1970-2023)	FRED, free
Survey of Consumer Finances (SCF, Federal Reserve)	Historical stock market participation rate (used as calibration target)	no page yet
NBER business cycle dates	Calibration of productivity shock Markov chain (recession/expansion probabilities)	NBER cycles
Public Plans Data / Social Security Administration data	Decomposition of DB replacement ratio vs. social security replacement ratio	no page yet

Sample: U.S. aggregate, 1929-2023 for returns and consumption; 1970-2023 for pension fund data.

When to read the full paper

Use the original article if you are: replicating or extending the quantitative model; examining Internet Appendix robustness checks (bequest motives, alternative bond supply, transition dynamics); doing a literature review of intermediary asset pricing or pension finance; or auditing a specific parameter value from the calibration (Table I, p. 160). The locators in the table above point to the exact figures and tables. For “what did this paper find,” the table above is the intended default.

Attribution and rights

Source: peer-reviewed, The Journal of Finance 81(1), February 2026. This distillation was extracted by an LLM on 2026-05-31 and augmented on 2026-06-01; it is not human-verified or independently reproduced.

Attribution (CC BY 4.0). Coimbra, Nuno, Francisco Gomes, Alexander Michaelides, and Jialu Shen. “Asset Pricing and Risk-Sharing Implications of Alternative Pension Plan Systems.” The Journal of Finance 81, no. 1 (February 2026): 143-188. DOI: 10.1111/jofi.13507. (c) 2025 The Author(s). Published by Wiley Periodicals LLC on behalf of the American Finance Association. Licensed under CC BY 4.0. This page is an adaptation by the Institute for Automated Research: core results extracted and re-expressed; changes were made. CC BY 4.0 permits mirroring; the PDF is not hosted in this batch.