Spot and forward interest rates

To compensate for these gaps and distortions, one can try to build a standard
representation of the term structure using observable interest rates. This is typically
done in terms of prices and interest rates of discount bonds, fixed-income investments
with only one payment at maturity, and spot or zero coupon interest rates,
or interest rates on notional discount bonds of different maturities. The spot interest
rate is the constant annual rate at which a fixed-income investment’s value must
grow starting at time to reach $1 at a future. The spot or zero coupon curve is a
function relating spot interest rates to the time to maturity. Most of the zero-coupon
curve cannot be observed directly, with two major exceptions: bank deposit rates
and short-term government bonds, which are generally discount paper.
A forward interest rate is an interest rate contracted today to be paid from one
future date called the settlement date to a still later future date called the maturity
date. The forward curve relates forward rates of a given time to maturity to the time
to settlement. There is thus a distinct forward curve for each time to maturity. For
example, the 3-month forward curve is the curve relating the rates on forward 3-
month deposits to the future date on which the deposits settle. Any forward rate can
be derived from a set of spot rates via arbitrage arguments by identifying the set of
deposits or discount bonds which will lock in a rate prevailing from one future date
to another, without any current cash outlay.