Quantitative Finance & Risk
Andrew Lo Autocorrelated Sharpe Ratio Annualization Formula
Replace naive SQRT(12) Sharpe annualization on smoothed or illiquid returns with Andrew Lo’s (2002) exact AR(1) serial-correlation scaling factor eta(q, rho_1).
Expose why SQRT(12) annualization overstates smoothed fund Sharpe ratios by 30%+
Multiplying a monthly Sharpe ratio SR_m by SQRT(12) = 3.4641 assumes monthly returns are serially uncorrelated (rho_1 = 0). Appraisal-smoothed private credit, real estate, and illiquid distressed funds exhibit positive first-order autocorrelation (e.g., rho_1 = 0.40), which dampens monthly volatility without dampening true 12-month variance!
Andrew Lo (2002) derived the exact q-period AR(1) variance sum factor V_mult(q, rho_1) = q + (2*rho_1 / (1 - rho_1)^2) * (q*(1 - rho_1) - (1 - rho_1^q)), making the true annualization multiplier eta(q, rho_1) = q / SQRT(V_mult). For monthly SR_m = 0.50 (B2), q = 12 months (B3), and rho_1 = 0.40 (B4): V_mult = 12 + (0.80 / 0.36) * (7.20 - (1 - 0.40^12)) = 25.7778, so eta = 12 / SQRT(25.7778) = 2.3635 — cutting true annualized Sharpe from naive 1.7321 down to 1.1818 (-31.8% haircut!).
| Annualization Metric (q = 12) | Formula Expression | Sample Output (rho_1 = 0.40) |
|---|---|---|
| Naive i.i.d. Sharpe (Bug) | SQRT(q) * SR_m = 3.4641 * 0.50 | 1.7321 (Overstated by +0.5503) |
| AR(1) Variance Multiplier V_mult | q + (2*rho/(1-rho)^2)*(q*(1-rho) - (1-rho^q)) | 25.7778 (vs 12.0000 i.i.d.) |
| Lo Annualization Factor eta | q / SQRT(V_mult) | 2.3635 (vs 3.4641 i.i.d.) |
| Lo True Annualized Sharpe | eta(q, rho_1) * SR_m | 1.1818 |
Lo AR(1) Annualization Factor eta(q, rho_1) Formula
Guard stationary AR(1) bounds (-1 < rho_1 < 1) and q >= 1; verify that rho_1 = 0 collapses V_mult to q and eta to SQRT(q).
=LET(q_per, B3, rho_a, B4, IF(OR(q_per < 1, ABS(rho_a) >= 1), "Invalid AR(1) Input", LET(v_mul, q_per + ((2 * rho_a) / ((1 - rho_a)^2)) * (q_per * (1 - rho_a) - (1 - (rho_a^q_per))), ROUND(q_per / SQRT(v_mul), 4))))Autocorrelation-Adjusted Annualized Sharpe Formula & Mean-Reversion Check
Multiply SR_m by Lo’s eta factor. Notice the flip side: if a market-making strategy exhibits mean-reverting negative autocorrelation (set B4 = -0.25), 12-month variance grows slower than 12x (V_mult = 7.5200), lifting true annualized Sharpe ABOVE naive SQRT(12) from 1.7321 to 2.1880!
=LET(sr_m, B2, q_per, B3, rho_a, B4, IF(OR(q_per < 1, ABS(rho_a) >= 1), "Invalid AR(1) Input", LET(v_mul, q_per + ((2 * rho_a) / ((1 - rho_a)^2)) * (q_per * (1 - rho_a) - (1 - (rho_a^q_per))), ROUND(sr_m * (q_per / SQRT(v_mul)), 4))))Try the example
Use the original values or change them to test your understanding.