Learning Module 4: Liability-Driven and Index-Based Strategies

Learning Outcome Statements (LOS)

1. Managing the Interest Rate Risk of a Single Liability

Goal: Immunization aims to minimize the variance of the realized rate of return over a specific horizon and “lock in” the cash flow yield (internal rate of return) of the portfolio.

Conditions for Immunization (Single Liability): 1. Market Value: Initial Market Value of Assets (\(P_A\)) \(\ge\) Present Value of Liability (\(P_L\)). 2. Duration: Portfolio Macaulay Duration = Investment Horizon (Liability Due Date). 3. Convexity: Minimize Portfolio Convexity. * Reasoning: Minimizing convexity minimizes Structural Risk (risk from non-parallel yield curve shifts and twists). * Ideally, use a Bullet Portfolio (cash flows concentrated around the horizon) rather than a Barbell Portfolio (cash flows dispersed) to reduce structural risk. * Ideally, use a zero-coupon bond matching the liability maturity (Zero Convexity dispersion).

Rebalancing: * A portfolio must be rebalanced regularly because duration changes as time passes and yields change. * There is a trade-off between transaction costs (frequent rebalancing) and duration mismatch risk (infrequent rebalancing).

2. Managing the Interest Rate Risk of Multiple Liabilities

A. Duration Matching Conditions for Immunization (Multiple Liabilities): 1. Market Value: \(P_A \ge P_L\). 2. Money Duration (BPV): Basis Point Value of Assets (\(BPV_A\)) = Basis Point Value of Liabilities (\(BPV_L\)). * Formula: \(BPV = \text{MD} \times \text{Market Value} \times 0.0001\). 3. Convexity: Asset Convexity (\(C_A\)) > Liability Convexity (\(C_L\)). * Note: While \(C_A\) must exceed \(C_L\) to ensure assets rise more than liabilities when rates fall (and fall less when rates rise), \(C_A\) should still be minimized to the extent possible to reduce structural risk.

B. Derivatives Overlay * Used to adjust the duration of a portfolio (e.g., if assets are short-term bonds for liquidity, but liabilities are long-term) without changing the underlying assets. * Futures BPV Formula: \(BPV_{\text{futures}} \approx \frac{BPV_{CTD}}{CF_{CTD}}\). * CTD: Cheapest-to-Deliver bond. * CF: Conversion Factor.

  • Number of Futures Contracts (\(N_f\)): \[N_f = \frac{BPV_L - BPV_A}{BPV_{\text{futures}}}\]
    • If \(N_f\) is positive: Buy (Long) futures.
    • If \(N_f\) is negative: Sell (Short) futures. The number of contracts calculation above immunizes the current liabilities. If there is a surplus and the aim is to improve portfolio performance once the surplus threshold is reached, the manager may choose to over-hedge or under-hedge ie. buy more or fewer futures contracts than calculated. C. Contingent Immunization
  • Concept: A hybrid active/passive strategy. As long as the Surplus (\(P_A - P_L\)) is positive and above a designated threshold, the manager can actively manage the portfolio to generate excess returns.
  • Trigger: If poor performance causes the surplus to drop to the threshold (safety net), the portfolio must be immediately immunized (switched to a passive duration-matching strategy) to ensure the liability can be met.
  • Application: Can be applied via over-hedging or under-hedging in a derivatives overlay strategy based on interest rate views.

3. Risks in Liability-Driven Investing (LDI)

  1. Model Risk: The risk that assumptions (e.g., parallel yield curve shifts) are wrong or approximations are inaccurate. LDI models often assume parallel shifts, but curves can steepen, flatten, or twist.
  2. Spread Risk: The risk that the yield on assets (e.g., corporate bonds) and the discount rate on liabilities (often swap rates or Treasury yields) do not move in sync. A widening spread reduces asset values relative to liabilities.
  3. Counterparty Credit Risk: Relevant for derivatives (e.g., interest rate swaps) used in overlays.
  4. Asset Liquidity Risk: The risk that assets cannot be sold (or positions unwound) quickly at fair value to meet liability outflows or rebalancing needs.

4. Bond Indexes

  • Complexity: Fixed-income indexes are harder to replicate than equity indexes due to the sheer number of securities (size/breadth), illiquidity of many issues, and the fact that bonds trade OTC (not on exchanges).
  • Drift: Index duration naturally drifts downward as bonds age, and composition changes frequently as bonds mature or are called and new bonds are issued.
  • Primary Risk Factors: To track an index, a manager must match its exposure to:
    1. Duration: Option-adjusted duration, convexity, and key rate durations.
    2. Portfolio Weights: Sector, Credit Quality, Maturity, Coupon, and Issuer.

5. Alternative Methods for Establishing Passive Bond Market Exposure

1. Full Replication * Method: Holding all securities in the index in exact weights. * Pros: Minimal tracking error (theoretically). * Cons: Often impossible or prohibitively expensive due to the large number of illiquid bonds in broad indexes.

2. Enhanced Indexing * Method: Buying a subset of the index (sampling) to match the primary risk factors (duration, sector, quality) of the index. * Goal: Match index performance with lower cost (higher tracking error than full replication, but lower than active management). * Techniques: * Cell Matching (Stratified Sampling): Divide index into cells (e.g., by duration and sector) and buy representative bonds for each cell. * Multifactor Models: Use risk factors to match exposures.

3. Pooled Investment Vehicles * Mutual Funds: Transact at end-of-day NAV; good for smaller investors. * Exchange-Traded Funds (ETFs): Trade intraday; liquid; typically use sampling; can trade at premium/discount to NAV.

4. Total Return Swaps (TRS) * Method: Receiver gets the total return of a bond index; Payer pays a reference rate (e.g., MRR) +/- spread. * Pros: Synthetic exposure with low initial cash outlay; access to difficult asset classes. * Cons: Counterparty credit risk; do not own the underlying assets.

6. Benchmark Selection

  • Criteria for a Valid Benchmark:
    • Unambiguous & Transparent: Rules for inclusion are clear.
    • Investable: The manager can actually hold the securities.
    • Measurable: Returns can be calculated frequently (daily/weekly).
    • Appropriate: Matches the manager’s style/mandate.
    • Reflect Current Investment Opinions: The manager has knowledge of the constituents.
  • Challenges: Custom benchmarks may be needed for specific LDI mandates (e.g., matching a specific liability stream) where standard broad market indexes are unsuitable.