Liability-Driven and Index-Based Strategies: formula sheet

Key formulas

Surplus

Surplus = Market value of assets − Present value of liabilities

Negative surplus means underfunded. LDI manages the variability of this number.

Funded ratio

Funded ratio = Assets ÷ PV of liabilities

Below 1 means a deficit. Use the same discounting basis for the liabilities each time.

Change in surplus

ΔSurplus = ΔAssets − ΔLiabilities

Surplus risk comes from assets and liabilities not moving together.

Duration approximation of value change

%ΔValue ≈ −Duration × ΔYield

Apply to assets and to liabilities separately, then convert to currency changes.

Duration gap (value basis)

Hedge works when Asset value × Asset duration = Liability value × Liability duration

Matching durations alone is not enough if values differ. Match the currency sensitivity (money duration).

Surplus variance

σ²(S) = σ²(A) + σ²(L) − 2 × Cov(A, L)

Higher correlation between assets and liabilities lowers surplus risk. Use currency amounts or consistent weights.

Portfolio Macaulay duration

D_portfolio = Σ (w_i × D_i)

Weights w_i are market value weights. Use this to solve for the mix of two bonds that hits a target duration.

Two-bond weight for a target duration

w_short × D_short + (1 − w_short) × D_long = D_target

Solve for w_short. The target must lie between the two durations.

Single liability immunization conditions

Asset Macaulay duration = liability horizon; PV(assets) ≥ PV(liability)

Classic immunization for a parallel yield shift. Zero-coupon bonds maturing on the liability date remove reinvestment risk.

Multiple liability immunization conditions

PV(assets) ≥ PV(liabilities); duration(assets) = duration(liabilities); dispersion/convexity(assets) ≥ liabilities

The third condition means asset cash flows are more dispersed than liability cash flows. Wider dispersion generally implies higher asset convexity, but verify convexity or the dispersion measure directly rather than inferring it from the range of maturities.

Money duration (BPV) matching

BPV(assets) = BPV(liabilities), where BPV ≈ modified duration × PV × 0.0001

Use when asset and liability values differ. It ensures the same currency change per basis point.

Key rate duration match

KRD_assets(k) = KRD_liabilities(k) for each key maturity k, with PV matched

Controls exposure to non-parallel shifts, twists and steepening at specified maturities.

Liability PV (single payment)

PV = Liability ÷ (1 + y)^t

Discount at the stated yield. Asset PV should be at least this amount.

Cash flow matching build order

Work backward: last liability → first liability

Each bond's principal plus final coupon covers its liability date. Earlier coupons reduce what is needed for earlier dates.

Cash flow matching condition

Bond cash flow at date t ≥ liability at date t, for every t

Payments must arrive on or before the liability date. Any excess is reinvested, which creates reinvestment risk.

Immunization conditions (for comparison)

PV assets = PV liabilities; duration of assets = duration of liabilities; asset convexity/dispersion ≥ liability

Assumes a parallel shift in rates at the start. Needs rebalancing and is exposed to non-parallel shifts.

Required value for safety net

Required portfolio value now = Minimum target value at horizon ÷ (1 + immunization rate)^T

Use the rate available on immunized bonds today and the time horizon T in years. This is the trigger value: the value that, if immunized, would just reach the safety net.

Surplus (cushion)

Surplus = Current portfolio value − Required value to secure safety net

Active management is allowed while surplus > 0. When surplus reaches zero, the trigger is hit and the manager immunizes.

Cushion spread

Cushion spread = Immunization rate available − Safety-net return

A larger cushion spread gives more room for active risk.

Duration gap in money terms

PVBP gap = PVBP(liabilities) − PVBP(assets)

A positive gap means liabilities are more rate-sensitive than assets. Add hedge PVBP equal to the gap.

PVBP of a portfolio

PVBP ≈ Modified duration × Market value × 0.0001

Use the same rate-change basis (1 bp) for assets, liabilities and the hedge.

Number of futures contracts

N = (PVBP gap) ÷ (PVBP per futures contract)

Use the cheapest-to-deliver bond's PVBP adjusted by the conversion factor where the question gives it. Round to a whole number.

Notional of swap needed

Swap notional = (PVBP gap) ÷ (PVBP per unit of swap notional)

A receive-fixed swap has positive PVBP like a long bond (value rises as rates fall). A pay-fixed swap has negative PVBP, so to add asset duration you receive fixed.

Funded ratio

Funded ratio = PV of assets ÷ PV of liabilities

Hedge aims to stabilise this ratio against rate moves.

Market-value weight of a bond

wᵢ = (Pᵢ × Parᵢ outstanding) ÷ Σ(Pⱼ × Parⱼ outstanding)

Price should include accrued interest (full price) for market value. Larger debt means larger weight.

Index return

R_index = Σ wᵢ × Rᵢ

Weights are at the start of the period. Rᵢ is the total return of bond i (price change, coupon, accrued interest and reinvestment).

Index duration

D_index = Σ wᵢ × Dᵢ

Market-value-weighted average of constituent durations. Use it to compare the benchmark with liability duration.

Qualities of a good benchmark

Unambiguous, investable, measurable, appropriate, reflective of current investment opinions, specified in advance, accountable, owned

Use as a checklist when evaluating a proposed benchmark.

Tracking error (tracking risk)

TE = standard deviation of (portfolio return − index return)

Measured over time on active returns. Higher TE means the portfolio departs more from the index.

Active return

Active return = Portfolio return − Index return

Net of fees and costs. The index itself has no costs, so an indexed portfolio usually lags slightly.

Cell matching rule

Portfolio weight in each cell ≈ index weight in that cell

Also match overall duration, key rate durations, yield and convexity to the index.

Approximate price change from a yield change

%ΔPrice ≈ −ModDur × Δy + ½ × Convexity × (Δy)²

Use the duration term for small moves. Add convexity for large moves. Δy is in decimal form.

Portfolio duration

Portfolio duration = Σ (weight of bond i × duration of bond i)

Weights are market-value weights. Use this to find barbell weights that match a bullet's duration.

Rolling yield

Rolling yield = (coupon income ÷ bond price) + rolldown return

Rolldown return is the price gain from the bond moving to a shorter maturity on an unchanged curve.

Credit spread price effect

%ΔPrice ≈ −SpreadDuration × ΔSpread

Use this for spread moves with benchmark yields unchanged. Add the duration effect for a benchmark yield move.

Total return approximation

Return ≈ yield income + rolldown + (−ModDur × Δy + ½ × Convexity × (Δy)²) + (−SpreadDur × ΔSpread) − expected credit losses

Use it to compare positions under a scenario. Keep each source of return on its own line.

Curve shape rules

Flattening: barbell beats bullet. Steepening: bullet beats barbell (duration-matched, before convexity).

Check which end of the curve moves. Do not rely on the word alone.

Quick revision

• LDI starts with the liability: its size, timing and sensitivity to rates.
• Surplus = asset value − liability value; the aim is to manage changes in surplus.
• Classical immunization requires three conditions: asset present value ≥ liability present value; asset Macaulay duration = liability Macaulay duration (the investment horizon for a single liability); and asset BPV (money duration) = liability BPV. Together these aim to lock in the return needed to meet the liability.
• Matching duration alone does not remove risk from non-parallel yield curve shifts. Asset convexity should be equal to or slightly greater than liability convexity, with low cash flow dispersion (a bullet-like portfolio), to limit this risk.
• Cash flow matching largely removes interest rate and reinvestment risk if asset cash flows match each liability date, but credit risk remains. It can also cost more and limits flexibility.
• Contingent immunization allows active management until the safety net is reached, then switches to immunization.
• Derivatives let you adjust duration or hedge liabilities without selling bonds, but add basis, counterparty and collateral issues.
• Leverage can extend asset duration to match long liabilities but increases risk in a rate or spread shock.
• Bond indexes hold many illiquid securities and change with issuance and maturity, so full replication is often impractical.
• Stratified sampling and enhanced indexing trade some tracking risk for lower cost.
• In active strategies, state the rate or curve view first, then the duration, convexity or curve positioning that expresses it.
• In essays, follow the command word, answer only what is asked, and show your working.

Common mistakes

• Matching asset and liability durations without checking values Fix: Compare money durations: value × duration. When assets are smaller than liabilities, the asset duration must exceed the liability duration so that the money durations are equal: asset duration = liability duration × PV liabilities ÷ asset value.
• Judging the plan by asset return or asset volatility only Fix: Measure risk and success by the surplus. A low-volatility asset mix can still produce high surplus risk.
• Matching asset duration to the liability maturity using modified duration instead of Macaulay duration for classic single liability immunization. Fix: For the horizon condition, use Macaulay duration equal to the time to the liability. Use modified duration or BPV when matching price sensitivity.
• Forgetting the PV condition and treating duration match alone as enough. Fix: Always check that asset PV is at least liability PV. A duration match on an underfunded portfolio leaves a shortfall.
• Building the cash flow matching portfolio from the first liability forward. Fix: Start from the last liability. The final bond's principal is the largest amount, and its earlier coupons then reduce earlier needs.
• Saying cash flow matching has no risk at all. Fix: It still has credit risk, call risk, and some reinvestment risk if cash arrives before it is needed. It is also hard to apply to uncertain liabilities.
• Matching percentage durations instead of money sensitivity. Fix: Always convert to PVBP or duration × market value. Hedge the money gap.
• Choosing the wrong swap side. Fix: To add asset duration, the plan receives fixed, which gains when rates fall. Pay-fixed reduces duration. Check by asking who gains if rates fall.
• Treating market-value weighting as a sign of a high-quality or well-diversified index. Fix: Remember that more borrowing means more weight. Say the index overweights the most indebted issuers and can concentrate in them.
• Ignoring that bonds mature and the index must be rebalanced. Fix: State that constituents drop out at maturity or when they fail rules, so turnover is high and duration drifts.

Exam tips

• When the command word is 'justify' or 'recommend', name the liability trait (duration, inflation link, certainty) and link it to your asset choice.
• Show surplus calculations in currency terms. A correct number typed alone earns full credit, but show a line of working in case of a slip.
• Contrast asset-only and liability-relative in one sentence: the first targets asset risk and return, the second targets surplus.
• For insurers and banks, give each a distinct focus: cash flow matching and spread risk for life insurers, liquidity for non-life, funding and duration gap for banks.
• Check the sign: falling yields raise both assets and liabilities, and the bigger money duration wins.
• Read the command word. For "calculate", type the number clearly with its unit. For "justify", give the condition and the reason in one or two short sentences.
• For multiple liability questions, state all three conditions in your answer. Examiners look for PV, duration and the dispersion or convexity requirement.
• Wider dispersion of asset cash flows generally implies higher asset convexity, and a barbell can meet the condition more easily. Do not assume it from the range of maturities alone. Verify convexity or the dispersion measure.