Can someone walk through surplus optimization for a pension fund step by step?

Question

AcadiFi · Accepted Answer

Surplus optimization is MVO applied to *surplus returns* instead of *asset returns*. The surplus is the difference between plan assets and plan liabilities.

**Setup:**
Let's say a pension fund has:
- Assets: $800M
- Liabilities (PV of obligations): $700M
- Surplus: $100M
- Funded ratio: 800/700 = 114%

**Surplus Return:**

R_surplus = R_assets − (L/A) × R_liabilities

Where L/A is the liability-to-asset ratio (700/800 = 0.875)

**How It Differs from Asset-Only MVO:**

In regular MVO, you maximize: E(R_portfolio) − (λ/2) × σ²_portfolio

In surplus optimization, you maximize: E(R_surplus) − (λ_s/2) × σ²_surplus

**The critical difference:** Surplus volatility depends not only on asset risk, but on the *correlation between assets and liabilities.* Assets that are highly correlated with liabilities *reduce* surplus volatility, even if they're risky in isolation.

**Practical Example:**

Consider two asset classes:
- **Long-duration government bonds:** 4% expected return, 8% volatility, but 0.9 correlation with pension liabilities
- **Emerging market equities:** 9% expected return, 22% volatility, 0.1 correlation with liabilities

In an asset-only MVO, EM equities look attractive (high return). But in surplus optimization, long bonds are far more valuable because they *hedge the liabilities.* When interest rates fall and liabilities increase, long bonds also increase, stabilizing the surplus.

**Three-Step Process:**
1. Estimate expected returns, volatilities, and correlations for all asset classes *and* the liability portfolio
2. Convert asset-space inputs to surplus-space inputs using the surplus return formula
3. Run MVO on surplus returns with the pension fund's surplus risk tolerance

**The Liability Glide Path:**
As the funded ratio improves, the fund should systematically shift from return-seeking assets (equities) to liability-hedging assets (long bonds matching liability duration). This is the "glide path." A fully funded plan should be nearly 100% in liability-hedging assets.

| Funded Ratio | Liability-Hedging | Return-Seeking |
|---|---|---|
| 80% (underfunded) | 40% | 60% |
| 100% (fully funded) | 70% | 30% |
| 120% (overfunded) | 90% | 10% |

**Exam Tip:** If a vignette describes a pension fund, expect to identify whether surplus optimization, hedging/return-seeking, or a glide path is the most appropriate technique.

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Can someone walk through surplus optimization for a pension fund step by step?

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