Portfolio Risk Management and Index Tracking

Fixed Income Trading and Risk Management by Alexander Düring (ISBN 9781119756354)

Previous: Bond Index Mechanics | Next: Hedging

Part VII turns from tickets to books. Chapter 35 asks how you neutralize a portfolio against pricing factors, and how passive managers fake holding “the market” without buying every bond.

Risk-neutral portfolios

Portfolio value V = Σ wᵢ Pᵢ. Each price depends on factors vₖ. Risk management solves ∂V/∂vₖ = 0 for all k.

Only the empty portfolio is truly risk-free if every bond price is its own factor. With k factors you need at least k assets to hedge. Institutional portfolios usually have N > k.

Risk stability problems:

• Convexity: DV01 falls as bonds age; rises/falls with rates (negative convexity in futures/RMBS). Hedges drift.
• Cross-gamma: FX or inflation moves rescale foreign exposure; second cross-partials bite.

Long-only portfolios need assets with opposite factor sensitivities. Bonds mostly have same-sign rate sensitivity, so long-only bond books are never flat. Equities can offset somewhat; still correlated via index futures. Multi-asset “all-weather” portfolios market low risk but face the same constraints.

Passive investing and logical gaps

Index tracking boomed because active managers rarely beat benchmarks net of fees. Argument: everyone combined holds the market, so average return is market return. Buy cheap beta via ETFs.

Bond markets break the story:

• Central banks and constrained investors hold huge slices not available to privates.
• EMH says prices are fair because active investors correct mispricings. If everyone is passive, who corrects?
• Passive rise might increase alpha for remaining active managers (test pending).

Replication mechanics

Start: assets = cash, liabilities = full index (like shorting every line). Goal: assets mimic liability performance.

• Synthetic: total return swap or index futures (roll risk, counterparty, cash drag in ETFs)
• Physical: buy bonds, usually subset

Full replication: all names, index weights, trade on index changes. Partial replication: optimize names vs tracking error vs costs. Stop adding bonds when marginal trading cost exceeds tracking error reduction.

Weights in partial replication need not match index weights. Rebalance may follow risk limits, not only index announcements.

Spanning sets and Lagrange

Pick k risk factors. Find N ≥ k assets whose factor sensitivities span R^k. Solve weights via Lagrangian:

• Hard constraints: portfolio factor exposure = market exposure Fⱼ
• Soft target: minimize Σ(wᵢ − w̄ᵢ)² for liquidity (stay near market weights)

Bonds use duration sensitivities forward-looking, not historical equity betas. Multi-bucket duration matching (1-5Y, 5-10Y, …) for curve risk. Corporate trackers may bucket by sector instead.

Variation: penalize expensive spline spreads or reward carry in the objective. More tracking error, more ex-ante carry. Whether it pays is market-specific.

Friction effects

Indices model some frictions (coupon reinvestment at next rebalance, bid for holdings, ask for entrants). Partial replication pays extra costs outside rebalance windows.

Cash drag dominates many trackers: June 2020 example, ECB 5Y at −11.3bp vs €STR near −55bp in act/act terms → ~44bp annual drag on cash. Five percent cash balance ≈ 2bp underperformance vs index. Repo financing to avoid cash is leverage, illegal for most funds, dangerous in sustained outflows.

Equity ETFs use futures to stay 100% invested despite cash. Illiquid index segments (Japan corporates, some Europe) force tracking error on inflows when old bonds unavailable.

Chapter 35 is the institutional mirror of Chapter 30’s trader discipline. Same math, different constraints: long-only, redemption risk, and the index rulebook you never see until tracking error bites.