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# 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.

Owais Siddiqui
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