Forward Rate

The mechanics rest on a single indifference condition: an investor who locks in a longer-dated spot rate today must end up with the same terminal wealth as one who invests at a shorter spot rate and simultaneously contracts today to reinvest at a forward rate. Written in words, one plus the two-period spot rate, raised to the power two, equals one plus the one-period spot rate, multiplied by one plus the one-period forward rate beginning one period from now. Solving for the unknown forward gives the rate the market has already embedded in the curve.

The same logic extends to any horizon. One plus the spot rate for the longer maturity, compounded over its full life, equals one plus the spot rate for the shorter maturity compounded over its life, multiplied by one plus the forward rate compounded over the remaining periods. Every forward rate is therefore a residual of two observable spot rates, not an independent input.

A spot rate is a geometric average of the sequence of forward rates covering the same horizon, so forwards behave like marginal rates while spot rates behave like averages. That relationship dictates the geometry of the two curves. When the spot curve slopes upward, each successive average must be pulled higher by the next marginal rate, so the forward curve lies above the spot curve and rises more steeply. When the spot curve is flat, the two coincide. When it is inverted, forwards lie below spot rates, and the implied forward path declines faster than the spot curve itself.

This also explains a common source of confusion in quoting conventions. A forward rate must be identified by two dates, the start of the period and its length, because the number changes materially with both. Periodicity and day-count matter as well: mixing annual compounding on one side of the no-arbitrage equation with semiannual on the other produces a rate that is internally inconsistent.

Forward rates are not merely descriptive. A bond can be valued by discounting each cash flow through the chain of implied one-period forwards rather than by a single spot rate, and the two approaches must give the same price, since the forwards were extracted from those spot rates. Forward rate agreements, interest rate futures and the floating legs of swaps all settle against rates for periods beginning after the trade date, so the implied forward curve is the natural pricing benchmark. Deviations from it are what dealers arbitrage.

The practical use is breakeven analysis. If an investor expects future short rates to be lower than the forwards imply, holding longer maturities and rolling down the curve should outperform repeated short-term investment. If realised rates exceed the forwards, the short strategy wins. Total-return investors therefore compare their own rate view against the forwards rather than against today's spot levels.

Expect three tasks. First, compute a forward rate from two spot rates, or invert the process and recover a spot rate from a shorter spot rate and a stated forward. Set up the compounding on both sides explicitly and confirm the exponents sum correctly before solving. Second, reason qualitatively about curve shape: given an upward-sloping, flat or inverted spot curve, state where the forward curve must lie and why. Third, use forwards for valuation, discounting a bond's cash flows through the implied forward path and reconciling the result with spot-rate discounting.

Also be prepared to discuss interpretation. Forwards reflect no-arbitrage relationships among traded prices, and any expectations content they carry is contaminated by term premia, liquidity effects and hedging demand. Distinguishing the arbitrage-free construction from the economic forecast is the conceptual point examiners test most often.