Hansen-Jagannathan Bound & Equity Premium Puzzle

See in 4 arithmetic lines why smooth aggregate consumption cannot explain historical equity Sharpe ratios!

Under zero arbitrage, every excess asset return R_e = R_i - R_f satisfies Euler equation E[M * R_e] = 0 for Stochastic Discount Factor (pricing kernel) M > 0 with E[M] = 1 / (1 + R_f). Applying Cauchy-Schwarz (|Corr(M, R_e)| <= 1), Lars Peter Hansen and Ravi Jagannathan (1991) proved the model-free SDF Volatility Lower Bound: sigma(M) / E[M] >= ABS(E[R_e]) / sigma(R_e) = |Sharpe_Ratio|! Under Rajnish Mehra and Edward Prescott’s (1985) standard power-utility Consumption CAPM (M = beta * (C_1 / C_0)^(-gamma)), log-normal approximation implies sigma(M) / E[M] approx gamma * sigma_c (where sigma_c is annual aggregate consumption growth volatility), forcing Minimum Implied Relative Risk Aversion gamma_min = |Sharpe_Ratio| / sigma_c!

Verify on postwar US stylized macro-finance numbers: Equity_Excess_Return E[R_e] = 0.0640 (6.40%/yr, B2), Equity_Volatility sigma_R = 0.1600 (16.00%/yr, B3), Risk_Free_Rate R_f = 0.0100 (1.00%, B4), Aggregate_Consumption_Growth_SD sigma_c = 0.0160 (1.60%/yr, B5), and Plausible_Household_CRRA gamma_test = 2.50 (B6): Observed Market Sharpe Ratio (and exact Hansen-Jagannathan SDF Volatility Bound sigma(M)/E[M]) is 0.0640 / 0.1600 = 0.4000 (requiring sigma(M) >= 0.4000 / 1.01 = 0.3960!), whereas CCAPM at gamma = 2.50 generates Pricing-Kernel Volatility of only 2.50 * 0.0160 = 0.0400 (10x too smooth!), forcing Minimum Implied Risk Aversion gamma_min = 0.4000 / 0.0160 = 25.00 — the famous Equity Premium Puzzle!

Hansen-Jagannathan Minimum Pricing-Kernel Volatility sigma(M)_min Formula

Validate sigma_R > 0, gross risk-free rate 1 + R_f > 0, and consumption volatility sigma_c > 0.

=LET(er_ex, B2, sg_r, B3, rf_rt, B4, IF(OR(sg_r <= 0, rf_rt <= -1), "Invalid Asset Pricing Input", ROUND((ABS(er_ex) / sg_r) / (1 + rf_rt), 4)))

Mehra-Prescott Implied Minimum CRRA Risk Aversion & Bound Pass/Fail Formula

Test disaster-risk or volatile-factor volatility sigma_c = 0.16 (e.g., Barro rare disasters or leveraged intermediary capital): gamma_min drops from 25.00 down to a plausible 2.50!

=LET(er_ex, B2, sg_r, B3, sg_c, B5, IF(OR(sg_r <= 0, sg_c <= 0), "Invalid Asset Pricing Input", ROUND((ABS(er_ex) / sg_r) / sg_c, 2)))

Further reading

• Arte Sheets Lesson — Nordhaus-Ramsey Social Cost of Carbon Discounting Formula
• Arte Sheets Lesson — Kelly Criterion Optimal Bet & Position Sizing