31  Forward Rates

Definition: Given the term structure of zero coupon bond yield rates, \(\{s_t\}_{t>0}\), the forward rate applicable from year \(t\) to \(t+k\), denoted by the symbol \(f_{[t,t+1]}\) should satisfy the relationship: \[(1 + f_{[t,t+k]})^k = \frac{(1+s_{t+k})^{t+k}}{(1+s_{t})^{t}}\] Unless we are told otherwise, we assume that forward rates are expressed as annual effective rates of interest.

When the unit of time is one year, this rate can also be described as follows: * k-year spot rate that comes into effect t years in the future * k-year forward rate, deferred for t years * k-year forward rate, starting in t years

When \(k = 1\), we drop the second subscript: \[f_t =f_{[t,t+1]}\] This rate is sometimes referred to as the one-year forward rate for the (t+1)th year.

31.1 Properties

The initial forward rate applies from time 0 ot time 1, so it is equal to the time-1 spot rate: \[f_0 = s_1\]

The relationship between spot rates and forward rates: \[(1+s_t)^t = (1+f_0)(1+f_1)...(1+f_{t-1}) \hskip7em \text{(t-terms)}\]

This can explain that while the forward rate curves may fluctuate strongly, the spot rate curve tends to be more stable because the effect is

Example: You are given the following information: * The 3-year spot rate is \(3\%\) * The 2-year forward rate, deferred 3 years, is \(5\%\) * The 3-year forward rate, deferred 2 years,is \(4\%\) Determine the price of a \(\$1,000\) 2-year zero-coupon bond.

Solution: We need to determine \(1000\times (1+s_2)^{-2}\). We already know: * \(s_3 = 0.03\) * \(f_{[3,5]} = 0.05\) * \(f_{[2,5]} = 0.04\) Then, \[1 + f_{[2,3]} =\frac{(1 + f_{[2,5]})^3}{(1 + f_{[3,5]})^2} = \frac{1.04^3}{1.05^2}\] and \[(1+s_2)^2 = \frac{(1+s_3)^3}{1+f_{[2,3]}} = \frac{1.03^3\times 1.05^2}{1.04^3} = 1.0672\] Thus the price of the bond is: \[P = \frac{1000}{(1+s_2)^2} = 933.71\]