Quantitative Finance & Portfolio Choice
Merton Optimal Portfolio Share & Consumption-Wealth Formula
Calculate Merton’s continuous-time risky-asset allocation pi* = (mu - r_f)/(gamma * sigma^2), certainty-equivalent return r_CE, and optimal CRRA payout propensity c*/W.
Size endowment equity weight and sustainable spending rate simultaneously via Merton (1969)
Robert C. Merton (1969, 1971) solved the joint continuous-time portfolio and consumption problem for an investor or endowment with CRRA relative risk aversion gamma > 0, subjective discount rate rho > 0, risk-free rate r_f, equity expected return mu, and volatility sigma > 0: optimal risky-asset weight is pi* = (mu - r_f) / (gamma * sigma^2) (generalizing the log-utility Kelly fraction at gamma = 1!), certainty-equivalent portfolio return is r_CE = r_f + (mu - r_f)^2 / (2 * gamma * sigma^2), and optimal consumption-to-wealth spending rate is c* / W = (rho + (gamma - 1) * r_CE) / gamma!
Verify on an university endowment calibration with equity drift mu = 0.10 (B2), T-bill rate r_f = 0.04 (B3 -> equity risk premium 6.0%), volatility sigma = 0.20 (B4 -> sigma^2 = 0.04), relative risk aversion gamma = 3.0 (B5), and time preference rho = 0.03 (B6): optimal equity weight is pi* = 0.06 / (3 * 0.04) = 0.5000 (50% equities / 50% risk-free!), expected portfolio return is 0.0700 while certainty-equivalent return is r_CE = 0.04 + 0.0036 / 0.24 = 0.0550 (5.50%/yr), and optimal spending rule is c* / W = (0.03 + 2 * 0.0550) / 3 = 0.0467 (4.67%/yr)! Check log utility gamma = 1.0 to see pi* = 1.5000 and c* / W = rho = 0.0300 independent of return!
| Merton Portfolio Metric (mu=10%, r_f=4%, sig=20%, rho=3%) | gamma = 3.0 (Endowment) | gamma = 1.0 (Log / Kelly) |
|---|---|---|
| Optimal Risky Weight (pi*) | 0.5000 (50.00%) | 1.5000 (150.00% Levered) |
| Certainty-Equivalent Return (r_CE) | 0.0550 (5.50%/yr) | 0.0850 (8.50%/yr) |
| Optimal Spending Rate (c* / W) | 0.0467 (4.67%/yr) | 0.0300 (= rho Exactly!) |
Merton Optimal Risky Portfolio Weight pi* = (mu - r_f) / (gamma * sigma^2) Formula
Validate volatility sigma > 0 and CRRA relative risk aversion gamma > 0.
=LET(mu_r, B2, rf_r, B3, sig_v, B4, gam_r, B5, IF(OR(sig_v <= 0, gam_r <= 0), "Invalid Merton Input", ROUND((mu_r - rf_r) / (gam_r * (sig_v^2)), 4)))Merton Infinite-Horizon Optimal Consumption-to-Wealth Ratio c* / W Formula
Enforce Merton’s transversality condition c* / W > 0 so wealth utility remains finite.
=LET(mu_r, B2, rf_r, B3, sig_v, B4, gam_r, B5, rho_d, B6, IF(OR(sig_v <= 0, gam_r <= 0, rho_d <= 0), "Invalid Merton Input", LET(r_ce, rf_r + ((mu_r - rf_r)^2) / (2 * gam_r * (sig_v^2)), cw_rt, (rho_d + (gam_r - 1) * r_ce) / gam_r, IF(cw_rt <= 0, "Transversality Violated (Require c*/W > 0)", ROUND(cw_rt, 4)))))Try the example
Use the original values or change them to test your understanding.