Problem to get forward rates using spot rates

An analyst collects the following information regarding spot rates:

  • 1-year rate = 4%.
  • 2-year rate = 5%.
  • 3-year rate = 6%.
  • 4-year rate = 7%.

The 2-year forward rate two years from today is closest to: … we are searching then 2y2y

((1.07)4/(1.05)^2)0.5 … my question is why would we at the end make it the squared root … if we were searching 3y3y … suppose that the 6-year rate was 9% … then 3y3y would be ((1.09)4/(1.06)2)1/3 ??? and also other question the 3 because they is a 3 year rate … (not because it is 3 years from now, right?)

Think of it this way. The key concept behind forward rates is that they are the implied rates that make an investor indifferent between investing at the long (in this case, 4-year) spot rate or investing at the short (2-year) spot rate followed by the “leftover” forward rate (in this case, the two-year forward in two years).

Indifferent means that the two approaches would result in the same terminal wealth. So, if you invested $1 at the 4-year spot, at the end of 4 years, you’d have 1.074 = $1.3108. Likewise, if you invested for 2 years at the 2-year spot of 5%, you’d have 1.052 = 1.1025.

The 2-year forward is the rate that would turn 1.1025 into 1.3108 in two years. time. Using the FV formula

1.1025 x (1+r)2 = 1.3108 ==> (1+r)2 = 1.3108/1.1025 ==> r + (1.3108/1.1025)0.50 - 1

Now just substitute 1.074 for 1.3108 and 1.052 for 1.1025, and you have your answer.

Thank you busprof.

Aslo, is my question with 3y3y is correct?

No, it’s not.

The correct calculation is (1.096 / 1.063)1/3 − 1 = 12.0849%

Also, note that we now have subscripts and superscripts. Using them makes your formulae much easier to read.

Hey S2000, thank you. Didnt understand that last part with subscripts. Let me do you another question please. The 1/ 3 … that 3 comes from a) rate 3 years from now ; b) 3 year rate. Thank you :slight_smile:

You’re quite welcome.

The 1/3 comes from it being a 3-year rate. You use the 3-year spot rate for the beginning and the 6-year spot rate for the end.

Thank you very very much. Analyst forum has been more than great for my preparation and thanks to people like you that answer questions.

You’re quite welcome.

I had a quick question regarding the forward rate problem as well.

The way Schweser has us do it is through a shortcut [(4 x .07) - (2 x .05)] / 2 = 9%

In this type of problem 2y2y do we always use the total time until forward (4 yrs) rate minus the 2 yr rate?

For example if it asked for the 1y2y would it be [(3 x .06) - (1 x .04)] / 2 ?

Thanks very much for any advice.