Forward and spot interest rates – introduction
A yield curve contains information about the market’s perception of interest rates over future periods of time.
These perceived future interest rates are known as forward interest rates. For instance, the overlap between the spot one-year interest rate and the spot two year interest rate implies an interest rate for the period of time between Year 1 and Year 2. This rate is known as the one-year forward interest rate, starting at the end of Year 1.
The forward interest rate is calculated from the two spot rates.
Consider two strategies
1) Invest £1 for two years.
2) Invest £1 for one year and then reinvest it for another year.
In Strategy 1, you would get £(1 + R2)2 at the end of the two years. In Strategy 2, you would have £(1 + R1) at the end of one year, which you would have to reinvest at the one year interest rate at that time. When considered today, this interest rate is called the forward rate that applies for the one year period starting at the end of Year 1. It is denoted as: 1F2 (or 1R2).
It is computed by solving the equation: (1 + R1)(1 +1F2) = (1 + R2)2
This is simply the rate that equates the values of Strategies 1 and 2, given today’s interest rates.
From any yield curve, you can calculate a set of implied forward rates. In general, the implied forward interest rate between-period “t” and Period “T”, calculated today, solves the equation:
(1 + Rt)t (1 + tFT)T-t = (1 + RT)T
Where:
t = End of first period and
T= End of second period
Example
Assume the following 3 year term structure of spot interest rates:
Rate 1+ Spot Rate
R1 1.04
R2 1.06
R3 1.08
From these spot rates, we can compute the implied set of spot and forward rates as follows:
Rate Spot / Forward Rates
0F1 0.04
1F2 0.0804
2F3 0.1211
The spot rates in the first table are the geometric averages of the spot / forward rates in the second table. To verify this, consider the general equation where t = 1 and T = 2:
(1 + R1)(1 + 1F2) = (1 + R2)2
This yields:
(1.04) (1 + 1F2) = (1.06)2
You can calculate the 1F2 rate by taking 1.062 / 1.04 = 1.0804.
Check back with your workings: 1.04 x 1.0804 = 1.1236
Square root of 1.1236 = 1.06.
Similarly, we can do the same for verifying the 3 year spot rate:
(1 + R1)(1 + 1F2)(1 + 2F3) = 1.083
To find the 2F3 rate:
(1.04) (1.0804) (1 + 2F3) = 1.083
(1 + 2 F3) = 1.083 /(1.04) (1.0804)
= 1.1211