Fixed-Income Active Management: Credit Strategies: formula sheet

Key formulas

Credit spread

Credit spread = Yield on credit bond − Yield on benchmark of similar maturity

Benchmark may be a government bond or a swap rate. Use the benchmark the question names.

Expected loss

Expected loss = Probability of default × Loss given default

Loss given default = 1 − recovery rate, expressed as a share of exposure.

Spread components

Spread ≈ Expected loss + Credit risk premium + Liquidity premium

A conceptual breakdown. Exam questions usually ask you to identify which part changed.

Price impact of spread change

%ΔPrice ≈ −Spread duration × ΔSpread

An approximation for small changes. Express ΔSpread in decimal form, so 25 bps = 0.0025.

Excess spread return

Excess spread return ≈ Spread × Time − Spread duration × ΔSpread − Expected loss

Carry from the spread, minus the price loss from widening, minus expected credit loss. Use the same period for all terms.

Yield spread

Credit spread = bond yield − benchmark yield

The benchmark is a government curve for G-spread and a swap curve for I-spread. Both use a same-maturity (interpolated) benchmark point.

Price change from spread change

%ΔPrice ≈ −SD × ΔSpread

SD is spread duration. Enter ΔSpread in percent (50 bp = 0.50%). Benchmark curve is held constant. Add a convexity term for large moves.

Portfolio spread duration

SD(portfolio) = Σ (wᵢ × SDᵢ)

wᵢ is the market-value weight. This holds for the weights you use and the same spread definition.

Z-spread definition

Price = Σ CFₜ ÷ (1 + sₜ + Z)ᵗ

sₜ are benchmark spot rates. Z is one constant spread over all spot rates.

OAS relationship

Callable: Z-spread = OAS + option cost, so OAS = Z-spread − option cost. Putable: Z-spread = OAS − option value, so OAS = Z-spread + option value.

Option cost or value is the spread-equivalent value of the option. A callable bond's Z-spread is above its OAS. A putable bond's Z-spread is below its OAS. For an option-free bond, OAS = Z-spread.

Loss given default

LGD = 1 − Recovery rate

Recovery rate is the share of exposure recovered. Both are usually a percentage of exposure.

Expected loss

EL = PD × LGD × Exposure

Use the same time period for PD as for the loss you want. Without exposure, EL is a percentage.

Expected loss rate (no exposure)

EL % = PD × (1 − Recovery rate)

A simple annual credit cost to compare against the spread.

Multi-period survival

Survival to T = (1 − PD) ^ T

Holds only when annual PD is constant and independent across years. Cumulative PD = 1 − (1 − PD) ^ T.

Price change from spread move

%ΔPrice ≈ −Spread duration × ΔSpread

Use for migration risk. Convert ΔSpread to decimal form.

Credit valuation adjustment

CVA = Σ [Expected exposure_t × PD_t × LGD × Discount factor_t]

PD_t is the probability of default in period t. It is a present value, so discount.

Approximate price change from spread change

%ΔPrice ≈ −Spread duration × ΔSpread

Use the spread change in decimals; for a 25 bps widening use 0.0025. Convexity is ignored in this approximation.

Excess return (approximate)

Excess return ≈ (Spread × Time) − (Spread duration × ΔSpread) − (Expected loss rate)

Carry minus spread-change loss minus credit losses. Expected loss = probability of default × loss given default (approx.).

Duration times spread (DTS) risk

DTS = Spread duration × Spread

A rough guide: a bond's spread volatility often scales with its spread level, so DTS compares risk across quality and maturity.

Active spread duration

Active spread duration = Portfolio spread duration − Benchmark spread duration

Positive means more exposure to spread widening than the benchmark.

Contribution to spread duration

Contribution = Weight × Spread duration (sum across bonds or sectors)

Use it to compare sector overweights on a risk basis, not just by weight.

Expected loss

Expected loss ≈ Probability of default × Loss given default

Use it for a single period.

Loss given default

LGD = 1 − Recovery rate

Recovery rate is a percentage of par or exposure.

Credit-adjusted spread view

Excess spread ≈ Spread − Expected loss

A rough guide to compensation for bearing credit risk. Use annual figures consistently.

Spread return approximation

Return from spread change ≈ −Spread duration × ΔSpread

Use for mark-to-market moves, with spread change in decimal form.

Absolute priority order

Senior secured > Senior unsecured > Subordinated > Equity

A guide to recoveries. Actual outcomes may deviate.

Approximate CDS price change

ΔCDS value ≈ −(Δ spread) × EffSpreadDur × notional

Sign is for the protection seller: if spreads widen, the seller loses and the buyer gains. Effective spread duration of the CDS is similar to that of a bond of the same maturity.

CDS-bond basis

Basis = CDS spread − bond credit spread

Negative basis: buy bond and buy protection. Positive basis: sell bond (or avoid) and sell protection. Requires matched issuer and maturity.

Hedge notional for credit exposure

Hedge notional = Position value × (Bond spread duration ÷ CDS spread duration)

Matches spread sensitivity so the hedge offsets spread moves. It does not remove default-timing or basis risk.

Synthetic credit position

Government bond + sold CDS protection ≈ corporate bond

Approximate. Differences come from basis, counterparty risk and funding.

Quick revision

• Credit spread is compensation for credit risk, liquidity and other factors over a benchmark.
• Spread duration measures price sensitivity to a change in spread.
• Approximate price change ≈ −spread duration × change in spread.
• Expected loss = probability of default × loss given default × exposure.
• Loss given default = 1 − recovery rate.
• Spread widening hurts prices; tightening helps them.
• Investment-grade returns rely more on spread changes and positioning than on default events.
• High-yield returns depend heavily on default and recovery outcomes and on security selection.
• Credit derivatives can adjust exposure without trading the bond.
• Structured credit offers tailored risk, but check complexity and liquidity.
• Tie every recommendation to the client's objectives and constraints.
• Show the calculation; a correct number alone earns credit in essays.

Common mistakes

• Treating the whole spread as compensation for default only. Fix: Always list expected loss, risk premium and liquidity premium. Spreads are usually larger than expected loss alone.
• Saying spreads tighten when the economy weakens. Fix: Weak economy raises default risk and risk aversion, so spreads widen. Keep rate moves and spread moves separate.
• Using interest rate duration to estimate the price effect of a spread change on a floating-rate note. Fix: Use spread duration. A floater has rate duration near zero but spread duration close to its remaining maturity.
• Forgetting to convert basis points to percent. Fix: Divide bp by 100 before multiplying, so 40 bp = 0.40%.
• Using the recovery rate in place of LGD in the expected loss formula. Fix: Always write LGD = 1 − recovery first, then multiply.
• Treating expected loss as the maximum or typical loss. Fix: Say it is a probability-weighted average. Actual outcomes are zero loss or a large loss.
• Mixing up interest rate duration and spread duration. Fix: Use spread duration for spread changes and effective duration for rate changes. Say which one you use.
• Calling a strategy top-down just because it uses sectors. Fix: Ask where the decision starts. If the macro or sector view comes first, it is top-down. If issuer picks drive weights, it is bottom-up.
• Treating HY bonds as just higher-yielding investment-grade bonds. Fix: State that default, recovery and liquidity drive HY returns, and that HY correlates more with equities.
• Mixing up maintenance and incurrence covenants. Fix: Maintenance is tested regularly regardless of action. Incurrence is tested only when the issuer acts, such as borrowing more.

Exam tips

• Read command words closely. If asked to explain, give the driver and the direction. If asked to calculate, show the formula and the number.
• Convert bps to decimals before you multiply. Show this step so you keep credit if you slip later.
• Tie IG versus HY choices to the client's risk tolerance, liquidity needs and any rating limits in the policy.
• When stress appears in a vignette, expect spreads to widen, HY to fall more than IG, and liquidity to worsen.
• Separate rate effects from spread effects in return questions. Most traps live in that gap.
• Read the command word: calculate needs a number with shown work; explain or justify needs a short reason tied to the client or the view.
• In item sets, check whether the question gives spread duration or only modified duration. Use the one asked for, and only add convexity if given.
• Always state the direction: widening hurts, tightening helps. This earns marks on explain questions.