Fixed Income
Term structure
Spot, par, and forward rates are linked: (1 + s2)^2 = (1 + s1)(1 + 1f1), so the implied one-year forward beginning in one year is (1 + s2)^2/(1 + s1) - 1. A par rate is the coupon that prices a bond at par given the spot curve.
Arbitrage-free valuation
Binomial interest rate trees are calibrated so the model reproduces the benchmark curve exactly. Backward induction values the bond at each node; the option-adjusted spread (OAS) is the constant spread added to all nodes to match the market price. Callable bonds have lower OAS than straight bonds; putables have higher.
Credit
Credit analysis weighs capacity, collateral, covenants, and character. Expected loss = probability of default x loss given default. Credit spreads compensate for expected loss and risk premia; credit default swaps isolate credit risk, and the CDS spread approximates the spread over the risk-free curve.
Traps
- Effective duration and convexity from a tree capture embedded options; modified duration does not.
- Riding the yield curve assumes an unchanged curve and positive roll-down.
- Z-spread equals OAS only for option-free bonds.
- Forward rates are break-even rates, not forecasts.
Worked example
One-year spot is 3% and two-year spot is 4%. The one-year forward rate one year ahead is (1.04^2/1.03) - 1 = 5.01%. An investor who expects one-year rates next year to be below 5.01% should buy the two-year bond rather than roll a one-year bond.
Rule of thumb: with embedded options, quote OAS; without them, quote Z-spread. Duration times spread gives an approximate spread-duration contribution to return.