joellewis/fixed-income-corporate

Fixed Income — Corporate

Core Concepts

Credit Spreads

Compensation for default risk, liquidity risk, and downgrade risk above the risk-free rate. Multiple spread measures exist with increasing precision:

G-spread (Government Spread): Bond yield minus interpolated Treasury yield of the same maturity. Simple but assumes a flat term structure between benchmark maturities.

Z-spread (Zero-Volatility Spread): The constant spread added to each point on the risk-free spot rate curve such that the sum of discounted cash flows equals the bond's market price. Superior to G-spread because it accounts for the full shape of the term structure.

OAS (Option-Adjusted Spread): For bonds with embedded options, OAS = Z-spread minus the value of the embedded option. OAS represents the "true" credit compensation after removing the option component. Requires an interest rate model to compute.

Credit Ratings

AAA/AA/A/BBB are investment grade. BB/B/CCC/CC/C/D are high yield (speculative grade). The BBB/BB boundary is the most consequential threshold — many institutional mandates prohibit sub-investment-grade holdings. A downgrade across this boundary ("fallen angel") forces selling by constrained investors.

Migration Matrix

A transition matrix shows the probability of moving from one rating to another over a 1-year horizon. A BBB-rated issuer has roughly 85-90% probability of remaining BBB, 4-5% chance of upgrade, 4-5% chance of downgrade, and a small probability (~0.2%) of default. Migration matrices are published annually by rating agencies.

Default Probability, Loss Given Default, and Recovery Rate

• PD = Probability of Default over a given horizon
• LGD = Loss Given Default (percentage of exposure lost)
• Recovery Rate (RR) = 1 - LGD
• Expected Loss: EL = PD × LGD × EAD (Exposure at Default)

Recovery rates vary by seniority: senior secured (60-65%), senior unsecured (40-50%), subordinated (20-30%).

Callable Bonds

The issuer can redeem the bond early. Call schedules specify prices and dates. Yield-to-call (YTC) is calculated using the call date and call price. Yield-to-worst (YTW) is the minimum of YTM and all possible YTCs. For callable bonds, OAS is the appropriate spread measure (not G-spread or Z-spread).

Covenants

Maintenance covenants: Tested periodically (e.g., quarterly). Issuer must maintain financial ratios at all times. Common in bank loans.

Incurrence covenants: Tested only when the issuer takes a specific action (e.g., issues new debt). Common in bond indentures. Key covenants include leverage ratio (Debt/EBITDA), interest coverage (EBITDA/Interest), and restricted payments.

Private Credit

Direct lending by non-bank lenders to middle-market companies. Offers an illiquidity premium of 150-400bp over comparable syndicated loans. Typically features stronger covenant protection than public market deals. Valuations are mark-based (quarterly), which smooths reported volatility.

CDS (Credit Default Swaps)

A derivative where the protection buyer pays a periodic spread and receives payment upon a credit event. CDS spreads can be used to derive market-implied default probabilities. CDS spreads are often more responsive to credit deterioration than bond spreads.

Key Formulas

| Formula | Expression | Use Case | |---------|-----------|----------| | G-spread | Bond Yield - Interpolated Treasury Yield | Simple spread measure | | Z-spread | Constant spread s: P = sum CF_t / (1+s_t+s)^t | Full curve spread | | OAS | Z-spread - Option Cost | Spread for callable bonds | | Expected Loss | EL = PD × LGD × EAD | Credit loss estimation | | Recovery Rate | RR = 1 - LGD | Recovery from default | | Yield-to-Worst | min(YTM, YTC_1, YTC_2, ...) | Conservative yield measure |

Worked Examples

Example 1: Compare Z-spread vs G-spread

Given: A 7-year corporate bond yields 5.8%. The 7-year interpolated Treasury yield is 4.5%. The Z-spread (computed using the full spot curve) is 118bp. Calculate: G-spread and compare to Z-spread Solution: G-spread = 5.8% - 4.5% = 1.30% = 130bp Z-spread = 118bp The G-spread (130bp) exceeds the Z-spread (118bp) by 12bp. This difference arises because the G-spread uses a single interpolated benchmark point while the Z-spread properly accounts for the shape of the entire yield curve. In a steep curve environment, G-spread tends to overstate the true spread.

Example 2: Expected Loss Calculation

Given: PD = 2% (annual), LGD = 60%, EAD = $1,000,000 Calculate: Expected annual loss Solution: EL = PD × LGD × EAD EL = 0.02 × 0.60 × $1,000,000 EL = $12,000

The expected annual credit loss is $12,000, or 1.2% of the exposure. This represents the actuarial cost of credit risk — the spread must at least cover this expected loss, with additional compensation for unexpected losses and risk aversion.

Common Pitfalls

• Using G-spread for callable bonds — use OAS instead, which removes the option component
• Ignoring liquidity premium in spread analysis — part of the spread compensates for illiquidity, not just default risk
• Rating agency lag vs market-implied credit quality — CDS spreads often move before rating actions
• Assuming recovery rates are constant — they vary significantly by seniority and economic cycle (lower in recessions)

Cross-References

• fixed-income-sovereign: the Treasury curve used as the risk-free benchmark
• fixed-income-structured: CLOs and structured credit products
• alternatives: private credit as an alternative investment
• asset-allocation: credit allocation in multi-asset portfolios

Running the Script

uv run scripts/fixed_income_corporate.py            # run the demo (uses PEP 723 inline deps)
uv run scripts/fixed_income_corporate.py --verify   # check demo outputs against the worked examples (exit 1 on mismatch)
python3 scripts/fixed_income_corporate.py            # alternative (requires: pip install numpy scipy)

The demo prints the calculations covered above; its values match the worked examples in this skill. Run --help for a list of the classes and functions. For programmatic use, import the module rather than running it — the demo only executes under python fixed_income_corporate.py.