Forward Rates

Forward Rates

A forward rate is the implied future interest rate between two time periods, derived from current spot rates.

No-Arbitrage Derivation

An investor can either:

1. Invest for $t_2$ years at spot rate $s_2$
2. Invest for $t_1$ years at $s_1$, then roll over at forward rate $f$

No-arbitrage requires both strategies to produce the same result:

$(1 + s_2)^{t_2} = (1 + s_1)^{t_1} \cdot (1 + f)^{t_2 - t_1}$

Forward Rate Formula

$f(t_1, t_2) = \left(\frac{(1 + s_2)^{t_2}}{(1 + s_1)^{t_1}}\right)^{\frac{1}{t_2 - t_1}} - 1$

Example

Spot rates: 1-year = 3%, 2-year = 4%

$f(1, 2) = \frac{1.04^2}{1.03^1} - 1 = \frac{1.0816}{1.03} - 1 \approx 5.01\%$

The market implies a 1-year rate of 5.01% starting one year from now.

Interpretation

If the forward rate exceeds current short rates, the market expects rates to rise.