The Formula for Converting Spot Rate to Forward Rate

The relationship between spot and forward rates is similar, like the relationship between discounted present value and future value. A forward interest rate acts as a discount rate for a single payment from one future date (say, five years from now) and discounts it to a closer future date (three years from now).

Theoretically, the forward rate should be equal to the spot rate plus any earnings from the security, plus any finance charges. You can see this principle in equity forward contracts, where the differences between forward and spot prices are based on dividends payable less interest payable during the period.

A spot rate is used by buyers and sellers looking to make an immediate purchase or sale, while a forward rate is considered to be the market's expectations for future prices. It can serve as an economic indicator of how the market expects the future to perform, while spot rates are not indicators of market expectations, and are instead the starting point to any financial transaction.

Therefore, it is normal for forward rates to be used by investors, who may believe that they have knowledge or information on how the prices of specific items will move over time. If a potential investor believes that real future rates will be higher or lower than the stated forward rates at the present date, it could signal an investment opportunity.

For simplicity, consider how to calculate the forward rates for zero-coupon bonds. A basic formula for calculating forward rates looks like this:

$\begin{aligned} &\text{Forward rate} = \frac{\left(1+r_a \right )^{t_a}}{\left(1+r_b \right )^{t_b}}-1\\ &\textbf{where:}\\ &r_a = \text{The spot rate for the bond of term } t_a \text{ periods}\\ &r_b = \text{The spot rate for the bond with a shorter term of } t_b \text{ periods} \end{aligned}$​Forward rate=(1+rb​)tb​(1+ra​)ta​​−1where:ra​=The spot rate for the bond of term ta​ periods​

In the formula, "x" is the end future date (say, 5 years), and "y" is the closer future date (three years), based on the spot rate curve.

Suppose a hypothetical two-year bond is yielding 10%, while a one-year bond is yielding 8%. The return produced from the two-year bond is the same as if an investor receives 8% for the one-year bond and then uses a rollover to roll it over into another one-year bond at 12.04%.

$\text{Forward rate} = \frac{\left(1+0.10 \right )^{2}}{\left(1+0.08 \right )^{1}}-1 = 0.1204 = 12.04\%$Forward rate=(1+0.08)1(1+0.10)2​−1=0.1204=12.04%

This hypothetical 12.04% is the forward rate of the investment.

To see the relationship again, suppose the spot rate for a three-year and four-year bond is 7% and 6%, respectively. A forward rate between years three and four—the equivalent rate required if the three-year bond is rolled over into a one-year bond after it matures—would be 3.06%.

To understand the differences and relationship between spot rates and forward rates, it helps to think of interest rates as the prices of financial transactions. Consider a $1,000 bond with an annual coupon of $50. The issuer is essentially paying 5% ($50) to borrow the $1,000.

A "spot" interest rate tells you what the price of a financial contract is on the spot date, which is normally within two days after a trade. A financial instrument with a spot rate of 2.5% is the agreed-upon market price of the transaction based on current buyer and seller action.

Forward rates are theorized prices of financial transactions that might take place at some point in the future. The spot rate answers the question, "How much would it cost to execute a financial transaction today?" The forward rate answers the question, "How much would it cost to execute a financial transaction at future date X?"

Note that both spot rates and forward rates are agreed to in the present. It's the timing of the execution that's different. A spot rate is used if the agreed trade occurs today or tomorrow. A forward rate is used if the agreed trade isn't set to occur until later in the future.