Forward Rate Calculator
Understanding Spot Rates and Forward Rates
In finance, understanding the relationship between current interest rates (spot rates) and future expected interest rates (forward rates) is crucial for investment decisions, risk management, and valuation. This calculator helps you determine a forward rate based on two known spot rates and their respective maturities.
What are Spot Rates?
A spot rate is the interest rate for a loan that begins today and matures at a specified future date. For example, a 1-year spot rate is the interest rate on a bond or loan that starts now and matures in one year. A 2-year spot rate is for a loan starting now and maturing in two years. Spot rates are observable in the market and reflect the current market consensus on interest rates for different maturities.
What are Forward Rates?
A forward rate is an interest rate that is agreed upon today for a loan or investment that will begin at some point in the future. For instance, a forward rate from year 1 to year 2 would be the rate agreed upon today for a loan that starts at the end of year 1 and matures at the end of year 2. Forward rates are not directly observable like spot rates; they are implied by the current spot rates.
The Relationship Between Spot and Forward Rates
The core principle is that investing for a longer period at the corresponding spot rate should yield the same return as investing for a shorter period at its spot rate and then reinvesting at the implied forward rate for the remaining period. This is known as the no-arbitrage principle.
Mathematically, if we denote:
- $S_1$ as the spot rate for maturity $t_1$
- $S_2$ as the spot rate for maturity $t_2$ (where $t_2 > t_1$)
- $f(t_1, t_2)$ as the forward rate from time $t_1$ to time $t_2$
Assuming simple interest for simplicity in this explanation (though the calculator uses a more common bond market convention that implies compounding over the period):
The total return from investing for $t_2$ years at the $t_2$-year spot rate should equal the total return from investing for $t_1$ years at the $t_1$-year spot rate and then reinvesting the proceeds at the forward rate $f(t_1, t_2)$ for the remaining $t_2 – t_1$ years.
A commonly used formula derived from bond market conventions (which implicitly assumes compounding over the period up to the maturity date) is:
$(1 + S_2 \times t_2) = (1 + S_1 \times t_1) \times (1 + f(t_1, t_2) \times (t_2 – t_1))$
Rearranging this to solve for the forward rate $f(t_1, t_2)$ gives:
$f(t_1, t_2) = \frac{(1 + S_2 \times t_2)}{(1 + S_1 \times t_1)} – 1$
This is the formula implemented in the calculator above, where $S_1$ is `spotRate1`, $t_1$ is `time1`, $S_2$ is `spotRate2`, and $t_2$ is `time2`. The result is the forward rate from the end of period $t_1$ to the end of period $t_2$. Note that this calculation assumes rates are expressed annually.
Example Calculation
Suppose the following spot rates are observed in the market:
- The 1-year spot rate ($S_1$) is 5.0% (0.05). So, $t_1 = 1$ year.
- The 2-year spot rate ($S_2$) is 6.0% (0.06). So, $t_2 = 2$ years.
We want to find the forward rate from year 1 to year 2 ($f(1, 2)$).
Using the formula:
$f(1, 2) = \frac{(1 + 0.06 \times 2)}{(1 + 0.05 \times 1)} – 1$
$f(1, 2) = \frac{(1 + 0.12)}{(1 + 0.05)} – 1$
$f(1, 2) = \frac{1.12}{1.05} – 1$
$f(1, 2) \approx 1.066667 – 1$
$f(1, 2) \approx 0.066667$
So, the forward rate from year 1 to year 2 is approximately 6.67%. This implies that the market expects interest rates to be around 6.67% for the period between the end of the first year and the end of the second year.