How to calculate forward rates from spot rates

Hi @pcunniff, @mikey might chip in later but meanwhile I’ll have a go 😀

I’m actually not sure what you’re unsure about since you did the “1 year forward 2 years from now” example correctly.

The formula to calculate forward rates from spot rates is:

$forward rate=\frac{{(1+r_a)}^{t_a}}{{(1+r_b)}^{t_b}}–1$

where:

• r_a​ = The spot rate for the bond of term t_a​ periods
• r_b​ = The spot rate for the bond with a shorter term of t_b​ periods​

The simple, brain-dead way to remember this is the numerator (a) is the ‘longer’ term, and the denominator (b) is the ‘shorter’ term.

To explain conceptually, using your example, if you want to quote the forward rate of a 6 year bond today, you would use:

${(1+0.11)}^6$

But if we want this rate 4 years from now, not today, we ‘discount’ it by 4 years. But what rate should we use? The 4-year spot rate:

$\frac{{(1+0.11)}^6}{{(1+0.085)}^4}$

Tack on a “-1” to the calculation gives you a nice pretty % number.

$\frac{{(1+0.11)}^6}{{(1+0.085)}^4}–1=35\%$

The mathematical way to argue this would be to say that:

$$\begin{gathered} 6yr forward rate = 4yr forward rate \times 6yr forward 4yrs from today \\ {(1+0.11)}^6={(1+0.085)}^4 \times 6yr forward 4yrs from today \\ 6yr forward 4yrs from today = \frac{{(1+0.11)}^6}{{(1+0.085)}^4} \end{gathered}$$