CFA Level III · Level III Core

Overview of Fixed-Income Portfolio Management: formula sheet

Full chapter guide

Key formulas

Real return (approximate): Real return ≈ Nominal return − Inflation; Exact form: (1 + nominal) ÷ (1 + inflation) − 1. Use the exact form when numbers are large.
Price change from yield change: %ΔPrice ≈ −Modified duration × ΔYield; Use for a quick estimate of rate risk. Ignores convexity, so it is least accurate for large yield moves.
Duration matching rule: Asset duration ≈ Liability duration, and PV assets ≥ PV liabilities; Matching duration alone does not fully protect against non-parallel yield shifts. Cash-flow matching is stronger.
Role-to-bond mapping: Diversification → high-quality government; Income → higher-yield/credit; Preservation → short, high-quality; Inflation → linkers/floaters; Liabilities → matched duration or cash flows; Always confirm against the client's constraints.
Bid-ask spread: Spread = Ask price − Bid price; A round trip (buy at ask, sell at bid) costs about one full spread, so the cost of a buy or sell alone is about half.
Relative (percentage) spread: Relative spread = (Ask − Bid) ÷ Mid price, where Mid = (Ask + Bid) ÷ 2; Use this to compare bonds with different price levels.
Round-trip cost of a position: Cost ≈ Position size × (Ask − Bid) ÷ Mid price; This is the cost of buying at the ask and selling at the bid, with no change in the mid price.
Market-value weight of a bond in the index: Weight = Market value of bond i ÷ Σ market values of all bonds in index; Market value usually includes accrued interest. Larger debt outstanding means a larger weight.
Index return (market-value weighted): Index return = Σ (weight_i × return_i); Use beginning-of-period weights.
Benchmark qualities checklist: Unambiguous, investable, measurable, appropriate, reflective of current investment opinions, specified in advance, accountable (owned by the manager); Seven qualities. Use this list to judge any candidate benchmark.
Bond index construction choices: Replication, stratified sampling (cell approach), optimization (enhanced indexing); Full replication is rarely practical for broad bond indexes.
Building-block expected return: E(R) ≈ Yield income + Rolldown return + E(change in price from yield view) − E(credit losses) + E(currency gains/losses); Use only the pieces the question gives. Keep all terms on the same horizon, usually one year.
Rolldown return: Rolldown return ≈ (Price at the new, shorter maturity with the unchanged curve yield − Beginning price) ÷ Beginning price; Assumes the yield curve stays unchanged. It is zero on a flat curve and negative on an inverted curve.
Price change from duration and convexity: %ΔPV ≈ −(ModDur × ΔYield) + ½ × Convexity × (ΔYield)²; Enter ΔYield as a decimal, for example 0.0050 for 50 bps. Some questions use effective duration and effective convexity; apply them the same way.
Portfolio duration: Portfolio duration = Σ (market value weight × bond duration); Weights use market value. The weighted average is a good approximation only for a parallel shift in the yield curve. It is not exact when yields differ across bonds.
Key rate duration: KRD(i) = −(% change in portfolio value) ÷ (change in key rate i); Only the one key rate shifts. The key rate durations sum to (approximately) the portfolio's effective duration.
Money duration and PVBP: Money duration = ModDur × Market value; PVBP ≈ Money duration × 0.0001; PVBP is the value change for a 1 bp yield change.
Duration gap against benchmark: Active duration = Portfolio duration − Benchmark duration; Positive means the portfolio gains more than the benchmark when yields fall.
Basic immunization conditions: Asset Macaulay duration = Liability Macaulay duration; PV of assets ≥ PV of liabilities; Assets must also have a dispersion of cash flows close to, but not much wider than, the liabilities to limit structural risk.
Money duration match (alternative): Asset money duration (or BPV) = Liability money duration (or BPV); Use this when asset and liability values differ. It matches the currency change for a 1 bp yield move.
Duration of a portfolio: D(portfolio) = Σ w(i) × D(i); w(i) is the market value weight of each bond. Use it to hit a target duration.
Cash flow matching rule: Bond cash inflow on or before each liability date ≥ liability due on that date; Work backward from the last liability. Reinvestment between dates is minimal, which keeps interest rate risk low. Cash flow matching still leaves credit and default risk, because a missed bond payment leaves a liability unfunded.
Tracking error (active risk): Tracking error = standard deviation of (portfolio return − benchmark return); Passive aims for very low tracking error. Active accepts more in return for expected excess return.
Return from riding the yield curve (approx.): Total return ≈ yield carry over the horizon + roll-down price gain, where roll-down gain ≈ (change in yield from rolling) × modified duration at the horizon × (−1); Assumes an unchanged curve. Yield falls as the bond rolls to a shorter maturity, so price rises. Use the duration of the bond at the horizon.
Price change approximation: %ΔP ≈ −ModDur × ΔY + ½ × Convexity × (ΔY)²; Shows why higher convexity (barbell) helps when yield changes are large.
Credit spread: Spread = bond yield − benchmark yield of similar maturity; Spread tightening raises bond prices. Spread widening lowers them.
Spread price impact (approx.): %ΔP ≈ −SpreadDur × ΔSpread; Use spread duration, not only interest rate duration, for credit exposure.
Duration of a portfolio: Portfolio duration = Σ (weight × duration of each holding); Use it to build a barbell with the same duration as a bullet.

Quick revision

Bond roles: income, diversification, capital preservation, liability hedging, inflation protection where linked.
Bond markets are mostly dealer-based and over the counter, so liquidity varies by issue and by time.
Index weakness: market-value weighting gives more weight to larger debtors.
Return sources: coupon, rolldown, price change from yield moves, credit losses and currency effect.
%ΔPrice ≈ −ModDur × Δyield + ½ × Convexity × (Δyield)²; use effective duration and convexity for bonds with options.
Classical immunization needs asset Macaulay duration equal to liability duration (the investment horizon for a single liability) and asset present value at least equal to liability present value. For multiple liabilities, asset convexity (cash flow dispersion) must also be at least that of the liabilities. Rebalance as rates and time change.
Cash flow matching reduces reinvestment risk but is costly and restrictive.
Contingent immunization is active until the safety margin is used up, then it switches to immunization.
Steepener, flattener and butterfly trades express yield curve views through duration or key rate positions.
Credit strategies depend on spread view, so state whether spreads widen or tighten and the effect on price.
Answer the command word: calculate, justify, identify, or recommend. For a calculation, a correct number typed on its own earns full credit, so work through the steps for your own accuracy.

Common mistakes

Treating income generation and capital preservation as the same role. Fix: Income seeks a high cash yield and may accept credit risk. Preservation seeks stable principal and accepts lower yield.
Claiming bonds always diversify equities. Fix: Say high-quality government bonds usually diversify, but correlation can turn positive, for example when inflation or rates rise. Credit-heavy portfolios diversify less.
Treating the bid-ask spread as the cost of a single trade. Fix: One side costs about half the spread from mid. A round trip costs about the full spread.
Assuming bonds trade on exchanges like shares. Fix: Say most bonds trade OTC through dealers, with electronic platforms growing alongside.
Treating bond indexes like equity indexes, with a fixed constituent list and stable risk. Fix: Remember that bonds mature, new issues keep entering, and duration drifts, so bond indexes need frequent rebalancing.
Saying market-value weighting is always good because it reflects the market. Fix: Point out that it gives the largest weights to the most indebted issuers, which can concentrate risk.
Treating rolldown as part of the price change from a yield view Fix: Rolldown assumes an unchanged curve. Any move in the curve goes in the separate yield-view term.
Forgetting the minus sign in the duration term Fix: Write −Duration × ΔY every time. A yield rise means a price fall.
Saying immunization removes all interest rate risk. Fix: State that it works mainly for small parallel shifts and needs rebalancing as durations drift. Non-parallel shifts leave structural risk.
Choosing cash flow matching when bonds that match every date are unavailable or costly. Fix: Check feasibility and cost. If matching is impractical, immunization is the better answer.

Exam tips

Match the role to the investor type first. Pension, insurer, retiree and endowment each have a typical primary role.
When asked to justify, cite one client fact per point, in the fewest words.
If a command word says contrast or distinguish, state both roles, not just one.
For a calculation, a correct number on its own earns full credit. Only the number of responses asked for is evaluated, in the order given, so do not add extra answers.
Mention that diversification benefits are not guaranteed when it asks for limitations.
Show the spread and mid price steps. A correct number on its own earns credit, but showing work protects you if you slip.
Answer only the number of points asked for, in the order given.
Always tie the justification to the client's constraint, such as liability timing or cash needs.