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 (wealth-management plugin): the Treasury curve used as the risk-free benchmark
- fixed-income-structured (wealth-management plugin): CLOs and structured credit products
- alternatives (wealth-management plugin): private credit as an alternative investment
- asset-allocation (wealth-management plugin): credit allocation in multi-asset portfolios
Running the Script
bashuv 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.
