CFA CFA Level 1 How to calculate forward rates from spot rates

# How to calculate forward rates from spot rates

Topic Resolution: Resolved
• Author
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• pcunniff
Participant
• CFA Level 1
5

Anyone out there (mikey – I know you’re reading this ðŸ˜‰), know an efficient way to calculate forward from spot rates? I understand the 3y1y (1 year spot 3 years from now) blah blah. My questions is if they give us say:

1 year spot – 5%

2 year spot – 6%

3 year spot – 7%

4 year spot -8.5%

5 year spot – 10%

6 year spot -11%

We can go 100 diff ways with this trying to calc the 2y2y or 4y2y etc. My understanding is the numerator is always the 2 added together. For example, what is the 4 year forward 2 years from now? This would be the 2y4y. My understanding is it would be (1+11%)^6 but I get confused on the denominator. Would you use (1+8.5%)^4? What are we raising it by?? This is where I get lost.

Another way to look at it is what is the 1 year forward 2 years from now? 2y1y, which is (1.07)^3/(1.06)^2 -1=9.02%.

Why do we take the denominator as the 1.06^2?

• Zee Tan
Keymaster
• CFA Charterholder
5

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

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:

where:

• raâ€‹ = The spot rate for the bond of term taâ€‹ periods
• rbâ€‹ = The spot rate for the bond with a shorter term of tbâ€‹ periodsâ€‹

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:

${\left(1+0.11\right)}^{6}$

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:

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

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

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

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

• pcunniff
Participant
• CFA Level 1
2

Thanks gents!