How is required probability of success used in asset allocation?

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AcadiFi · Accepted Answer

Required probability of success (POS) is the minimum likelihood that a portfolio fully funds a goal at its target horizon. Different goal tiers warrant different POS thresholds: essential needs ~95-99%, wants ~75-85%, wishes ~50-65%. Higher POS requires more conservative allocations because the distribution of outcomes must have a narrow left tail.

Math connection: for a geometric Brownian motion model, the probability that terminal wealth W_T exceeds a goal G is POS = Φ[(ln(W_0/G) + (μ - σ²/2)T) / (σ√T)], where μ is expected return, σ is volatility, T is horizon. Increasing POS requires either lower σ (less risk), higher μ (accepting unrealistic assumptions), larger W_0 (more initial wealth), or longer T.

Example: Zennalie Marchwell has $500K to fund $1.2M retirement goal in 20 years.
- Portfolio A (aggressive): μ=7%, σ=16% → POS = Φ[(ln(500/1200) + (0.07 - 0.0128)·20) / (0.16·√20)] = Φ[(−0.875 + 1.144) / 0.715] = Φ(0.376) ≈ 65%.
- Portfolio B (moderate): μ=5.5%, σ=11% → Φ[(−0.875 + 0.979) / 0.492] = Φ(0.211) ≈ 58%.
- Portfolio C (conservative): μ=4%, σ=7% → Φ[(−0.875 + 0.751) / 0.313] = Φ(−0.396) ≈ 35%.

If Zennalie treats this as an essential need (95% POS), no module achieves it — she must increase savings, extend horizon, or accept lower target. If treated as a want (70% POS), Portfolio A nearly qualifies with modest adjustment.

Practitioners use Monte Carlo rather than closed-form to handle fat tails, sequence risk, and withdrawal schedules. Firms like Henmere Wealth Systems show clients a 'probability dial' for each goal, making trade-offs tangible. CFA candidates should understand POS is the bridge between risk tolerance conversations and quantitative allocation decisions.

How is required probability of success used in asset allocation?

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