Fixed Income

A spot rate (zero rate) is the yield on a single cash flow received at one future date, with no intervening coupons. YTM is a single internal rate of return that discounts all of a coupon bond's cash flows, so it is a complex average of the relevant spot rates and only equals the spot rate for a zero-coupon bond.

Each cash flow is discounted at the spot rate matching its own maturity: P = sum of CF_t / (1 + z_t)^t. Using a single YTM for all cash flows can mis-price the bond relative to discounting each flow at its arbitrage-free spot rate.

A forward rate is an interest rate agreed today for a loan that begins at a future date. The notation f(A,B-A) or 'AyByr' denotes the rate for a B-A year loan starting A years from now (e.g., 2y1y is the one-year rate two years forward).

(1 + z_A)^A * (1 + f(A,B-A))^(B-A) = (1 + z_B)^B. This no-arbitrage condition says investing to time A then rolling at the forward rate must equal investing directly to time B.

When the spot curve is upward sloping, forward rates lie above spot rates; when the spot curve is downward sloping, forwards lie below spots. Forward rates being above spot rates is consistent with a rising spot curve.

Forward rates are the future reinvestment rates that make an investor indifferent between rolling a series of short maturities and buying one longer maturity. If realized future rates differ from the forwards, total returns diverge from the breakeven.