How Drawdown-Recovery Markov Simulator works

Cornish-Fisher quantile expansion

For a target distribution with skew γ₃ and excess kurtosis (γ₄ − 3), the moment-matched quantile from a standard normal z is:

q ≈ z + (z² − 1)·γ₃/6 + (z³ − 3z)·(γ₄ − 3)/24 − (2z³ − 5z)·γ₃²/36

This is the standard fourth-order expansion (Cornish-Fisher 1937). It is exact at z = 0 and accurate for moderate skew (|γ₃| ≲ 1) and excess kurtosis (γ₄ − 3 ≲ 6). Beyond those bounds the approximation distorts.

Simulation procedure

μ_m = monthly_sharpe × 0.04                           ← assume σ_m = 4%
For each path p of P:
  value = 1 − drawdown_threshold
  For t = 1..max_months:
    z = standard_normal_sample()
    q = cornish_fisher(z, γ₃, γ₄ − 3)
    r = μ_m + 0.04 · q
    value *= 1 + r
    If value ≥ 1:
      record recovery_time = t; break
  If never recovers, record never += 1

Outputs

• 25th / 50th / 75th / 95th percentile of recovery times.
• Probability of recovering within 12 months.
• Probability of never recovering within the simulation cap (240 months).

Why σ_m is fixed

Time-to-recover for a given drawdown threshold depends on μ_m / σ_m (the monthly Sharpe), not σ_m alone — fixing σ_m = 4% is convenient and does not affect the result. Doubling both μ and σ yields the same recovery times.

References

• Cornish, E. A., Fisher, R. A. (1937). "Moments and cumulants in the specification of distributions." Revue de l'Institut International de Statistique 5(4): 307–320. DOI: 10.2307/1400905.
• Maillard, D. (2012). "A user's guide to the Cornish-Fisher expansion." SSRN 1997178.
• Magdon-Ismail, M., Atiya, A. F. (2004). "Maximum drawdown." Risk Magazine 17(10): 99–102.
• Burghardt, G., Duncan, R., Liu, L. (2003). "Deciphering Drawdown." Risk Magazine, September.

Limitations

• Returns are i.i.d. by assumption — no regime switching, no autocorrelation.
• Cornish-Fisher accuracy degrades for extreme skew/kurtosis. For very fat tails consider a t-distribution sampler.
• Simulation cap of 240 months means very-low-Sharpe inputs will report ∞-by-cap.

External resources

• Drawdowns and Rallies in a Finite Time-Horizon (Pospisil & Vecer 2010)