Zero-Coupon Bonds, Market Rates, Swaps, and Conventions

The discount factor $P(t,T)$ is the single primitive that anchors all fixed-income pricing. Every rate you see quoted in the market — overnight, three-month SOFR, a two-year swap rate — is simply a way of re-expressing the same underlying discount curve under a specific compounding convention. This note establishes the exact algebraic relationships among zero-coupon bonds, spot rates, forward rates, and the instantaneous short rate, then shows how coupon bonds, floating rate notes, and interest rate swaps are all priced from this same curve.

The swap market is the dominant rate-setting mechanism for maturities from 2 to 30 years. Understanding the swap pricing formula — how the fixed rate is set so the swap has zero initial value — is essential for reading market rates and for bootstrapping the curve in the next lesson.

• define the zero-coupon bond $P(t,T)$ and state its boundary conditions;
• derive the simple forward rate, continuously compounded forward rate, and instantaneous forward rate from $P(t,T)$;
• state the relationship $P(t,T) = e^{-\int_t^T f(t,u)\,du}$;
• price a floating rate note and show its value equals $P(t,T_0)$;
• derive the par swap rate $R_{\text{swap}}(t)$ as a ratio of discount factors;
• name three common day-count conventions and explain why they matter for exact pricing.

1. 1. The zero-coupon bond as the primitive
2. 1. The family of interest rates
3. 1. The money-market account and the short rate
4. 1. Coupon bonds and floating rate notes
5. 1. Interest rate swaps
6. 1. Market conventions