Forward Rates

What are Forward Rates?

The future spot rates implied by today’s spot rates are forward rates. Consider the following scenario: the offered one-year rate is 3%, and the offered two-year rate is 4%. (both with annual compounding). We can estimate that the rate provided for the second year is 5%. This is because averaging 3 per cent for the first year with 5 per cent for the second year results in a total of 4 per cent for the two years.

Example of Forward Rates:

Suppose F is the forward rate for the second year. The forward rate is such that USD 100, if invested at 3% for the first year and at a rate of F for the second year, gives the same outcome as 4% for two years.
This means that:

$ 100\, \times \, 1.03\, \times \, \left ( 1+F \right )\, =\, 100\, \times \, 1.042^{2} $

so that:

$ F\, = \, \frac{1.042^{2}}{1.03}\, = \, 1.03-1\, = 0.051 $
or 5.01%.

This is close to the 5% given by our approximate argument. However, it is not the same because there are non-linearities when rates are expressed with annual compounding.
When rates are expressed with semi-annual compounding (as is frequently the case in fixed-income markets), an extension of this analysis shows that the forward rate per six months for six months starting at the time T is

$ \left ( 1+\frac{R_{2}}{2} \right )\,t\, +\, 0-5 $

$ \left ( 1+\frac{R_{,}}{2} \right )T\; \left ( 10.5 \right ) $

where and R2 are the spot rates for maturities T and T + 0.5
(respectively) with semi-annual compounding. Thus, the annualised forward rate expressed with semi-annual compounding is twice this.

Why are Forward Rates important?

Knowing the future rate is valuable regardless of whatever version is used. It allows the investor to choose the investment option (purchasing one T-bill or two) that offers the best chance of profit.