The Time Value of Money: Compounding Conventions, Zero Rates, and Forward Rates

Every derivative price is a present value — a future cash flow discounted at the appropriate rate. But "discounting" is not a single operation: whether you say an interest rate is 10% per annum depends entirely on how frequently the interest compounds, and using the wrong convention can introduce pricing errors of several basis points on a large notional. The machinery introduced here — continuous compounding, zero rates stripped from the yield curve via bootstrap, and forward rates locked in by a replication strategy — is the arithmetic infrastructure that every subsequent pricing argument in this unit will use.

• explain why quoting "10% per annum" is ambiguous without specifying a compounding frequency;
• convert any rate between annual, semiannual, quarterly, monthly, and continuous compounding using the standard conversion formulas;
• define a zero-coupon rate (spot rate) and explain why it is the correct discount rate for a single cash flow at that maturity;
• price a coupon bond using a schedule of zero rates, and find the bond's yield to maturity;
• apply the bootstrap method to strip zero rates from a sequence of Treasury bonds;
• derive and apply the forward rate formula $R_F = (R_2 T_2 - R_1 T_1) / (T_2 - T_1)$;
• compute the payoff and mark-to-market value of a forward rate agreement.

1. Types of rates
2. Measuring interest rates: compounding conventions
3. Zero rates
4. Bond pricing
5. Bootstrap: stripping zero rates from coupon bonds
6. Forward rates
7. Forward rate agreements